Subgroups — C₆₇×#34267 #30713518

Order 17152.

3154 conjugacy classes of subgroups.

The subgroup lattice has 848 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#537×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2364	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2363	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2362	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2361	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2361	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2360	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2324	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2323	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2311	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
20	#537×C₆₇	C₄	1	16	yes	complemented
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	C₂×#121×C₆₇	C₂²	1	2	yes	—
25	C₂×#121×C₆₇	C₂²	1	4	yes	—
26	#520×C₆₇	—	2	8	no	—
27	C₄×#41×C₆₇	—	2	8	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
38	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#117×C₆₇	C₂²	1	2	yes	—
40	C₂×#117×C₆₇	C₂²	1	2	yes	—
41	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
42	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
44	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
45	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
46	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
47	C₂²×C₄²×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	#390×C₆₇	C₂²	1	1	yes	characteristic
52	#390×C₆₇	C₂²	1	1	yes	characteristic
53	#390×C₆₇	C₂²	1	2	yes	—
54	#390×C₆₇	C₂²	1	4	yes	—
55	#389×C₆₇	C₂²	1	2	yes	—
56	#389×C₆₇	C₂²	1	2	yes	—
57	#389×C₆₇	C₂²	1	2	yes	—
58	#389×C₆₇	C₂²	1	2	yes	—
59	#389×C₆₇	C₂²	1	4	yes	—
60	#388×C₆₇	C₂²	1	4	yes	—
61	#388×C₆₇	C₂²	1	4	yes	—
62	#388×C₆₇	C₂²	1	4	yes	—
63	#388×C₆₇	C₂²	1	4	yes	—
64	#387×C₆₇	C₂²	1	4	yes	—
65	#387×C₆₇	C₂²	1	4	yes	—
66	#386×C₆₇	C₂²	1	2	yes	—
67	#386×C₆₇	C₂²	1	2	yes	—
68	#386×C₆₇	C₂²	1	4	yes	—
69	#386×C₆₇	C₂²	1	4	yes	—
70	#386×C₆₇	C₂²	1	4	yes	—
71	#385×C₆₇	C₂²	1	4	yes	—
72	#385×C₆₇	C₂²	1	4	yes	—
73	#380×C₆₇	C₂²	1	2	yes	—
74	#380×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
80	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
81	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
82	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
83	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
84	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
85	C₂×#95×C₆₇	C₂²	1	2	yes	—
86	C₂×#41×C₆₇	—	2	4	no	—
87	C₂×#41×C₆₇	—	2	4	no	—
88	C₂×#41×C₆₇	—	2	4	no	—
89	C₂×#41×C₆₇	—	2	4	no	—
90	C₂×#41×C₆₇	C₂×C₄	1	8	yes	complemented
91	C₂×#41×C₆₇	C₂×C₄	1	8	yes	complemented
92	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
95	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
96	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
99	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
101	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
103	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
104	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
105	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
106	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
107	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
108	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
110	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
111	#121×C₆₇	C₂×C₄	1	16	yes	complemented
112	#121×C₆₇	C₂×C₄	1	16	yes	complemented
113	#121×C₆₇	C₂×C₄	1	32	yes	complemented
114	C₄×Q₈×C₆₇	—	2	4	no	—
115	C₄×Q₈×C₆₇	—	2	4	no	—
116	C₄×Q₈×C₆₇	—	2	8	no	—
117	C₄×D₄×C₆₇	—	2	4	no	—
118	C₄×D₄×C₆₇	—	2	4	no	—
119	C₄×D₄×C₆₇	—	2	8	no	—
120	C₄×D₄×C₆₇	—	2	8	no	—
121	C₄×D₄×C₆₇	—	2	16	no	—
122	C₄×D₄×C₆₇	—	2	8	no	—
123	C₄×D₄×C₆₇	—	4	8	no	—
124	#117×C₆₇	—	2	4	no	—
125	#117×C₆₇	—	2	8	no	—
126	#117×C₆₇	—	2	8	no	—
127	#117×C₆₇	D₄	1	8	yes	complemented
128	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
131	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
132	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
134	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
