Subgroups — C₆₇×#34384 #30713635

Order 17152.

2318 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2383	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2361	C₂	1	2	yes	maximal
4	C₆₇×#2360	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2352	C₂	1	2	yes	maximal, complemented
8	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
10	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2330	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2326	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2322	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2317	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2314	C₂	1	2	yes	maximal, complemented
16	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
18	#529×C₆₇	C₄	1	16	yes	complemented
19	C₂×#126×C₆₇	C₂²	1	2	yes	—
20	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#123×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#117×C₆₇	C₂²	1	2	yes	—
32	C₂×#117×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	#391×C₆₇	C₂²	1	2	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	4	yes	—
41	#389×C₆₇	C₂²	1	4	yes	—
42	#389×C₆₇	C₂²	1	4	yes	—
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	4	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#388×C₆₇	C₂²	1	4	yes	—
47	#388×C₆₇	C₂²	1	4	yes	—
48	#388×C₆₇	C₂²	1	4	yes	—
49	#388×C₆₇	C₂²	1	4	yes	—
50	#387×C₆₇	C₂²	1	4	yes	—
51	#387×C₆₇	C₂²	1	4	yes	—
52	#387×C₆₇	C₂²	1	4	yes	—
53	#387×C₆₇	C₂²	1	4	yes	—
54	#386×C₆₇	C₂²	1	4	yes	—
55	#385×C₆₇	C₂²	1	4	yes	—
56	#385×C₆₇	C₂²	1	4	yes	—
57	#385×C₆₇	C₂²	1	4	yes	—
58	#385×C₆₇	C₂²	1	4	yes	—
59	#385×C₆₇	C₂²	1	4	yes	—
60	#384×C₆₇	C₂²	1	2	yes	—
61	#384×C₆₇	C₂²	1	2	yes	—
62	#384×C₆₇	C₂²	1	1	yes	characteristic
63	#384×C₆₇	C₂²	1	1	yes	characteristic
64	#384×C₆₇	C₂²	1	2	yes	—
65	#384×C₆₇	C₂²	1	2	yes	—
66	#383×C₆₇	C₂²	1	2	yes	—
67	#382×C₆₇	C₂²	1	1	yes	characteristic
68	#382×C₆₇	C₂²	1	1	yes	characteristic
69	#382×C₆₇	C₂²	1	2	yes	—
70	#381×C₆₇	C₂²	1	2	yes	—
71	#380×C₆₇	C₂²	1	1	yes	characteristic
72	#380×C₆₇	C₂²	1	1	yes	characteristic
73	#380×C₆₇	C₂²	1	2	yes	—
74	#380×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
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	2	yes	—
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
82	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
83	#376×C₆₇	C₂²	1	2	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
91	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
92	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
93	#126×C₆₇	C₂×C₄	1	16	yes	complemented
94	#125×C₆₇	C₂×C₄	1	8	yes	complemented
95	#124×C₆₇	C₂×C₄	1	8	yes	complemented
96	#123×C₆₇	C₂×C₄	1	16	yes	complemented
97	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
98	#121×C₆₇	C₂×C₄	1	8	yes	—
99	C₄×Q₈×C₆₇	—	4	4	no	—
100	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
101	C₄×D₄×C₆₇	—	4	4	no	—
102	C₄×D₄×C₆₇	—	4	8	no	—
103	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
104	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
105	#117×C₆₇	—	4	4	no	—
106	#117×C₆₇	C₂×C₄	1	16	yes	complemented
107	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
108	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
109	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
110	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
136	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
137	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
139	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
143	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
144	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
145	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
146	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
147	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
148	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
149	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	2	yes	—
156	#95×C₆₇	C₂³	1	2	yes	—
157	#95×C₆₇	C₂³	1	2	yes	—
158	#95×C₆₇	C₂³	1	4	yes	—
159	#95×C₆₇	C₂³	1	2	yes	—
160	#95×C₆₇	C₂³	1	2	yes	—
161	#95×C₆₇	C₂³	1	4	yes	—
162	#95×C₆₇	C₂³	1	4	yes	—
163	#95×C₆₇	C₂³	1	2	yes	—
164	#95×C₆₇	C₂³	1	2	yes	—
165	#95×C₆₇	C₂³	1	4	yes	—
166	#95×C₆₇	C₂³	1	4	yes	—
167	#95×C₆₇	C₂³	1	4	yes	—
168	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
169	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂×Q₈×C₆₇	—	2	2	no	—
171	C₂×Q₈×C₆₇	—	2	2	no	—
172	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
