Subgroups — C₆₇×#34409 #30713660

Order 17152.

2314 conjugacy classes of subgroups.

The subgroup lattice has 894 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2381	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2379	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2378	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2372	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2368	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2359	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2359	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2343	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2339	C₂	1	2	yes	maximal
12	C₆₇×#2337	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2332	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2331	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2329	C₂	1	2	yes	maximal, complemented
16	C₆₇×#2322	C₂	1	2	yes	maximal, complemented
17	C₆₇×#2310	C₂	1	1	yes	maximal, characteristic, complemented
18	C₄×#123×C₆₇	C₂	1	2	yes	maximal, complemented
19	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	#536×C₆₇	C₄	1	16	yes	complemented
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#123×C₆₇	C₂²	1	2	yes	—
28	C₂×#123×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	#391×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#391×C₆₇	C₂²	1	1	yes	characteristic
41	#391×C₆₇	C₂²	1	1	yes	characteristic
42	#391×C₆₇	C₂²	1	1	yes	characteristic
43	#391×C₆₇	C₂²	1	1	yes	characteristic
44	#391×C₆₇	C₂²	1	1	yes	characteristic
45	#391×C₆₇	C₂²	1	2	yes	—
46	#391×C₆₇	C₂²	1	1	yes	characteristic
47	#391×C₆₇	C₂²	1	2	yes	—
48	#391×C₆₇	C₂²	1	1	yes	characteristic
49	#391×C₆₇	C₂²	1	1	yes	characteristic
50	#391×C₆₇	C₂²	1	2	yes	—
51	#391×C₆₇	C₂²	1	2	yes	—
52	#389×C₆₇	C₂²	1	2	yes	—
53	#389×C₆₇	C₂²	1	2	yes	—
54	#389×C₆₇	C₂²	1	2	yes	—
55	#389×C₆₇	C₂²	1	2	yes	—
56	#388×C₆₇	C₂²	1	2	yes	—
57	#388×C₆₇	C₂²	1	2	yes	—
58	#388×C₆₇	C₂²	1	2	yes	—
59	#388×C₆₇	C₂²	1	2	yes	—
60	#388×C₆₇	C₂²	1	2	yes	—
61	#388×C₆₇	C₂²	1	2	yes	—
62	#388×C₆₇	C₂²	1	2	yes	—
63	#388×C₆₇	C₂²	1	2	yes	—
64	#387×C₆₇	C₂²	1	2	yes	—
65	#387×C₆₇	C₂²	1	2	yes	—
66	#387×C₆₇	C₂²	1	2	yes	—
67	#387×C₆₇	C₂²	1	2	yes	—
68	#387×C₆₇	C₂²	1	2	yes	—
69	#387×C₆₇	C₂²	1	2	yes	—
70	#387×C₆₇	C₂²	1	2	yes	—
71	#387×C₆₇	C₂²	1	2	yes	—
72	#387×C₆₇	C₂²	1	2	yes	—
73	#387×C₆₇	C₂²	1	2	yes	—
74	#387×C₆₇	C₂²	1	2	yes	—
75	#387×C₆₇	C₂²	1	2	yes	—
76	#386×C₆₇	C₂²	1	2	yes	—
77	#386×C₆₇	C₂²	1	2	yes	—
78	#386×C₆₇	C₂²	1	2	yes	—
79	#386×C₆₇	C₂²	1	2	yes	—
80	#385×C₆₇	C₂²	1	2	yes	—
81	#385×C₆₇	C₂²	1	2	yes	—
82	#385×C₆₇	C₂²	1	2	yes	—
83	#385×C₆₇	C₂²	1	2	yes	—
84	#385×C₆₇	C₂²	1	2	yes	—
85	#385×C₆₇	C₂²	1	2	yes	—
86	#383×C₆₇	C₂²	1	2	yes	—
87	#382×C₆₇	C₂²	1	2	yes	—
88	#381×C₆₇	C₂²	1	2	yes	—
89	#381×C₆₇	C₂²	1	2	yes	—
90	#381×C₆₇	C₂²	1	2	yes	—
91	#380×C₆₇	C₂²	1	2	yes	—
92	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
93	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
94	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
95	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
96	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
97	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
98	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
99	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
100	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
101	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
102	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
103	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
104	#376×C₆₇	C₂²	1	2	yes	—
105	C₂×#95×C₆₇	C₂²	1	2	yes	—
106	C₂×#95×C₆₇	C₂²	1	2	yes	—
107	C₂×#95×C₆₇	C₂²	1	2	yes	—
108	C₂×#41×C₆₇	—	4	2	no	—
109	C₂×#41×C₆₇	—	4	2	no	—
110	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
111	C₂²×Q₈×C₆₇	—	4	2	no	—
112	C₂²×Q₈×C₆₇	—	4	2	no	—
113	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
114	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
115	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
116	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
117	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
118	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
120	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
122	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
125	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
