Subgroups — #13100232

Order 1600.

344 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717194	C₂	1	1	yes	maximal, characteristic
2	#11716867	C₂	1	2	yes	maximal, complemented
3	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
4	C₂×C₄⋊(C₅⋊D₅)	C₄	1	1	yes	characteristic, complemented
5	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
6	C₂²⋊#1043	—	2	2	no	—
7	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
8	#64853	—	5	6	no	maximal
9	C₂²×C₅⋊D₅	—	2	1	no	—
10	C₂²×C₅⋊D₅	D₄	1	2	yes	—
11	C₂×#1043	—	4	2	no	—
12	C₂×C₄×C₅²	—	2	1	no	—
13	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₄⋊(C₅⋊D₅)	—	4	1	no	—
15	C₄⋊(C₅⋊D₅)	—	4	4	no	—
16	C₄⋊(C₅⋊D₅)	—	4	1	no	—
17	#4020	—	5	6	no	—
18	#3999	—	5	12	no	—
19	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₅⋊D₅	—	2	2	no	—
21	C₂×C₅⋊D₅	—	4	2	no	—
22	C₂×C₅⋊D₅	—	4	2	no	—
23	C₂×C₅⋊D₅	—	4	1	no	—
24	#1043	—	8	2	no	—
25	C₄×C₅²	—	2	2	no	—
26	C₄×C₅²	—	2	1	no	—
27	C₂×D₂₀	—	5	6	no	—
28	C₂×D₂₀	—	5	6	no	—
29	C₂×D₂₀	—	10	12	no	—
30	C₂²⋊AGL(1,𝔽₅)	—	10	12	no	—
31	C₄²×C₅	AGL(1,𝔽₅)	1	6	yes	complemented
32	#353	—	25	1	no	Sylow
33	C₂×C₅²	—	2	1	no	—
34	C₂×C₅²	#99	1	1	yes	characteristic
35	C₅⋊D₅	—	4	2	no	—
36	C₅⋊D₅	—	8	1	no	—
37	C₂²×D₅	—	5	12	no	—
38	C₂²×D₅	—	10	6	no	—
39	C₂×AGL(1,𝔽₅)	—	20	12	no	—
40	C₂×C₄×C₅	—	2	6	no	—
41	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
42	D₂₀	—	20	6	no	—
43	D₂₀	—	20	6	no	—
44	D₂₀	—	20	24	no	—
45	#127	—	25	1	no	—
46	#99	—	25	2	no	—
47	C₅²	#353	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	6	yes	—
49	C₂×D₅	—	10	12	no	—
50	C₂×D₅	—	20	12	no	—
51	C₂×D₅	—	20	12	no	—
52	C₂×D₅	—	20	6	no	—
53	AGL(1,𝔽₅)	—	40	12	no	—
54	C₄×C₅	—	2	12	no	—
55	C₄×C₅	—	2	6	no	—
56	C₂×D₄	—	25	1	no	—
57	C₂×D₄	—	25	1	no	—
58	C₂×D₄	—	50	2	no	—
59	C₂²⋊C₄	—	50	2	no	—
60	C₄²	#1043	1	1	yes	characteristic, complemented
61	C₂×C₅	—	2	6	no	—
62	C₂×C₅	#3999	1	6	yes	—
63	D₅	—	20	12	no	—
64	D₅	—	40	6	no	—
65	C₂³	—	25	2	no	—
66	C₂³	—	50	1	no	—
67	D₄	—	100	1	no	—
68	D₄	—	100	1	no	—
69	D₄	—	100	4	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	100	2	no	—
72	C₂×C₄	C₂×#1043	1	1	yes	characteristic
73	C₅	#64853	1	6	yes	complemented
74	C₂²	—	50	2	no	—
75	C₂²	—	100	2	no	—
76	C₂²	—	100	2	no	—
77	C₂²	—	100	1	no	—
78	C₂²	C₂²⋊#1043	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	200	2	no	—
82	C₂	—	2	1	no	—
83	C₂	—	100	2	no	—
84	C₂	—	200	1	no	—
85	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	#11717194	C₂
cosocle	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₄²	#1043
hypocenter	C₅²	#353
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. #11716867	C₂	semidirect
4. C₂×C₄⋊(C₅⋊D₅)	C₄	semidirect
7. C₄²×C₅²	C₄	semidirect
31. C₄²×C₅	AGL(1,𝔽₅)	semidirect
47. C₅²	#353	semidirect
60. C₄²	#1043	semidirect
73. C₅	#64853	semidirect

Maximal subgroups

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

subgroup	index
#11717194	2
#11716867	2
#64853	5