135	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
136	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
141	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
143	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
144	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
146	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
147	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
148	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
149	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
150	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
151	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
152	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
153	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
154	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
155	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
156	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
157	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
158	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄²×C₆₇	—	2	4	no	—
160	C₂×C₄²×C₆₇	—	2	8	no	—
161	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
162	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
164	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
165	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
166	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
167	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
168	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
169	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
170	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
171	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
172	#95×C₆₇	C₂³	1	4	yes	—
173	#95×C₆₇	C₂³	1	4	yes	—
174	#95×C₆₇	C₂³	1	2	yes	—
175	#95×C₆₇	C₂³	1	4	yes	—
176	#95×C₆₇	C₂³	1	2	yes	—
177	#95×C₆₇	C₂³	1	4	yes	—
178	#95×C₆₇	C₂³	1	4	yes	—
179	#95×C₆₇	C₂³	1	2	yes	—
180	#95×C₆₇	C₂³	1	2	yes	—
181	#95×C₆₇	C₂³	1	4	yes	—
182	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
183	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
184	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
185	#41×C₆₇	—	2	32	no	—
186	#41×C₆₇	—	2	32	no	—
187	C₂×Q₈×C₆₇	—	2	2	no	—
188	C₂×Q₈×C₆₇	—	2	2	no	—
189	C₂×Q₈×C₆₇	—	2	2	no	—
190	C₂×Q₈×C₆₇	—	2	2	no	—
191	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
192	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
193	C₂×D₄×C₆₇	—	2	4	no	—
194	C₂×D₄×C₆₇	—	2	4	no	—
195	C₂×D₄×C₆₇	—	2	4	no	—
196	C₂×D₄×C₆₇	—	2	4	no	—
197	C₂×D₄×C₆₇	—	2	2	no	—
198	C₂×D₄×C₆₇	—	2	2	no	—
199	C₂×D₄×C₆₇	—	2	2	no	—
200	C₂×D₄×C₆₇	—	2	2	no	—
201	C₂×D₄×C₆₇	—	4	4	no	—
202	C₂×D₄×C₆₇	—	4	4	no	—
203	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
204	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
205	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
206	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
207	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
208	C₂²×C₄×C₆₇	—	2	4	no	—
209	C₂²×C₄×C₆₇	—	2	4	no	—
210	C₂²×C₄×C₆₇	—	2	4	no	—
211	C₂²×C₄×C₆₇	—	2	4	no	—
212	C₂²×C₄×C₆₇	—	2	2	no	—
213	C₂²×C₄×C₆₇	—	2	2	no	—
214	C₂²×C₄×C₆₇	—	2	4	no	—
215	C₂²×C₄×C₆₇	—	2	4	no	—
216	C₂²×C₄×C₆₇	—	2	4	no	—
217	C₂²×C₄×C₆₇	—	2	2	no	—
218	C₂²×C₄×C₆₇	—	2	4	no	—
219	C₂²×C₄×C₆₇	—	2	4	no	—
220	C₂²×C₄×C₆₇	—	2	2	no	—
221	C₂²×C₄×C₆₇	—	2	8	no	—
222	C₂²×C₄×C₆₇	—	4	2	no	—
223	C₂²×C₄×C₆₇	—	4	2	no	—
224	C₂²×C₄×C₆₇	—	4	2	no	—
225	C₂²×C₄×C₆₇	—	4	2	no	—
226	C₂²×C₄×C₆₇	—	4	1	no	—
227	C₂²×C₄×C₆₇	—	4	1	no	—
228	C₂²×C₄×C₆₇	—	4	2	no	—
229	C₂²×C₄×C₆₇	—	4	4	no	—
230	C₂²×C₄×C₆₇	—	4	4	no	—
231	C₂²×C₄×C₆₇	—	4	2	no	—
232	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
233	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
234	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
235	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
236	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
237	C₂²×C₄×C₆₇	#41	1	4	yes	—
238	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
239	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
240	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
