Subgroups — #37582735

Order 24448.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133273	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133146	C₂	1	1	yes	maximal, characteristic
3	#27132999	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132991	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132972	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132961	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132871	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132866	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132825	C₂	1	1	yes	maximal, characteristic
10	#27132767	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132764	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132754	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132721	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₁₉₁	C₂	1	1	yes	maximal, characteristic
16	C₂×#15690277	C₂²	1	1	yes	characteristic
17	C₂×#15690275	C₂²	1	1	yes	characteristic
18	#20259320	C₂²	1	1	yes	characteristic
19	#20259315	—	2	1	no	—
20	#20259315	—	2	1	no	—
21	#20259313	—	2	1	no	—
22	#20259313	C₂²	1	1	yes	characteristic
23	#20259312	—	2	1	no	—
24	#20259301	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
28	#20259287	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₉₁	—	2	1	no	—
30	D₄×Dic₁₉₁	—	2	1	no	—
31	D₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
33	#20259281	—	2	1	no	—
34	#20259281	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
37	#20259256	C₂²	1	1	yes	characteristic
38	#20259256	C₂²	1	1	yes	characteristic
39	#20259253	C₂²	1	1	yes	characteristic
40	#20259253	C₂²	1	1	yes	characteristic
41	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
43	#20259247	C₂²	1	1	yes	characteristic
44	#20259247	C₂²	1	1	yes	characteristic
45	#20259246	C₂²	1	1	yes	characteristic
46	#20259246	C₂²	1	1	yes	characteristic
47	#20259246	C₂²	1	1	yes	characteristic
48	#20259243	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
50	#20259232	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
52	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
56	#103×C₁₉₁	C₂²	1	1	yes	characteristic
57	#102×C₁₉₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₉₁	—	2	1	no	—
61	C₂²×Dic₁₉₁	—	2	1	no	—
62	C₂²×Dic₁₉₁	—	2	1	no	—
63	#15690277	—	2	2	no	—
64	#15690275	—	2	2	no	—
65	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₉₁	—	2	1	no	—
68	C₄⋊Dic₁₉₁	—	2	1	no	—
69	C₄⋊Dic₁₉₁	—	2	1	no	—
70	C₄⋊Dic₁₉₁	—	2	1	no	—
71	C₄⋊Dic₁₉₁	—	2	1	no	—
72	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
73	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
77	Dic₁₉₁⋊C₄	—	2	1	no	—
78	Dic₁₉₁⋊C₄	—	2	1	no	—
79	Dic₁₉₁⋊C₄	—	2	1	no	—
80	Dic₁₉₁⋊C₄	—	2	1	no	—
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	C₂²⋊Dic₁₉₁	—	2	1	no	—
83	C₂²⋊Dic₁₉₁	—	2	1	no	—
84	C₂²⋊Dic₁₉₁	—	2	1	no	—
85	C₂²⋊Dic₁₉₁	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₉₁	D₄	1	1	yes	characteristic
87	C₄×Dic₁₉₁	—	2	1	no	—
88	C₄×Dic₁₉₁	—	2	1	no	—
89	C₄×Dic₁₉₁	—	2	1	no	—
90	C₄×Dic₁₉₁	—	2	1	no	—
91	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
93	#41×C₁₉₁	—	2	2	no	—
94	#41×C₁₉₁	—	2	2	no	—
95	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₉₁	—	2	1	no	—
