Subgroups — #30826372

Order 17152.

630 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	#22569185	C₂	1	2	yes	maximal, complemented
2	#22559106	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559100	C₂	1	2	yes	maximal, complemented
4	#22559094	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	SD₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	D₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558669	C₂	1	2	yes	maximal, complemented
10	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3168	C₂	1	1	yes	maximal, characteristic
12	#16863527	C₂²	1	2	yes	—
13	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
14	#16863474	C₂²	1	2	yes	—
15	#16863468	—	2	2	no	—
16	D₄×C₆₇⋊C₈	—	2	2	no	—
17	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	#16863457	—	2	2	no	—
19	#16863453	C₂²	1	2	yes	—
20	#16863352	—	2	2	no	—
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	#16863149	C₂²	1	1	yes	characteristic
24	#16863149	C₂²	1	1	yes	characteristic
25	#16863147	C₂²	1	2	yes	—
26	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
27	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
28	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
29	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
30	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
34	#529×C₆₇	C₂²	1	2	yes	—
35	#499×C₆₇	C₄	1	1	yes	characteristic
36	#495×C₆₇	C₄	1	1	yes	characteristic
37	#488×C₆₇	C₄	1	2	yes	—
38	#467×C₆₇	C₄	1	1	yes	characteristic
39	#466×C₆₇	C₄	1	1	yes	characteristic
40	#465×C₆₇	C₄	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
43	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761612	—	2	2	no	—
46	C₂²×C₆₇⋊C₈	—	2	2	no	—
47	C₂²×C₆₇⋊C₈	—	2	1	no	—
48	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
49	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
53	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
54	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
55	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
56	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
57	#14523954	—	2	2	no	—
58	#14523954	—	2	2	no	—
59	C₄×C₆₇⋊C₈	—	2	2	no	—
60	C₄×C₆₇⋊C₈	—	2	1	no	—
61	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
63	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	#126×C₆₇	—	2	2	no	—
73	#125×C₆₇	C₂×C₄	1	2	yes	—
74	#124×C₆₇	C₂×C₄	1	2	yes	—
75	#123×C₆₇	—	2	2	no	—
76	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
77	#121×C₆₇	C₂×C₄	1	2	yes	—
78	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
79	C₄×D₄×C₆₇	—	2	2	no	—
80	C₄×D₄×C₆₇	C₂³	1	2	yes	—
81	#117×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
84	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
85	#106×C₆₇	C₂×C₄	1	2	yes	—
86	#103×C₆₇	C₂×C₄	1	2	yes	—
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	#102×C₆₇	C₂×C₄	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	#11761612	—	4	4	no	—
95	C₂×C₆₇⋊C₈	—	2	4	no	—
96	C₂×C₆₇⋊C₈	—	2	1	no	—
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	—	4	4	no	—
103	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
114	Q₁₆×C₆₇	—	2	2	no	—
115	SD₁₆×C₆₇	—	2	4	no	—
116	D₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₂²⋊C₄×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₆₇	#41	1	1	yes	characteristic
138	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₆₇	—	2	2	no	—
142	Q₈×C₆₇	—	2	2	no	—
143	D₄×C₆₇	—	2	2	no	—
144	D₄×C₆₇	—	2	2	no	—
145	D₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	#131	1	2	yes	—
165	C₈×C₆₇	#131	1	2	yes	—
166	C₆₇⋊C₈	—	2	2	no	—
167	C₆₇⋊C₈	—	2	2	no	—
168	C₆₇⋊C₈	—	2	1	no	—
169	C₆₇⋊C₈	—	2	1	no	—
170	C₆₇⋊C₈	—	4	2	no	—
171	C₆₇⋊C₈	—	4	2	no	—
172	C₆₇⋊C₈	#135	1	2	yes	—
173	C₆₇⋊C₈	#135	1	2	yes	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	1	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	#443	1	2	yes	—
180	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	—	2	1	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
189	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
190	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
191	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
192	#17959	—	67	1	no	maximal, Sylow
193	C₂×C₆₇	—	2	2	no	—
194	C₂×C₆₇	—	8	2	no	—
195	C₂×C₆₇	#2931	1	1	yes	characteristic
196	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
198	#3168	D₆₇	1	1	yes	characteristic
199	#2966	—	67	2	no	—
200	#1591	—	67	1	no	—
201	#1590	—	67	2	no	—
202	#1589	—	67	1	no	—
203	C₈×Q₁₆	—	67	1	no	—
204	C₈×SD₁₆	—	67	2	no	—
205	C₈×D₈	—	67	1	no	—
206	#1568	—	67	1	no	—
207	#1494	—	67	2	no	—
208	C₂×C₈²	—	67	1	no	—
209	C₆₇	#17959	1	1	yes	characteristic, complemented, Sylow
210	#529	C₂×D₆₇	1	2	yes	—
211	#499	Dic₆₇	1	1	yes	characteristic
212	#495	Dic₆₇	1	1	yes	characteristic
213	#488	Dic₆₇	1	2	yes	—
214	#467	Dic₆₇	1	1	yes	characteristic
215	#466	Dic₆₇	1	1	yes	characteristic
216	#465	Dic₆₇	1	2	yes	—
217	#446	—	67	2	no	—
218	C₈×Q₈	—	67	2	no	—