241	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
242	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
243	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
244	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
245	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
246	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
247	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
248	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
249	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
250	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
251	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
252	C₄⋊C₄×C₆₇	—	2	4	no	—
253	C₄⋊C₄×C₆₇	—	2	8	no	—
254	C₄⋊C₄×C₆₇	—	2	4	no	—
255	C₄⋊C₄×C₆₇	—	2	4	no	—
256	C₄⋊C₄×C₆₇	—	2	4	no	—
257	C₄⋊C₄×C₆₇	—	2	2	no	—
258	C₄⋊C₄×C₆₇	—	2	2	no	—
259	C₄⋊C₄×C₆₇	—	2	8	no	—
260	C₄⋊C₄×C₆₇	—	4	4	no	—
261	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
262	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
263	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
264	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
265	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
266	C₂²⋊C₄×C₆₇	—	2	4	no	—
267	C₂²⋊C₄×C₆₇	—	2	4	no	—
268	C₂²⋊C₄×C₆₇	—	2	4	no	—
269	C₂²⋊C₄×C₆₇	—	2	8	no	—
270	C₂²⋊C₄×C₆₇	—	2	4	no	—
271	C₂²⋊C₄×C₆₇	—	2	4	no	—
272	C₂²⋊C₄×C₆₇	—	4	2	no	—
273	C₂²⋊C₄×C₆₇	—	4	2	no	—
274	C₂²⋊C₄×C₆₇	—	4	4	no	—
275	C₂²⋊C₄×C₆₇	—	4	4	no	—
276	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
277	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
278	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
279	C₄²×C₆₇	—	2	2	no	—
280	C₄²×C₆₇	—	2	2	no	—
281	C₄²×C₆₇	—	2	4	no	—
282	C₄²×C₆₇	—	2	4	no	—
283	C₄²×C₆₇	—	2	2	no	—
284	C₄²×C₆₇	—	2	2	no	—
285	C₄²×C₆₇	—	2	8	no	—
286	C₄²×C₆₇	—	2	4	no	—
287	C₄²×C₆₇	—	4	4	no	—
288	C₄²×C₆₇	—	4	4	no	—
289	C₄²×C₆₇	C₂×D₄	1	8	yes	—
290	C₂³×C₆₇	—	2	1	no	—
291	C₂³×C₆₇	—	2	1	no	—
292	C₂³×C₆₇	—	2	1	no	—
293	C₂³×C₆₇	—	2	1	no	—
294	C₂³×C₆₇	—	2	2	no	—
295	C₂³×C₆₇	—	2	2	no	—
296	C₂³×C₆₇	—	2	2	no	—
297	C₂³×C₆₇	—	2	2	no	—
298	C₂³×C₆₇	—	2	2	no	—
299	C₂³×C₆₇	—	2	2	no	—
300	C₂³×C₆₇	—	2	2	no	—
301	C₂³×C₆₇	—	2	2	no	—
302	C₂³×C₆₇	—	2	2	no	—
303	C₂³×C₆₇	—	2	2	no	—
304	C₂³×C₆₇	—	2	2	no	—
305	C₂³×C₆₇	—	2	2	no	—
306	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
307	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
308	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
309	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
310	C₂³×C₆₇	C₂×#41	1	2	yes	—
311	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
312	Q₈×C₆₇	—	2	8	no	—
313	Q₈×C₆₇	—	2	8	no	—
314	D₄×C₆₇	—	2	16	no	—
315	D₄×C₆₇	—	2	16	no	—
316	D₄×C₆₇	—	2	8	no	—
317	D₄×C₆₇	—	2	8	no	—
318	D₄×C₆₇	—	4	16	no	—
319	C₂×C₄×C₆₇	—	2	1	no	—
320	C₂×C₄×C₆₇	—	2	1	no	—
321	C₂×C₄×C₆₇	—	2	2	no	—
322	C₂×C₄×C₆₇	—	2	2	no	—
323	C₂×C₄×C₆₇	—	2	2	no	—
324	C₂×C₄×C₆₇	—	2	2	no	—
325	C₂×C₄×C₆₇	—	2	2	no	—
326	C₂×C₄×C₆₇	—	2	4	no	—
327	C₂×C₄×C₆₇	—	2	2	no	—
328	C₂×C₄×C₆₇	—	2	2	no	—
329	C₂×C₄×C₆₇	—	2	4	no	—
330	C₂×C₄×C₆₇	—	2	1	no	—
331	C₂×C₄×C₆₇	—	2	2	no	—
332	C₂×C₄×C₆₇	—	2	1	no	—
333	C₂×C₄×C₆₇	—	2	16	no	—
334	C₂×C₄×C₆₇	—	2	16	no	—
335	C₂×C₄×C₆₇	—	2	1	no	—
336	C₂×C₄×C₆₇	—	2	1	no	—
337	C₂×C₄×C₆₇	—	2	4	no	—
338	C₂×C₄×C₆₇	—	2	2	no	—
339	C₂×C₄×C₆₇	—	2	2	no	—
340	C₂×C₄×C₆₇	—	2	2	no	—
341	C₂×C₄×C₆₇	—	2	4	no	—
342	C₂×C₄×C₆₇	—	2	1	no	—
343	C₂×C₄×C₆₇	—	2	2	no	—
344	C₂×C₄×C₆₇	—	2	1	no	—
345	C₂×C₄×C₆₇	—	2	2	no	—
346	C₂×C₄×C₆₇	—	2	2	no	—
347	C₂×C₄×C₆₇	—	2	1	no	—
348	C₂×C₄×C₆₇	—	2	1	no	—
349	C₂×C₄×C₆₇	—	2	4	no	—
350	C₂×C₄×C₆₇	—	4	8	no	—
351	C₂×C₄×C₆₇	—	4	8	no	—
352	C₂×C₄×C₆₇	—	4	8	no	—
353	C₂×C₄×C₆₇	—	4	4	no	—
354	C₂×C₄×C₆₇	—	4	4	no	—
355	C₂×C₄×C₆₇	—	8	8	no	—
356	C₂×C₄×C₆₇	—	8	4	no	—
357	C₂×C₄×C₆₇	C₄×D₄	1	16	yes	complemented
358	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
359	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
360	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
361	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
362	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
