Subgroups — #34091877

Order 19904.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23755738	C₂	1	2	yes	maximal, complemented
2	#23755706	C₂	1	1	yes	maximal, characteristic, complemented
3	#23755635	C₂	1	2	yes	maximal, complemented
4	#23755627	C₂	1	1	yes	maximal, characteristic, complemented
5	2₋¹⁺⁴×C₃₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₂₂	C₂²	1	1	yes	characteristic
7	#19097305	—	2	2	no	—
8	#19097303	—	2	2	no	—
9	#19097299	C₂²	1	2	yes	—
10	Dic₃₁₁⋊C₄	—	2	2	no	—
11	C₄×Dic₃₁₁	C₂²	1	1	yes	characteristic
12	#41×C₃₁₁	—	2	2	no	—
13	#41×C₃₁₁	—	2	2	no	—
14	#41×C₃₁₁	C₂²	1	2	yes	—
15	C₂×Q₈×C₃₁₁	—	2	2	no	—
16	C₂×Q₈×C₃₁₁	C₂²	1	1	yes	characteristic
17	Dic₆₂₂	—	2	2	no	—
18	C₂×Dic₃₁₁	—	2	1	no	—
19	C₂×Dic₃₁₁	—	2	1	no	—
20	C₃₁₁⋊C₈	—	2	2	no	—
21	Q₈×C₃₁₁	—	2	2	no	—
22	Q₈×C₃₁₁	—	2	2	no	—
23	Q₈×C₃₁₁	D₄	1	2	yes	—
24	D₄×C₃₁₁	—	2	2	no	—
25	D₄×C₃₁₁	—	2	2	no	—
26	D₄×C₃₁₁	D₄	1	2	yes	—
27	C₂×C₄×C₃₁₁	—	2	4	no	—
28	C₂×C₄×C₃₁₁	—	2	2	no	—
29	C₂×C₄×C₃₁₁	D₄	1	2	yes	—
30	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₃₁₁	—	2	2	no	—
32	C₂²×C₃₁₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₃₁₁	—	2	2	no	—
34	C₄×C₃₁₁	—	2	2	no	—
35	C₄×C₃₁₁	C₂×D₄	1	2	yes	—
36	Dic₃₁₁	—	2	2	no	—
37	Dic₃₁₁	—	4	1	no	—
38	C₂×C₃₁₁	—	2	1	no	—
39	C₂×C₃₁₁	—	4	2	no	—
40	C₂×C₃₁₁	#120	1	1	yes	characteristic
41	C₃₁₁	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	311	1	no	maximal, Sylow
43	2₋¹⁺⁴	D₃₁₁	1	1	yes	characteristic
44	#137	—	311	2	no	—
45	C₄⋊Q₈	—	311	1	no	—
46	#104	—	311	2	no	—
47	#101	—	311	1	no	—
48	#41	—	2	2	no	—
49	#41	—	2	2	no	—
50	#41	C₂×D₃₁₁	1	2	yes	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	311	1	no	—
53	C₂×Q₈	C₂×D₃₁₁	1	1	yes	characteristic
54	Q₁₆	—	622	2	no	—
55	SD₁₆	—	622	2	no	—
56	M₁₆	—	311	2	no	—
57	C₄⋊C₄	—	622	2	no	—
58	C₄²	—	311	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	622	2	no	—
62	Q₈	C₂²⋊D₃₁₁	1	2	yes	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	C₂²⋊D₃₁₁	1	2	yes	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	622	1	no	—
69	C₂×C₄	—	622	1	no	—
70	C₂×C₄	C₂²⋊D₃₁₁	1	2	yes	—
71	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
72	C₈	—	622	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₂×C₂²⋊D₃₁₁	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	622	2	no	—
78	C₄	—	1244	1	no	—
79	C₄	C₂×C₂²⋊D₃₁₁	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	#23755677	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23755677
commutator, cosocle	C₂×C₄×C₃₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₁
Fitting, prime radical	2₋¹⁺⁴×C₃₁₁	C₂
socle	C₂×C₃₁₁	#120
hypercenter	2₋¹⁺⁴	D₃₁₁
hypocenter	C₃₁₁	#456
LCS	G ▷ C₂×C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #23755738	C₂	semidirect
2. #23755706	C₂	semidirect
3. #23755635	C₂	semidirect
4. #23755627	C₂	semidirect
41. C₃₁₁	#456	semidirect

Maximal subgroups

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

subgroup	index
#23755738	2
#23755706	2
#23755635	2
#23755627	2
2₋¹⁺⁴×C₃₁₁	2
#456	311