126	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
127	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
128	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
129	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
130	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
131	#124×C₆₇	C₂×C₄	1	8	yes	complemented
132	#124×C₆₇	C₂×C₄	1	8	yes	complemented
133	#123×C₆₇	C₂×C₄	1	16	yes	complemented
134	#123×C₆₇	C₂×C₄	1	16	yes	complemented
135	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
136	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
137	C₄×D₄×C₆₇	—	4	4	no	—
138	C₄×D₄×C₆₇	—	4	4	no	—
139	#117×C₆₇	—	4	2	no	—
140	#117×C₆₇	—	4	2	no	—
141	#117×C₆₇	C₂×C₄	1	8	yes	complemented
142	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
143	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
144	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
147	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
151	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
152	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
160	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
161	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
163	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
164	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
165	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
166	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
167	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
168	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
169	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
170	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
171	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
172	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
173	C₂×C₄²×C₆₇	—	4	2	no	—
174	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
175	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
176	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
177	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
178	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
179	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
180	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
181	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
182	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
183	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
184	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
185	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
186	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
187	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
188	#95×C₆₇	C₂³	1	2	yes	—
189	#95×C₆₇	C₂³	1	2	yes	—
190	#95×C₆₇	C₂³	1	2	yes	—
191	#95×C₆₇	C₂³	1	2	yes	—
192	#95×C₆₇	C₂³	1	2	yes	—
193	#95×C₆₇	C₂³	1	2	yes	—
194	#95×C₆₇	C₂³	1	2	yes	—
195	#95×C₆₇	C₂³	1	2	yes	—
196	#95×C₆₇	C₂³	1	2	yes	—
197	#95×C₆₇	C₂³	1	2	yes	—
198	#95×C₆₇	C₂³	1	2	yes	—
199	#95×C₆₇	C₂³	1	2	yes	—
200	#95×C₆₇	C₂³	1	2	yes	—
201	#95×C₆₇	C₂³	1	2	yes	—
202	#95×C₆₇	C₂³	1	2	yes	—
203	#95×C₆₇	C₂³	1	2	yes	—
204	#95×C₆₇	C₂³	1	2	yes	—
205	#95×C₆₇	C₂³	1	2	yes	—
206	#95×C₆₇	C₂³	1	2	yes	—
207	#95×C₆₇	C₂³	1	2	yes	—
208	#95×C₆₇	C₂³	1	2	yes	—
209	#95×C₆₇	C₂³	1	2	yes	—
210	#95×C₆₇	C₂³	1	2	yes	—
211	#95×C₆₇	C₂³	1	2	yes	—
212	#95×C₆₇	C₂³	1	2	yes	—
213	#95×C₆₇	C₂³	1	2	yes	—
214	#95×C₆₇	C₂³	1	2	yes	—
215	#95×C₆₇	C₂³	1	2	yes	—
216	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
217	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
218	#41×C₆₇	—	4	16	no	—
219	C₂×Q₈×C₆₇	—	4	16	no	—
220	C₂×Q₈×C₆₇	—	4	1	no	—
221	C₂×Q₈×C₆₇	—	4	1	no	—
222	C₂×Q₈×C₆₇	—	4	1	no	—
223	C₂×Q₈×C₆₇	—	4	1	no	—
224	C₂×Q₈×C₆₇	—	4	1	no	—
225	C₂×Q₈×C₆₇	—	4	1	no	—
226	C₂×Q₈×C₆₇	—	4	1	no	—
227	C₂×Q₈×C₆₇	—	4	1	no	—
228	C₂×Q₈×C₆₇	—	4	1	no	—
229	C₂×Q₈×C₆₇	—	4	1	no	—
230	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
231	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
232	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
233	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
234	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
235	C₂×D₄×C₆₇	—	2	4	no	—
236	C₂×D₄×C₆₇	—	2	4	no	—
237	C₂×D₄×C₆₇	—	4	1	no	—
238	C₂×D₄×C₆₇	—	4	1	no	—
239	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
240	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
