Subgroups — #31772590

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
2	C₂×C₄×D₅₄₁	C₂	1	2	yes	maximal, complemented
3	#22575813	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₄×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₄₁	C₂²	1	2	yes	—
7	D₂₁₆₄	C₄	1	4	yes	complemented
8	C₂×Dic₅₄₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₄₁	C₂²	1	1	yes	characteristic
10	C₄×D₅₄₁	—	2	4	no	—
11	C₂×C₄×C₅₄₁	C₂²	1	2	yes	—
12	C₂×C₄×C₅₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₅₄₁	C₂³	1	1	yes	characteristic
14	C₂×D₅₄₁	—	2	2	no	—
15	C₂×D₅₄₁	—	2	2	no	—
16	C₂×D₅₄₁	C₂×C₄	1	4	yes	complemented
17	C₄×C₅₄₁	—	2	2	no	—
18	C₄×C₅₄₁	C₂×C₄	1	2	yes	complemented
19	Dic₅₄₁	—	2	1	no	—
20	Dic₅₄₁	D₄	1	2	yes	complemented
21	C₂×C₅₄₁	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₅₄₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₅₄₁	#41	1	1	yes	characteristic
24	D₅₄₁	—	2	4	no	—
25	C₅₄₁	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
26	C₄×D₄	—	541	1	no	maximal, Sylow
27	C₂×D₄	—	541	1	no	—
28	C₂²×C₄	—	541	2	no	—
29	C₄⋊C₄	D₅₄₁	1	1	yes	characteristic, complemented
30	C₂²⋊C₄	—	541	2	no	—
31	C₄²	—	541	1	no	—
32	C₂³	—	541	2	no	—
33	D₄	—	541	4	no	—
34	C₂×C₄	—	541	1	no	—
35	C₂×C₄	—	541	1	no	—
36	C₂×C₄	—	1082	4	no	—
37	C₂×C₄	C₂×D₅₄₁	1	2	yes	—
38	C₂×C₄	C₂×D₅₄₁	1	1	yes	characteristic
39	C₂²	—	541	4	no	—
40	C₂²	—	1082	2	no	—
41	C₂²	—	1082	2	no	—
42	C₂²	C₂²×D₅₄₁	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	541	2	no	—
45	C₄	—	1082	1	no	—
46	C₄	C₄×D₅₄₁	1	2	yes	complemented
47	C₂	—	1082	4	no	—
48	C₂	C₂×C₄×D₅₄₁	1	1	yes	characteristic
49	C₂	D₄×D₅₄₁	1	1	yes	characteristic
50	C₂	#22575837	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅₄₁
commutator	C₂×C₅₄₁	C₂²×C₄
Fitting	C₄⋊C₄×C₅₄₁	C₂
socle, cosocle	C₂²×C₅₄₁	C₂³
prime radical	C₂×D₂₁₆₄	C₂
hypercenter	C₄⋊C₄	D₅₄₁
hypocenter	C₅₄₁	C₄×D₄
LCS	G ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₄₁	C₂	semidirect
3. #22575813	C₂	semidirect
4. C₄×Dic₅₄₁	C₂	semidirect
5. C₄⋊C₄×C₅₄₁	C₂	semidirect
7. D₂₁₆₄	C₄	semidirect
16. C₂×D₅₄₁	C₂×C₄	semidirect
18. C₄×C₅₄₁	C₂×C₄	semidirect
20. Dic₅₄₁	D₄	semidirect
25. C₅₄₁	C₄×D₄	semidirect
29. C₄⋊C₄	D₅₄₁	semidirect
46. C₄	C₄×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×C₄×D₅₄₁	2
#22575813	2
C₄×Dic₅₄₁	2
C₄⋊C₄×C₅₄₁	2
C₄×D₄	541