Subgroups — #37543896

Order 23168.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041816	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041613	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041599	C₂	1	1	yes	maximal, characteristic
5	#26041589	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041588	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041504	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041503	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041433	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041419	C₂	1	1	yes	maximal, characteristic
11	#26041384	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041381	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041377	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041334	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₈₁	C₂²	1	1	yes	characteristic
17	C₂×#15668439	C₂²	1	1	yes	characteristic
18	C₂×#15668438	C₂²	1	1	yes	characteristic
19	C₂×#15668436	C₂²	1	1	yes	characteristic
20	#19153811	—	2	1	no	—
21	#19153807	C₂²	1	1	yes	characteristic
22	#19153805	C₂²	1	1	yes	characteristic
23	#19153801	—	2	1	no	—
24	#19153792	C₂²	1	1	yes	characteristic
25	#19153791	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₆₂	—	2	1	no	—
28	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
29	#19153781	—	2	1	no	—
30	#19153781	C₂²	1	1	yes	characteristic
31	C₄×D₇₂₄	—	2	1	no	—
32	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
33	#19153773	—	2	1	no	—
34	C₄⋊C₄×D₁₈₁	—	2	1	no	—
35	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
36	C₄²×D₁₈₁	—	2	1	no	—
37	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
38	#19153750	C₂²	1	1	yes	characteristic
39	#19153749	C₂²	1	1	yes	characteristic
40	#19153747	C₂²	1	1	yes	characteristic
41	#19153745	C₂²	1	1	yes	characteristic
42	#19153745	C₂²	1	1	yes	characteristic
43	#19153741	C₂²	1	1	yes	characteristic
44	#19153740	C₂²	1	1	yes	characteristic
45	#19153740	C₂²	1	1	yes	characteristic
46	#19153739	C₂²	1	1	yes	characteristic
47	#19153738	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153728	C₂²	1	1	yes	characteristic
50	#19153728	C₂²	1	1	yes	characteristic
51	#19153727	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×C₁₈₁	C₂²	1	1	yes	characteristic
56	#103×C₁₈₁	C₂²	1	1	yes	characteristic
57	#103×C₁₈₁	C₂²	1	1	yes	characteristic
58	#97×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
61	Q₈×D₁₈₁	—	2	2	no	—
62	Q₈×D₁₈₁	—	2	2	no	—
63	C₂×D₇₂₄	—	2	1	no	—
64	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	C₂×C₄×D₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
71	#15668439	—	2	2	no	—
72	#15668438	—	2	2	no	—
73	#15668436	—	2	2	no	—
74	C₂×#13089678	C₂³	1	1	yes	characteristic
75	C₂×#13089678	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	—	2	1	no	—
77	C₄⋊Dic₁₈₁	—	2	1	no	—
78	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668428	—	2	1	no	—
85	#15668426	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	C₄×Dic₁₈₁	—	2	1	no	—
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₄×Dic₁₈₁	—	2	1	no	—
93	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₈₁	—	2	2	no	—
99	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	—	2	1	no	—
102	C₄⋊C₄×C₁₈₁	—	2	1	no	—