98	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₉₁	—	2	1	no	—
100	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₉₁	—	2	2	no	—
102	SD₁₆×C₁₉₁	—	2	2	no	—
103	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉₁	—	2	1	no	—
106	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
108	Dic₃₈₂	—	2	1	no	—
109	Dic₃₈₂	—	2	1	no	—
110	C₂×Dic₁₉₁	—	2	1	no	—
111	C₂×Dic₁₉₁	—	2	1	no	—
112	C₂×Dic₁₉₁	—	2	1	no	—
113	C₂×Dic₁₉₁	—	2	1	no	—
114	C₂×Dic₁₉₁	—	8	1	no	—
115	C₂×Dic₁₉₁	—	8	1	no	—
116	C₂×Dic₁₉₁	—	8	1	no	—
117	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
121	C₁₉₁⋊C₈	—	2	1	no	—
122	C₁₉₁⋊C₈	—	2	1	no	—
123	C₂³×C₁₉₁	—	2	1	no	—
124	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₁₉₁	—	2	1	no	—
126	Q₈×C₁₉₁	—	2	1	no	—
127	Q₈×C₁₉₁	—	2	1	no	—
128	Q₈×C₁₉₁	#41	1	2	yes	—
129	D₄×C₁₉₁	—	2	4	no	—
130	D₄×C₁₉₁	—	2	1	no	—
131	D₄×C₁₉₁	#41	1	2	yes	—
132	C₂×C₄×C₁₉₁	—	2	4	no	—
133	C₂×C₄×C₁₉₁	—	2	1	no	—
134	C₂×C₄×C₁₉₁	—	2	1	no	—
135	C₂×C₄×C₁₉₁	—	2	1	no	—
136	C₂×C₄×C₁₉₁	—	2	1	no	—
137	C₂×C₄×C₁₉₁	—	2	1	no	—
138	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₉₁	—	2	1	no	—
141	C₈×C₁₉₁	—	2	1	no	—
142	C₂²×C₁₉₁	—	2	1	no	—
143	C₂²×C₁₉₁	—	2	2	no	—
144	C₂²×C₁₉₁	—	2	1	no	—
145	C₂²×C₁₉₁	—	2	1	no	—
146	C₂²×C₁₉₁	—	4	1	no	—
147	C₂²×C₁₉₁	—	4	1	no	—
148	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₁₉₁	—	2	1	no	—
150	C₄×C₁₉₁	—	2	2	no	—
151	C₄×C₁₉₁	—	4	1	no	—
152	C₄×C₁₉₁	—	4	1	no	—
153	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
155	Dic₁₉₁	—	2	1	no	—
156	Dic₁₉₁	—	2	1	no	—
157	Dic₁₉₁	—	2	1	no	—
158	Dic₁₉₁	—	2	1	no	—
159	Dic₁₉₁	—	4	1	no	—
160	Dic₁₉₁	—	4	1	no	—
161	Dic₁₉₁	—	4	1	no	—
162	Dic₁₉₁	—	4	1	no	—
163	C₂×C₁₉₁	—	4	1	no	—
164	C₂×C₁₉₁	—	4	2	no	—
165	C₂×C₁₉₁	#525	1	1	yes	characteristic
166	C₂×C₁₉₁	#577	1	1	yes	characteristic
167	C₂×C₁₉₁	#578	1	1	yes	characteristic
168	C₁₉₁	#3120	1	1	yes	characteristic, complemented, Sylow
169	#3120	—	191	1	no	maximal, Sylow
170	#541	—	191	1	no	—
171	#520	—	191	1	no	—
172	#484	—	191	1	no	—
173	#483	—	191	1	no	—
174	#479	—	191	1	no	—
175	#478	—	191	1	no	—
176	#462	—	191	1	no	—
177	#461	—	191	1	no	—
178	#452	—	191	1	no	—
179	#452	D₁₉₁	1	1	yes	characteristic
180	#441	—	191	1	no	—
181	#440	—	191	1	no	—
182	C₄×Q₁₆	—	191	1	no	—
183	C₄×SD₁₆	—	191	1	no	—
184	#433	—	191	1	no	—
185	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
186	C₂×Q₁₆	—	191	1	no	—
187	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
188	C₂×SD₁₆	—	191	1	no	—
189	C₂×SD₁₆	C₂×D₁₉₁	1	1	yes	characteristic
190	C₄⋊Q₈	—	191	1	no	—
191	#126	—	382	1	no	—
192	#126	—	382	1	no	—
193	#125	—	382	1	no	—
194	#125	—	382	1	no	—
195	#125	—	191	1	no	—
196	C₂²⋊Q₈	—	191	1	no	—
197	C₂²⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
198	C₄×Q₈	—	191	1	no	—
199	C₄×Q₈	—	191	1	no	—
200	C₄×Q₈	—	191	1	no	—
201	C₄×Q₈	—	191	1	no	—
202	C₄×D₄	—	382	1	no	—
203	C₄×D₄	—	382	1	no	—
204	C₄×D₄	—	191	1	no	—
205	C₄×D₄	—	191	1	no	—
206	#117	—	382	1	no	—
207	#117	—	191	1	no	—
208	C₂×C₄⋊C₄	—	382	1	no	—
209	C₂×C₄⋊C₄	—	191	1	no	—
210	#107	—	191	1	no	—
211	#107	—	191	1	no	—
212	#106	—	191	1	no	—
213	#106	—	191	1	no	—
214	C₄⋊C₈	—	191	1	no	—
215	C₄⋊C₈	—	191	1	no	—
216	#103	—	191	1	no	—
217	#103	—	191	1	no	—
218	#103	—	191	1	no	—
