Subgroups — #37583932

Order 24448.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133280	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133266	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132973	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132961	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132952	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132941	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132909	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132897	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132890	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132877	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132876	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132862	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132856	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132691	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690276	C₂²	1	1	yes	characteristic
17	C₂×#15690275	C₂²	1	1	yes	characteristic
18	C₂×#15690272	C₂²	1	1	yes	characteristic
19	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
20	#20259321	C₂²	1	1	yes	characteristic
21	#20259318	C₂²	1	1	yes	characteristic
22	#20259315	—	2	1	no	—
23	#20259313	C₂²	1	1	yes	characteristic
24	#20259309	—	2	1	no	—
25	#20259309	C₂²	1	1	yes	characteristic
26	#20259306	—	2	1	no	—
27	#20259304	—	2	1	no	—
28	#20259303	—	2	1	no	—
29	#20259301	C₂²	1	1	yes	characteristic
30	#20259299	—	2	1	no	—
31	#20259296	C₂²	1	1	yes	characteristic
32	#20259294	—	2	1	no	—
33	#20259293	—	2	1	no	—
34	#20259287	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
37	#20259284	C₂²	1	1	yes	characteristic
38	#20259281	C₂²	1	1	yes	characteristic
39	#20259256	C₂²	1	1	yes	characteristic
40	#20259255	C₂²	1	1	yes	characteristic
41	#20259253	C₂²	1	1	yes	characteristic
42	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
46	#20259247	C₂²	1	1	yes	characteristic
47	#20259245	C₂²	1	1	yes	characteristic
48	#20259244	C₂²	1	1	yes	characteristic
49	#20259243	C₂²	1	1	yes	characteristic
50	#20259243	C₂²	1	1	yes	characteristic
51	#20259242	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
54	#121×C₁₉₁	C₂²	1	1	yes	characteristic
55	#121×C₁₉₁	C₂²	1	1	yes	characteristic
56	#106×C₁₉₁	C₂²	1	1	yes	characteristic
57	#102×C₁₉₁	C₂²	1	1	yes	characteristic
58	#102×C₁₉₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₈₂	—	2	1	no	—
60	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
61	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
63	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
64	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
65	C₂²×Dic₁₉₁	—	2	1	no	—
66	C₂²×Dic₁₉₁	—	2	1	no	—
67	C₂×C₄×D₁₉₁	—	2	1	no	—
68	#15690276	—	4	1	no	—
69	#15690275	—	4	1	no	—
70	#15690272	—	4	1	no	—
71	C₂×C₁₉₁⋊C₈	—	4	1	no	—
72	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₉₁	—	2	1	no	—
75	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₁₉₁	—	2	1	no	—
84	C₂²⋊Dic₁₉₁	—	2	1	no	—
85	C₂²⋊Dic₁₉₁	—	2	1	no	—
86	C₂²⋊Dic₁₉₁	—	2	1	no	—
87	C₂²⋊Dic₁₉₁	—	2	1	no	—
88	C₂²⋊Dic₁₉₁	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₉₁	D₄	1	1	yes	characteristic
90	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
92	C₄×Dic₁₉₁	—	2	1	no	—
93	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₉₁	—	2	1	no	—
96	C₂×D₄×C₁₉₁	—	2	1	no	—
97	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
100	D₈×C₁₉₁	—	4	1	no	—
