Subgroups — #41003744

Order 26944.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212425	C₂	1	2	yes	maximal, complemented
2	#27212362	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#13132997	—	2	2	no	—
4	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊D₄₂₁	—	2	2	no	—
6	C₄×Dic₄₂₁	C₄	1	1	yes	characteristic, complemented
7	C₂×D₄×C₄₂₁	C₄	1	1	yes	characteristic, complemented
8	C₂×#13132997	—	4	2	no	—
9	C₂²×D₄₂₁	D₄	1	2	yes	—
10	D₁₆₈₄	—	4	1	no	—
11	C₂²⋊D₄₂₁	—	4	4	no	—
12	C₂×Dic₄₂₁	—	2	1	no	—
13	C₂³×C₄₂₁	—	2	1	no	—
14	D₄×C₄₂₁	—	4	1	no	—
15	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄₂₁	—	4	2	no	—
17	C₂²×C₄₂₁	—	4	1	no	—
18	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×D₄₂₁	—	2	2	no	—
20	C₂×D₄₂₁	—	4	2	no	—
21	#13132997	—	8	2	no	—
22	C₄×C₄₂₁	—	2	1	no	—
23	Dic₄₂₁	—	2	2	no	—
24	C₂×C₄₂₁	—	2	1	no	—
25	C₂×C₄₂₁	—	8	1	no	—
26	C₂×C₄₂₁	#99	1	1	yes	characteristic
27	D₄₂₁	—	4	2	no	—
28	C₄₂₁	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	421	1	no	maximal, Sylow
30	#127	—	421	1	no	—
31	#99	—	421	2	no	—
32	C₂×D₄	—	842	2	no	—
33	C₂×D₄	—	421	1	no	—
34	C₂×D₄	#13132997	1	1	yes	characteristic, complemented
35	C₂²⋊C₄	—	842	2	no	—
36	C₄²	—	421	1	no	—
37	C₂³	—	2	1	no	—
38	C₂³	—	421	2	no	—
39	D₄	—	4	1	no	—
40	D₄	—	1684	1	no	—
41	D₄	—	1684	4	no	—
42	C₂×C₄	—	842	1	no	—
43	C₂×C₄	—	1684	2	no	—
44	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
45	C₂²	—	4	2	no	—
46	C₂²	—	4	1	no	—
47	C₂²	—	842	2	no	—
48	C₂²	—	1684	2	no	—
49	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	3368	2	no	—
52	C₄	—	842	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	8	1	no	—
55	C₂	—	1684	2	no	—
56	C₂	#27212424	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212424
commutator	C₂×C₄×C₄₂₁	C₂×C₄
Frattini	C₂×C₄	C₂×#13132997
Fitting	C₂×D₄×C₄₂₁	C₄
socle	C₂×C₄₂₁	#99
prime radical	#27212362	C₂
cosocle	C₂×D₁₆₈₄	C₂²
hypercenter	C₂×D₄	#13132997
hypocenter	C₄₂₁	#353
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #27212425	C₂	semidirect
6. C₄×Dic₄₂₁	C₄	semidirect
7. C₂×D₄×C₄₂₁	C₄	semidirect
28. C₄₂₁	#353	semidirect
34. C₂×D₄	#13132997	semidirect

Maximal subgroups

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

subgroup	index
#27212425	2
#27212362	2
#353	421