Subgroups — #41011561

Order 27008.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212939	C₂	1	2	yes	maximal, complemented
2	#493×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269892	—	2	2	no	—
4	C₂×D₈×C₂₁₁	—	2	2	no	—
5	#127×C₂₁₁	C₄	1	1	yes	characteristic, complemented
6	#127×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₂₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₂₁₁	—	4	2	no	—
9	C₂×D₄×C₂₁₁	—	2	1	no	—
10	C₂×D₄×C₂₁₁	—	4	1	no	—
11	C₂×D₄×C₂₁₁	—	4	1	no	—
12	C₂×D₄×C₂₁₁	D₄	1	2	yes	—
13	D₈×C₂₁₁	—	4	4	no	—
14	C₂×C₈×C₂₁₁	—	2	1	no	—
15	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₁₁	—	8	2	no	—
17	C₂³×C₂₁₁	—	2	2	no	—
18	C₂³×C₂₁₁	—	4	1	no	—
19	D₄×C₂₁₁	—	4	2	no	—
20	D₄×C₂₁₁	—	4	2	no	—
21	D₄×C₂₁₁	—	8	2	no	—
22	D₄×C₂₁₁	—	8	2	no	—
23	C₂×C₄×C₂₁₁	—	2	1	no	—
24	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₂₁₁	—	2	2	no	—
26	C₂²×C₂₁₁	—	4	2	no	—
27	C₂²×C₂₁₁	—	8	2	no	—
28	C₂²×C₂₁₁	—	8	2	no	—
29	C₂²×C₂₁₁	—	8	1	no	—
30	C₂²×C₂₁₁	#99	1	1	yes	characteristic
31	C₄×C₂₁₁	—	2	2	no	—
32	C₄×C₂₁₁	—	2	1	no	—
33	Dic₂₁₁	—	16	2	no	—
34	C₂×C₂₁₁	—	2	1	no	—
35	C₂×C₂₁₁	—	8	2	no	—
36	C₂×C₂₁₁	—	16	1	no	—
37	C₂×C₂₁₁	#353	1	1	yes	characteristic
38	C₂₁₁	#1408	1	1	yes	characteristic, complemented, Sylow
39	#1408	—	211	1	no	maximal, Sylow
40	#493	D₂₁₁	1	1	yes	characteristic
41	#353	—	211	2	no	—
42	C₂×D₈	—	2	2	no	—
43	#127	Dic₂₁₁	1	1	yes	characteristic, complemented
44	#127	C₂×D₂₁₁	1	1	yes	characteristic
45	#99	—	422	2	no	—
46	C₄×C₈	Dic₂₁₁	1	1	yes	characteristic, complemented
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	4	1	no	—
50	C₂×D₄	C₂²⋊D₂₁₁	1	2	yes	—
51	D₈	—	4	4	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂²⋊C₄	—	844	2	no	—
54	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
55	C₂³	—	2	2	no	—
56	C₂³	—	4	1	no	—
57	D₄	—	4	2	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	8	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	1688	2	no	—
63	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	2	no	—
68	C₂²	—	8	1	no	—
69	C₂²	#20269892	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	3376	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	8	2	no	—
75	C₂	—	16	1	no	—
76	C₂	#27212939	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212939
commutator	C₄²×C₂₁₁	C₂×C₄
Frattini	#127	C₂×D₂₁₁
Fitting, prime radical	#493×C₂₁₁	C₂
socle	C₂×C₂₁₁	#353
cosocle	#127×C₂₁₁	C₂²
hypercenter	#493	D₂₁₁
hypocenter	C₂₁₁	#1408
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #493

Complements

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

N	H ≅ G/N	type
1. #27212939	C₂	semidirect
5. #127×C₂₁₁	C₄	semidirect
7. C₄×C₈×C₂₁₁	C₄	semidirect
38. C₂₁₁	#1408	semidirect
43. #127	Dic₂₁₁	semidirect
46. C₄×C₈	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27212939	2
#493×C₂₁₁	2
#1408	211