173	C₂×D₄×C₆₇	—	2	4	no	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	2	no	—
176	C₂×D₄×C₆₇	—	2	2	no	—
177	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
178	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
179	C₂²×C₄×C₆₇	—	2	2	no	—
180	C₂²×C₄×C₆₇	—	2	2	no	—
181	C₂²×C₄×C₆₇	—	2	2	no	—
182	C₂²×C₄×C₆₇	—	2	2	no	—
183	C₂²×C₄×C₆₇	—	2	4	no	—
184	C₂²×C₄×C₆₇	—	2	2	no	—
185	C₂²×C₄×C₆₇	—	2	2	no	—
186	C₂²×C₄×C₆₇	—	2	4	no	—
187	C₂²×C₄×C₆₇	—	2	8	no	—
188	C₂²×C₄×C₆₇	—	4	4	no	—
189	C₂²×C₄×C₆₇	—	4	4	no	—
190	C₂²×C₄×C₆₇	—	4	4	no	—
191	C₂²×C₄×C₆₇	—	4	4	no	—
192	C₂²×C₄×C₆₇	—	4	4	no	—
193	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
194	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
195	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
196	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
197	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
209	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₄⋊C₄×C₆₇	—	2	2	no	—
213	C₄⋊C₄×C₆₇	—	2	4	no	—
214	C₄⋊C₄×C₆₇	—	2	4	no	—
215	C₄⋊C₄×C₆₇	—	2	4	no	—
216	C₄⋊C₄×C₆₇	—	2	4	no	—
217	C₄⋊C₄×C₆₇	—	2	2	no	—
218	C₄⋊C₄×C₆₇	—	2	2	no	—
219	C₄⋊C₄×C₆₇	—	2	2	no	—
220	C₄⋊C₄×C₆₇	—	4	4	no	—
221	C₄⋊C₄×C₆₇	—	4	4	no	—
222	C₄⋊C₄×C₆₇	—	4	4	no	—
223	C₄⋊C₄×C₆₇	—	4	2	no	—
224	C₄⋊C₄×C₆₇	—	4	4	no	—
225	C₄⋊C₄×C₆₇	—	4	2	no	—
226	C₄⋊C₄×C₆₇	—	4	4	no	—
227	C₄⋊C₄×C₆₇	—	4	4	no	—
228	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
229	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
230	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
231	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
232	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
233	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
234	C₂²⋊C₄×C₆₇	—	2	4	no	—
235	C₂²⋊C₄×C₆₇	—	2	4	no	—
236	C₂²⋊C₄×C₆₇	—	4	4	no	—
237	C₂²⋊C₄×C₆₇	—	4	4	no	—
238	C₂²⋊C₄×C₆₇	—	4	4	no	—
239	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
240	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
241	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
242	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
243	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
244	C₄²×C₆₇	—	2	4	no	—
245	C₄²×C₆₇	—	2	4	no	—
246	C₄²×C₆₇	—	4	4	no	—
247	C₄²×C₆₇	—	4	4	no	—
248	C₄²×C₆₇	—	4	4	no	—
249	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
250	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
251	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
252	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
253	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
254	C₂³×C₆₇	—	2	1	no	—
255	C₂³×C₆₇	—	2	1	no	—
256	C₂³×C₆₇	—	2	1	no	—
257	C₂³×C₆₇	—	2	1	no	—
258	C₂³×C₆₇	—	2	2	no	—
259	C₂³×C₆₇	—	2	2	no	—
260	C₂³×C₆₇	—	2	2	no	—
261	C₂³×C₆₇	—	2	2	no	—
262	C₂³×C₆₇	—	2	2	no	—
263	C₂³×C₆₇	—	2	2	no	—
264	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
265	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
266	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
267	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
268	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
269	Q₈×C₆₇	—	2	8	no	—
270	D₄×C₆₇	—	2	16	no	—
271	D₄×C₆₇	—	2	8	no	—
272	C₂×C₄×C₆₇	—	2	8	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	1	no	—
277	C₂×C₄×C₆₇	—	2	1	no	—
278	C₂×C₄×C₆₇	—	2	1	no	—
279	C₂×C₄×C₆₇	—	2	1	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	1	no	—
282	C₂×C₄×C₆₇	—	2	1	no	—
283	C₂×C₄×C₆₇	—	2	1	no	—
284	C₂×C₄×C₆₇	—	2	1	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	2	8	no	—
288	C₂×C₄×C₆₇	—	2	16	no	—
289	C₂×C₄×C₆₇	—	2	8	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	2	2	no	—
293	C₂×C₄×C₆₇	—	2	2	no	—
294	C₂×C₄×C₆₇	—	2	4	no	—
295	C₂×C₄×C₆₇	—	2	4	no	—
296	C₂×C₄×C₆₇	—	2	4	no	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	4	no	—
299	C₂×C₄×C₆₇	—	2	4	no	—
300	C₂×C₄×C₆₇	—	2	2	no	—
301	C₂×C₄×C₆₇	—	2	4	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	2	no	—
304	C₂×C₄×C₆₇	—	2	2	no	—
305	C₂×C₄×C₆₇	—	2	2	no	—
306	C₂×C₄×C₆₇	—	2	2	no	—
307	C₂×C₄×C₆₇	—	2	2	no	—
308	C₂×C₄×C₆₇	—	8	8	no	—
309	C₂×C₄×C₆₇	—	8	8	no	—
310	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
311	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
312	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
313	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
314	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
315	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