241	C₂²×C₄×C₆₇	—	2	2	no	—
242	C₂²×C₄×C₆₇	—	2	2	no	—
243	C₂²×C₄×C₆₇	—	2	2	no	—
244	C₂²×C₄×C₆₇	—	2	2	no	—
245	C₂²×C₄×C₆₇	—	2	2	no	—
246	C₂²×C₄×C₆₇	—	2	2	no	—
247	C₂²×C₄×C₆₇	—	2	4	no	—
248	C₂²×C₄×C₆₇	—	2	4	no	—
249	C₂²×C₄×C₆₇	—	2	4	no	—
250	C₂²×C₄×C₆₇	—	2	4	no	—
251	C₂²×C₄×C₆₇	—	4	2	no	—
252	C₂²×C₄×C₆₇	—	4	2	no	—
253	C₂²×C₄×C₆₇	—	4	2	no	—
254	C₂²×C₄×C₆₇	—	4	2	no	—
255	C₂²×C₄×C₆₇	—	4	2	no	—
256	C₂²×C₄×C₆₇	—	4	2	no	—
257	C₂²×C₄×C₆₇	—	4	2	no	—
258	C₂²×C₄×C₆₇	—	4	2	no	—
259	C₂²×C₄×C₆₇	—	4	2	no	—
260	C₂²×C₄×C₆₇	—	4	2	no	—
261	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
262	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
263	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
264	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
265	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
266	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
267	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
268	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
269	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
270	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
271	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
272	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
273	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
274	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
275	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
276	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
277	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
278	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
279	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
280	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
281	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
282	C₄⋊C₄×C₆₇	—	2	4	no	—
283	C₄⋊C₄×C₆₇	—	4	1	no	—
284	C₄⋊C₄×C₆₇	—	4	2	no	—
285	C₄⋊C₄×C₆₇	—	4	1	no	—
286	C₄⋊C₄×C₆₇	—	4	2	no	—
287	C₄⋊C₄×C₆₇	—	4	2	no	—
288	C₄⋊C₄×C₆₇	—	4	2	no	—
289	C₄⋊C₄×C₆₇	—	4	1	no	—
290	C₄⋊C₄×C₆₇	—	4	2	no	—
291	C₄⋊C₄×C₆₇	—	4	1	no	—
292	C₄⋊C₄×C₆₇	—	4	2	no	—
293	C₄⋊C₄×C₆₇	—	4	2	no	—
294	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
295	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
296	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
297	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
298	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
299	C₂²⋊C₄×C₆₇	—	2	4	no	—
300	C₂²⋊C₄×C₆₇	—	2	4	no	—
301	C₂²⋊C₄×C₆₇	—	2	4	no	—
302	C₂²⋊C₄×C₆₇	—	4	2	no	—
303	C₂²⋊C₄×C₆₇	—	4	2	no	—
304	C₂²⋊C₄×C₆₇	—	4	2	no	—
305	C₂²⋊C₄×C₆₇	—	4	2	no	—
306	C₂²⋊C₄×C₆₇	—	4	2	no	—
307	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
308	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
309	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
310	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
311	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
312	C₄²×C₆₇	—	2	4	no	—
313	C₄²×C₆₇	—	4	1	no	—
314	C₄²×C₆₇	—	4	2	no	—
315	C₄²×C₆₇	—	4	1	no	—
316	C₄²×C₆₇	—	4	2	no	—
317	C₄²×C₆₇	—	4	2	no	—
318	C₄²×C₆₇	—	4	2	no	—
319	C₄²×C₆₇	—	4	1	no	—
320	C₄²×C₆₇	—	4	2	no	—
321	C₄²×C₆₇	—	4	1	no	—
322	C₄²×C₆₇	—	4	2	no	—
323	C₄²×C₆₇	—	4	2	no	—
324	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
325	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
326	C₂³×C₆₇	—	2	2	no	—
327	C₂³×C₆₇	—	2	2	no	—
328	C₂³×C₆₇	—	2	2	no	—
329	C₂³×C₆₇	—	2	2	no	—
330	C₂³×C₆₇	—	2	2	no	—
331	C₂³×C₆₇	—	2	2	no	—
332	C₂³×C₆₇	—	2	2	no	—
333	C₂³×C₆₇	—	2	2	no	—
334	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
335	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
336	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
337	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
338	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
339	Q₈×C₆₇	—	4	4	no	—
340	Q₈×C₆₇	—	4	4	no	—
341	Q₈×C₆₇	—	4	4	no	—
342	Q₈×C₆₇	—	4	4	no	—
343	Q₈×C₆₇	—	4	4	no	—
344	D₄×C₆₇	—	2	16	no	—
345	D₄×C₆₇	—	4	4	no	—
346	C₂×C₄×C₆₇	—	2	8	no	—
347	C₂×C₄×C₆₇	—	2	8	no	—
348	C₂×C₄×C₆₇	—	2	8	no	—
349	C₂×C₄×C₆₇	—	2	2	no	—
350	C₂×C₄×C₆₇	—	2	1	no	—
351	C₂×C₄×C₆₇	—	2	1	no	—
352	C₂×C₄×C₆₇	—	2	1	no	—
353	C₂×C₄×C₆₇	—	2	1	no	—
354	C₂×C₄×C₆₇	—	2	2	no	—
355	C₂×C₄×C₆₇	—	2	2	no	—
356	C₂×C₄×C₆₇	—	2	1	no	—
357	C₂×C₄×C₆₇	—	2	1	no	—
358	C₂×C₄×C₆₇	—	2	1	no	—
359	C₂×C₄×C₆₇	—	2	1	no	—
360	C₂×C₄×C₆₇	—	2	1	no	—