363	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
364	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
365	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
366	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
367	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
368	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
369	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
370	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
371	C₂²×C₆₇	—	2	2	no	—
372	C₂²×C₆₇	—	2	2	no	—
373	C₂²×C₆₇	—	2	2	no	—
374	C₂²×C₆₇	—	2	2	no	—
375	C₂²×C₆₇	—	2	2	no	—
376	C₂²×C₆₇	—	2	2	no	—
377	C₂²×C₆₇	—	2	4	no	—
378	C₂²×C₆₇	—	2	4	no	—
379	C₂²×C₆₇	—	2	4	no	—
380	C₂²×C₆₇	—	2	4	no	—
381	C₂²×C₆₇	—	2	4	no	—
382	C₂²×C₆₇	—	2	4	no	—
383	C₂²×C₆₇	—	4	2	no	—
384	C₂²×C₆₇	—	4	2	no	—
385	C₂²×C₆₇	—	4	2	no	—
386	C₂²×C₆₇	—	4	2	no	—
387	C₂²×C₆₇	—	4	2	no	—
388	C₂²×C₆₇	—	4	2	no	—
389	C₂²×C₆₇	—	4	2	no	—
390	C₂²×C₆₇	—	4	2	no	—
391	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
392	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
393	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
394	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
395	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
396	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
397	C₂²×C₆₇	#585	1	1	yes	characteristic
398	C₂²×C₆₇	#585	1	1	yes	characteristic
399	C₄×C₆₇	—	2	4	no	—
400	C₄×C₆₇	—	2	4	no	—
401	C₄×C₆₇	—	2	8	no	—
402	C₄×C₆₇	—	2	2	no	—
403	C₄×C₆₇	—	2	4	no	—
404	C₄×C₆₇	—	2	2	no	—
405	C₄×C₆₇	—	2	4	no	—
406	C₄×C₆₇	—	4	1	no	—
407	C₄×C₆₇	—	4	4	no	—
408	C₄×C₆₇	—	4	1	no	—
409	C₄×C₆₇	—	4	2	no	—
410	C₄×C₆₇	—	4	2	no	—
411	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
412	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
413	#34267	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
414	C₂×C₆₇	—	2	4	no	—
415	C₂×C₆₇	—	4	4	no	—
416	C₂×C₆₇	—	4	4	no	—
417	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
418	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
420	C₂×C₆₇	#3488	1	1	yes	characteristic
421	C₂×C₆₇	#3489	1	1	yes	characteristic
422	C₂×C₆₇	#3511	1	1	yes	characteristic
423	C₂×C₆₇	#3512	1	1	yes	characteristic
424	C₂×#537	C₂×C₆₇	1	1	yes	characteristic
425	C₂×#520	C₂×C₆₇	1	1	yes	characteristic, complemented
426	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
427	#2364	C₂×C₆₇	1	4	yes	complemented
428	#2363	C₂×C₆₇	1	2	yes	complemented
429	#2362	C₂×C₆₇	1	2	yes	complemented
430	#2361	C₂×C₆₇	1	2	yes	complemented
431	#2361	C₂×C₆₇	1	4	yes	complemented
432	#2360	C₂×C₆₇	1	2	yes	complemented
433	#2324	C₂×C₆₇	1	2	yes	complemented
434	#2323	C₂×C₆₇	1	2	yes	complemented
435	#2311	C₂×C₆₇	1	1	yes	characteristic, complemented
436	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
437	C₄×#121	C₂×C₆₇	1	2	yes	complemented
438	#2297	C₂×C₆₇	1	1	yes	characteristic, complemented
439	C₂×#389	C₂×C₆₇	1	1	yes	characteristic, complemented
440	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
441	C₆₇	#34267	1	1	yes	characteristic, direct product factor, Sylow
442	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
443	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
444	#537	C₄×C₆₇	1	16	yes	complemented
445	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
446	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
447	C₂×#121	C₂²×C₆₇	1	2	yes	—
448	C₂×#121	C₂²×C₆₇	1	2	yes	—
449	C₂×#121	C₂²×C₆₇	1	4	yes	—
450	#520	—	2	8	no	—
451	C₄×#41	—	2	8	no	—
452	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
453	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
454	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
455	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
456	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
457	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
458	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
459	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
460	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
461	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
462	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
463	C₂×#117	C₂²×C₆₇	1	2	yes	—
