Subgroups — #13100161

Order 1600.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#672	C₂	1	1	yes	maximal, characteristic
2	#11716579	C₂	1	2	yes	maximal, complemented
3	C₂×C₅²×C₈	C₄	1	1	yes	characteristic
4	C₅×#672	C₄	1	1	yes	characteristic
5	C₅×#672	C₂²	1	1	yes	characteristic
6	#64877	—	5	1	no	maximal
7	#64719	—	5	1	no	maximal
8	C₂×C₄×C₅²	C₈	1	1	yes	characteristic
9	C₅²×C₈	C₈	1	1	yes	characteristic
10	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅×C₁₆	—	5	1	no	—
12	C₂×#672	D₅	1	1	yes	characteristic
13	#3928	—	5	2	no	—
14	#3926	—	5	2	no	—
15	C₂²×C₅²	C₁₆	1	1	yes	characteristic
16	C₄×C₅²	C₁₆	1	1	yes	characteristic
17	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
18	C₂×C₅×C₈	—	4	1	no	—
19	C₂×C₅×C₈	Dic₅	1	1	yes	characteristic
20	C₂×C₅×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
21	C₅×C₁₆	—	5	1	no	—
22	C₅×C₁₆	—	5	1	no	—
23	#672	Dic₅	1	1	yes	characteristic
24	#672	C₂×D₅	1	1	yes	characteristic
25	M₆₄	—	25	1	no	Sylow
26	C₂×C₅²	—	2	1	no	—
27	C₂×C₅²	C₂×C₁₆	1	1	yes	characteristic
28	C₂×C₄×C₅	—	4	1	no	—
29	C₂×C₄×C₅	#168	1	1	yes	characteristic
30	C₂×C₄×C₅	#170	1	1	yes	characteristic
31	C₅×C₈	—	4	1	no	—
32	C₅×C₈	—	4	1	no	—
33	C₅×C₈	#168	1	1	yes	characteristic
34	C₅×C₈	#170	1	1	yes	characteristic
35	C₅×C₈	C₂×Dic₅	1	1	yes	characteristic
36	C₅×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₂×C₁₆	—	5	1	no	—
38	C₃₂	—	25	2	no	—
39	C₅²	M₆₄	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₅	—	4	1	no	—
41	C₂²×C₅	#672	1	1	yes	characteristic
42	C₂²×C₅	#674	1	1	yes	characteristic
43	C₄×C₅	—	4	1	no	—
44	C₄×C₅	—	4	1	no	—
45	C₄×C₅	#672	1	1	yes	characteristic
46	C₄×C₅	#674	1	1	yes	characteristic
47	C₄×C₅	C₂×#168	1	1	yes	characteristic
48	C₄×C₅	C₂×#170	1	1	yes	characteristic
49	C₂×C₈	#1042	1	1	yes	characteristic
50	C₁₆	—	5	1	no	—
51	C₁₆	—	5	1	no	—
52	C₂×C₅	—	2	1	no	—
53	C₂×C₅	—	2	1	no	—
54	C₂×C₅	—	4	2	no	—
55	C₂×C₅	—	4	1	no	—
56	C₂×C₅	C₂×#672	1	1	yes	characteristic
57	C₂×C₅	C₂×#674	1	1	yes	characteristic
58	C₂×C₄	#6032	1	1	yes	characteristic
59	C₈	#6032	1	1	yes	characteristic
60	C₈	C₂×#1042	1	1	yes	characteristic
61	C₅	—	4	1	no	—
62	C₅	#64719	1	1	yes	characteristic, complemented
63	C₅	#64877	1	1	yes	characteristic, complemented
64	C₂²	#87763	1	1	yes	characteristic
65	C₄	#87763	1	1	yes	characteristic
66	C₄	C₂×#6032	1	1	yes	characteristic
67	C₂	—	2	1	no	—
68	C₂	C₂×#87763	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×#1042
commutator, socle	C₂×C₅²	C₂×C₁₆
Fitting	C₂×C₅²×C₈	C₄
prime radical	C₂²×C₅²	C₁₆
cosocle	C₅×#672	C₂²
hypercenter	C₂×C₈	#1042
hypocenter	C₅²	M₆₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #11716579	C₂	semidirect
39. C₅²	M₆₄	semidirect
62. C₅	#64719	semidirect
63. C₅	#64877	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#672	2
#11716579	2
#64877	5
#64719	5