Subgroups — #37550498

Order 23168.

452 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042358	C₂	1	2	yes	maximal, complemented
2	#26042324	C₂	1	4	yes	maximal, complemented
3	#26042323	C₂	1	2	yes	maximal, complemented
4	#26042321	C₂	1	2	yes	maximal, complemented
5	C₂×#19153887	C₂	1	1	yes	maximal, characteristic, complemented
6	#26042276	C₂	1	2	yes	maximal, complemented
7	#26042242	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₈₁	C₂	1	1	yes	maximal, characteristic
9	#19153904	C₂²	1	4	yes	complemented
10	#19153903	—	2	4	no	—
11	#19153903	C₂²	1	4	yes	complemented
12	C₂×#15668459	—	2	2	no	—
13	C₂×#15668459	C₂²	1	2	yes	—
14	D₁₈₁⋊M₁₆	—	2	2	no	—
15	D₁₈₁⋊M₁₆	C₂²	1	1	yes	characteristic
16	#19153895	C₂²	1	1	yes	characteristic
17	#19153887	C₂²	1	2	yes	complemented
18	#19153887	C₂²	1	2	yes	complemented
19	#19153881	C₂²	1	4	yes	complemented
20	#19153880	C₂²	1	4	yes	complemented
21	#19153870	C₂²	1	2	yes	complemented
22	C₈×#10624382	C₂²	1	2	yes	complemented
23	#41×D₁₈₁	C₂²	1	2	yes	—
24	#19153839	C₂²	1	1	yes	characteristic
25	#19153838	C₄	1	1	yes	characteristic, complemented
26	#19153837	C₂²	1	2	yes	—
27	#19153836	C₄	1	2	yes	complemented
28	#19153835	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₈₁	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₁₈₁	C₄	1	2	yes	complemented
31	D₈×D₁₈₁	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
33	#135×C₁₈₁	C₄	1	1	yes	characteristic, complemented
34	#15668459	—	2	4	no	—
35	#15668459	—	2	2	no	—
36	#15668459	D₄	1	1	yes	characteristic
37	#15668459	D₄	1	1	yes	characteristic
38	#15668459	C₂³	1	4	yes	—
39	#15668458	—	2	4	no	—
40	#15668458	D₄	1	2	yes	—
41	D₁₈₁⋊C₈	—	2	4	no	—
42	D₁₈₁⋊C₈	C₂³	1	2	yes	—
43	C₂×#13089685	—	2	4	no	—
44	C₂×#13089685	—	2	2	no	—
45	C₄⋊#10624382	—	2	1	no	—
46	C₂²⋊#10624382	—	2	1	no	—
47	C₄×#10624382	C₂³	1	2	yes	—
48	#15668450	C₂³	1	2	yes	—
49	#15668449	—	2	2	no	—
50	#15668449	C₂×C₄	1	2	yes	—
51	#15668448	—	2	2	no	—
52	#15668448	C₂³	1	2	yes	—
53	Q₈×D₁₈₁	C₂×C₄	1	2	yes	—
54	D₄×D₁₈₁	—	2	2	no	—
55	D₄×D₁₈₁	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₁₈₁	—	2	2	no	—
57	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
58	Dic₇₂₄	C₂×C₄	1	1	yes	characteristic
59	#15668439	C₂×C₄	1	2	yes	—
60	#15668438	C₂×C₄	1	2	yes	—
61	#15668437	C₂×C₄	1	2	yes	—
62	#15668436	C₂×C₄	1	2	yes	—
63	D₁₄₄₈	C₂×C₄	1	1	yes	characteristic
64	#15668434	C₂×C₄	1	2	yes	—
65	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₈₁	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₈₁	C₂×C₄	1	1	yes	characteristic
68	C₈×D₁₈₁	C₂³	1	1	yes	characteristic
69	C₈×D₁₈₁	C₂³	1	1	yes	characteristic
70	#41×C₁₈₁	C₂×C₄	1	2	yes	—
71	Q₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
72	SD₁₆×C₁₈₁	C₂×C₄	1	2	yes	—
73	D₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
75	C₂×#10624382	—	2	2	no	—
76	#13089685	—	2	4	no	—
77	#13089685	C₂×D₄	1	2	yes	—
78	#13089685	C₂×D₄	1	2	yes	—
79	C₂²×D₁₈₁	—	2	2	no	—
