Subgroups — C₆₇×#35151 #30714402

Order 17152.

2358 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2855	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2626	C₂	1	4	yes	maximal
5	C₆₇×#2441	C₂	1	8	yes	maximal, complemented
6	C₆₇×#2439	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2317	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2316	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2280	C₂	1	2	yes	maximal
10	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
11	#544×C₆₇	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
13	C₂×#125×C₆₇	C₂²	1	4	yes	complemented
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂×#117×C₆₇	—	2	8	no	—
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₂²	1	2	yes	—
18	C₂×#117×C₆₇	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
23	#399×C₆₇	C₂²	1	8	yes	—
24	#396×C₆₇	C₂²	1	16	yes	—
25	#393×C₆₇	C₂²	1	8	yes	—
26	#391×C₆₇	C₂²	1	8	yes	complemented
27	#389×C₆₇	C₂²	1	8	yes	complemented
28	#389×C₆₇	C₂²	1	8	yes	complemented
29	#389×C₆₇	C₂²	1	8	yes	complemented
30	#387×C₆₇	C₂²	1	16	yes	complemented
31	#385×C₆₇	C₂²	1	8	yes	—
32	#384×C₆₇	C₂²	1	4	yes	complemented
33	#382×C₆₇	C₂²	1	4	yes	complemented
34	#381×C₆₇	C₂²	1	4	yes	—
35	#380×C₆₇	C₂²	1	4	yes	—
36	#380×C₆₇	C₂²	1	4	yes	—
37	#380×C₆₇	C₂²	1	4	yes	—
38	#379×C₆₇	C₂²	1	4	yes	—
39	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
40	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
42	C₂³×C₄×C₆₇	—	2	4	no	—
43	C₂³×C₄×C₆₇	—	2	4	no	—
44	C₂³×C₄×C₆₇	Q₈	1	8	yes	complemented
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	—	2	16	no	—
51	#125×C₆₇	—	2	16	no	—
52	C₂²⋊Q₈×C₆₇	—	2	16	no	—
53	#117×C₆₇	—	2	8	no	—
54	#117×C₆₇	—	2	8	no	—
55	#117×C₆₇	—	2	32	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
68	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₄²×C₆₇	—	2	8	no	—
74	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
77	#95×C₆₇	C₂³	1	8	yes	—
78	#95×C₆₇	C₂³	1	8	yes	—
79	#95×C₆₇	C₂³	1	8	yes	—
80	#95×C₆₇	C₂³	1	16	yes	—
81	C₂⁴×C₆₇	—	2	2	no	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	4	2	no	—
84	C₂⁴×C₆₇	—	4	2	no	—
85	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
86	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
87	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂×Q₈×C₆₇	—	2	8	no	—
89	C₂×Q₈×C₆₇	—	4	8	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	32	no	—
93	C₂²×C₄×C₆₇	—	2	8	no	—
94	C₂²×C₄×C₆₇	—	2	8	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	4	4	no	—
97	C₂²×C₄×C₆₇	—	4	4	no	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	—
102	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
103	C₂²×C₄×C₆₇	#41	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
108	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
111	C₄⋊C₄×C₆₇	—	2	8	no	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	8	no	—
115	C₄⋊C₄×C₆₇	—	2	16	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	4	8	no	—
118	C₄⋊C₄×C₆₇	—	4	16	no	—
119	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
120	C₂²⋊C₄×C₆₇	—	2	8	no	—
121	C₂²⋊C₄×C₆₇	—	2	8	no	—
122	C₂²⋊C₄×C₆₇	—	2	16	no	—
123	C₂²⋊C₄×C₆₇	—	4	8	no	—
124	C₂²⋊C₄×C₆₇	—	4	8	no	—
125	C₄²×C₆₇	—	2	16	no	—
126	C₄²×C₆₇	—	2	8	no	—
127	C₄²×C₆₇	—	2	8	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	8	no	—
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	2	4	no	—
135	C₂³×C₆₇	—	4	8	no	—
136	C₂³×C₆₇	—	4	4	no	—
137	C₂³×C₆₇	—	4	4	no	—
138	C₂³×C₆₇	—	4	8	no	—
139	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
140	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
141	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
142	C₂³×C₆₇	C₂×#41	1	2	yes	—
143	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
144	Q₈×C₆₇	—	8	8	no	—
145	C₂×C₄×C₆₇	—	2	8	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	8	no	—
149	C₂×C₄×C₆₇	—	2	32	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	8	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	4	8	no	—
160	C₂×C₄×C₆₇	—	4	8	no	—
161	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
165	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
166	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
167	C₂²×C₆₇	—	2	4	no	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	2	4	no	—
171	C₂²×C₆₇	—	2	4	no	—
172	C₂²×C₆₇	—	2	8	no	—
173	C₂²×C₆₇	—	4	16	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	—	4	4	no	—
177	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
