Subgroups — #37583122

Order 24448.

364 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	#27133217	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133195	C₂	1	1	yes	maximal, characteristic
3	#27133070	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133054	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133030	C₂	1	1	yes	maximal, characteristic, complemented
6	#27133021	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132922	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132920	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132908	C₂	1	1	yes	maximal, characteristic
10	#27132906	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#15690277	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#15690275	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#15690270	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
17	C₂×#15690277	C₂²	1	1	yes	characteristic
18	C₂×#15690275	C₂²	1	1	yes	characteristic
19	C₂×#15690270	C₂²	1	1	yes	characteristic
20	C₂×#15690270	C₂²	1	1	yes	characteristic
21	#20259321	C₂²	1	1	yes	characteristic
22	#20259320	C₂²	1	1	yes	characteristic
23	#20259315	—	2	1	no	—
24	#20259313	C₂²	1	1	yes	characteristic
25	#20259312	—	2	1	no	—
26	#20259309	C₂²	1	1	yes	characteristic
27	#20259304	—	2	1	no	—
28	#20259302	C₂²	1	1	yes	characteristic
29	#20259302	C₂²	1	1	yes	characteristic
30	#20259300	—	2	1	no	—
31	#20259298	C₂²	1	1	yes	characteristic
32	#20259293	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₈₂	—	2	1	no	—
34	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
35	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₉₁	—	2	1	no	—
37	#20259281	—	2	1	no	—
38	#20259279	—	2	1	no	—
39	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
40	#20259256	C₂²	1	1	yes	characteristic
41	#20259255	C₂²	1	1	yes	characteristic
42	#20259253	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	#20259245	C₂²	1	1	yes	characteristic
47	#20259245	C₂²	1	1	yes	characteristic
48	#20259243	C₂²	1	1	yes	characteristic
49	#20259232	C₂²	1	1	yes	characteristic
50	#20259232	C₂²	1	1	yes	characteristic
51	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
55	#106×C₁₉₁	C₂²	1	1	yes	characteristic
56	#103×C₁₉₁	C₂²	1	1	yes	characteristic
57	#102×C₁₉₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₈₂	—	2	1	no	—
60	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
63	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
64	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
66	Dic₇₆₄	—	2	2	no	—
67	#15690277	—	2	2	no	—
68	#15690275	—	2	2	no	—
69	#15690270	—	2	2	no	—
70	#15690270	—	2	2	no	—
71	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₉₁	—	2	1	no	—
74	C₄⋊Dic₁₉₁	—	2	1	no	—
75	C₄⋊Dic₁₉₁	—	2	1	no	—
76	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
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	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	—	2	1	no	—
84	Dic₁₉₁⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₁₉₁	—	2	1	no	—
86	C₂²⋊Dic₁₉₁	—	2	1	no	—
87	C₂²⋊Dic₁₉₁	—	2	1	no	—