80	C₂²×D₁₈₁	#41	1	1	yes	characteristic
81	Dic₃₆₂	—	2	2	no	—
82	Dic₃₆₂	C₂²×C₄	1	2	yes	—
83	D₇₂₄	—	2	2	no	—
84	D₇₂₄	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₈₁	—	2	2	no	—
86	C₂²⋊D₁₈₁	—	2	2	no	—
87	C₂²⋊D₁₈₁	—	2	2	no	—
88	C₂×Dic₁₈₁	—	2	2	no	—
89	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
90	C₄×D₁₈₁	—	2	2	no	—
91	C₄×D₁₈₁	—	2	2	no	—
92	C₄×D₁₈₁	—	2	2	no	—
93	C₄×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₈₁	C₂×D₄	1	1	yes	characteristic
95	C₄×D₁₈₁	#41	1	1	yes	characteristic
96	C₄×D₁₈₁	C₂⁴	1	1	yes	characteristic
97	#13089678	C₂²×C₄	1	1	yes	characteristic
98	#13089678	C₂²×C₄	1	1	yes	characteristic
99	Q₈×C₁₈₁	C₂²×C₄	1	2	yes	—
100	D₄×C₁₈₁	—	2	2	no	—
101	D₄×C₁₈₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₈₁	—	2	2	no	—
103	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₁₈₁	—	2	2	no	—
107	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
108	C₂×D₁₈₁	—	2	2	no	—
109	C₂×D₁₈₁	—	4	1	no	—
110	C₂×D₁₈₁	—	8	2	no	—
111	C₂×D₁₈₁	C₄×D₄	1	1	yes	characteristic
112	C₂×D₁₈₁	C₂×#41	1	1	yes	characteristic
113	#10624382	—	4	2	no	—
114	C₄×C₁₈₁	—	2	2	no	—
115	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
116	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
117	Dic₁₈₁	—	2	2	no	—
118	Dic₁₈₁	C₄×D₄	1	1	yes	characteristic
119	Dic₁₈₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₁₈₁	—	2	1	no	—
121	C₂×C₁₈₁	—	4	2	no	—
122	C₂×C₁₈₁	C₂×C₄×D₄	1	1	yes	characteristic
123	D₁₈₁	—	2	1	no	—
124	D₁₈₁	—	2	1	no	—
125	D₁₈₁	—	4	2	no	—
126	C₁₈₁	#2957	1	1	yes	characteristic, complemented, Sylow
127	#2957	—	181	1	no	maximal, Sylow
128	C₂×#135	—	181	1	no	—
129	#568	—	181	2	no	—
130	#444	—	181	4	no	—
131	#443	—	181	2	no	—
132	#443	—	181	2	no	—
133	C₂×#108	—	181	1	no	—
134	#421	—	181	2	no	—
135	#405	—	181	1	no	—
136	C₂×#41	—	181	2	no	—
137	#135	—	181	2	no	—
138	#135	—	181	1	no	—
139	#135	—	181	2	no	—
140	#135	—	181	1	no	—
141	#135	—	181	1	no	—
142	#135	#10624382	1	1	yes	characteristic, complemented
143	C₂×Q₁₆	—	181	1	no	—
144	C₂×SD₁₆	—	181	2	no	—
145	C₂×D₈	—	181	1	no	—
146	#131	—	362	4	no	—
147	#131	—	181	4	no	—
148	#131	—	181	4	no	—
149	C₂×M₁₆	—	362	2	no	—
150	C₂×M₁₆	—	362	2	no	—
151	C₂×M₁₆	—	181	2	no	—
152	C₂×M₁₆	—	181	1	no	—
153	C₂²×C₈	—	181	1	no	—
154	#117	—	181	1	no	—
155	#108	—	181	2	no	—
156	#108	—	181	2	no	—
157	#104	—	181	4	no	—
158	#104	—	181	4	no	—
159	#97	—	181	2	no	—
160	C₄×C₈	—	181	2	no	—
161	#41	—	362	2	no	—
162	#41	—	362	2	no	—
163	#41	—	181	2	no	—
164	#41	—	181	2	no	—
165	#41	—	181	2	no	—
166	#41	C₂×#10624382	1	2	yes	—
167	C₂×Q₈	—	181	2	no	—
168	C₂×D₄	—	362	2	no	—
169	C₂×D₄	—	181	2	no	—
170	C₂²×C₄	—	362	2	no	—
171	C₂²×C₄	—	181	1	no	—
172	Q₁₆	—	181	2	no	—
173	Q₁₆	—	181	1	no	—
174	Q₁₆	C₂×#10624382	1	1	yes	characteristic
175	SD₁₆	—	181	2	no	—
176	SD₁₆	—	181	2	no	—
177	SD₁₆	—	181	2	no	—
178	SD₁₆	C₂×#10624382	1	2	yes	—
179	D₈	—	181	2	no	—
180	D₈	—	181	1	no	—
181	D₈	C₂×#10624382	1	1	yes	characteristic
