Subgroups — #41011569

Order 27008.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212939	C₂	1	2	yes	maximal
2	#490×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269892	—	2	2	no	—
4	#127×C₂₁₁	C₂²	1	1	yes	characteristic
5	#125×C₂₁₁	C₄	1	1	yes	characteristic, complemented
6	#102×C₂₁₁	—	2	2	no	—
7	#97×C₂₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₂₁₁	—	4	2	no	—
9	C₂×D₄×C₂₁₁	—	4	1	no	—
10	C₂×D₄×C₂₁₁	D₄	1	2	yes	—
11	C₂×C₈×C₂₁₁	—	2	1	no	—
12	C₄⋊C₄×C₂₁₁	—	2	1	no	—
13	C₄⋊C₄×C₂₁₁	—	4	1	no	—
14	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₂₁₁	—	8	2	no	—
16	C₂³×C₂₁₁	—	2	2	no	—
17	D₄×C₂₁₁	—	4	2	no	—
18	D₄×C₂₁₁	—	8	2	no	—
19	C₂×C₄×C₂₁₁	—	2	1	no	—
20	C₂×C₄×C₂₁₁	—	4	1	no	—
21	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₂₁₁	—	4	1	no	—
23	C₂²×C₂₁₁	—	4	2	no	—
24	C₂²×C₂₁₁	—	8	2	no	—
25	C₂²×C₂₁₁	#99	1	1	yes	characteristic
26	C₄×C₂₁₁	—	2	1	no	—
27	C₄×C₂₁₁	—	4	1	no	—
28	C₄×C₂₁₁	—	8	1	no	—
29	Dic₂₁₁	—	16	2	no	—
30	C₂×C₂₁₁	—	2	1	no	—
31	C₂×C₂₁₁	—	8	2	no	—
32	C₂×C₂₁₁	#354	1	1	yes	characteristic
33	C₂₁₁	#1412	1	1	yes	characteristic, complemented, Sylow
34	#1412	—	211	1	no	maximal, Sylow
35	#490	D₂₁₁	1	1	yes	characteristic
36	#353	—	211	2	no	—
37	#127	C₂×D₂₁₁	1	1	yes	characteristic
38	#125	Dic₂₁₁	1	1	yes	characteristic, complemented
39	#102	—	2	2	no	—
40	#99	—	422	2	no	—
41	#97	Dic₂₁₁	1	1	yes	characteristic, complemented
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	C₂²⋊D₂₁₁	1	2	yes	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	844	2	no	—
48	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	D₄	—	4	2	no	—
51	D₄	—	8	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	1688	2	no	—
55	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₂²	—	4	2	no	—
58	C₂²	—	8	2	no	—
59	C₂²	#20269892	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	3376	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	#27212941	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212941
commutator	C₄²×C₂₁₁	C₂×C₄
Frattini	#127	C₂×D₂₁₁
Fitting	#490×C₂₁₁	C₂
socle	C₂×C₂₁₁	#354
prime radical, cosocle	#127×C₂₁₁	C₂²
hypercenter	#490	D₂₁₁
hypocenter	C₂₁₁	#1412
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #490

Complements

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

N	H ≅ G/N	type
5. #125×C₂₁₁	C₄	semidirect
7. #97×C₂₁₁	C₄	semidirect
33. C₂₁₁	#1412	semidirect
38. #125	Dic₂₁₁	semidirect
41. #97	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
#490×C₂₁₁	2
#1412	211