464	C₂×#117	C₂²×C₆₇	1	2	yes	—
465	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
466	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
467	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
468	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
469	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
470	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
471	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
472	#391	C₂²×C₆₇	1	2	yes	—
473	#391	C₂²×C₆₇	1	1	yes	characteristic
474	#391	C₂²×C₆₇	1	1	yes	characteristic
475	#390	C₂²×C₆₇	1	2	yes	—
476	#390	C₂²×C₆₇	1	1	yes	characteristic
477	#390	C₂²×C₆₇	1	4	yes	—
478	#390	C₂²×C₆₇	1	1	yes	characteristic
479	#389	C₂²×C₆₇	1	2	yes	—
480	#389	C₂²×C₆₇	1	4	yes	—
481	#389	C₂²×C₆₇	1	2	yes	—
482	#389	C₂²×C₆₇	1	2	yes	—
483	#389	C₂²×C₆₇	1	2	yes	—
484	#388	C₂²×C₆₇	1	4	yes	—
485	#388	C₂²×C₆₇	1	4	yes	—
486	#388	C₂²×C₆₇	1	4	yes	—
487	#388	C₂²×C₆₇	1	4	yes	—
488	#387	C₂²×C₆₇	1	4	yes	—
489	#387	C₂²×C₆₇	1	4	yes	—
490	#386	C₂²×C₆₇	1	4	yes	—
491	#386	C₂²×C₆₇	1	4	yes	—
492	#386	C₂²×C₆₇	1	2	yes	—
493	#386	C₂²×C₆₇	1	4	yes	—
494	#386	C₂²×C₆₇	1	2	yes	—
495	#385	C₂²×C₆₇	1	4	yes	—
496	#385	C₂²×C₆₇	1	4	yes	—
497	#380	C₂²×C₆₇	1	2	yes	—
498	#380	C₂²×C₆₇	1	2	yes	—
499	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
500	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
501	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
502	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
503	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
504	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
505	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
506	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
507	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
508	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
509	C₂×#95	C₂²×C₆₇	1	2	yes	—
510	C₂×#41	—	2	4	no	—
511	C₂×#41	—	2	4	no	—
512	C₂×#41	—	2	4	no	—
513	C₂×#41	—	2	4	no	—
514	C₂×#41	C₂×C₄×C₆₇	1	8	yes	complemented
515	C₂×#41	C₂×C₄×C₆₇	1	8	yes	complemented
516	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
517	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
518	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
519	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
520	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
521	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
522	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
523	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
524	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
525	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
526	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
527	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
528	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
529	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
530	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
531	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
532	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
533	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
534	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
535	#121	C₂×C₄×C₆₇	1	16	yes	complemented
536	#121	C₂×C₄×C₆₇	1	16	yes	complemented
537	#121	C₂×C₄×C₆₇	1	32	yes	complemented
538	C₄×Q₈	—	2	4	no	—
539	C₄×Q₈	—	2	4	no	—
540	C₄×Q₈	—	2	8	no	—
541	C₄×D₄	—	2	4	no	—
542	C₄×D₄	—	2	4	no	—
543	C₄×D₄	—	2	8	no	—
544	C₄×D₄	—	2	8	no	—
545	C₄×D₄	—	2	16	no	—
546	C₄×D₄	—	2	8	no	—
547	C₄×D₄	—	4	8	no	—
548	#117	—	2	4	no	—
549	#117	—	2	8	no	—
550	#117	—	2	8	no	—
551	#117	D₄×C₆₇	1	8	yes	complemented
552	C₂×C₄⋊C₄	—	2	4	no	—
553	C₂×C₄⋊C₄	—	2	8	no	—
554	C₂×C₄⋊C₄	—	4	4	no	—
555	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
556	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
557	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
558	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
559	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
560	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
561	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