182	M₁₆	—	362	2	no	—
183	M₁₆	—	362	4	no	—
184	M₁₆	—	362	4	no	—
185	M₁₆	—	181	4	no	—
186	M₁₆	—	181	1	no	—
187	M₁₆	—	181	1	no	—
188	M₁₆	—	181	2	no	—
189	C₂×C₈	—	362	2	no	—
190	C₂×C₈	—	362	4	no	—
191	C₂×C₈	—	362	4	no	—
192	C₂×C₈	—	181	2	no	—
193	C₂×C₈	—	181	1	no	—
194	C₂×C₈	—	181	1	no	—
195	C₂×C₈	—	181	1	no	—
196	C₂×C₈	—	181	1	no	—
197	C₂×C₈	—	181	1	no	—
198	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
199	C₄⋊C₄	—	362	1	no	—
200	C₂²⋊C₄	—	362	1	no	—
201	C₄²	—	181	2	no	—
202	C₂³	—	362	2	no	—
203	C₂³	—	181	1	no	—
204	Q₈	—	362	2	no	—
205	Q₈	—	181	2	no	—
206	Q₈	C₂²×#10624382	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	362	2	no	—
209	D₄	—	362	2	no	—
210	D₄	—	362	2	no	—
211	D₄	—	362	2	no	—
212	D₄	—	181	2	no	—
213	D₄	C₂²×#10624382	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	362	2	no	—
216	C₂×C₄	—	362	2	no	—
217	C₂×C₄	—	362	2	no	—
218	C₂×C₄	—	362	2	no	—
219	C₂×C₄	—	362	2	no	—
220	C₂×C₄	—	181	1	no	—
221	C₂×C₄	—	181	1	no	—
222	C₂×C₄	—	181	1	no	—
223	C₂×C₄	—	181	1	no	—
224	C₂×C₄	—	181	1	no	—
225	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
226	C₈	—	362	4	no	—
227	C₈	—	181	2	no	—
228	C₈	—	181	2	no	—
229	C₈	—	181	1	no	—
230	C₈	—	181	1	no	—
231	C₈	C₂²×#10624382	1	1	yes	characteristic
232	C₈	C₂²×#10624382	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	362	2	no	—
235	C₂²	—	1448	2	no	—
236	C₂²	—	181	1	no	—
237	C₂²	—	181	1	no	—
238	C₂²	—	724	1	no	—
239	C₂²	D₄×#10624382	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	362	2	no	—
242	C₄	—	181	1	no	—
243	C₄	—	181	1	no	—
244	C₄	—	724	2	no	—
245	C₄	D₄×#10624382	1	1	yes	characteristic
246	C₄	C₂³×#10624382	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	—	362	1	no	—
250	C₂	—	362	1	no	—
251	C₂	—	724	2	no	—
252	C₂	C₂×D₄×#10624382	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10624382
commutator	C₄×C₁₈₁	C₂³×C₄
Frattini	C₄	C₂³×#10624382
Fitting	#135×C₁₈₁	C₄
socle	C₂×C₁₈₁	C₂×C₄×D₄
prime radical	#135×D₁₈₁	C₂
cosocle	C₄×D₁₈₁	C₂⁴
hypercenter	#135	#10624382
hypocenter	C₁₈₁	#2957
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #26042358	C₂	semidirect
2. #26042324	C₂	semidirect
3. #26042323	C₂	semidirect
4. #26042321	C₂	semidirect
5. C₂×#19153887	C₂	semidirect
6. #26042276	C₂	semidirect
7. #26042242	C₂	semidirect
9. #19153904	C₂²	semidirect
11. #19153903	C₂²	semidirect
17. #19153887	C₂²	semidirect
18. #19153887	C₂²	semidirect
19. #19153881	C₂²	semidirect
20. #19153880	C₂²	semidirect
21. #19153870	C₂²	semidirect
22. C₈×#10624382	C₂²	semidirect
25. #19153838	C₄	semidirect
27. #19153836	C₄	semidirect
29. Q₁₆×D₁₈₁	C₄	semidirect
30. SD₁₆×D₁₈₁	C₄	semidirect
31. D₈×D₁₈₁	C₄	semidirect
33. #135×C₁₈₁	C₄	semidirect
126. C₁₈₁	#2957	semidirect
142. #135	#10624382	semidirect

Maximal subgroups

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

subgroup	index
#26042358	2
#26042324	2
#26042323	2
#26042321	2
C₂×#19153887	2
#26042276	2
#26042242	2
#135×D₁₈₁	2
#2957	181