361	C₂×C₄×C₆₇	—	2	1	no	—
362	C₂×C₄×C₆₇	—	2	1	no	—
363	C₂×C₄×C₆₇	—	2	1	no	—
364	C₂×C₄×C₆₇	—	2	2	no	—
365	C₂×C₄×C₆₇	—	2	2	no	—
366	C₂×C₄×C₆₇	—	2	2	no	—
367	C₂×C₄×C₆₇	—	2	2	no	—
368	C₂×C₄×C₆₇	—	2	2	no	—
369	C₂×C₄×C₆₇	—	2	2	no	—
370	C₂×C₄×C₆₇	—	2	2	no	—
371	C₂×C₄×C₆₇	—	2	2	no	—
372	C₂×C₄×C₆₇	—	2	2	no	—
373	C₂×C₄×C₆₇	—	2	2	no	—
374	C₂×C₄×C₆₇	—	2	2	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	2	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	2	no	—
379	C₂×C₄×C₆₇	—	2	2	no	—
380	C₂×C₄×C₆₇	—	2	2	no	—
381	C₂×C₄×C₆₇	—	2	2	no	—
382	C₂×C₄×C₆₇	—	2	2	no	—
383	C₂×C₄×C₆₇	—	2	2	no	—
384	C₂×C₄×C₆₇	—	2	2	no	—
385	C₂×C₄×C₆₇	—	2	2	no	—
386	C₂×C₄×C₆₇	—	2	2	no	—
387	C₂×C₄×C₆₇	—	2	2	no	—
388	C₂×C₄×C₆₇	—	2	2	no	—
389	C₂×C₄×C₆₇	—	2	2	no	—
390	C₂×C₄×C₆₇	—	4	8	no	—
391	C₂×C₄×C₆₇	—	4	8	no	—
392	C₂×C₄×C₆₇	—	8	4	no	—
393	C₂×C₄×C₆₇	—	8	4	no	—
394	C₂×C₄×C₆₇	—	8	4	no	—
395	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
396	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
397	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
398	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
399	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
400	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
401	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
402	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
403	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
404	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
405	C₂²×C₆₇	—	2	4	no	—
406	C₂²×C₆₇	—	2	4	no	—
407	C₂²×C₆₇	—	2	2	no	—
408	C₂²×C₆₇	—	2	2	no	—
409	C₂²×C₆₇	—	2	4	no	—
410	C₂²×C₆₇	—	2	4	no	—
411	C₂²×C₆₇	—	2	4	no	—
412	C₂²×C₆₇	—	4	2	no	—
413	C₂²×C₆₇	—	4	2	no	—
414	C₂²×C₆₇	—	4	2	no	—
415	C₂²×C₆₇	—	4	2	no	—
416	C₂²×C₆₇	#518	1	4	yes	—
417	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
418	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
420	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
421	C₂²×C₆₇	#585	1	2	yes	—
422	C₄×C₆₇	—	2	2	no	—
423	C₄×C₆₇	—	2	2	no	—
424	C₄×C₆₇	—	2	2	no	—
425	C₄×C₆₇	—	2	2	no	—
426	C₄×C₆₇	—	2	2	no	—
427	C₄×C₆₇	—	2	2	no	—
428	C₄×C₆₇	—	2	4	no	—
429	C₄×C₆₇	—	2	4	no	—
430	C₄×C₆₇	—	2	4	no	—
431	C₄×C₆₇	—	2	4	no	—
432	C₄×C₆₇	—	4	2	no	—
433	C₄×C₆₇	—	4	2	no	—
434	C₄×C₆₇	—	4	2	no	—
435	C₄×C₆₇	—	4	2	no	—
436	C₄×C₆₇	—	4	2	no	—
437	C₄×C₆₇	—	4	2	no	—
438	C₄×C₆₇	—	4	2	no	—
439	#34409	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
440	C₂×C₆₇	—	2	4	no	—
441	C₂×C₆₇	—	4	4	no	—
442	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
443	C₂×C₆₇	C₄×2₋¹⁺⁴	1	2	yes	—
444	C₂×C₆₇	#3535	1	2	yes	—
445	C₂×C₆₇	#3544	1	1	yes	characteristic
446	C₂×C₆₇	#3545	1	1	yes	characteristic
447	C₂×#536	C₂×C₆₇	1	1	yes	characteristic
448	#2381	C₂×C₆₇	1	1	yes	characteristic, complemented
449	#2379	C₂×C₆₇	1	1	yes	characteristic, complemented
450	#2378	C₂×C₆₇	1	2	yes	complemented
451	#2372	C₂×C₆₇	1	1	yes	characteristic, complemented
452	#2369	C₂×C₆₇	1	1	yes	characteristic, complemented
453	#2368	C₂×C₆₇	1	2	yes	complemented
454	#2359	C₂×C₆₇	1	2	yes	complemented
455	#2359	C₂×C₆₇	1	2	yes	complemented
456	#2343	C₂×C₆₇	1	2	yes	complemented
457	#2339	C₂×C₆₇	1	2	yes	—
458	#2337	C₂×C₆₇	1	2	yes	complemented
459	#2332	C₂×C₆₇	1	2	yes	complemented
460	#2331	C₂×C₆₇	1	2	yes	complemented
461	#2329	C₂×C₆₇	1	2	yes	complemented
462	#2322	C₂×C₆₇	1	2	yes	complemented
463	#2310	C₂×C₆₇	1	1	yes	characteristic, complemented
464	C₄×#123	C₂×C₆₇	1	2	yes	complemented
465	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
466	C₆₇	#34409	1	1	yes	characteristic, direct product factor, Sylow
467	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
468	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
469	#536	C₄×C₆₇	1	16	yes	complemented
470	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
471	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
472	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
473	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
474	C₂×#123	C₂²×C₆₇	1	2	yes	—
475	C₂×#123	C₂²×C₆₇	1	2	yes	—
476	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
477	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
478	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
479	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
480	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