562	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
563	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
564	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
565	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
566	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
567	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
568	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
569	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
570	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
571	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
572	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
573	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
574	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
575	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
576	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
577	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
578	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
579	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
580	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
581	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
582	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
583	C₂×C₄²	—	2	4	no	—
584	C₂×C₄²	—	2	8	no	—
585	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
586	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
587	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
588	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
589	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
590	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
591	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
592	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
593	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
594	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
595	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
596	#95	C₂³×C₆₇	1	4	yes	—
597	#95	C₂³×C₆₇	1	4	yes	—
598	#95	C₂³×C₆₇	1	2	yes	—
599	#95	C₂³×C₆₇	1	4	yes	—
600	#95	C₂³×C₆₇	1	2	yes	—
601	#95	C₂³×C₆₇	1	4	yes	—
602	#95	C₂³×C₆₇	1	4	yes	—
603	#95	C₂³×C₆₇	1	2	yes	—
604	#95	C₂³×C₆₇	1	2	yes	—
605	#95	C₂³×C₆₇	1	4	yes	—
606	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
607	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
608	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
609	#41	—	2	32	no	—
610	#41	—	2	32	no	—
611	C₂×Q₈	—	2	2	no	—
612	C₂×Q₈	—	2	2	no	—
613	C₂×Q₈	—	2	2	no	—
614	C₂×Q₈	—	2	2	no	—
615	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
616	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
617	C₂×D₄	—	2	4	no	—
618	C₂×D₄	—	2	4	no	—
619	C₂×D₄	—	2	4	no	—
620	C₂×D₄	—	2	4	no	—
621	C₂×D₄	—	2	2	no	—
622	C₂×D₄	—	2	2	no	—
623	C₂×D₄	—	2	2	no	—
624	C₂×D₄	—	2	2	no	—
625	C₂×D₄	—	4	4	no	—
626	C₂×D₄	—	4	4	no	—
627	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
628	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
629	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
630	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
631	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
632	C₂²×C₄	—	2	4	no	—
633	C₂²×C₄	—	2	4	no	—
634	C₂²×C₄	—	2	4	no	—
635	C₂²×C₄	—	2	4	no	—
636	C₂²×C₄	—	2	2	no	—
637	C₂²×C₄	—	2	2	no	—
638	C₂²×C₄	—	2	4	no	—
639	C₂²×C₄	—	2	4	no	—
640	C₂²×C₄	—	2	4	no	—
641	C₂²×C₄	—	2	2	no	—
642	C₂²×C₄	—	2	4	no	—
643	C₂²×C₄	—	2	4	no	—
644	C₂²×C₄	—	2	2	no	—
645	C₂²×C₄	—	2	8	no	—
646	C₂²×C₄	—	4	2	no	—
647	C₂²×C₄	—	4	2	no	—
648	C₂²×C₄	—	4	2	no	—
649	C₂²×C₄	—	4	2	no	—
650	C₂²×C₄	—	4	1	no	—
651	C₂²×C₄	—	4	1	no	—
652	C₂²×C₄	—	4	2	no	—
653	C₂²×C₄	—	4	4	no	—
654	C₂²×C₄	—	4	4	no	—
655	C₂²×C₄	—	4	2	no	—
656	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
657	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
658	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
659	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
660	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
661	C₂²×C₄	#41×C₆₇	1	4	yes	—
662	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
663	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