103	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₄²×C₁₈₁	—	2	1	no	—
106	C₄²×C₁₈₁	—	2	1	no	—
107	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₈₁	—	2	1	no	—
109	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
110	Dic₃₆₂	—	2	4	no	—
111	Dic₃₆₂	—	2	1	no	—
112	Dic₃₆₂	#41	1	2	yes	—
113	D₇₂₄	—	2	1	no	—
114	D₇₂₄	—	2	1	no	—
115	D₇₂₄	—	4	2	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	—	2	1	no	—
118	C₂×Dic₁₈₁	—	2	1	no	—
119	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₈₁	—	2	4	no	—
123	C₄×D₁₈₁	—	2	2	no	—
124	C₄×D₁₈₁	—	2	1	no	—
125	C₄×D₁₈₁	—	2	1	no	—
126	C₄×D₁₈₁	—	4	1	no	—
127	C₄×D₁₈₁	—	4	2	no	—
128	C₄×D₁₈₁	—	4	2	no	—
129	#13089678	—	2	1	no	—
130	#13089678	—	2	1	no	—
131	Q₈×C₁₈₁	—	2	1	no	—
132	Q₈×C₁₈₁	—	2	1	no	—
133	Q₈×C₁₈₁	—	2	1	no	—
134	Q₈×C₁₈₁	#41	1	2	yes	—
135	C₂×C₄×C₁₈₁	—	2	1	no	—
136	C₂×C₄×C₁₈₁	—	2	1	no	—
137	C₂×C₄×C₁₈₁	—	2	1	no	—
138	C₂×C₄×C₁₈₁	—	2	1	no	—
139	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₈₁	—	2	1	no	—
143	C₈×C₁₈₁	—	2	1	no	—
144	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₈₁	—	2	1	no	—
146	C₂×D₁₈₁	—	2	1	no	—
147	C₂×D₁₈₁	—	4	1	no	—
148	C₂×D₁₈₁	—	4	1	no	—
149	C₂×D₁₈₁	—	4	1	no	—
150	C₂×D₁₈₁	#137	1	2	yes	—
151	C₄×C₁₈₁	—	2	2	no	—
152	C₄×C₁₈₁	—	2	1	no	—
153	C₄×C₁₈₁	—	2	1	no	—
154	C₄×C₁₈₁	—	4	1	no	—
155	C₄×C₁₈₁	—	4	1	no	—
156	C₄×C₁₈₁	—	4	1	no	—
157	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
159	Dic₁₈₁	—	2	2	no	—
160	Dic₁₈₁	—	2	1	no	—
161	Dic₁₈₁	—	2	1	no	—
162	Dic₁₈₁	—	2	1	no	—
163	Dic₁₈₁	—	4	1	no	—
164	Dic₁₈₁	—	4	1	no	—
165	C₂×C₁₈₁	#525	1	1	yes	characteristic
166	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
167	C₂×C₁₈₁	#575	1	1	yes	characteristic
168	D₁₈₁	—	2	2	no	—
169	D₁₈₁	—	8	1	no	—
170	C₁₈₁	#3117	1	1	yes	characteristic, complemented, Sylow
171	#3117	—	181	1	no	maximal, Sylow
172	#529	—	181	1	no	—
173	C₂×C₄×Q₈	—	181	1	no	—
174	#484	—	181	1	no	—
175	#481	—	181	1	no	—
176	#479	—	181	1	no	—
177	#479	—	181	1	no	—
178	#464	—	181	1	no	—
179	#464	—	181	1	no	—
180	#451	—	181	1	no	—
181	#448	—	181	1	no	—
182	#441	—	181	1	no	—
183	#441	D₁₈₁	1	1	yes	characteristic, complemented
184	#440	—	181	1	no	—
185	#440	—	181	1	no	—
186	#432	—	181	1	no	—
187	C₂²×Q₈	—	181	1	no	—
188	C₂×Q₁₆	—	181	1	no	—
189	C₂×Q₁₆	C₂×D₁₈₁	1	1	yes	characteristic
190	C₂×SD₁₆	—	181	1	no	—
191	C₂×SD₁₆	—	181	1	no	—
192	#126	—	362	1	no	—
193	#125	—	181	1	no	—
194	#124	—	181	1	no	—
195	#123	—	362	1	no	—
196	C₂²⋊Q₈	—	181	1	no	—
197	#121	—	181	1	no	—
198	C₄×Q₈	—	362	1	no	—
199	C₄×Q₈	—	362	1	no	—
200	C₄×Q₈	—	181	1	no	—
201	C₄×Q₈	—	181	1	no	—
202	C₄×Q₈	—	181	1	no	—
203	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
204	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
205	C₄×D₄	—	362	1	no	—
206	C₄×D₄	—	181	1	no	—
207	#117	—	362	1	no	—
208	C₂×C₄⋊C₄	—	362	1	no	—
209	C₂×C₄⋊C₄	—	181	1	no	—
210	C₂×C₄²	—	362	1	no	—
211	C₂×C₄²	—	181	1	no	—
212	#107	—	181	1	no	—
213	#107	—	181	1	no	—
214	#106	—	181	1	no	—
215	#106	C₂×D₁₈₁	1	1	yes	characteristic
216	C₄⋊C₈	—	181	1	no	—
217	C₄⋊C₈	—	181	1	no	—
218	#103	—	181	1	no	—
219	#103	—	181	1	no	—
220	#103	—	181	1	no	—
221	#103	—	181	1	no	—
222	#103	C₂×D₁₈₁	1	1	yes	characteristic
223	#103	C₂×D₁₈₁	1	1	yes	characteristic
224	#102	—	181	1	no	—
225	#102	—	181	1	no	—
226	C₂²⋊C₈	—	181	1	no	—