88	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
91	C₄×Dic₁₉₁	—	2	1	no	—
92	C₄×Dic₁₉₁	—	2	1	no	—
93	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₉₁	—	2	1	no	—
99	SD₁₆×C₁₉₁	—	2	2	no	—
100	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉₁	—	2	1	no	—
103	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
106	C₄²×C₁₉₁	—	2	1	no	—
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
109	Dic₃₈₂	—	2	1	no	—
110	Dic₃₈₂	—	2	1	no	—
111	Dic₃₈₂	—	2	1	no	—
112	Dic₃₈₂	—	2	1	no	—
113	Dic₃₈₂	—	4	2	no	—
114	C₂²⋊D₁₉₁	—	4	2	no	—
115	C₂×Dic₁₉₁	—	2	1	no	—
116	C₂×Dic₁₉₁	—	2	1	no	—
117	C₂×Dic₁₉₁	—	2	1	no	—
118	C₂×Dic₁₉₁	—	2	1	no	—
119	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₉₁	—	2	2	no	—
123	C₄×D₁₉₁	—	2	1	no	—
124	C₄×D₁₉₁	—	2	1	no	—
125	C₄×D₁₉₁	—	4	1	no	—
126	C₁₉₁⋊C₈	—	2	1	no	—
127	C₁₉₁⋊C₈	#41	1	2	yes	—
128	C₂³×C₁₉₁	—	2	1	no	—
129	Q₈×C₁₉₁	—	2	1	no	—
130	Q₈×C₁₉₁	—	2	1	no	—
131	D₄×C₁₉₁	—	2	1	no	—
132	D₄×C₁₉₁	—	2	1	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	—	4	2	no	—
137	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
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	#41	1	2	yes	—
142	C₂²×C₁₉₁	—	4	1	no	—
143	C₂²×C₁₉₁	—	4	1	no	—
144	C₂²×C₁₉₁	—	4	1	no	—
145	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₉₁	—	2	1	no	—
147	C₂×D₁₉₁	—	2	1	no	—
148	C₂×D₁₉₁	—	2	1	no	—
149	C₄×C₁₉₁	—	2	1	no	—
150	C₄×C₁₉₁	—	4	1	no	—
151	C₄×C₁₉₁	—	4	1	no	—
152	C₄×C₁₉₁	—	4	1	no	—
153	C₄×C₁₉₁	#137	1	2	yes	—
154	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
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	—	8	1	no	—
164	C₂×C₁₉₁	#532	1	1	yes	characteristic
165	C₂×C₁₉₁	C₂×#137	1	1	yes	characteristic
166	C₂×C₁₉₁	#575	1	1	yes	characteristic
167	D₁₉₁	—	4	1	no	—
168	C₁₉₁	#3172	1	1	yes	characteristic, complemented, Sylow
169	#3172	—	191	1	no	maximal, Sylow
170	#533	—	191	1	no	—
171	#529	—	191	1	no	—
172	#501	—	191	1	no	—
173	#497	—	191	1	no	—
174	#491	—	191	1	no	—
175	#489	—	191	1	no	—
176	#470	—	191	1	no	—
177	#470	—	191	1	no	—
178	#468	—	191	1	no	—
179	#468	—	191	1	no	—
180	C₄×Q₁₆	—	191	1	no	—
181	C₄×Q₁₆	—	191	1	no	—
182	C₄×SD₁₆	—	191	1	no	—
183	C₄×SD₁₆	D₁₉₁	1	1	yes	characteristic, complemented
184	C₄×M₁₆	—	191	1	no	—
185	C₂×Q₁₆	—	191	1	no	—
186	C₂×Q₁₆	—	191	1	no	—
187	C₂×SD₁₆	—	191	1	no	—
188	C₂×SD₁₆	C₂×D₁₉₁	1	1	yes	characteristic
189	C₂×M₁₆	—	191	1	no	—
190	C₂×M₁₆	—	191	1	no	—
191	C₄⋊Q₈	—	191	1	no	—
192	C₄⋊Q₈	—	191	1	no	—
193	#126	—	382	1	no	—
194	#125	—	382	1	no	—
195	#125	—	191	1	no	—
196	#124	—	191	1	no	—
197	#123	—	382	1	no	—
198	C₂²⋊Q₈	—	382	1	no	—
199	C₂²⋊Q₈	—	191	1	no	—
200	C₂²⋊Q₈	—	191	1	no	—
201	C₂²⋊Q₈	—	191	1	no	—
202	#121	—	191	1	no	—
203	C₄×Q₈	—	382	1	no	—
204	C₄×Q₈	—	191	1	no	—
205	C₄×Q₈	—	191	1	no	—
206	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
207	C₄×D₄	—	382	1	no	—
208	C₄×D₄	C₂×D₁₉₁	1	1	yes	characteristic
209	#117	—	382	1	no	—
210	#117	—	382	1	no	—
211	C₂×C₄²	—	191	1	no	—
212	#107	—	191	1	no	—
213	#107	—	191	1	no	—
214	#106	—	191	1	no	—
215	#106	C₂×D₁₉₁	1	1	yes	characteristic
