Subgroups — #14580712

Order 2624.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10620282	C₂	1	4	yes	maximal, complemented
2	#13050161	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#66491	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₄₁	—	2	2	no	—
6	C₂×D₄×C₄₁	C₂²	1	2	yes	—
7	C₂×D₁₆₄	—	2	2	no	—
8	C₂×D₁₆₄	C₂²	1	2	yes	—
9	C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	#10620282	—	2	8	no	—
11	C₂×#66491	C₂²	1	2	yes	—
12	C₂³×C₄₁	—	2	2	no	—
13	C₂²×D₄₁	—	2	2	no	—
14	D₄×C₄₁	—	2	4	no	—
15	D₄×C₄₁	—	4	4	no	—
16	C₂×C₄×C₄₁	D₄	1	2	yes	—
17	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
18	D₁₆₄	—	2	4	no	—
19	D₁₆₄	—	4	4	no	—
20	#66491	D₄	1	4	yes	complemented
21	C₂²×C₄₁	—	4	2	no	—
22	C₂²×C₄₁	—	4	4	no	—
23	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₁	—	4	2	no	—
25	C₂×D₄₁	—	4	4	no	—
26	C₄×C₄₁	D₈	1	4	yes	complemented
27	C₄×C₄₁	C₂×D₄	1	2	yes	—
28	C₂×C₄₁	—	8	2	no	—
29	C₂×C₄₁	#127	1	1	yes	characteristic
30	C₂×C₄₁	C₂×D₈	1	2	yes	—
31	D₄₁	—	8	2	no	—
32	#493	—	41	1	no	maximal, Sylow
33	C₄₁	#493	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₈	—	41	4	no	—
35	#127	—	41	1	no	—
36	#127	D₄₁	1	1	yes	characteristic, complemented
37	C₄×C₈	—	41	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	41	2	no	—
40	C₂×D₄	—	82	2	no	—
41	C₂×D₄	C₂×D₄₁	1	2	yes	—
42	D₈	—	82	8	no	—
43	C₂×C₈	—	41	2	no	—
44	C₄²	C₂×D₄₁	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	82	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	82	4	no	—
50	D₄	—	164	4	no	—
51	C₂×C₄	C₂²⋊D₄₁	1	2	yes	—
52	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
53	C₈	—	41	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	164	2	no	—
57	C₂²	—	164	4	no	—
58	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
59	C₄	#10620282	1	4	yes	complemented
60	C₄	D₄×D₄₁	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	328	2	no	—
63	C₂	#13050160	1	1	yes	characteristic
64	C₂	C₂×#10620282	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	#127×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
hypercenter	#127	D₄₁
hypocenter	C₄₁	#493
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#10620282	C₂	semidirect
2. #13050161	C₂	semidirect
3. C₄×#66491	C₂	semidirect
4. #127×C₄₁	C₂	semidirect
20. #66491	D₄	semidirect
26. C₄×C₄₁	D₈	semidirect
33. C₄₁	#493	semidirect
36. #127	D₄₁	semidirect
59. C₄	#10620282	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10620282	2
#13050161	2
C₄×#66491	2
#127×C₄₁	2
#493	41