481	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
482	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
483	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
484	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
485	#391	C₂²×C₆₇	1	1	yes	characteristic
486	#391	C₂²×C₆₇	1	2	yes	—
487	#391	C₂²×C₆₇	1	1	yes	characteristic
488	#391	C₂²×C₆₇	1	2	yes	—
489	#391	C₂²×C₆₇	1	1	yes	characteristic
490	#391	C₂²×C₆₇	1	1	yes	characteristic
491	#391	C₂²×C₆₇	1	2	yes	—
492	#391	C₂²×C₆₇	1	2	yes	—
493	#391	C₂²×C₆₇	1	2	yes	—
494	#391	C₂²×C₆₇	1	2	yes	—
495	#391	C₂²×C₆₇	1	1	yes	characteristic
496	#391	C₂²×C₆₇	1	1	yes	characteristic
497	#391	C₂²×C₆₇	1	1	yes	characteristic
498	#391	C₂²×C₆₇	1	1	yes	characteristic
499	#389	C₂²×C₆₇	1	2	yes	—
500	#389	C₂²×C₆₇	1	2	yes	—
501	#389	C₂²×C₆₇	1	2	yes	—
502	#389	C₂²×C₆₇	1	2	yes	—
503	#388	C₂²×C₆₇	1	2	yes	—
504	#388	C₂²×C₆₇	1	2	yes	—
505	#388	C₂²×C₆₇	1	2	yes	—
506	#388	C₂²×C₆₇	1	2	yes	—
507	#388	C₂²×C₆₇	1	2	yes	—
508	#388	C₂²×C₆₇	1	2	yes	—
509	#388	C₂²×C₆₇	1	2	yes	—
510	#388	C₂²×C₆₇	1	2	yes	—
511	#387	C₂²×C₆₇	1	2	yes	—
512	#387	C₂²×C₆₇	1	2	yes	—
513	#387	C₂²×C₆₇	1	2	yes	—
514	#387	C₂²×C₆₇	1	2	yes	—
515	#387	C₂²×C₆₇	1	2	yes	—
516	#387	C₂²×C₆₇	1	2	yes	—
517	#387	C₂²×C₆₇	1	2	yes	—
518	#387	C₂²×C₆₇	1	2	yes	—
519	#387	C₂²×C₆₇	1	2	yes	—
520	#387	C₂²×C₆₇	1	2	yes	—
521	#387	C₂²×C₆₇	1	2	yes	—
522	#387	C₂²×C₆₇	1	2	yes	—
523	#386	C₂²×C₆₇	1	2	yes	—
524	#386	C₂²×C₆₇	1	2	yes	—
525	#386	C₂²×C₆₇	1	2	yes	—
526	#386	C₂²×C₆₇	1	2	yes	—
527	#385	C₂²×C₆₇	1	2	yes	—
528	#385	C₂²×C₆₇	1	2	yes	—
529	#385	C₂²×C₆₇	1	2	yes	—
530	#385	C₂²×C₆₇	1	2	yes	—
531	#385	C₂²×C₆₇	1	2	yes	—
532	#385	C₂²×C₆₇	1	2	yes	—
533	#383	C₂²×C₆₇	1	2	yes	—
534	#382	C₂²×C₆₇	1	2	yes	—
535	#381	C₂²×C₆₇	1	2	yes	—
536	#381	C₂²×C₆₇	1	2	yes	—
537	#381	C₂²×C₆₇	1	2	yes	—
538	#380	C₂²×C₆₇	1	2	yes	—
539	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
540	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
541	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
542	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
543	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
544	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
545	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
546	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
547	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
548	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
549	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
550	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
551	#376	C₂²×C₆₇	1	2	yes	—
552	C₂×#95	C₂²×C₆₇	1	2	yes	—
553	C₂×#95	C₂²×C₆₇	1	2	yes	—
554	C₂×#95	C₂²×C₆₇	1	2	yes	—
555	C₂×#41	—	4	2	no	—
556	C₂×#41	—	4	2	no	—
557	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
558	C₂²×Q₈	—	4	2	no	—
559	C₂²×Q₈	—	4	2	no	—
560	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
561	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
562	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
563	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
564	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
565	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
566	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
567	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
568	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
569	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
570	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
571	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
572	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
573	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
574	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
575	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
576	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
577	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
578	#124	C₂×C₄×C₆₇	1	8	yes	complemented
579	#124	C₂×C₄×C₆₇	1	8	yes	complemented
580	#123	C₂×C₄×C₆₇	1	16	yes	complemented
581	#123	C₂×C₄×C₆₇	1	16	yes	complemented
582	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
583	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
584	C₄×D₄	—	4	4	no	—
585	C₄×D₄	—	4	4	no	—
586	#117	—	4	2	no	—
587	#117	—	4	2	no	—
588	#117	C₂×C₄×C₆₇	1	8	yes	complemented
589	C₂×C₄⋊C₄	—	4	2	no	—
590	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