316	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
317	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
318	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
319	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
320	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
321	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
322	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
323	C₂²×C₆₇	—	2	2	no	—
324	C₂²×C₆₇	—	2	2	no	—
325	C₂²×C₆₇	—	2	2	no	—
326	C₂²×C₆₇	—	2	2	no	—
327	C₂²×C₆₇	—	2	2	no	—
328	C₂²×C₆₇	—	2	2	no	—
329	C₂²×C₆₇	—	2	4	no	—
330	C₂²×C₆₇	—	2	4	no	—
331	C₂²×C₆₇	—	2	4	no	—
332	C₂²×C₆₇	—	4	2	no	—
333	C₂²×C₆₇	—	4	2	no	—
334	C₂²×C₆₇	—	4	2	no	—
335	C₂²×C₆₇	—	4	2	no	—
336	C₂²×C₆₇	#520	1	4	yes	—
337	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
338	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
340	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
341	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
342	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
343	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
344	C₄×C₆₇	—	2	2	no	—
345	C₄×C₆₇	—	2	2	no	—
346	C₄×C₆₇	—	2	4	no	—
347	C₄×C₆₇	—	2	2	no	—
348	C₄×C₆₇	—	2	2	no	—
349	C₄×C₆₇	—	2	4	no	—
350	C₄×C₆₇	—	2	4	no	—
351	C₄×C₆₇	—	4	4	no	—
352	C₄×C₆₇	—	4	2	no	—
353	C₄×C₆₇	—	4	4	no	—
354	C₄×C₆₇	—	4	2	no	—
355	C₄×C₆₇	—	4	2	no	—
356	C₄×C₆₇	—	4	2	no	—
357	C₄×C₆₇	#520	1	4	yes	—
358	C₄×C₆₇	#520	1	4	yes	—
359	#34384	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
360	C₂×C₆₇	—	2	4	no	—
361	C₂×C₆₇	—	4	4	no	—
362	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
363	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
364	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
365	C₂×C₆₇	#3543	1	1	yes	characteristic
366	C₂×C₆₇	#3544	1	1	yes	characteristic
367	C₂×C₆₇	#3552	1	1	yes	characteristic
368	C₂×C₆₇	#3552	1	1	yes	characteristic
369	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
370	#2383	C₂×C₆₇	1	4	yes	complemented
371	#2361	C₂×C₆₇	1	2	yes	—
372	#2360	C₂×C₆₇	1	2	yes	complemented
373	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
374	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
375	#2352	C₂×C₆₇	1	2	yes	complemented
376	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
377	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
378	D₄×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
379	#2330	C₂×C₆₇	1	2	yes	complemented
380	#2326	C₂×C₆₇	1	2	yes	complemented
381	#2322	C₂×C₆₇	1	4	yes	complemented
382	#2317	C₂×C₆₇	1	2	yes	complemented
383	#2314	C₂×C₆₇	1	2	yes	complemented
384	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
385	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
386	C₆₇	#34384	1	1	yes	characteristic, direct product factor, Sylow
387	#529	C₄×C₆₇	1	16	yes	complemented
388	C₂×#126	C₂²×C₆₇	1	2	yes	—
389	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
390	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
391	C₂×#123	C₂²×C₆₇	1	2	yes	—
392	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
393	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
394	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
395	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
396	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
397	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
398	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
399	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
400	C₂×#117	C₂²×C₆₇	1	2	yes	—
401	C₂×#117	C₂²×C₆₇	1	2	yes	—
402	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
403	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
404	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
405	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
406	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
407	#391	C₂²×C₆₇	1	2	yes	—
408	#390	C₂²×C₆₇	1	2	yes	—
409	#389	C₂²×C₆₇	1	4	yes	—
410	#389	C₂²×C₆₇	1	4	yes	—
411	#389	C₂²×C₆₇	1	4	yes	—
412	#389	C₂²×C₆₇	1	2	yes	—
413	#389	C₂²×C₆₇	1	4	yes	—
414	#389	C₂²×C₆₇	1	2	yes	—
415	#388	C₂²×C₆₇	1	4	yes	—
416	#388	C₂²×C₆₇	1	4	yes	—
417	#388	C₂²×C₆₇	1	4	yes	—
418	#388	C₂²×C₆₇	1	4	yes	—
419	#387	C₂²×C₆₇	1	4	yes	—
420	#387	C₂²×C₆₇	1	4	yes	—
421	#387	C₂²×C₆₇	1	4	yes	—
422	#387	C₂²×C₆₇	1	4	yes	—
423	#386	C₂²×C₆₇	1	4	yes	—
424	#385	C₂²×C₆₇	1	4	yes	—
425	#385	C₂²×C₆₇	1	4	yes	—
426	#385	C₂²×C₆₇	1	4	yes	—
427	#385	C₂²×C₆₇	1	4	yes	—
428	#385	C₂²×C₆₇	1	4	yes	—
429	#384	C₂²×C₆₇	1	2	yes	—