219	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
220	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
221	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
222	#436	—	67	2	no	—
223	#435	—	134	2	no	—
224	C₈×D₄	—	67	2	no	—
225	C₈×D₄	—	134	2	no	—
226	#433	—	134	2	no	—
227	#432	—	67	2	no	—
228	#405	—	134	2	no	—
229	C₂×C₄×C₈	—	67	1	no	—
230	C₂×C₄×C₈	—	67	1	no	—
231	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
232	#335	—	67	1	no	—
233	#335	—	67	1	no	—
234	#334	—	67	2	no	—
235	Q₈⋊C₈	—	67	2	no	—
236	Q₈⋊C₈	—	67	2	no	—
237	D₄⋊C₈	—	67	2	no	—
238	D₄⋊C₈	—	67	2	no	—
239	C₈²	—	67	1	no	—
240	C₈²	—	67	1	no	—
241	C₈²	—	67	1	no	—
242	C₈²	—	67	1	no	—
243	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
244	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
245	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
246	C₂×M₁₆	—	134	2	no	—
247	C₂²×C₈	—	67	1	no	—
248	C₂²×C₈	—	67	1	no	—
249	C₂²×C₈	—	134	2	no	—
250	C₂²×C₈	—	134	1	no	—
251	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
252	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
253	#126	—	2	2	no	—
254	#125	C₂×Dic₆₇	1	2	yes	—
255	#124	C₂×Dic₆₇	1	2	yes	—
256	#123	—	2	2	no	—
257	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
258	#121	C₂×Dic₆₇	1	2	yes	—
259	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
260	C₄×D₄	—	2	2	no	—
261	C₄×D₄	C₂²×D₆₇	1	2	yes	—
262	#117	—	2	2	no	—
263	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
264	#107	C₂×Dic₆₇	1	1	yes	characteristic
265	#107	C₂×Dic₆₇	1	1	yes	characteristic
266	#106	C₂×Dic₆₇	1	2	yes	—
267	C₄⋊C₈	—	67	2	no	—
268	C₄⋊C₈	—	67	2	no	—
269	C₄⋊C₈	—	134	2	no	—
270	C₄⋊C₈	—	134	2	no	—
271	#103	C₂×Dic₆₇	1	2	yes	—
272	#103	C₂×Dic₆₇	1	2	yes	—
273	#102	C₂×Dic₆₇	1	2	yes	—
274	#102	C₂×Dic₆₇	1	2	yes	—
275	C₂²⋊C₈	—	134	2	no	—
276	C₂²⋊C₈	—	134	2	no	—
277	C₂²⋊C₈	—	134	2	no	—
278	#97	—	134	2	no	—
279	#97	—	134	2	no	—
280	C₄×C₈	—	67	1	no	—
281	C₄×C₈	—	67	1	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	—	67	1	no	—
285	C₄×C₈	—	67	1	no	—
286	C₄×C₈	—	134	2	no	—
287	C₄×C₈	—	134	1	no	—
288	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
289	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
292	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
297	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
298	C₂²×C₄	#11761627	1	1	yes	characteristic
299	Q₁₆	—	2	2	no	—
300	SD₁₆	—	2	4	no	—
301	D₈	—	2	2	no	—
302	M₁₆	—	268	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	268	4	no	—
306	C₂×C₈	—	67	1	no	—
307	C₂×C₈	—	67	1	no	—
308	C₂×C₈	—	67	1	no	—
309	C₂×C₈	—	67	1	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	—	134	2	no	—
312	C₂×C₈	—	134	2	no	—
313	C₂×C₈	—	134	2	no	—
314	C₂×C₈	—	134	4	no	—
315	C₂×C₈	—	134	1	no	—
316	C₂×C₈	—	134	1	no	—
317	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
318	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
319	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
320	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
326	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
334	C₄²	D₄×D₆₇	1	1	yes	characteristic
335	C₄²	#11761627	1	1	yes	characteristic
336	C₄²	C₂³×D₆₇	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
355	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
356	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
359	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	268	2	no	—
363	C₈	—	268	2	no	—
364	C₈	—	67	2	no	—
365	C₈	—	67	2	no	—
366	C₈	—	134	2	no	—
367	C₈	—	134	2	no	—
368	C₈	—	134	1	no	—
369	C₈	—	134	1	no	—
370	C₈	#14524050	1	2	yes	—
371	C₈	#14524050	1	2	yes	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	#16863513	1	2	yes	—
378	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	4	2	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	C₂×#14524050	1	1	yes	characteristic
387	C₄	C₂×#14524050	1	1	yes	characteristic
388	C₄	#135×D₆₇	1	1	yes	characteristic
389	C₄	#135×D₆₇	1	1	yes	characteristic
390	C₂	—	2	2	no	—
391	C₂	—	8	2	no	—
392	C₂	#22568921	1	1	yes	characteristic
393	C₂	#135×Dic₆₇	1	1	yes	characteristic
394	C₂	C₂×#16863513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3168	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#466×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3168	D₆₇
hypocenter	C₆₇	#17959
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #22569185	C₂	semidirect
2. #22559106	C₂	semidirect
3. #22559100	C₂	semidirect
4. #22559094	C₂	semidirect
5. Q₁₆×C₆₇⋊C₈	C₂	semidirect
6. SD₁₆×C₆₇⋊C₈	C₂	semidirect
7. D₈×C₆₇⋊C₈	C₂	semidirect
8. #22558972	C₂	semidirect
9. #22558669	C₂	semidirect
10. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
209. C₆₇	#17959	semidirect

Maximal subgroups

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

subgroup	index
#22569185	2
#22559106	2
#22559100	2
#22559094	2
Q₁₆×C₆₇⋊C₈	2
SD₁₆×C₆₇⋊C₈	2
D₈×C₆₇⋊C₈	2
#22558972	2
#22558669	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#3168	2
#17959	67