Subgroups — #41002633

Order 26944.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212281	C₂	1	2	yes	maximal, complemented
2	#127×C₄₂₁	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₄₂₁	—	2	2	no	—
4	C₂×D₄×C₄₂₁	—	2	2	no	—
5	C₂×D₄×C₄₂₁	C₄	1	1	yes	characteristic, complemented
6	C₂×D₄×C₄₂₁	C₂²	1	1	yes	characteristic
7	C₄²×C₄₂₁	C₄	1	1	yes	characteristic, complemented
8	C₂×Dic₄₂₁	—	4	2	no	—
9	C₂³×C₄₂₁	—	2	1	no	—
10	C₂³×C₄₂₁	D₄	1	2	yes	—
11	D₄×C₄₂₁	—	4	1	no	—
12	D₄×C₄₂₁	—	4	1	no	—
13	D₄×C₄₂₁	—	4	4	no	—
14	C₂×C₄×C₄₂₁	—	2	1	no	—
15	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄₂₁	—	2	2	no	—
17	C₂²×C₄₂₁	—	4	2	no	—
18	C₂²×C₄₂₁	—	4	2	no	—
19	C₂²×C₄₂₁	—	4	1	no	—
20	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₄₂₁	—	2	2	no	—
22	C₄×C₄₂₁	—	2	1	no	—
23	Dic₄₂₁	—	8	2	no	—
24	C₂×C₄₂₁	—	2	1	no	—
25	C₂×C₄₂₁	—	4	2	no	—
26	C₂×C₄₂₁	—	8	1	no	—
27	C₂×C₄₂₁	#99	1	1	yes	characteristic
28	C₄₂₁	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	421	1	no	maximal, Sylow
30	#127	D₄₂₁	1	1	yes	characteristic
31	#99	—	421	2	no	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	Dic₄₂₁	1	1	yes	characteristic, complemented
34	C₂×D₄	C₂×D₄₂₁	1	1	yes	characteristic
35	C₂²⋊C₄	—	842	2	no	—
36	C₄²	Dic₄₂₁	1	1	yes	characteristic, complemented
37	C₂³	—	2	1	no	—
38	C₂³	C₂²⋊D₄₂₁	1	2	yes	—
39	D₄	—	4	1	no	—
40	D₄	—	4	1	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	1684	2	no	—
44	C₂×C₄	C₂×Dic₄₂₁	1	1	yes	characteristic
45	C₂²	—	2	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	C₂²⋊Dic₄₂₁	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	3368	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	4	2	no	—
55	C₂	—	8	1	no	—
56	C₂	#27212281	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212281
commutator	C₂×C₄×C₄₂₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₄₂₁
Fitting, prime radical	#127×C₄₂₁	C₂
socle	C₂×C₄₂₁	#99
cosocle	C₂×D₄×C₄₂₁	C₂²
hypercenter	#127	D₄₂₁
hypocenter	C₄₂₁	#353
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
1. #27212281	C₂	semidirect
5. C₂×D₄×C₄₂₁	C₄	semidirect
7. C₄²×C₄₂₁	C₄	semidirect
28. C₄₂₁	#353	semidirect
33. C₂×D₄	Dic₄₂₁	semidirect
36. C₄²	Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212281	2
#127×C₄₂₁	2
#353	421