430	#384	C₂²×C₆₇	1	2	yes	—
431	#384	C₂²×C₆₇	1	2	yes	—
432	#384	C₂²×C₆₇	1	1	yes	characteristic
433	#384	C₂²×C₆₇	1	1	yes	characteristic
434	#384	C₂²×C₆₇	1	2	yes	—
435	#383	C₂²×C₆₇	1	2	yes	—
436	#382	C₂²×C₆₇	1	1	yes	characteristic
437	#382	C₂²×C₆₇	1	1	yes	characteristic
438	#382	C₂²×C₆₇	1	2	yes	—
439	#381	C₂²×C₆₇	1	2	yes	—
440	#380	C₂²×C₆₇	1	2	yes	—
441	#380	C₂²×C₆₇	1	2	yes	—
442	#380	C₂²×C₆₇	1	1	yes	characteristic
443	#380	C₂²×C₆₇	1	1	yes	characteristic
444	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
445	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
446	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
447	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
448	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
449	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
450	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
451	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
452	#376	C₂²×C₆₇	1	2	yes	—
453	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
454	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
455	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
456	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
457	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
458	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
459	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
460	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
461	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
462	#126	C₂×C₄×C₆₇	1	16	yes	complemented
463	#125	C₂×C₄×C₆₇	1	8	yes	complemented
464	#124	C₂×C₄×C₆₇	1	8	yes	complemented
465	#123	C₂×C₄×C₆₇	1	16	yes	complemented
466	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
467	#121	C₂×C₄×C₆₇	1	8	yes	—
468	C₄×Q₈	—	4	4	no	—
469	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
470	C₄×D₄	—	4	4	no	—
471	C₄×D₄	—	4	8	no	—
472	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
473	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
474	#117	—	4	4	no	—
475	#117	C₂×C₄×C₆₇	1	16	yes	complemented
476	C₂×C₄⋊C₄	—	4	4	no	—
477	C₂×C₄⋊C₄	—	4	4	no	—
478	C₂×C₄⋊C₄	—	4	4	no	—
479	C₂×C₄⋊C₄	—	4	4	no	—
480	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
481	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
483	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
484	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
485	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
486	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
487	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
488	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
489	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
490	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
491	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
492	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
494	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
495	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
496	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
497	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
498	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
499	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
500	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
501	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
502	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
503	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
504	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
505	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
506	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
507	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
508	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
509	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
510	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
511	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
512	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
513	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
514	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
515	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
516	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
517	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
518	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
519	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
520	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