664	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
665	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
666	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
667	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
668	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
669	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
670	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
671	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
672	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
673	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
674	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
675	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
676	C₄⋊C₄	—	2	4	no	—
677	C₄⋊C₄	—	2	8	no	—
678	C₄⋊C₄	—	2	4	no	—
679	C₄⋊C₄	—	2	4	no	—
680	C₄⋊C₄	—	2	4	no	—
681	C₄⋊C₄	—	2	2	no	—
682	C₄⋊C₄	—	2	2	no	—
683	C₄⋊C₄	—	2	8	no	—
684	C₄⋊C₄	—	4	4	no	—
685	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
686	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
687	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
688	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
689	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
690	C₂²⋊C₄	—	2	4	no	—
691	C₂²⋊C₄	—	2	4	no	—
692	C₂²⋊C₄	—	2	4	no	—
693	C₂²⋊C₄	—	2	8	no	—
694	C₂²⋊C₄	—	2	4	no	—
695	C₂²⋊C₄	—	2	4	no	—
696	C₂²⋊C₄	—	4	4	no	—
697	C₂²⋊C₄	—	4	2	no	—
698	C₂²⋊C₄	—	4	2	no	—
699	C₂²⋊C₄	—	4	4	no	—
700	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
701	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
702	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
703	C₄²	—	2	2	no	—
704	C₄²	—	2	2	no	—
705	C₄²	—	2	4	no	—
706	C₄²	—	2	4	no	—
707	C₄²	—	2	2	no	—
708	C₄²	—	2	2	no	—
709	C₄²	—	2	8	no	—
710	C₄²	—	2	4	no	—
711	C₄²	—	4	4	no	—
712	C₄²	—	4	4	no	—
713	C₄²	C₂×D₄×C₆₇	1	8	yes	—
714	C₂³	—	2	1	no	—
715	C₂³	—	2	1	no	—
716	C₂³	—	2	1	no	—
717	C₂³	—	2	1	no	—
718	C₂³	—	2	2	no	—
719	C₂³	—	2	2	no	—
720	C₂³	—	2	2	no	—
721	C₂³	—	2	2	no	—
722	C₂³	—	2	2	no	—
723	C₂³	—	2	2	no	—
724	C₂³	—	2	2	no	—
725	C₂³	—	2	2	no	—
726	C₂³	—	2	2	no	—
727	C₂³	—	2	2	no	—
728	C₂³	—	2	2	no	—
729	C₂³	—	2	2	no	—
730	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
731	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
732	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
733	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
734	C₂³	C₂×#41×C₆₇	1	2	yes	—
735	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
736	Q₈	—	2	8	no	—
737	Q₈	—	2	8	no	—
738	D₄	—	2	16	no	—
739	D₄	—	2	16	no	—
740	D₄	—	2	8	no	—
741	D₄	—	2	8	no	—
742	D₄	—	4	16	no	—
743	C₂×C₄	—	2	1	no	—
744	C₂×C₄	—	2	1	no	—
745	C₂×C₄	—	2	2	no	—
746	C₂×C₄	—	2	2	no	—
747	C₂×C₄	—	2	2	no	—
748	C₂×C₄	—	2	2	no	—
749	C₂×C₄	—	2	2	no	—
750	C₂×C₄	—	2	4	no	—
751	C₂×C₄	—	2	2	no	—
752	C₂×C₄	—	2	2	no	—
753	C₂×C₄	—	2	4	no	—
754	C₂×C₄	—	2	1	no	—
755	C₂×C₄	—	2	2	no	—
756	C₂×C₄	—	2	1	no	—
757	C₂×C₄	—	2	16	no	—
758	C₂×C₄	—	2	16	no	—
759	C₂×C₄	—	2	1	no	—
760	C₂×C₄	—	2	1	no	—
761	C₂×C₄	—	2	4	no	—
762	C₂×C₄	—	2	2	no	—
763	C₂×C₄	—	2	2	no	—
764	C₂×C₄	—	2	2	no	—
765	C₂×C₄	—	2	4	no	—
766	C₂×C₄	—	2	1	no	—
767	C₂×C₄	—	2	2	no	—
768	C₂×C₄	—	2	1	no	—
769	C₂×C₄	—	2	2	no	—
770	C₂×C₄	—	2	2	no	—
771	C₂×C₄	—	2	1	no	—
772	C₂×C₄	—	2	1	no	—
773	C₂×C₄	—	2	4	no	—
774	C₂×C₄	—	4	8	no	—
775	C₂×C₄	—	4	8	no	—
776	C₂×C₄	—	4	8	no	—
777	C₂×C₄	—	4	4	no	—
778	C₂×C₄	—	4	4	no	—
779	C₂×C₄	—	8	8	no	—
780	C₂×C₄	—	8	4	no	—
781	C₂×C₄	C₄×D₄×C₆₇	1	16	yes	complemented
782	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
783	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
784	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
785	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
786	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
787	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
788	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
789	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
790	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
791	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
792	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