591	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
592	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
593	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
594	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
595	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
596	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
597	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
598	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
599	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
600	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
601	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
602	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
603	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
604	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
605	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
606	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
607	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
608	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
609	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
610	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
611	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
612	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
613	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
614	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
615	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
616	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
617	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
618	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
619	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
620	C₂×C₄²	—	4	2	no	—
621	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
622	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
623	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
624	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
625	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
626	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
627	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
628	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
629	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
630	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
631	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
632	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
633	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
634	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
635	#95	C₂³×C₆₇	1	2	yes	—
636	#95	C₂³×C₆₇	1	2	yes	—
637	#95	C₂³×C₆₇	1	2	yes	—
638	#95	C₂³×C₆₇	1	2	yes	—
639	#95	C₂³×C₆₇	1	2	yes	—
640	#95	C₂³×C₆₇	1	2	yes	—
641	#95	C₂³×C₆₇	1	2	yes	—
642	#95	C₂³×C₆₇	1	2	yes	—
643	#95	C₂³×C₆₇	1	2	yes	—
644	#95	C₂³×C₆₇	1	2	yes	—
645	#95	C₂³×C₆₇	1	2	yes	—
646	#95	C₂³×C₆₇	1	2	yes	—
647	#95	C₂³×C₆₇	1	2	yes	—
648	#95	C₂³×C₆₇	1	2	yes	—
649	#95	C₂³×C₆₇	1	2	yes	—
650	#95	C₂³×C₆₇	1	2	yes	—
651	#95	C₂³×C₆₇	1	2	yes	—
652	#95	C₂³×C₆₇	1	2	yes	—
653	#95	C₂³×C₆₇	1	2	yes	—
654	#95	C₂³×C₆₇	1	2	yes	—
655	#95	C₂³×C₆₇	1	2	yes	—
656	#95	C₂³×C₆₇	1	2	yes	—
657	#95	C₂³×C₆₇	1	2	yes	—
658	#95	C₂³×C₆₇	1	2	yes	—
659	#95	C₂³×C₆₇	1	2	yes	—
660	#95	C₂³×C₆₇	1	2	yes	—
661	#95	C₂³×C₆₇	1	2	yes	—
662	#95	C₂³×C₆₇	1	2	yes	—
663	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
664	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
665	#41	—	4	16	no	—
666	C₂×Q₈	—	4	16	no	—
667	C₂×Q₈	—	4	1	no	—
668	C₂×Q₈	—	4	1	no	—
669	C₂×Q₈	—	4	1	no	—
670	C₂×Q₈	—	4	1	no	—
671	C₂×Q₈	—	4	1	no	—
672	C₂×Q₈	—	4	1	no	—
673	C₂×Q₈	—	4	1	no	—
674	C₂×Q₈	—	4	1	no	—
675	C₂×Q₈	—	4	1	no	—
676	C₂×Q₈	—	4	1	no	—
677	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
678	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
679	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
680	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
681	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
682	C₂×D₄	—	2	4	no	—
683	C₂×D₄	—	2	4	no	—
684	C₂×D₄	—	4	1	no	—
685	C₂×D₄	—	4	1	no	—
686	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
687	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
688	C₂²×C₄	—	2	2	no	—
689	C₂²×C₄	—	2	2	no	—
690	C₂²×C₄	—	2	2	no	—
691	C₂²×C₄	—	2	2	no	—
692	C₂²×C₄	—	2	2	no	—