521	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
522	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
523	#95	C₂³×C₆₇	1	2	yes	—
524	#95	C₂³×C₆₇	1	2	yes	—
525	#95	C₂³×C₆₇	1	2	yes	—
526	#95	C₂³×C₆₇	1	2	yes	—
527	#95	C₂³×C₆₇	1	4	yes	—
528	#95	C₂³×C₆₇	1	2	yes	—
529	#95	C₂³×C₆₇	1	2	yes	—
530	#95	C₂³×C₆₇	1	4	yes	—
531	#95	C₂³×C₆₇	1	4	yes	—
532	#95	C₂³×C₆₇	1	2	yes	—
533	#95	C₂³×C₆₇	1	2	yes	—
534	#95	C₂³×C₆₇	1	4	yes	—
535	#95	C₂³×C₆₇	1	4	yes	—
536	#95	C₂³×C₆₇	1	4	yes	—
537	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
538	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
539	C₂×Q₈	—	2	2	no	—
540	C₂×Q₈	—	2	2	no	—
541	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
542	C₂×D₄	—	2	4	no	—
543	C₂×D₄	—	2	4	no	—
544	C₂×D₄	—	2	2	no	—
545	C₂×D₄	—	2	2	no	—
546	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
547	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	2	2	no	—
555	C₂²×C₄	—	2	4	no	—
556	C₂²×C₄	—	2	8	no	—
557	C₂²×C₄	—	4	4	no	—
558	C₂²×C₄	—	4	4	no	—
559	C₂²×C₄	—	4	4	no	—
560	C₂²×C₄	—	4	4	no	—
561	C₂²×C₄	—	4	4	no	—
562	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
563	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
564	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
565	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
566	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
568	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
569	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
570	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
571	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
572	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
573	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
574	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
575	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
576	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
577	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
578	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
579	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
580	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	4	4	no	—
590	C₄⋊C₄	—	4	4	no	—
591	C₄⋊C₄	—	4	4	no	—
592	C₄⋊C₄	—	4	2	no	—
593	C₄⋊C₄	—	4	4	no	—
594	C₄⋊C₄	—	4	2	no	—
595	C₄⋊C₄	—	4	4	no	—
596	C₄⋊C₄	—	4	4	no	—
597	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
598	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
599	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
600	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
601	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
602	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	—	2	4	no	—
605	C₂²⋊C₄	—	4	4	no	—
606	C₂²⋊C₄	—	4	4	no	—
607	C₂²⋊C₄	—	4	4	no	—
608	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
609	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
610	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
611	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
612	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
613	C₄²	—	2	4	no	—
614	C₄²	—	2	4	no	—
615	C₄²	—	4	4	no	—
616	C₄²	—	4	4	no	—
617	C₄²	—	4	4	no	—
618	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
619	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
620	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
621	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
622	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
623	C₂³	—	2	1	no	—
624	C₂³	—	2	1	no	—
625	C₂³	—	2	1	no	—
626	C₂³	—	2	1	no	—
627	C₂³	—	2	2	no	—
628	C₂³	—	2	2	no	—
629	C₂³	—	2	2	no	—
630	C₂³	—	2	2	no	—
631	C₂³	—	2	2	no	—
632	C₂³	—	2	2	no	—
633	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
634	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
635	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
636	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
637	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
638	Q₈	—	2	8	no	—
639	D₄	—	2	16	no	—
640	D₄	—	2	8	no	—
641	C₂×C₄	—	2	8	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	8	no	—
657	C₂×C₄	—	2	16	no	—
658	C₂×C₄	—	2	8	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	4	no	—
664	C₂×C₄	—	2	4	no	—
665	C₂×C₄	—	2	4	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	4	no	—
668	C₂×C₄	—	2	4	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	4	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	8	8	no	—
678	C₂×C₄	—	8	8	no	—
