Subgroups — #27212413

Order 13472.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269771	C₂	1	3	yes	maximal, complemented
2	#20269770	C₂	1	6	yes	maximal, complemented
3	C₂×Dic₈₄₂	C₂	1	3	yes	maximal, complemented
4	Q₈×D₄₂₁	C₂	1	2	yes	maximal, complemented
5	#41×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₈₄₂	C₂²	1	6	yes	—
7	Dic₈₄₂	C₂²	1	3	yes	—
8	D₁₆₈₄	C₂²	1	1	yes	characteristic
9	C₂²⋊D₄₂₁	C₂²	1	6	yes	—
10	C₂×Dic₄₂₁	C₂²	1	6	yes	—
11	C₄×D₄₂₁	C₂²	1	6	yes	—
12	Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	D₄×C₄₂₁	C₂²	1	3	yes	—
14	C₂×C₄×C₄₂₁	C₂²	1	3	yes	—
15	C₂²×C₄₂₁	C₂³	1	3	yes	—
16	C₂×D₄₂₁	C₂³	1	2	yes	—
17	C₄×C₄₂₁	C₂³	1	1	yes	characteristic
18	C₄×C₄₂₁	C₂³	1	3	yes	—
19	Dic₄₂₁	C₂³	1	6	yes	—
20	C₂×C₄₂₁	—	2	3	no	—
21	C₂×C₄₂₁	C₂⁴	1	1	yes	characteristic
22	D₄₂₁	—	2	2	no	—
23	C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
24	2₋¹⁺⁴	—	421	1	no	maximal, Sylow
25	#41	—	421	6	no	—
26	#41	—	421	3	no	—
27	#41	D₄₂₁	1	1	yes	characteristic, complemented
28	C₂×Q₈	—	421	2	no	—
29	C₂×Q₈	—	421	3	no	—
30	Q₈	—	421	6	no	—
31	Q₈	—	421	3	no	—
32	Q₈	C₂×D₄₂₁	1	1	yes	characteristic
33	D₄	—	421	6	no	—
34	D₄	—	421	1	no	—
35	D₄	C₂×D₄₂₁	1	3	yes	—
36	C₂×C₄	—	421	6	no	—
37	C₂×C₄	—	421	6	no	—
38	C₂×C₄	C₂×D₄₂₁	1	3	yes	—
39	C₂²	—	421	2	no	—
40	C₂²	C₂²×D₄₂₁	1	3	yes	—
41	C₄	—	421	6	no	—
42	C₄	C₂²×D₄₂₁	1	3	yes	—
43	C₄	C₂²×D₄₂₁	1	1	yes	characteristic
44	C₂	—	2	3	no	—
45	C₂	—	842	2	no	—
46	C₂	C₂³×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₄₂₁
commutator, socle, cosocle	C₂×C₄₂₁	C₂⁴
Fitting	#41×C₄₂₁	C₂
hypercenter	#41	D₄₂₁
hypocenter	C₄₂₁	2₋¹⁺⁴
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #20269771	C₂	semidirect
2. #20269770	C₂	semidirect
3. C₂×Dic₈₄₂	C₂	semidirect
4. Q₈×D₄₂₁	C₂	semidirect
5. #41×C₄₂₁	C₂	semidirect
23. C₄₂₁	2₋¹⁺⁴	semidirect
27. #41	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#20269771	2
#20269770	2
C₂×Dic₈₄₂	2
Q₈×D₄₂₁	2
#41×C₄₂₁	2
2₋¹⁺⁴	421