Subgroups — #27213329

Order 13504.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270007	C₂	1	2	yes	maximal, complemented
2	#20269975	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269904	C₂	1	2	yes	maximal, complemented
4	#20269896	C₂	1	1	yes	maximal, characteristic, complemented
5	2₋¹⁺⁴×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	#16809952	—	2	2	no	—
8	#16809950	—	2	2	no	—
9	#16809946	C₂²	1	2	yes	—
10	Dic₂₁₁⋊C₄	—	2	2	no	—
11	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
12	#41×C₂₁₁	—	2	2	no	—
13	#41×C₂₁₁	—	2	2	no	—
14	#41×C₂₁₁	C₂²	1	2	yes	—
15	C₂×Q₈×C₂₁₁	—	2	2	no	—
16	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₂₁₁	—	2	1	no	—
19	C₂×Dic₂₁₁	—	2	1	no	—
20	C₂₁₁⋊C₈	—	2	2	no	—
21	Q₈×C₂₁₁	—	2	2	no	—
22	Q₈×C₂₁₁	—	2	2	no	—
23	Q₈×C₂₁₁	D₄	1	2	yes	—
24	D₄×C₂₁₁	—	2	2	no	—
25	D₄×C₂₁₁	—	2	2	no	—
26	D₄×C₂₁₁	D₄	1	2	yes	—
27	C₂×C₄×C₂₁₁	—	2	4	no	—
28	C₂×C₄×C₂₁₁	—	2	2	no	—
29	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₁₁	—	2	2	no	—
34	C₄×C₂₁₁	—	2	2	no	—
35	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
36	Dic₂₁₁	—	2	2	no	—
37	Dic₂₁₁	—	4	1	no	—
38	C₂×C₂₁₁	—	2	1	no	—
39	C₂×C₂₁₁	—	4	2	no	—
40	C₂×C₂₁₁	#120	1	1	yes	characteristic
41	C₂₁₁	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	211	1	no	maximal, Sylow
43	2₋¹⁺⁴	D₂₁₁	1	1	yes	characteristic
44	#137	—	211	2	no	—
45	C₄⋊Q₈	—	211	1	no	—
46	#104	—	211	2	no	—
47	#101	—	211	1	no	—
48	#41	—	2	2	no	—
49	#41	—	2	2	no	—
50	#41	C₂×D₂₁₁	1	2	yes	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	211	1	no	—
53	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
54	Q₁₆	—	422	2	no	—
55	SD₁₆	—	422	2	no	—
56	M₁₆	—	211	2	no	—
57	C₄⋊C₄	—	422	2	no	—
58	C₄²	—	211	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	422	2	no	—
62	Q₈	C₂²⋊D₂₁₁	1	2	yes	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	C₂²⋊D₂₁₁	1	2	yes	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	422	1	no	—
69	C₂×C₄	—	422	1	no	—
70	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
71	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₈	—	422	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	844	1	no	—
78	C₄	—	422	2	no	—
79	C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	#20269946	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269946
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting, prime radical	2₋¹⁺⁴×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
hypercenter	2₋¹⁺⁴	D₂₁₁
hypocenter	C₂₁₁	#456
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
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. #20270007	C₂	semidirect
2. #20269975	C₂	semidirect
3. #20269904	C₂	semidirect
4. #20269896	C₂	semidirect
41. C₂₁₁	#456	semidirect

Maximal subgroups

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

subgroup	index
#20270007	2
#20269975	2
#20269904	2
#20269896	2
2₋¹⁺⁴×C₂₁₁	2
#456	211