Subgroups — #31772602

Order 17312.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₅₄₁	C₂	1	2	yes	maximal, complemented
3	C₄⋊Dic₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575813	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₄₁	C₂²	1	2	yes	—
7	D₂₁₆₄	—	2	2	no	—
8	C₂²⋊D₅₄₁	—	2	4	no	—
9	C₂×Dic₅₄₁	C₂²	1	2	yes	—
10	C₂³×C₅₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅₄₁	—	2	2	no	—
12	C₂×C₄×C₅₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₅₄₁	—	2	1	no	—
15	C₂²×C₅₄₁	—	2	1	no	—
16	C₂²×C₅₄₁	D₄	1	2	yes	complemented
17	C₂²×C₅₄₁	C₂³	1	1	yes	characteristic
18	C₂×D₅₄₁	—	2	2	no	—
19	C₂×D₅₄₁	—	2	2	no	—
20	C₂×D₅₄₁	—	2	2	no	—
21	C₄×C₅₄₁	—	2	1	no	—
22	C₄×C₅₄₁	D₄	1	2	yes	complemented
23	Dic₅₄₁	—	2	2	no	—
24	C₂×C₅₄₁	—	2	2	no	—
25	C₂×C₅₄₁	C₂×D₄	1	1	yes	characteristic
26	C₂×C₅₄₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₅₄₁	#41	1	1	yes	characteristic
28	D₅₄₁	—	4	2	no	—
29	C₅₄₁	#121	1	1	yes	characteristic, complemented, Sylow
30	#121	—	541	1	no	maximal, Sylow
31	C₂×D₄	—	541	2	no	—
32	C₂×D₄	—	541	1	no	—
33	C₂²×C₄	D₅₄₁	1	1	yes	characteristic, complemented
34	C₄⋊C₄	—	541	1	no	—
35	C₂²⋊C₄	—	541	2	no	—
36	C₂³	—	541	2	no	—
37	C₂³	C₂×D₅₄₁	1	1	yes	characteristic
38	D₄	—	1082	4	no	—
39	D₄	—	1082	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	541	2	no	—
42	C₂×C₄	C₂×D₅₄₁	1	1	yes	characteristic
43	C₂×C₄	C₂×D₅₄₁	1	1	yes	characteristic
44	C₂²	—	2	1	no	—
45	C₂²	—	2	1	no	—
46	C₂²	—	1082	2	no	—
47	C₂²	—	1082	2	no	—
48	C₂²	—	1082	2	no	—
49	C₂²	D₂₁₆₄	1	2	yes	complemented
50	C₂²	C₂²×D₅₄₁	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	1082	2	no	—
53	C₄	C₂²⋊D₅₄₁	1	2	yes	complemented
54	C₂	—	2	2	no	—
55	C₂	—	2164	2	no	—
56	C₂	C₂×C₂²⋊D₅₄₁	1	1	yes	characteristic
57	C₂	C₂×D₂₁₆₄	1	1	yes	characteristic
58	C₂	#22575836	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₄₁
commutator, socle, cosocle	C₂²×C₅₄₁	C₂³
Fitting	C₂²×C₄×C₅₄₁	C₂
hypercenter	C₂²×C₄	D₅₄₁
hypocenter	C₅₄₁	#121
LCS	G ▷ C₂²×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₁₆₄	C₂	semidirect
2. C₂×C₂²⋊D₅₄₁	C₂	semidirect
3. C₄⋊Dic₅₄₁	C₂	semidirect
4. #22575813	C₂	semidirect
5. C₂²×C₄×C₅₄₁	C₂	semidirect
16. C₂²×C₅₄₁	D₄	semidirect
22. C₄×C₅₄₁	D₄	semidirect
29. C₅₄₁	#121	semidirect
33. C₂²×C₄	D₅₄₁	semidirect
49. C₂²	D₂₁₆₄	semidirect
53. C₄	C₂²⋊D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₁₆₄	2
C₂×C₂²⋊D₅₄₁	2
C₄⋊Dic₅₄₁	2
#22575813	2
C₂²×C₄×C₅₄₁	2
#121	541