101	M₁₆×C₁₉₁	—	4	1	no	—
102	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
107	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
108	C₂²×D₁₉₁	—	2	1	no	—
109	Dic₃₈₂	—	2	1	no	—
110	Dic₃₈₂	—	2	1	no	—
111	Dic₃₈₂	—	8	1	no	—
112	D₇₆₄	—	2	1	no	—
113	D₇₆₄	—	2	1	no	—
114	C₂²⋊D₁₉₁	—	8	1	no	—
115	C₂²⋊D₁₉₁	—	8	1	no	—
116	C₂²⋊D₁₉₁	—	8	1	no	—
117	C₂×Dic₁₉₁	—	2	1	no	—
118	C₂×Dic₁₉₁	—	2	1	no	—
119	C₂×Dic₁₉₁	—	2	1	no	—
120	C₂×Dic₁₉₁	—	8	1	no	—
121	C₂×Dic₁₉₁	—	8	1	no	—
122	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₉₁	—	8	1	no	—
127	C₁₉₁⋊C₈	—	2	1	no	—
128	C₁₉₁⋊C₈	—	2	1	no	—
129	C₂³×C₁₉₁	—	2	1	no	—
130	C₂³×C₁₉₁	—	2	1	no	—
131	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
132	D₄×C₁₉₁	—	2	1	no	—
133	D₄×C₁₉₁	—	2	1	no	—
134	D₄×C₁₉₁	—	2	1	no	—
135	D₄×C₁₉₁	—	2	1	no	—
136	D₄×C₁₉₁	—	8	1	no	—
137	D₄×C₁₉₁	—	8	1	no	—
138	C₂×C₄×C₁₉₁	—	2	1	no	—
139	C₂×C₄×C₁₉₁	—	2	1	no	—
140	C₂×C₄×C₁₉₁	—	2	1	no	—
141	C₂×C₄×C₁₉₁	—	2	1	no	—
142	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₉₁	—	2	1	no	—
145	C₈×C₁₉₁	—	2	1	no	—
146	C₂²×C₁₉₁	—	2	1	no	—
147	C₂²×C₁₉₁	—	2	1	no	—
148	C₂²×C₁₉₁	—	2	1	no	—
149	C₂²×C₁₉₁	—	4	1	no	—
150	C₂²×C₁₉₁	—	4	1	no	—
151	C₂²×C₁₉₁	—	4	1	no	—
152	C₂²×C₁₉₁	—	4	1	no	—
153	C₂²×C₁₉₁	—	4	1	no	—
154	C₂²×C₁₉₁	—	4	1	no	—
155	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₉₁	—	4	1	no	—
157	C₂×D₁₉₁	—	4	1	no	—
158	C₂×D₁₉₁	—	4	1	no	—
159	C₄×C₁₉₁	—	2	1	no	—
160	C₄×C₁₉₁	—	4	1	no	—
161	C₄×C₁₉₁	—	4	1	no	—
162	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₁₉₁	—	2	1	no	—
165	Dic₁₉₁	—	2	1	no	—
166	Dic₁₉₁	—	2	1	no	—
167	Dic₁₉₁	—	2	1	no	—
168	Dic₁₉₁	—	4	1	no	—
169	Dic₁₉₁	—	4	1	no	—
170	Dic₁₉₁	—	4	1	no	—
171	C₂×C₁₉₁	—	4	1	no	—
172	C₂×C₁₉₁	—	8	1	no	—
173	C₂×C₁₉₁	—	8	1	no	—
174	C₂×C₁₉₁	#537	1	1	yes	characteristic
175	C₂×C₁₉₁	#576	1	1	yes	characteristic
176	C₂×C₁₉₁	#577	1	1	yes	characteristic
177	D₁₉₁	—	8	1	no	—
178	C₁₉₁	#3257	1	1	yes	characteristic, complemented, Sylow
179	#3257	—	191	1	no	maximal, Sylow
180	#542	—	191	1	no	—
181	#540	—	191	1	no	—
182	#479	—	191	1	no	—
183	#478	—	191	1	no	—
184	#476	—	191	1	no	—
185	#474	—	191	1	no	—
186	#469	D₁₉₁	1	1	yes	characteristic, complemented
187	#468	—	191	1	no	—
188	#466	—	191	1	no	—
189	#465	—	191	1	no	—
190	#463	—	191	1	no	—
191	#463	—	191	1	no	—
192	#460	—	191	1	no	—
193	#459	—	191	1	no	—
194	#424	—	191	1	no	—
195	C₂×SD₁₆	—	191	1	no	—
196	C₂×SD₁₆	—	191	1	no	—
197	C₂×D₈	—	191	1	no	—
198	C₂×D₈	C₂×D₁₉₁	1	1	yes	characteristic
199	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
200	C₂²×C₈	—	191	1	no	—
201	C₄⋊Q₈	—	191	1	no	—
202	#127	—	191	1	no	—
203	#126	—	382	1	no	—
204	#125	—	191	1	no	—
205	#124	—	382	1	no	—
206	#124	—	191	1	no	—
207	#123	—	382	1	no	—
208	#123	—	382	1	no	—
209	#123	—	382	1	no	—
210	C₂²⋊Q₈	—	382	1	no	—
211	C₂²⋊Q₈	—	191	1	no	—
212	#121	—	382	1	no	—
213	#121	—	382	1	no	—
214	#121	—	191	1	no	—
215	#121	C₂×D₁₉₁	1	1	yes	characteristic
216	#121	C₂×D₁₉₁	1	1	yes	characteristic
217	C₄×Q₈	—	191	1	no	—
218	C₄×D₄	—	191	1	no	—
219	C₄×D₄	—	191	1	no	—
220	C₄×D₄	—	191	1	no	—
221	#117	—	191	1	no	—
222	#107	—	191	1	no	—
223	#107	—	191	1	no	—
224	#106	—	191	1	no	—
225	#106	C₂×D₁₉₁	1	1	yes	characteristic
226	C₄⋊C₈	—	191	1	no	—
227	C₄⋊C₈	—	191	1	no	—
228	C₄⋊C₈	—	191	1	no	—
229	C₄⋊C₈	—	191	1	no	—
230	#103	—	191	1	no	—
231	#103	—	191	1	no	—
232	#102	—	191	1	no	—
233	#102	—	191	1	no	—
234	#102	—	191	1	no	—
235	#102	—	191	1	no	—
236	#102	C₂×D₁₉₁	1	1	yes	characteristic