693	C₂²×C₄	—	2	2	no	—
694	C₂²×C₄	—	2	4	no	—
695	C₂²×C₄	—	2	4	no	—
696	C₂²×C₄	—	2	4	no	—
697	C₂²×C₄	—	2	4	no	—
698	C₂²×C₄	—	4	2	no	—
699	C₂²×C₄	—	4	2	no	—
700	C₂²×C₄	—	4	2	no	—
701	C₂²×C₄	—	4	2	no	—
702	C₂²×C₄	—	4	2	no	—
703	C₂²×C₄	—	4	2	no	—
704	C₂²×C₄	—	4	2	no	—
705	C₂²×C₄	—	4	2	no	—
706	C₂²×C₄	—	4	2	no	—
707	C₂²×C₄	—	4	2	no	—
708	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
709	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
710	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
711	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
712	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
713	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
714	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
715	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
716	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
717	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
718	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
719	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
720	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
721	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
722	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
723	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
724	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
725	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
726	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
727	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
728	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
729	C₄⋊C₄	—	2	4	no	—
730	C₄⋊C₄	—	4	1	no	—
731	C₄⋊C₄	—	4	2	no	—
732	C₄⋊C₄	—	4	1	no	—
733	C₄⋊C₄	—	4	1	no	—
734	C₄⋊C₄	—	4	2	no	—
735	C₄⋊C₄	—	4	1	no	—
736	C₄⋊C₄	—	4	2	no	—
737	C₄⋊C₄	—	4	2	no	—
738	C₄⋊C₄	—	4	2	no	—
739	C₄⋊C₄	—	4	2	no	—
740	C₄⋊C₄	—	4	2	no	—
741	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
742	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
743	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
744	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
745	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
746	C₂²⋊C₄	—	2	4	no	—
747	C₂²⋊C₄	—	2	4	no	—
748	C₂²⋊C₄	—	2	4	no	—
749	C₂²⋊C₄	—	4	2	no	—
750	C₂²⋊C₄	—	4	2	no	—
751	C₂²⋊C₄	—	4	2	no	—
752	C₂²⋊C₄	—	4	2	no	—
753	C₂²⋊C₄	—	4	2	no	—
754	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
755	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
756	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
757	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
758	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
759	C₄²	—	2	4	no	—
760	C₄²	—	4	1	no	—
761	C₄²	—	4	2	no	—
762	C₄²	—	4	1	no	—
763	C₄²	—	4	1	no	—
764	C₄²	—	4	2	no	—
765	C₄²	—	4	1	no	—
766	C₄²	—	4	2	no	—
767	C₄²	—	4	2	no	—
768	C₄²	—	4	2	no	—
769	C₄²	—	4	2	no	—
770	C₄²	—	4	2	no	—
771	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
772	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
773	C₂³	—	2	2	no	—
774	C₂³	—	2	2	no	—
775	C₂³	—	2	2	no	—
776	C₂³	—	2	2	no	—
777	C₂³	—	2	2	no	—
778	C₂³	—	2	2	no	—
779	C₂³	—	2	2	no	—
780	C₂³	—	2	2	no	—
781	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
782	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
783	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
784	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
785	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
786	Q₈	—	4	4	no	—
787	Q₈	—	4	4	no	—
788	Q₈	—	4	4	no	—
789	Q₈	—	4	4	no	—
790	Q₈	—	4	4	no	—
791	D₄	—	2	16	no	—
792	D₄	—	4	4	no	—
793	C₂×C₄	—	2	8	no	—
794	C₂×C₄	—	2	8	no	—
795	C₂×C₄	—	2	8	no	—
796	C₂×C₄	—	2	2	no	—
797	C₂×C₄	—	2	1	no	—
798	C₂×C₄	—	2	1	no	—
799	C₂×C₄	—	2	1	no	—
800	C₂×C₄	—	2	1	no	—
801	C₂×C₄	—	2	2	no	—
802	C₂×C₄	—	2	2	no	—
803	C₂×C₄	—	2	1	no	—
804	C₂×C₄	—	2	1	no	—
805	C₂×C₄	—	2	1	no	—
806	C₂×C₄	—	2	1	no	—
807	C₂×C₄	—	2	1	no	—
808	C₂×C₄	—	2	1	no	—
809	C₂×C₄	—	2	1	no	—
810	C₂×C₄	—	2	1	no	—
811	C₂×C₄	—	2	2	no	—
812	C₂×C₄	—	2	2	no	—
813	C₂×C₄	—	2	2	no	—
814	C₂×C₄	—	2	2	no	—
815	C₂×C₄	—	2	2	no	—
816	C₂×C₄	—	2	2	no	—
817	C₂×C₄	—	2	2	no	—
818	C₂×C₄	—	2	2	no	—
819	C₂×C₄	—	2	2	no	—
820	C₂×C₄	—	2	2	no	—
821	C₂×C₄	—	2	2	no	—
822	C₂×C₄	—	2	2	no	—
823	C₂×C₄	—	2	2	no	—
824	C₂×C₄	—	2	2	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	2	2	no	—
827	C₂×C₄	—	2	2	no	—
828	C₂×C₄	—	2	2	no	—