178	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
179	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
180	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
181	C₂²×C₆₇	#585	1	2	yes	—
182	C₂²×C₆₇	#585	1	2	yes	—
183	C₄×C₆₇	—	2	4	no	—
184	C₄×C₆₇	—	2	8	no	—
185	C₄×C₆₇	—	4	4	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	4	4	no	—
188	C₄×C₆₇	—	4	8	no	—
189	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
190	#35151	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂×C₆₇	—	2	4	no	—
192	C₂×C₆₇	—	4	4	no	—
193	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
194	C₂×C₆₇	#3458	1	2	yes	—
195	C₂×C₆₇	#3485	1	1	yes	characteristic
196	C₂×C₆₇	#3487	1	1	yes	characteristic
197	C₂×C₆₇	#3493	1	2	yes	—
198	C₂×#544	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic
200	#2855	C₂×C₆₇	1	4	yes	complemented
201	#2626	C₂×C₆₇	1	4	yes	—
202	#2441	C₂×C₆₇	1	8	yes	complemented
203	#2439	C₂×C₆₇	1	8	yes	complemented
204	#2317	C₂×C₆₇	1	2	yes	complemented
205	#2316	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#2280	C₂×C₆₇	1	2	yes	—
207	C₆₇	#35151	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
209	#544	—	2	8	no	—
210	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
211	C₂×#125	C₂²×C₆₇	1	4	yes	complemented
212	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
213	C₂×#117	—	2	8	no	—
214	C₂×#117	C₂²×C₆₇	1	2	yes	—
215	C₂×#117	C₂²×C₆₇	1	2	yes	—
216	C₂×#117	C₂²×C₆₇	1	8	yes	—
217	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
218	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
219	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
220	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
221	#399	C₂²×C₆₇	1	8	yes	—
222	#396	C₂²×C₆₇	1	16	yes	—
223	#393	C₂²×C₆₇	1	8	yes	—
224	#391	C₂²×C₆₇	1	8	yes	complemented
225	#389	C₂²×C₆₇	1	8	yes	complemented
226	#389	C₂²×C₆₇	1	8	yes	complemented
227	#389	C₂²×C₆₇	1	8	yes	complemented
228	#387	C₂²×C₆₇	1	16	yes	complemented
229	#385	C₂²×C₆₇	1	8	yes	—
230	#384	C₂²×C₆₇	1	4	yes	complemented
231	#382	C₂²×C₆₇	1	4	yes	complemented
232	#381	C₂²×C₆₇	1	4	yes	—
233	#380	C₂²×C₆₇	1	4	yes	—
234	#380	C₂²×C₆₇	1	4	yes	—
235	#380	C₂²×C₆₇	1	4	yes	—
236	#379	C₂²×C₆₇	1	4	yes	—
237	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
238	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
239	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
240	C₂³×C₄	—	2	4	no	—
241	C₂³×C₄	—	2	4	no	—
242	C₂³×C₄	Q₈×C₆₇	1	8	yes	complemented
243	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
244	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
246	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
247	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	16	no	—
249	#125	—	2	16	no	—
250	C₂²⋊Q₈	—	2	16	no	—
251	#117	—	2	8	no	—
252	#117	—	2	8	no	—
253	#117	—	2	32	no	—
254	C₂×C₄⋊C₄	—	2	4	no	—
255	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	—
256	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
259	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
261	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
262	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
263	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
264	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
265	C₂×C₂²⋊C₄	—	2	8	no	—
266	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
267	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
268	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
269	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
270	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
271	C₂×C₄²	—	2	8	no	—
272	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
273	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
274	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
275	#95	C₂³×C₆₇	1	8	yes	—
276	#95	C₂³×C₆₇	1	8	yes	—
277	#95	C₂³×C₆₇	1	8	yes	—
278	#95	C₂³×C₆₇	1	16	yes	—
279	C₂⁴	—	2	2	no	—
280	C₂⁴	—	2	2	no	—
281	C₂⁴	—	4	2	no	—
282	C₂⁴	—	4	2	no	—
283	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
284	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
285	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
286	C₂×Q₈	—	2	8	no	—
287	C₂×Q₈	—	4	8	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	32	no	—
291	C₂²×C₄	—	2	8	no	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	4	4	no	—
295	C₂²×C₄	—	4	4	no	—
296	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