793	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
794	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
795	C₂²	—	2	2	no	—
796	C₂²	—	2	2	no	—
797	C₂²	—	2	2	no	—
798	C₂²	—	2	2	no	—
799	C₂²	—	2	2	no	—
800	C₂²	—	2	2	no	—
801	C₂²	—	2	4	no	—
802	C₂²	—	2	4	no	—
803	C₂²	—	2	4	no	—
804	C₂²	—	2	4	no	—
805	C₂²	—	2	4	no	—
806	C₂²	—	2	4	no	—
807	C₂²	—	4	2	no	—
808	C₂²	—	4	2	no	—
809	C₂²	—	4	2	no	—
810	C₂²	—	4	2	no	—
811	C₂²	—	4	2	no	—
812	C₂²	—	4	2	no	—
813	C₂²	—	4	2	no	—
814	C₂²	—	4	2	no	—
815	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
816	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
817	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
818	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
819	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
820	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
821	C₂²	#585×C₆₇	1	1	yes	characteristic
822	C₂²	#585×C₆₇	1	1	yes	characteristic
823	C₄	—	2	4	no	—
824	C₄	—	2	4	no	—
825	C₄	—	2	8	no	—
826	C₄	—	2	2	no	—
827	C₄	—	2	4	no	—
828	C₄	—	2	2	no	—
829	C₄	—	2	4	no	—
830	C₄	—	4	1	no	—
831	C₄	—	4	4	no	—
832	C₄	—	4	1	no	—
833	C₄	—	4	2	no	—
834	C₄	—	4	2	no	—
835	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
836	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
837	C₂	—	2	4	no	—
838	C₂	—	4	4	no	—
839	C₂	—	4	4	no	—
840	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
841	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
842	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
843	C₂	C₆₇×#3488	1	1	yes	characteristic
844	C₂	C₆₇×#3489	1	1	yes	characteristic
845	C₂	C₆₇×#3511	1	1	yes	characteristic
846	C₂	C₆₇×#3512	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#537×C₆₇	C₂
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. C₂×#520×C₆₇	C₂	semidirect
3. C₂×C₄×#41×C₆₇	C₂	semidirect
4. C₆₇×#2364	C₂	semidirect
5. C₆₇×#2363	C₂	semidirect
6. C₆₇×#2362	C₂	semidirect
7. C₆₇×#2361	C₂	semidirect
8. C₆₇×#2361	C₂	semidirect
9. C₆₇×#2360	C₂	semidirect
10. C₆₇×#2324	C₂	semidirect
11. C₆₇×#2323	C₂	semidirect
12. C₆₇×#2311	C₂	semidirect
13. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
14. C₄×#121×C₆₇	C₂	semidirect
15. C₆₇×#2297	C₂	semidirect
16. C₂×#389×C₆₇	C₂	semidirect
17. C₆₇×#2274	C₂	semidirect
20. #537×C₆₇	C₄	semidirect
90. C₂×#41×C₆₇	C₂×C₄	semidirect
91. C₂×#41×C₆₇	C₂×C₄	semidirect
109. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
110. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
111. #121×C₆₇	C₂×C₄	semidirect
112. #121×C₆₇	C₂×C₄	semidirect
113. #121×C₆₇	C₂×C₄	semidirect
127. #117×C₆₇	D₄	semidirect
131. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
357. C₂×C₄×C₆₇	C₄×D₄	semidirect
413. #34267	C₆₇	direct (N × H)
425. C₂×#520	C₂×C₆₇	semidirect
426. C₂×C₄×#41	C₂×C₆₇	semidirect
427. #2364	C₂×C₆₇	semidirect
428. #2363	C₂×C₆₇	semidirect
429. #2362	C₂×C₆₇	semidirect
430. #2361	C₂×C₆₇	semidirect
431. #2361	C₂×C₆₇	semidirect
432. #2360	C₂×C₆₇	semidirect
433. #2324	C₂×C₆₇	semidirect
434. #2323	C₂×C₆₇	semidirect
435. #2311	C₂×C₆₇	semidirect
436. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
437. C₄×#121	C₂×C₆₇	semidirect
438. #2297	C₂×C₆₇	semidirect
439. C₂×#389	C₂×C₆₇	semidirect
440. #2274	C₂×C₆₇	semidirect
441. C₆₇	#34267	direct (N × H)
444. #537	C₄×C₆₇	semidirect
514. C₂×#41	C₂×C₄×C₆₇	semidirect
515. C₂×#41	C₂×C₄×C₆₇	semidirect
533. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
534. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
535. #121	C₂×C₄×C₆₇	semidirect
536. #121	C₂×C₄×C₆₇	semidirect
537. #121	C₂×C₄×C₆₇	semidirect
551. #117	D₄×C₆₇	semidirect
555. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
781. C₂×C₄	C₄×D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#537×C₆₇	2
C₂×#520×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2364	2
C₆₇×#2363	2
C₆₇×#2362	2
C₆₇×#2361	2
C₆₇×#2360	2
C₆₇×#2324	2
C₆₇×#2323	2
C₆₇×#2311	2
C₄×C₂²⋊Q₈×C₆₇	2
C₄×#121×C₆₇	2
C₆₇×#2297	2
C₂×#389×C₆₇	2
C₆₇×#2274	2
#34267	67

Subgroups — C67×#34267 #30713518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34267 #30713518