227	C₂²⋊C₈	—	181	1	no	—
228	#97	—	181	1	no	—
229	#97	C₂×D₁₈₁	1	1	yes	characteristic
230	C₂×Q₈	—	362	1	no	—
231	C₂×Q₈	—	362	2	no	—
232	C₂×Q₈	—	362	2	no	—
233	C₂×Q₈	—	181	1	no	—
234	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
236	C₂×D₄	—	362	1	no	—
237	C₂×D₄	—	181	1	no	—
238	C₂²×C₄	—	362	1	no	—
239	C₂²×C₄	—	362	1	no	—
240	C₂²×C₄	—	362	1	no	—
241	C₂²×C₄	—	181	1	no	—
242	C₂²×C₄	—	181	1	no	—
243	C₂²×C₄	—	181	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	362	2	no	—
246	SD₁₆	—	362	2	no	—
247	SD₁₆	—	362	2	no	—
248	C₂×C₈	—	181	1	no	—
249	C₂×C₈	—	181	1	no	—
250	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	362	1	no	—
255	C₄⋊C₄	—	362	1	no	—
256	C₄⋊C₄	—	362	1	no	—
257	C₄⋊C₄	—	362	1	no	—
258	C₄⋊C₄	—	362	1	no	—
259	C₄⋊C₄	—	362	1	no	—
260	C₄⋊C₄	—	362	1	no	—
261	C₄⋊C₄	—	181	1	no	—
262	C₄⋊C₄	—	181	1	no	—
263	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	362	1	no	—
266	C₂²⋊C₄	—	362	1	no	—
267	C₂²⋊C₄	—	362	1	no	—
268	C₂²⋊C₄	—	362	1	no	—
269	C₂²⋊C₄	—	362	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	362	1	no	—
273	C₄²	—	362	1	no	—
274	C₄²	—	362	1	no	—
275	C₄²	—	181	1	no	—
276	C₄²	—	181	1	no	—
277	C₄²	—	181	1	no	—
278	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
279	C₂³	—	362	1	no	—
280	C₂³	—	181	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	362	4	no	—
285	Q₈	—	362	1	no	—
286	Q₈	—	181	2	no	—
287	Q₈	#15668449	1	2	yes	—
288	D₄	—	362	1	no	—
289	D₄	—	362	1	no	—
290	D₄	—	724	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	362	4	no	—
296	C₂×C₄	—	362	2	no	—
297	C₂×C₄	—	362	1	no	—
298	C₂×C₄	—	362	1	no	—
299	C₂×C₄	—	362	1	no	—
300	C₂×C₄	—	362	1	no	—
301	C₂×C₄	—	362	1	no	—
302	C₂×C₄	—	181	1	no	—
303	C₂×C₄	—	181	1	no	—
304	C₂×C₄	—	181	1	no	—
305	C₂×C₄	—	724	1	no	—
306	C₂×C₄	—	724	2	no	—
307	C₂×C₄	—	724	2	no	—
308	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	362	1	no	—
314	C₈	—	362	1	no	—
315	C₂²	—	362	1	no	—
316	C₂²	—	362	1	no	—
317	C₂²	—	181	2	no	—
318	C₂²	—	724	1	no	—
319	C₂²	—	724	1	no	—
320	C₂²	—	724	1	no	—
321	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	362	2	no	—
329	C₄	—	362	1	no	—
330	C₄	—	362	1	no	—
331	C₄	—	362	1	no	—
332	C₄	—	724	1	no	—
333	C₄	—	724	1	no	—
334	C₄	#41×D₁₈₁	1	1	yes	characteristic
335	C₄	C₂×#15668449	1	1	yes	characteristic
336	C₂	—	362	2	no	—
337	C₂	—	1448	1	no	—
338	C₂	#26041788	1	1	yes	characteristic
339	C₂	#137×D₁₈₁	1	1	yes	characteristic
340	C₂	#26042078	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#441×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	#19153791	C₂²
hypercenter	#441	D₁₈₁
hypocenter	C₁₈₁	#3117
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₁₈₁	C₂	semidirect
3. #26041613	C₂	semidirect
5. #26041589	C₂	semidirect
6. #26041588	C₂	semidirect
7. #26041504	C₂	semidirect
8. #26041503	C₂	semidirect
9. #26041433	C₂	semidirect
11. #26041384	C₂	semidirect
12. #26041381	C₂	semidirect
13. #26041377	C₂	semidirect
14. #26041334	C₂	semidirect
15. #441×C₁₈₁	C₂	semidirect
170. C₁₈₁	#3117	semidirect
183. #441	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041816	2
C₄×Q₈×D₁₈₁	2
#26041613	2
#26041599	2
#26041589	2
#26041588	2
#26041504	2
#26041503	2
#26041433	2
#26041419	2
#26041384	2
#26041381	2
#26041377	2
#26041334	2
#441×C₁₈₁	2
#3117	181