Subgroups — #37536008

Order 23168.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041684	C₂	1	1	yes	maximal, characteristic
2	#26041106	C₂	1	2	yes	maximal
3	#26041104	C₂	1	2	yes	maximal
4	#26041055	C₂	1	1	yes	maximal, characteristic
5	#499×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	#19153816	—	2	1	no	—
7	C₂×#15668424	C₂²	1	2	yes	—
8	#19153750	—	2	1	no	—
9	#19153750	C₂²	1	2	yes	—
10	#125×C₁₈₁	—	2	1	no	—
11	#107×C₁₈₁	C₂²	1	1	yes	characteristic
12	#107×C₁₈₁	C₂²	1	1	yes	characteristic
13	#106×C₁₈₁	—	2	1	no	—
14	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₆₂	—	4	1	no	—
16	C₄⋊Dic₁₈₁	—	2	2	no	—
17	C₄⋊Dic₁₈₁	—	4	1	no	—
18	#15668424	Q₈	1	4	yes	complemented
19	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₈₁	—	2	1	no	—
22	C₄⋊C₄×C₁₈₁	—	2	1	no	—
23	C₄⋊C₄×C₁₈₁	—	4	1	no	—
24	C₄⋊C₄×C₁₈₁	—	4	1	no	—
25	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
26	Dic₃₆₂	—	8	2	no	—
27	C₂×Dic₁₈₁	—	4	2	no	—
28	C₂×C₄×C₁₈₁	—	4	1	no	—
29	C₂×C₄×C₁₈₁	—	4	1	no	—
30	C₂×C₄×C₁₈₁	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₈₁	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₈₁	Q₁₆	1	2	yes	—
34	C₈×C₁₈₁	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₈₁	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₈₁	—	2	1	no	—
37	C₄×C₁₈₁	—	2	1	no	—
38	C₄×C₁₈₁	—	8	1	no	—
39	C₄×C₁₈₁	—	8	1	no	—
40	C₄×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
41	C₄×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
42	Dic₁₈₁	—	8	2	no	—
43	C₂×C₁₈₁	#500	1	1	yes	characteristic
44	C₂×C₁₈₁	#509	1	1	yes	characteristic
45	C₂×C₁₈₁	#510	1	1	yes	characteristic
46	C₁₈₁	#2255	1	1	yes	characteristic, complemented, Sylow
47	#2255	—	181	1	no	maximal, Sylow
48	#500	—	181	1	no	—
49	#499	D₁₈₁	1	1	yes	characteristic
50	#367	—	181	2	no	—
51	#366	—	181	2	no	—
52	#346	—	181	1	no	—
53	C₄⋊Q₈	—	362	1	no	—
54	#125	—	2	1	no	—
55	C₂×C₁₆	—	181	2	no	—
56	#107	—	362	1	no	—
57	#107	—	181	2	no	—
58	#107	C₂×D₁₈₁	1	1	yes	characteristic
59	#107	C₂×D₁₈₁	1	1	yes	characteristic
60	#106	—	2	1	no	—
61	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
62	C₂×Q₈	—	724	1	no	—
63	C₂×C₈	C₂²⋊D₁₈₁	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄⋊C₄	—	362	2	no	—
70	C₄⋊C₄	—	724	1	no	—
71	C₄²	C₂²⋊D₁₈₁	1	1	yes	characteristic
72	C₁₆	—	181	4	no	—
73	Q₈	—	1448	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	724	2	no	—
77	C₂×C₄	#15668434	1	1	yes	characteristic
78	C₂×C₄	#15668434	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
80	C₈	#15668439	1	2	yes	—
81	C₈	Q₈×D₁₈₁	1	2	yes	—
82	C₂²	C₂×#15668434	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	1448	2	no	—
88	C₄	#19153814	1	1	yes	characteristic
89	C₄	C₂×#15668439	1	1	yes	characteristic
90	C₂	#26041682	1	1	yes	characteristic
91	C₂	#26041714	1	1	yes	characteristic
92	C₂	#26041719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15668434
commutator, cosocle	C₂×C₈×C₁₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₈₁
Fitting	#499×C₁₈₁	C₂
socle, prime radical	C₂²×C₁₈₁	C₂×D₈
hypercenter	#499	D₁₈₁
hypocenter	C₁₈₁	#2255
LCS	G ▷ C₂×C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
18. #15668424	Q₈	semidirect
46. C₁₈₁	#2255	semidirect

Maximal subgroups

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

subgroup	index
#26041684	2
#26041106	2
#26041104	2
#26041055	2
#499×C₁₈₁	2
#2255	181