237	#102	C₂×D₁₉₁	1	1	yes	characteristic
238	C₂×Q₈	—	382	1	no	—
239	C₂×Q₈	—	191	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	382	1	no	—
243	C₂×D₄	—	382	1	no	—
244	C₂×D₄	—	382	1	no	—
245	C₂×D₄	—	191	1	no	—
246	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
247	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
248	C₂²×C₄	—	382	1	no	—
249	C₂²×C₄	—	382	1	no	—
250	C₂²×C₄	—	382	1	no	—
251	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
252	SD₁₆	—	764	1	no	—
253	SD₁₆	—	764	1	no	—
254	D₈	—	4	1	no	—
255	D₈	—	764	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	191	1	no	—
258	C₂×C₈	—	191	1	no	—
259	C₂×C₈	—	764	1	no	—
260	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
262	C₄⋊C₄	—	382	1	no	—
263	C₄⋊C₄	—	382	1	no	—
264	C₄⋊C₄	—	382	1	no	—
265	C₄⋊C₄	—	382	1	no	—
266	C₄⋊C₄	—	382	1	no	—
267	C₄⋊C₄	—	191	1	no	—
268	C₄⋊C₄	—	191	1	no	—
269	C₄⋊C₄	—	191	1	no	—
270	C₄⋊C₄	—	191	1	no	—
271	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	382	1	no	—
276	C₂²⋊C₄	—	382	1	no	—
277	C₂²⋊C₄	—	382	1	no	—
278	C₂²⋊C₄	—	382	1	no	—
279	C₂²⋊C₄	—	382	1	no	—
280	C₂²⋊C₄	—	382	1	no	—
281	C₂²⋊C₄	—	382	1	no	—
282	C₂²⋊C₄	—	191	1	no	—
283	C₂²⋊C₄	—	191	1	no	—
284	C₄²	—	382	1	no	—
285	C₄²	—	191	1	no	—
286	C₄²	—	191	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	382	1	no	—
290	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
291	Q₈	—	382	1	no	—
292	Q₈	—	382	1	no	—
293	Q₈	—	1528	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	382	1	no	—
301	D₄	—	382	1	no	—
302	D₄	—	1528	1	no	—
303	D₄	—	1528	1	no	—
304	D₄	—	1528	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	382	1	no	—
310	C₂×C₄	—	382	1	no	—
311	C₂×C₄	—	382	1	no	—
312	C₂×C₄	—	191	1	no	—
313	C₂×C₄	—	191	1	no	—
314	C₂×C₄	—	191	1	no	—
315	C₂×C₄	—	191	1	no	—
316	C₂×C₄	—	1528	1	no	—
317	C₂×C₄	—	1528	1	no	—
318	C₂×C₄	—	1528	1	no	—
319	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	382	1	no	—
324	C₈	—	382	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	764	1	no	—
335	C₂²	—	764	1	no	—
336	C₂²	—	764	1	no	—
337	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	382	1	no	—
342	C₄	—	382	1	no	—
343	C₄	—	382	1	no	—
344	C₄	—	382	1	no	—
345	C₄	—	764	1	no	—
346	C₄	—	764	1	no	—
347	C₄	—	764	1	no	—
348	C₄	#20259367	1	1	yes	characteristic
349	C₄	#20259369	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	8	1	no	—
352	C₂	—	8	1	no	—
353	C₂	—	1528	1	no	—
354	C₂	#27133240	1	1	yes	characteristic
355	C₂	#27133467	1	1	yes	characteristic
356	C₂	#27133473	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#469×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
hypercenter	#469	D₁₉₁
hypocenter	C₁₉₁	#3257
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #27133280	C₂	semidirect
2. #27133266	C₂	semidirect
3. #27132973	C₂	semidirect
4. #27132961	C₂	semidirect
5. #27132952	C₂	semidirect
6. #27132941	C₂	semidirect
7. #27132909	C₂	semidirect
8. #27132897	C₂	semidirect
9. #27132890	C₂	semidirect
10. #27132877	C₂	semidirect
11. #27132876	C₂	semidirect
12. #27132862	C₂	semidirect
13. #27132856	C₂	semidirect
14. #27132691	C₂	semidirect
15. #469×C₁₉₁	C₂	semidirect
178. C₁₉₁	#3257	semidirect
186. #469	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133280	2
#27133266	2
#27132973	2
#27132961	2
#27132952	2
#27132941	2
#27132909	2
#27132897	2
#27132890	2
#27132877	2
#27132876	2
#27132862	2
#27132856	2
#27132691	2
#469×C₁₉₁	2
#3257	191