Subgroups — #37544878

Order 23168.

326 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041832	C₂	1	1	yes	maximal, characteristic
2	#26041831	C₂	1	1	yes	maximal, characteristic
3	#26041683	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041676	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041628	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041628	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041621	C₂	1	1	yes	maximal, characteristic
8	#26041620	C₂	1	1	yes	maximal, characteristic
9	#26041617	C₂	1	1	yes	maximal, characteristic
10	#26041589	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041588	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041585	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041584	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041329	C₂	1	1	yes	maximal, characteristic
15	#485×C₁₈₁	C₂	1	1	yes	maximal, characteristic
16	#19153815	C₂²	1	1	yes	characteristic
17	#19153815	C₂²	1	1	yes	characteristic
18	#19153815	C₂²	1	1	yes	characteristic
19	#19153814	C₂²	1	1	yes	characteristic
20	#19153807	C₂²	1	1	yes	characteristic
21	#19153807	C₂²	1	1	yes	characteristic
22	#19153806	—	2	1	no	—
23	#19153795	C₂²	1	1	yes	characteristic
24	#19153793	—	2	1	no	—
25	#19153793	—	2	1	no	—
26	#19153793	—	2	1	no	—
27	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
28	C₄×Dic₃₆₂	—	2	1	no	—
29	C₄×Dic₃₆₂	—	2	1	no	—
30	#19153781	—	2	1	no	—
31	#19153781	C₂²	1	1	yes	characteristic
32	#19153775	C₂²	1	1	yes	characteristic
33	#19153772	—	2	1	no	—
34	C₂×C₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
35	#19153750	C₂²	1	1	yes	characteristic
36	#19153749	C₂²	1	1	yes	characteristic
37	#19153749	C₂²	1	1	yes	characteristic
38	#19153748	C₂²	1	1	yes	characteristic
39	#19153747	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	#19153741	C₂²	1	1	yes	characteristic
45	#19153740	C₂²	1	1	yes	characteristic
46	#19153740	C₂²	1	1	yes	characteristic
47	#19153739	C₂²	1	1	yes	characteristic
48	#19153739	C₂²	1	1	yes	characteristic
49	C₂²⋊#13089678	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	#117×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×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	C₂²⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	—	2	1	no	—
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₈₁	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₈₁	D₄	1	1	yes	characteristic
65	C₂×#13089678	C₂³	1	1	yes	characteristic
66	C₂×#13089678	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₈₁	—	2	1	no	—
68	C₄⋊Dic₁₈₁	—	2	1	no	—
69	C₄⋊Dic₁₈₁	—	2	1	no	—
70	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
73	#15668428	—	2	1	no	—
74	#15668428	—	2	1	no	—
75	#15668428	—	2	1	no	—
76	#15668428	—	2	1	no	—
77	#15668428	—	2	1	no	—
78	#15668428	—	2	1	no	—
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	C₂²⋊Dic₁₈₁	—	2	1	no	—
83	C₂²⋊Dic₁₈₁	—	2	1	no	—
84	C₄×Dic₁₈₁	—	2	1	no	—
85	C₄×Dic₁₈₁	—	2	1	no	—
86	C₄×Dic₁₈₁	—	2	1	no	—