297	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
298	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
299	C₂²×C₄	C₂×Q₈×C₆₇	1	16	yes	—
300	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
301	C₂²×C₄	#41×C₆₇	1	4	yes	—
302	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
303	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
304	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
305	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
306	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
307	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
309	C₄⋊C₄	—	2	8	no	—
310	C₄⋊C₄	—	2	4	no	—
311	C₄⋊C₄	—	2	4	no	—
312	C₄⋊C₄	—	2	8	no	—
313	C₄⋊C₄	—	2	16	no	—
314	C₄⋊C₄	—	2	8	no	—
315	C₄⋊C₄	—	4	8	no	—
316	C₄⋊C₄	—	4	16	no	—
317	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
318	C₂²⋊C₄	—	2	8	no	—
319	C₂²⋊C₄	—	2	8	no	—
320	C₂²⋊C₄	—	2	16	no	—
321	C₂²⋊C₄	—	4	8	no	—
322	C₂²⋊C₄	—	4	8	no	—
323	C₄²	—	2	16	no	—
324	C₄²	—	2	8	no	—
325	C₄²	—	2	8	no	—
326	C₂³	—	2	2	no	—
327	C₂³	—	2	2	no	—
328	C₂³	—	2	8	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	4	no	—
332	C₂³	—	2	4	no	—
333	C₂³	—	4	8	no	—
334	C₂³	—	4	4	no	—
335	C₂³	—	4	4	no	—
336	C₂³	—	4	8	no	—
337	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
338	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
339	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
340	C₂³	C₂×#41×C₆₇	1	2	yes	—
341	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
342	Q₈	—	8	8	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	2	32	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	8	no	—
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	8	no	—
354	C₂×C₄	—	2	8	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	4	8	no	—
358	C₂×C₄	—	4	8	no	—
359	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
360	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
361	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
362	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
363	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
364	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
365	C₂²	—	2	4	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	—	2	4	no	—
369	C₂²	—	2	4	no	—
370	C₂²	—	2	8	no	—
371	C₂²	—	4	16	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	4	no	—
375	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
376	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
377	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
378	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
379	C₂²	#585×C₆₇	1	2	yes	—
380	C₂²	#585×C₆₇	1	2	yes	—
381	C₄	—	2	4	no	—
382	C₄	—	2	8	no	—
383	C₄	—	4	4	no	—
384	C₄	—	4	4	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	8	no	—
387	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
388	C₂	—	2	4	no	—
389	C₂	—	4	4	no	—
390	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
391	C₂	C₆₇×#3458	1	2	yes	—
392	C₂	C₆₇×#3485	1	1	yes	characteristic
393	C₂	C₆₇×#3487	1	1	yes	characteristic
394	C₂	C₆₇×#3493	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#544×C₆₇	C₂	semidirect
3. C₆₇×#2855	C₂	semidirect
5. C₆₇×#2441	C₂	semidirect
6. C₆₇×#2439	C₂	semidirect
7. C₆₇×#2317	C₂	semidirect
8. C₆₇×#2316	C₂	semidirect
12. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
13. C₂×#125×C₆₇	C₂²	semidirect
26. #391×C₆₇	C₂²	semidirect
27. #389×C₆₇	C₂²	semidirect
28. #389×C₆₇	C₂²	semidirect
29. #389×C₆₇	C₂²	semidirect
30. #387×C₆₇	C₂²	semidirect
32. #384×C₆₇	C₂²	semidirect
33. #382×C₆₇	C₂²	semidirect
44. C₂³×C₄×C₆₇	Q₈	semidirect
190. #35151	C₆₇	direct (N × H)
198. C₂×#544	C₂×C₆₇	semidirect
200. #2855	C₂×C₆₇	semidirect
202. #2441	C₂×C₆₇	semidirect
203. #2439	C₂×C₆₇	semidirect
204. #2317	C₂×C₆₇	semidirect
205. #2316	C₂×C₆₇	semidirect
207. C₆₇	#35151	direct (N × H)
210. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
211. C₂×#125	C₂²×C₆₇	semidirect
224. #391	C₂²×C₆₇	semidirect
225. #389	C₂²×C₆₇	semidirect
226. #389	C₂²×C₆₇	semidirect
227. #389	C₂²×C₆₇	semidirect
228. #387	C₂²×C₆₇	semidirect
230. #384	C₂²×C₆₇	semidirect
231. #382	C₂²×C₆₇	semidirect
242. C₂³×C₄	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#544×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2855	2
C₆₇×#2626	2
C₆₇×#2441	2
C₆₇×#2439	2
C₆₇×#2317	2
C₆₇×#2316	2
C₆₇×#2280	2
#35151	67

Subgroups — C67×#35151 #30714402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35151 #30714402