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	#97	—	191	1	no	—
225	#97	—	191	1	no	—
226	C₄×C₈	—	191	1	no	—
227	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
228	C₂×Q₈	—	382	1	no	—
229	C₂×Q₈	—	191	1	no	—
230	C₂×Q₈	—	191	1	no	—
231	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
232	C₂×D₄	—	382	1	no	—
233	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	191	1	no	—
236	C₂²×C₄	—	191	1	no	—
237	C₂²×C₄	—	191	1	no	—
238	Q₁₆	—	382	2	no	—
239	Q₁₆	—	382	2	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	382	2	no	—
242	M₁₆	—	382	2	no	—
243	M₁₆	—	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	C₂²×D₁₉₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	382	1	no	—
265	C₂²⋊C₄	—	382	1	no	—
266	C₂²⋊C₄	—	382	1	no	—
267	C₂²⋊C₄	—	382	1	no	—
268	C₂²⋊C₄	—	382	1	no	—
269	C₂²⋊C₄	—	382	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	382	1	no	—
272	C₄²	—	382	1	no	—
273	C₄²	—	191	1	no	—
274	C₄²	—	191	1	no	—
275	C₄²	—	191	1	no	—
276	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	191	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	382	1	no	—
282	Q₈	—	382	1	no	—
283	Q₈	—	382	1	no	—
284	Q₈	—	382	1	no	—
285	Q₈	—	764	2	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	764	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	382	2	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	—	382	1	no	—
299	C₂×C₄	—	382	1	no	—
300	C₂×C₄	—	191	1	no	—
301	C₂×C₄	—	191	1	no	—
302	C₂×C₄	—	191	1	no	—
303	C₂×C₄	—	764	1	no	—
304	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
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	—	382	1	no	—
309	C₈	—	191	2	no	—
310	C₈	#15690287	1	2	yes	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	382	1	no	—
315	C₂²	—	382	1	no	—
316	C₂²	—	382	1	no	—
317	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	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	—	764	1	no	—
326	C₄	—	764	1	no	—
327	C₄	—	764	1	no	—
328	C₄	—	764	1	no	—
329	C₄	#20259351	1	2	yes	—
330	C₄	#41×D₁₉₁	1	1	yes	characteristic
331	C₄	C₂×#15690287	1	1	yes	characteristic
332	C₂	—	8	1	no	—
333	C₂	—	764	1	no	—
334	C₂	#27133210	1	1	yes	characteristic
335	C₂	C₂×#20259351	1	1	yes	characteristic
336	C₂	#27133460	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₄×SD₁₆×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	#20259293	C₂²
hypercenter	C₄×SD₁₆	D₁₉₁
hypocenter	C₁₉₁	#3172
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #27133217	C₂	semidirect
3. #27133070	C₂	semidirect
4. #27133054	C₂	semidirect
5. #27133030	C₂	semidirect
6. #27133021	C₂	semidirect
7. #27132922	C₂	semidirect
8. #27132920	C₂	semidirect
11. C₄×Dic₇₆₄	C₂	semidirect
12. C₄×#15690277	C₂	semidirect
13. C₄×#15690275	C₂	semidirect
14. C₄×#15690270	C₂	semidirect
15. C₄×SD₁₆×C₁₉₁	C₂	semidirect
168. C₁₉₁	#3172	semidirect
183. C₄×SD₁₆	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133217	2
#27133195	2
#27133070	2
#27133054	2
#27133030	2
#27133021	2
#27132922	2
#27132920	2
#27132908	2
#27132906	2
C₄×Dic₇₆₄	2
C₄×#15690277	2
C₄×#15690275	2
C₄×#15690270	2
C₄×SD₁₆×C₁₉₁	2
#3172	191