87	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
88	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
89	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₈₁	—	2	1	no	—
96	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
100	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
101	C₄²×C₁₈₁	—	2	1	no	—
102	Dic₃₆₂	—	2	1	no	—
103	Dic₃₆₂	—	2	1	no	—
104	Dic₃₆₂	—	4	2	no	—
105	Dic₃₆₂	—	4	2	no	—
106	Dic₃₆₂	—	4	2	no	—
107	C₂×Dic₁₈₁	—	2	2	no	—
108	C₂×Dic₁₈₁	—	2	2	no	—
109	C₂×Dic₁₈₁	—	2	1	no	—
110	C₂×Dic₁₈₁	—	2	1	no	—
111	C₂×Dic₁₈₁	—	2	1	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
117	#13089678	—	2	1	no	—
118	#13089678	—	2	1	no	—
119	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
120	Q₈×C₁₈₁	—	2	1	no	—
121	Q₈×C₁₈₁	—	2	1	no	—
122	C₂×C₄×C₁₈₁	—	2	1	no	—
123	C₂×C₄×C₁₈₁	—	2	1	no	—
124	C₂×C₄×C₁₈₁	—	2	1	no	—
125	C₂×C₄×C₁₈₁	—	2	1	no	—
126	C₂×C₄×C₁₈₁	—	2	1	no	—
127	C₂×C₄×C₁₈₁	—	2	1	no	—
128	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
130	C₈×C₁₈₁	—	2	1	no	—
131	C₈×C₁₈₁	—	2	1	no	—
132	C₂²×C₁₈₁	—	2	1	no	—
133	C₂²×C₁₈₁	—	2	1	no	—
134	C₂²×C₁₈₁	—	2	1	no	—
135	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₁₈₁	—	2	1	no	—
137	C₄×C₁₈₁	—	4	1	no	—
138	C₄×C₁₈₁	—	4	1	no	—
139	C₄×C₁₈₁	—	4	1	no	—
140	C₄×C₁₈₁	—	4	1	no	—
141	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
143	Dic₁₈₁	—	2	1	no	—
144	Dic₁₈₁	—	2	1	no	—
145	Dic₁₈₁	—	4	1	no	—
146	Dic₁₈₁	—	4	1	no	—
147	Dic₁₈₁	—	4	1	no	—
148	Dic₁₈₁	—	4	1	no	—
149	Dic₁₈₁	#135	1	2	yes	—
150	Dic₁₈₁	#137	1	2	yes	—
151	C₂×C₁₈₁	—	4	1	no	—
152	C₂×C₁₈₁	#536	1	1	yes	characteristic
153	C₂×C₁₈₁	C₂×#135	1	1	yes	characteristic
154	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
155	C₁₈₁	#3231	1	1	yes	characteristic, complemented, Sylow
156	#3231	—	181	1	no	maximal, Sylow
157	#533	—	181	1	no	—
158	#533	—	181	1	no	—
159	#500	—	181	1	no	—
160	#498	—	181	1	no	—
161	#487	—	181	1	no	—
162	#487	—	181	1	no	—
163	#485	—	181	1	no	—
164	#485	—	181	1	no	—
165	#485	—	181	1	no	—
166	#485	D₁₈₁	1	1	yes	characteristic
167	#479	—	181	1	no	—
168	#479	—	181	1	no	—
169	#479	—	181	1	no	—
170	#479	—	181	1	no	—
171	#431	—	181	1	no	—
172	C₄⋊Q₈	—	181	1	no	—
173	C₄⋊Q₈	—	181	1	no	—
174	C₄⋊Q₈	—	181	1	no	—
175	C₄⋊Q₈	—	181	1	no	—
176	#125	—	362	1	no	—
177	#125	—	181	1	no	—
178	#125	—	181	1	no	—
179	C₂²⋊Q₈	—	362	1	no	—
180	C₂²⋊Q₈	—	362	1	no	—
181	C₂²⋊Q₈	—	362	1	no	—
182	C₂²⋊Q₈	—	181	1	no	—
183	C₂²⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
184	C₄×Q₈	—	362	1	no	—
185	C₄×Q₈	—	362	1	no	—
186	C₄×Q₈	—	362	1	no	—
187	C₄×Q₈	—	181	1	no	—
188	C₄×Q₈	—	181	1	no	—
189	#117	—	362	1	no	—
190	#117	—	181	1	no	—
191	#117	C₂×D₁₈₁	1	1	yes	characteristic
192	C₂×C₄²	—	181	1	no	—
193	#107	—	181	1	no	—
194	#107	—	181	1	no	—
195	#107	—	181	1	no	—
196	#107	C₂×D₁₈₁	1	1	yes	characteristic
197	#106	—	181	1	no	—
198	#106	—	181	1	no	—
199	#106	—	181	1	no	—
200	#106	C₂×D₁₈₁	1	1	yes	characteristic
201	C₄⋊C₈	—	181	1	no	—
202	C₄⋊C₈	—	181	1	no	—
203	#103	—	181	1	no	—
204	#103	—	181	1	no	—
