Subgroups — #13100233

Order 1600.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717195	C₂	1	1	yes	maximal, characteristic
2	#11716873	C₂	1	2	yes	maximal, complemented
3	C₂×C₄⋊(C₅⋊D₅)	C₄	1	1	yes	characteristic
4	C₂×#6045	C₂²	1	1	yes	characteristic
5	#87860	—	2	2	no	—
6	C₄²×C₅²	C₄	1	1	yes	characteristic
7	#87808	—	2	2	no	—
8	#64856	—	5	6	no	maximal
9	C₂²×C₅⋊D₅	—	2	1	no	—
10	C₂×C₄×C₅²	—	2	1	no	—
11	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
12	C₂×#1039	D₄	1	2	yes	—
13	C₄⋊(C₅⋊D₅)	—	4	1	no	—
14	#6045	—	4	1	no	—
15	#6033	—	4	2	no	—
16	#4021	—	5	6	no	—
17	#4005	—	5	12	no	—
18	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₅⋊D₅	—	4	2	no	—
20	C₂×C₅⋊D₅	—	4	1	no	—
21	C₄×C₅²	—	2	1	no	—
22	C₄×C₅²	—	4	1	no	—
23	#1039	—	2	2	no	—
24	C₂×D₂₀	—	5	6	no	—
25	C₂×Dic₁₀	—	5	6	no	—
26	#704	—	10	12	no	—
27	C₄²×C₅	AGL(1,𝔽₅)	1	6	yes	—
28	#685	—	10	12	no	—
29	#355	—	25	1	no	Sylow
30	C₂×C₅²	—	2	1	no	—
31	C₂×C₅²	#99	1	1	yes	characteristic
32	C₅⋊D₅	—	8	1	no	—
33	C₂²×D₅	—	10	6	no	—
34	C₂×C₄×C₅	—	2	6	no	—
35	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
36	C₂×Dic₅	—	5	12	no	—
37	D₂₀	—	20	6	no	—
38	Dic₁₀	—	20	6	no	—
39	#170	—	20	12	no	—
40	#124	—	25	1	no	—
41	#101	—	25	2	no	—
42	C₅²	#355	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	6	yes	—
44	C₂×D₅	—	20	12	no	—
45	C₂×D₅	—	20	6	no	—
46	C₄×C₅	—	2	6	no	—
47	C₄×C₅	—	4	6	no	—
48	Dic₅	—	10	12	no	—
49	C₂×Q₈	—	25	1	no	—
50	C₂×D₄	—	25	1	no	—
51	M₁₆	—	50	2	no	—
52	C₂²⋊C₄	—	50	2	no	—
53	C₄²	#1043	1	1	yes	characteristic
54	C₂×C₅	—	2	6	no	—
55	C₂×C₅	#3999	1	6	yes	—
56	D₅	—	40	6	no	—
57	C₂³	—	50	1	no	—
58	Q₈	—	100	1	no	—
59	D₄	—	100	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	25	2	no	—
62	C₂×C₄	C₂×#1043	1	1	yes	characteristic
63	C₈	—	100	2	no	—
64	C₅	#64856	1	6	yes	complemented
65	C₂²	—	100	2	no	—
66	C₂²	—	100	1	no	—
67	C₂²	C₂²⋊#1043	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	50	2	no	—
71	C₂	—	2	1	no	—
72	C₂	—	200	1	no	—
73	C₂	#11716867	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716867
commutator	C₂×C₄×C₅²	C₂×C₄
Frattini	C₂×C₄	C₂×#1043
Fitting	C₄²×C₅²	C₄
socle	C₂×C₅²	#99
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₄
cosocle	C₂×#6045	C₂²
hypercenter	C₄²	#1043
hypocenter	C₅²	#355
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #11716873	C₂	semidirect
42. C₅²	#355	semidirect
64. C₅	#64856	semidirect

Maximal subgroups

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

subgroup	index
#11717195	2
#11716873	2
#64856	5