Subgroups — #13097972

Order 1600.

243 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₀	C₂	1	4	yes	maximal, complemented
2	#11716436	C₂	1	2	yes	maximal, complemented
3	C₄×C₈×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₀	—	2	4	no	—
5	C₂×D₁₀₀	C₂²	1	4	yes	—
6	C₂×C₈×C₂₅	C₂²	1	2	yes	—
7	C₄²×C₂₅	C₂²	1	1	yes	characteristic
8	D₂₀₀	—	2	8	no	—
9	#64984	—	5	1	no	maximal
10	C₂²×D₂₅	—	2	4	no	—
11	C₂×C₄×C₂₅	D₄	1	2	yes	—
12	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
13	D₁₀₀	—	2	8	no	—
14	D₁₀₀	—	4	8	no	—
15	C₈×C₂₅	D₄	1	4	yes	complemented
16	C₂×D₄₀	—	5	4	no	—
17	#4020	—	5	2	no	—
18	C₄×C₅×C₈	D₅	1	1	yes	characteristic
19	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
20	C₂×D₂₅	—	4	4	no	—
21	C₂×D₂₅	—	4	8	no	—
22	C₄×C₂₅	D₈	1	4	yes	complemented
23	C₄×C₂₅	C₂×D₄	1	2	yes	—
24	C₂×D₂₀	—	5	4	no	—
25	C₂×D₂₀	—	10	4	no	—
26	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
27	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
28	D₄₀	—	10	8	no	—
29	#493	—	25	1	no	Sylow
30	C₂×C₂₅	#127	1	1	yes	characteristic
31	C₂×C₂₅	C₂×D₈	1	2	yes	—
32	D₂₅	—	8	4	no	—
33	C₂²×D₅	—	10	4	no	—
34	C₂×C₄×C₅	D₂₀	1	2	yes	—
35	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
36	D₂₀	—	10	8	no	—
37	D₂₀	—	20	8	no	—
38	C₅×C₈	D₂₀	1	4	yes	—
39	C₂×D₈	—	25	4	no	—
40	#127	—	25	2	no	—
41	C₄×C₈	D₂₅	1	1	yes	characteristic, complemented
42	C₂₅	#493	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
44	C₂×D₅	—	20	4	no	—
45	C₂×D₅	—	20	8	no	—
46	C₄×C₅	D₄₀	1	4	yes	—
47	C₄×C₅	C₂×D₂₀	1	2	yes	—
48	C₂×D₄	—	25	4	no	—
49	C₂×D₄	—	50	4	no	—
50	D₈	—	50	8	no	—
51	C₂×C₈	C₂×D₂₅	1	2	yes	—
52	C₄²	C₂×D₂₅	1	1	yes	characteristic
53	C₂×C₅	#4020	1	1	yes	characteristic
54	C₂×C₅	C₂×D₄₀	1	2	yes	—
55	D₅	—	40	4	no	—
56	C₂³	—	50	4	no	—
57	D₄	—	50	8	no	—
58	D₄	—	100	8	no	—
59	C₂×C₄	D₁₀₀	1	2	yes	—
60	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
61	C₈	D₁₀₀	1	4	yes	complemented
62	C₅	#64984	1	1	yes	characteristic
63	C₂²	—	100	4	no	—
64	C₂²	—	100	8	no	—
65	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
66	C₄	D₂₀₀	1	4	yes	complemented
67	C₄	C₂×D₁₀₀	1	2	yes	—
68	C₂	—	200	4	no	—
69	C₂	#11716436	1	1	yes	characteristic
70	C₂	C₂×D₂₀₀	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₀
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₄×C₈×C₂₅	C₂
socle	C₂²×C₅	C₂×D₂₀
hypercenter	C₄×C₈	D₂₅
hypocenter	C₂₅	#493
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₀₀	C₂	semidirect
2. #11716436	C₂	semidirect
3. C₄×C₈×C₂₅	C₂	semidirect
15. C₈×C₂₅	D₄	semidirect
22. C₄×C₂₅	D₈	semidirect
41. C₄×C₈	D₂₅	semidirect
42. C₂₅	#493	semidirect
61. C₈	D₁₀₀	semidirect
66. C₄	D₂₀₀	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₀	2
#11716436	2
C₄×C₈×C₂₅	2
#64984	5