205	#103	—	181	1	no	—
206	#103	—	181	1	no	—
207	#103	—	181	1	no	—
208	#103	—	181	1	no	—
209	#103	C₂×D₁₈₁	1	1	yes	characteristic
210	#103	C₂×D₁₈₁	1	1	yes	characteristic
211	C₂²⋊C₈	—	181	1	no	—
212	C₂²⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
213	C₄×C₈	—	181	1	no	—
214	C₄×C₈	—	181	1	no	—
215	C₂×Q₈	—	362	1	no	—
216	C₂×Q₈	—	362	1	no	—
217	C₂×Q₈	—	362	1	no	—
218	C₂×Q₈	—	181	1	no	—
219	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
220	C₂²×C₄	—	181	1	no	—
221	C₂²×C₄	—	181	1	no	—
222	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
223	C₂×C₈	—	181	1	no	—
224	C₂×C₈	—	181	1	no	—
225	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	362	1	no	—
229	C₄⋊C₄	—	362	1	no	—
230	C₄⋊C₄	—	362	1	no	—
231	C₄⋊C₄	—	362	1	no	—
232	C₄⋊C₄	—	362	1	no	—
233	C₄⋊C₄	—	362	1	no	—
234	C₄⋊C₄	—	362	1	no	—
235	C₄⋊C₄	—	362	1	no	—
236	C₄⋊C₄	—	362	1	no	—
237	C₄⋊C₄	—	362	1	no	—
238	C₄⋊C₄	—	362	1	no	—
239	C₄⋊C₄	—	362	1	no	—
240	C₄⋊C₄	—	181	1	no	—
241	C₄⋊C₄	—	181	1	no	—
242	C₄⋊C₄	—	181	1	no	—
243	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	362	1	no	—
249	C₂²⋊C₄	—	362	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	362	1	no	—
252	C₄²	—	362	1	no	—
253	C₄²	—	362	1	no	—
254	C₄²	—	181	1	no	—
255	C₄²	—	181	1	no	—
256	C₄²	—	181	1	no	—
257	C₄²	—	181	1	no	—
258	C₂³	D₄×D₁₈₁	1	1	yes	characteristic
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	362	1	no	—
262	Q₈	—	362	1	no	—
263	Q₈	—	724	2	no	—
264	Q₈	—	724	2	no	—
265	Q₈	—	724	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	362	2	no	—
273	C₂×C₄	—	362	2	no	—
274	C₂×C₄	—	362	1	no	—
275	C₂×C₄	—	362	1	no	—
276	C₂×C₄	—	362	1	no	—
277	C₂×C₄	—	362	1	no	—
278	C₂×C₄	—	181	1	no	—
279	C₂×C₄	—	181	1	no	—
280	C₂×C₄	—	181	1	no	—
281	C₂×C₄	—	181	1	no	—
282	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	362	1	no	—
287	C₈	—	362	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	362	1	no	—
298	C₄	—	362	1	no	—
299	C₄	—	181	2	no	—
300	C₄	—	181	2	no	—
301	C₄	—	724	1	no	—
302	C₄	—	724	1	no	—
303	C₄	—	724	1	no	—
304	C₄	—	724	1	no	—
305	C₄	#19153863	1	1	yes	characteristic
306	C₄	#19153863	1	1	yes	characteristic
307	C₂	—	4	1	no	—
308	C₂	#26041851	1	1	yes	characteristic
309	C₂	#135×D₁₈₁	1	1	yes	characteristic
310	C₂	#137×D₁₈₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#485×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂³×C₁₈₁	C₂×D₄
hypercenter	#485	D₁₈₁
hypocenter	C₁₈₁	#3231
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #26041683	C₂	semidirect
4. #26041676	C₂	semidirect
5. #26041628	C₂	semidirect
6. #26041628	C₂	semidirect
10. #26041589	C₂	semidirect
11. #26041588	C₂	semidirect
12. #26041585	C₂	semidirect
13. #26041584	C₂	semidirect
155. C₁₈₁	#3231	semidirect

Maximal subgroups

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

subgroup	index
#26041832	2
#26041831	2
#26041683	2
#26041676	2
#26041628	2
#26041621	2
#26041620	2
#26041617	2
#26041589	2
#26041588	2
#26041585	2
#26041584	2
#26041329	2
#485×C₁₈₁	2
#3231	181