Subgroups — #37548860

Order 23168.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042184	C₂	1	2	yes	maximal
2	#26041645	C₂	1	1	yes	maximal, characteristic
3	#19153875	—	2	2	no	—
4	#19153813	C₂²	1	1	yes	characteristic
5	#19153736	—	2	2	no	—
6	#19153727	C₄	1	1	yes	characteristic, complemented
7	#125×C₁₈₁	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10624382	—	4	2	no	—
9	C₂×D₇₂₄	—	4	1	no	—
10	C₂×D₇₂₄	D₄	1	2	yes	—
11	C₂×#13089678	—	2	1	no	—
12	C₄⋊C₄×C₁₈₁	—	2	1	no	—
13	C₄⋊C₄×C₁₈₁	—	4	1	no	—
14	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
15	C₂×#10624382	—	8	2	no	—
16	C₂²×D₁₈₁	—	2	2	no	—
17	D₇₂₄	—	4	2	no	—
18	D₇₂₄	—	8	2	no	—
19	#13089678	—	4	1	no	—
20	C₂×C₄×C₁₈₁	—	2	1	no	—
21	C₂×C₄×C₁₈₁	—	4	1	no	—
22	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₁₈₁	#99	1	1	yes	characteristic
24	C₂×D₁₈₁	—	4	2	no	—
25	C₂×D₁₈₁	—	8	2	no	—
26	#10624382	—	16	2	no	—
27	C₄×C₁₈₁	—	2	1	no	—
28	C₄×C₁₈₁	—	4	1	no	—
29	C₄×C₁₈₁	—	8	1	no	—
30	C₂×C₁₈₁	—	2	1	no	—
31	C₂×C₁₈₁	#354	1	1	yes	characteristic
32	D₁₈₁	—	8	2	no	—
33	C₁₈₁	#1412	1	1	yes	characteristic, complemented, Sylow
34	#1412	—	181	1	no	maximal, Sylow
35	#490	—	181	1	no	—
36	#353	—	181	2	no	—
37	#127	—	181	1	no	—
38	#125	#10624382	1	1	yes	characteristic, complemented
39	#102	—	362	2	no	—
40	#99	—	362	2	no	—
41	#97	—	181	1	no	—
42	C₂×D₄	—	181	2	no	—
43	C₂×D₄	—	724	1	no	—
44	C₂×C₈	—	362	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	724	2	no	—
48	C₄²	C₂×#10624382	1	1	yes	characteristic
49	C₂³	—	362	2	no	—
50	D₄	—	1448	2	no	—
51	D₄	—	724	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	1448	2	no	—
55	C₂×C₄	C₂²⋊#10624382	1	1	yes	characteristic
56	C₈	—	724	1	no	—
57	C₂²	—	1448	2	no	—
58	C₂²	—	724	2	no	—
59	C₂²	#19153874	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	2896	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	1448	2	no	—
66	C₂	#26042185	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26042185
commutator	C₄²×C₁₈₁	C₂×C₄
Frattini	C₄²	C₂×#10624382
Fitting	#125×C₁₈₁	C₄
socle	C₂×C₁₈₁	#354
prime radical, cosocle	#19153813	C₂²
hypercenter	#125	#10624382
hypocenter	C₁₈₁	#1412
LCS	G ▷ C₄²×C₁₈₁ ▷ C₂×C₄×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
6. #19153727	C₄	semidirect
7. #125×C₁₈₁	C₄	semidirect
33. C₁₈₁	#1412	semidirect
38. #125	#10624382	semidirect

Maximal subgroups

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

subgroup	index
#26042184	2
#26041645	2
#1412	181