Subgroups — #27212445

Order 13472.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13132997	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13132997	C₂	1	2	yes	maximal
3	C₄×#13132997	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄₂₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13132997	C₂²	1	4	yes	—
6	C₂²×D₄₂₁	C₂²	1	1	yes	characteristic
7	C₂×Dic₄₂₁	C₄	1	1	yes	characteristic, complemented
8	C₄×D₄₂₁	C₄	1	1	yes	characteristic, complemented
9	C₄×D₄₂₁	C₄	1	1	yes	characteristic, complemented
10	C₄×D₄₂₁	C₂²	1	1	yes	characteristic
11	C₄×D₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₄₂₁	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₄₂₁	C₂×C₄	1	1	yes	characteristic
14	C₂×D₄₂₁	—	2	2	no	—
15	C₂×D₄₂₁	C₂×C₄	1	1	yes	characteristic
16	C₂×D₄₂₁	C₂³	1	1	yes	characteristic
17	#13132997	—	2	4	no	—
18	C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
20	Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
21	Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄₂₁	—	2	1	no	—
23	C₂×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
24	D₄₂₁	—	2	1	no	—
25	D₄₂₁	#41	1	2	yes	—
26	C₄₂₁	#117	1	1	yes	characteristic, complemented, Sylow
27	#117	—	421	1	no	maximal, Sylow
28	C₂²×C₄	—	421	1	no	—
29	C₄⋊C₄	—	421	2	no	—
30	C₂²⋊C₄	—	421	2	no	—
31	C₄²	—	421	2	no	—
32	C₂³	—	421	1	no	—
33	C₂×C₄	—	421	4	no	—
34	C₂×C₄	—	421	1	no	—
35	C₂×C₄	—	421	1	no	—
36	C₂×C₄	—	421	1	no	—
37	C₂×C₄	—	421	1	no	—
38	C₂×C₄	—	421	1	no	—
39	C₂×C₄	#13132997	1	1	yes	characteristic, complemented
40	C₂²	—	842	2	no	—
41	C₂²	—	421	1	no	—
42	C₂²	—	421	1	no	—
43	C₂²	C₂×#13132997	1	1	yes	characteristic
44	C₄	—	842	4	no	—
45	C₄	—	421	1	no	—
46	C₄	—	421	1	no	—
47	C₄	C₂×#13132997	1	1	yes	characteristic
48	C₄	C₂×#13132997	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	842	1	no	—
51	C₂	—	421	2	no	—
52	C₂	C₂²×#13132997	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13132997
commutator, socle	C₂×C₄₂₁	C₂²×C₄
Frattini	C₂	C₂²×#13132997
Fitting	C₂×C₄×C₄₂₁	C₄
prime radical	C₂²×D₄₂₁	C₂²
cosocle	C₂×D₄₂₁	C₂³
hypercenter	C₂×C₄	#13132997
hypocenter	C₄₂₁	#117
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13132997	C₂	semidirect
3. C₄×#13132997	C₂	semidirect
7. C₂×Dic₄₂₁	C₄	semidirect
8. C₄×D₄₂₁	C₄	semidirect
9. C₄×D₄₂₁	C₄	semidirect
12. C₂×C₄×C₄₂₁	C₄	semidirect
26. C₄₂₁	#117	semidirect
39. C₂×C₄	#13132997	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13132997	2
C₂²⋊#13132997	2
C₄×#13132997	2
C₂×C₄×D₄₂₁	2
#117	421