829	C₂×C₄	—	2	2	no	—
830	C₂×C₄	—	2	2	no	—
831	C₂×C₄	—	2	2	no	—
832	C₂×C₄	—	2	2	no	—
833	C₂×C₄	—	2	2	no	—
834	C₂×C₄	—	2	2	no	—
835	C₂×C₄	—	2	2	no	—
836	C₂×C₄	—	2	2	no	—
837	C₂×C₄	—	4	8	no	—
838	C₂×C₄	—	4	8	no	—
839	C₂×C₄	—	8	4	no	—
840	C₂×C₄	—	8	4	no	—
841	C₂×C₄	—	8	4	no	—
842	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
843	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
844	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
845	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
846	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
847	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
848	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
849	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
850	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
851	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
852	C₂²	—	2	4	no	—
853	C₂²	—	2	2	no	—
854	C₂²	—	2	2	no	—
855	C₂²	—	2	4	no	—
856	C₂²	—	2	4	no	—
857	C₂²	—	2	4	no	—
858	C₂²	—	2	4	no	—
859	C₂²	—	4	2	no	—
860	C₂²	—	4	2	no	—
861	C₂²	—	4	2	no	—
862	C₂²	—	4	2	no	—
863	C₂²	#518×C₆₇	1	4	yes	—
864	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
865	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
866	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
867	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
868	C₂²	#585×C₆₇	1	2	yes	—
869	C₄	—	2	4	no	—
870	C₄	—	2	2	no	—
871	C₄	—	2	2	no	—
872	C₄	—	2	2	no	—
873	C₄	—	2	2	no	—
874	C₄	—	2	2	no	—
875	C₄	—	2	2	no	—
876	C₄	—	2	4	no	—
877	C₄	—	2	4	no	—
878	C₄	—	2	4	no	—
879	C₄	—	4	2	no	—
880	C₄	—	4	2	no	—
881	C₄	—	4	2	no	—
882	C₄	—	4	2	no	—
883	C₄	—	4	2	no	—
884	C₄	—	4	2	no	—
885	C₄	—	4	2	no	—
886	C₂	—	2	4	no	—
887	C₂	—	4	4	no	—
888	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
889	C₂	C₄×2₋¹⁺⁴×C₆₇	1	2	yes	—
890	C₂	C₆₇×#3535	1	2	yes	—
891	C₂	C₆₇×#3544	1	1	yes	characteristic
892	C₂	C₆₇×#3545	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2381	C₂	semidirect
3. C₆₇×#2379	C₂	semidirect
4. C₆₇×#2378	C₂	semidirect
5. C₆₇×#2372	C₂	semidirect
6. C₆₇×#2369	C₂	semidirect
7. C₆₇×#2368	C₂	semidirect
8. C₆₇×#2359	C₂	semidirect
9. C₆₇×#2359	C₂	semidirect
10. C₆₇×#2343	C₂	semidirect
12. C₆₇×#2337	C₂	semidirect
13. C₆₇×#2332	C₂	semidirect
14. C₆₇×#2331	C₂	semidirect
15. C₆₇×#2329	C₂	semidirect
16. C₆₇×#2322	C₂	semidirect
17. C₆₇×#2310	C₂	semidirect
18. C₄×#123×C₆₇	C₂	semidirect
19. C₂×#391×C₆₇	C₂	semidirect
22. #536×C₆₇	C₄	semidirect
113. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
129. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
130. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
131. #124×C₆₇	C₂×C₄	semidirect
132. #124×C₆₇	C₂×C₄	semidirect
133. #123×C₆₇	C₂×C₄	semidirect
134. #123×C₆₇	C₂×C₄	semidirect
135. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
136. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
141. #117×C₆₇	C₂×C₄	semidirect
439. #34409	C₆₇	direct (N × H)
448. #2381	C₂×C₆₇	semidirect
449. #2379	C₂×C₆₇	semidirect
450. #2378	C₂×C₆₇	semidirect
451. #2372	C₂×C₆₇	semidirect
452. #2369	C₂×C₆₇	semidirect
453. #2368	C₂×C₆₇	semidirect
454. #2359	C₂×C₆₇	semidirect
455. #2359	C₂×C₆₇	semidirect
456. #2343	C₂×C₆₇	semidirect
458. #2337	C₂×C₆₇	semidirect
459. #2332	C₂×C₆₇	semidirect
460. #2331	C₂×C₆₇	semidirect
461. #2329	C₂×C₆₇	semidirect
462. #2322	C₂×C₆₇	semidirect
463. #2310	C₂×C₆₇	semidirect
464. C₄×#123	C₂×C₆₇	semidirect
465. C₂×#391	C₂×C₆₇	semidirect
466. C₆₇	#34409	direct (N × H)
469. #536	C₄×C₆₇	semidirect
560. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
576. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
577. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
578. #124	C₂×C₄×C₆₇	semidirect
579. #124	C₂×C₄×C₆₇	semidirect
580. #123	C₂×C₄×C₆₇	semidirect
581. #123	C₂×C₄×C₆₇	semidirect
582. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
583. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
588. #117	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#536×C₆₇	2
C₆₇×#2381	2
C₆₇×#2379	2
C₆₇×#2378	2
C₆₇×#2372	2
C₆₇×#2369	2
C₆₇×#2368	2
C₆₇×#2359	2
C₆₇×#2343	2
C₆₇×#2339	2
C₆₇×#2337	2
C₆₇×#2332	2
C₆₇×#2331	2
C₆₇×#2329	2
C₆₇×#2322	2
C₆₇×#2310	2
C₄×#123×C₆₇	2
C₂×#391×C₆₇	2
#34409	67

Subgroups — C67×#34409 #30713660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34409 #30713660