679	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
680	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
681	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
682	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
683	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
684	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
685	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
686	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
687	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
688	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
689	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
690	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
691	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
692	C₂²	—	2	2	no	—
693	C₂²	—	2	2	no	—
694	C₂²	—	2	2	no	—
695	C₂²	—	2	2	no	—
696	C₂²	—	2	2	no	—
697	C₂²	—	2	2	no	—
698	C₂²	—	2	4	no	—
699	C₂²	—	2	4	no	—
700	C₂²	—	2	4	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	2	no	—
704	C₂²	—	4	2	no	—
705	C₂²	#520×C₆₇	1	4	yes	—
706	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
707	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
708	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
709	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
710	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
711	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
712	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
713	C₄	—	2	2	no	—
714	C₄	—	2	2	no	—
715	C₄	—	2	4	no	—
716	C₄	—	2	2	no	—
717	C₄	—	2	2	no	—
718	C₄	—	2	4	no	—
719	C₄	—	2	4	no	—
720	C₄	—	4	2	no	—
721	C₄	—	4	4	no	—
722	C₄	—	4	4	no	—
723	C₄	—	4	2	no	—
724	C₄	—	4	2	no	—
725	C₄	—	4	2	no	—
726	C₄	#520×C₆₇	1	4	yes	—
727	C₄	#520×C₆₇	1	4	yes	—
728	C₂	—	2	4	no	—
729	C₂	—	4	4	no	—
730	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
731	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
732	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
733	C₂	C₆₇×#3543	1	1	yes	characteristic
734	C₂	C₆₇×#3544	1	1	yes	characteristic
735	C₂	C₆₇×#3552	1	1	yes	characteristic
736	C₂	C₆₇×#3552	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×#121×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×#2383	C₂	semidirect
4. C₆₇×#2360	C₂	semidirect
5. C₆₇×#2355	C₂	semidirect
6. C₆₇×#2354	C₂	semidirect
7. C₆₇×#2352	C₂	semidirect
8. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
9. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
10. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
11. C₆₇×#2330	C₂	semidirect
12. C₆₇×#2326	C₂	semidirect
13. C₆₇×#2322	C₂	semidirect
14. C₆₇×#2317	C₂	semidirect
15. C₆₇×#2314	C₂	semidirect
16. C₂×#384×C₆₇	C₂	semidirect
17. C₆₇×#2283	C₂	semidirect
18. #529×C₆₇	C₄	semidirect
93. #126×C₆₇	C₂×C₄	semidirect
94. #125×C₆₇	C₂×C₄	semidirect
95. #124×C₆₇	C₂×C₄	semidirect
96. #123×C₆₇	C₂×C₄	semidirect
97. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
100. C₄×Q₈×C₆₇	C₂×C₄	semidirect
103. C₄×D₄×C₆₇	C₂×C₄	semidirect
104. C₄×D₄×C₆₇	C₂×C₄	semidirect
106. #117×C₆₇	C₂×C₄	semidirect
143. C₂×C₄²×C₆₇	C₂×C₄	semidirect
359. #34384	C₆₇	direct (N × H)
370. #2383	C₂×C₆₇	semidirect
372. #2360	C₂×C₆₇	semidirect
373. #2355	C₂×C₆₇	semidirect
374. #2354	C₂×C₆₇	semidirect
375. #2352	C₂×C₆₇	semidirect
376. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
377. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
378. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
379. #2330	C₂×C₆₇	semidirect
380. #2326	C₂×C₆₇	semidirect
381. #2322	C₂×C₆₇	semidirect
382. #2317	C₂×C₆₇	semidirect
383. #2314	C₂×C₆₇	semidirect
384. C₂×#384	C₂×C₆₇	semidirect
385. #2283	C₂×C₆₇	semidirect
386. C₆₇	#34384	direct (N × H)
387. #529	C₄×C₆₇	semidirect
462. #126	C₂×C₄×C₆₇	semidirect
463. #125	C₂×C₄×C₆₇	semidirect
464. #124	C₂×C₄×C₆₇	semidirect
465. #123	C₂×C₄×C₆₇	semidirect
466. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
469. C₄×Q₈	C₂×C₄×C₆₇	semidirect
472. C₄×D₄	C₂×C₄×C₆₇	semidirect
473. C₄×D₄	C₂×C₄×C₆₇	semidirect
475. #117	C₂×C₄×C₆₇	semidirect
512. C₂×C₄²	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×#529×C₆₇	2
C₆₇×#2383	2
C₆₇×#2361	2
C₆₇×#2360	2
C₆₇×#2355	2
C₆₇×#2354	2
C₆₇×#2352	2
Q₈×C₄⋊C₄×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2330	2
C₆₇×#2326	2
C₆₇×#2322	2
C₆₇×#2317	2
C₆₇×#2314	2
C₂×#384×C₆₇	2
C₆₇×#2283	2
#34384	67

Subgroups — C67×#34384 #30713635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34384 #30713635