Subgroups — #39911243

Order 26272.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₈₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#27190128	C₂	1	1	yes	maximal, characteristic, complemented
3	#27190126	C₂	1	1	yes	maximal, characteristic
4	#27190126	C₂	1	1	yes	maximal, characteristic
5	#27190126	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₈₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×C₈₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈₂₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₈₂₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₈₂₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₈₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈₂₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈₂₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈₂₁	C₂³	1	1	yes	characteristic
16	C₂×D₈₂₁	—	2	1	no	—
17	C₂×D₈₂₁	—	2	1	no	—
18	C₂×D₈₂₁	—	2	1	no	—
19	C₄×C₈₂₁	—	2	1	no	—
20	C₄×C₈₂₁	—	2	1	no	—
21	C₄×C₈₂₁	—	2	1	no	—
22	Dic₈₂₁	—	2	1	no	—
23	Dic₈₂₁	—	2	1	no	—
24	Dic₈₂₁	—	2	1	no	—
25	C₂×C₈₂₁	#41	1	1	yes	characteristic
26	C₂×C₈₂₁	#41	1	1	yes	characteristic
27	C₂×C₈₂₁	#41	1	1	yes	characteristic
28	D₈₂₁	—	4	1	no	—
29	C₈₂₁	#126	1	1	yes	characteristic, complemented, Sylow
30	#126	—	821	1	no	maximal, Sylow
31	C₄⋊C₄	—	821	1	no	—
32	C₄⋊C₄	—	821	1	no	—
33	C₄⋊C₄	D₈₂₁	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	821	1	no	—
35	C₂²⋊C₄	—	821	1	no	—
36	C₂²⋊C₄	—	821	1	no	—
37	C₄²	—	821	1	no	—
38	C₂³	—	821	1	no	—
39	C₂×C₄	—	821	1	no	—
40	C₂×C₄	—	821	1	no	—
41	C₂×C₄	—	821	1	no	—
42	C₂×C₄	C₂×D₈₂₁	1	1	yes	characteristic
43	C₂×C₄	C₂×D₈₂₁	1	1	yes	characteristic
44	C₂×C₄	C₂×D₈₂₁	1	1	yes	characteristic
45	C₂²	—	1642	1	no	—
46	C₂²	—	1642	1	no	—
47	C₂²	—	1642	1	no	—
48	C₂²	C₂²×D₈₂₁	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	1642	1	no	—
53	C₄	—	1642	1	no	—
54	C₄	—	1642	1	no	—
55	C₂	—	3284	1	no	—
56	C₂	#27190148	1	1	yes	characteristic
57	C₂	#27190149	1	1	yes	characteristic
58	C₂	#27190150	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₈₂₁
commutator, socle, cosocle	C₂²×C₈₂₁	C₂³
Fitting	C₄⋊C₄×C₈₂₁	C₂
prime radical	C₂²×D₈₂₁	C₂²
hypercenter	C₄⋊C₄	D₈₂₁
hypocenter	C₈₂₁	#126
LCS	G ▷ C₂²×C₈₂₁ ▷ C₈₂₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊Dic₈₂₁	C₂	semidirect
2. #27190128	C₂	semidirect
6. C₄×Dic₈₂₁	C₂	semidirect
7. C₄⋊C₄×C₈₂₁	C₂	semidirect
29. C₈₂₁	#126	semidirect
33. C₄⋊C₄	D₈₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₈₂₁	2
#27190128	2
#27190126	2
C₄×Dic₈₂₁	2
C₄⋊C₄×C₈₂₁	2
#126	821