219	#103	—	191	1	no	—
220	#103	—	191	1	no	—
221	#103	C₂×D₁₉₁	1	1	yes	characteristic
222	#102	—	191	1	no	—
223	#102	C₂×D₁₉₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	191	1	no	—
225	C₂²⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
226	#97	—	191	1	no	—
227	C₄×C₈	—	191	1	no	—
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	C₂×Q₈	—	191	1	no	—
231	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	382	1	no	—
237	C₂²×C₄	—	382	1	no	—
238	C₂²×C₄	—	382	1	no	—
239	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	382	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	382	2	no	—
244	C₂×C₈	—	191	1	no	—
245	C₂×C₈	—	191	1	no	—
246	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	382	1	no	—
250	C₄⋊C₄	—	382	1	no	—
251	C₄⋊C₄	—	382	1	no	—
252	C₄⋊C₄	—	382	1	no	—
253	C₄⋊C₄	—	382	1	no	—
254	C₄⋊C₄	—	382	1	no	—
255	C₄⋊C₄	—	382	1	no	—
256	C₄⋊C₄	—	382	1	no	—
257	C₄⋊C₄	—	382	1	no	—
258	C₄⋊C₄	—	382	1	no	—
259	C₄⋊C₄	—	191	1	no	—
260	C₄⋊C₄	—	191	1	no	—
261	C₄⋊C₄	—	191	1	no	—
262	C₄⋊C₄	—	191	1	no	—
263	C₄⋊C₄	—	191	1	no	—
264	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	382	1	no	—
267	C₂²⋊C₄	—	382	1	no	—
268	C₂²⋊C₄	—	382	1	no	—
269	C₂²⋊C₄	—	191	1	no	—
270	C₂²⋊C₄	—	191	1	no	—
271	C₄²	—	382	1	no	—
272	C₄²	—	382	1	no	—
273	C₄²	—	382	1	no	—
274	C₄²	—	382	1	no	—
275	C₄²	—	191	1	no	—
276	C₄²	—	191	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	382	1	no	—
283	Q₈	—	382	1	no	—
284	Q₈	#15690286	1	2	yes	—
285	D₄	—	2	4	no	—
286	D₄	—	2	1	no	—
287	D₄	#15690286	1	2	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	382	1	no	—
295	C₂×C₄	—	382	1	no	—
296	C₂×C₄	—	382	1	no	—
297	C₂×C₄	—	382	1	no	—
298	C₂×C₄	—	191	1	no	—
299	C₂×C₄	—	191	1	no	—
300	C₂×C₄	—	191	1	no	—
301	C₂×C₄	—	191	1	no	—
302	C₂×C₄	—	1528	1	no	—
303	C₂×C₄	—	1528	1	no	—
304	C₂×C₄	—	1528	1	no	—
305	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	382	1	no	—
310	C₈	—	382	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	382	1	no	—
323	C₄	—	382	1	no	—
324	C₄	—	382	1	no	—
325	C₄	—	382	1	no	—
326	C₄	—	764	1	no	—
327	C₄	—	764	1	no	—
328	C₄	—	764	1	no	—
329	C₄	—	764	1	no	—
330	C₄	C₂×#15690286	1	1	yes	characteristic
331	C₄	C₂×#15690286	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	2	no	—
334	C₂	#27133169	1	1	yes	characteristic
335	C₂	#27133472	1	1	yes	characteristic
336	C₂	#27133477	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#452×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#41×C₁₉₁	C₂²
hypercenter	#452	D₁₉₁
hypocenter	C₁₉₁	#3120
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #27133273	C₂	semidirect
3. #27132999	C₂	semidirect
4. #27132991	C₂	semidirect
5. #27132972	C₂	semidirect
6. #27132961	C₂	semidirect
7. #27132871	C₂	semidirect
8. #27132866	C₂	semidirect
10. #27132767	C₂	semidirect
11. #27132764	C₂	semidirect
12. Q₁₆×Dic₁₉₁	C₂	semidirect
13. #27132754	C₂	semidirect
14. #27132721	C₂	semidirect
168. C₁₉₁	#3120	semidirect

Maximal subgroups

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

subgroup	index
#27133273	2
#27133146	2
#27132999	2
#27132991	2
#27132972	2
#27132961	2
#27132871	2
#27132866	2
#27132825	2
#27132767	2
#27132764	2
Q₁₆×Dic₁₉₁	2
#27132754	2
#27132721	2
#452×C₁₉₁	2
#3120	191