Subgroups — #66031

Order 320.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4044	C₂	1	4	yes	maximal
3	#4043	C₂	1	2	yes	maximal
4	#4036	C₂	1	4	yes	maximal, complemented
5	#4033	C₂	1	2	yes	maximal, complemented
6	#4018	C₂	1	1	yes	maximal, characteristic
7	#4015	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀	C₂²	1	2	yes	—
10	C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₄²×C₅	C₂²	1	1	yes	characteristic
13	#685	C₂²	1	2	yes	—
14	#685	C₂²	1	4	yes	—
15	C₄⋊Dic₅	C₂²	1	4	yes	—
16	C₄⋊Dic₅	C₂²	1	4	yes	—
17	#683	C₂²	1	2	yes	—
18	#683	C₂²	1	4	yes	—
19	C₄×Dic₅	C₂²	1	1	yes	characteristic
20	C₄×Dic₅	C₂²	1	4	yes	—
21	#541	—	5	1	no	maximal, Sylow
22	C₂²×D₅	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂³	1	4	yes	—
24	C₂×C₄×C₅	C₂³	1	2	yes	—
25	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₅	C₂³	1	2	yes	—
27	C₂×Dic₅	C₂³	1	4	yes	—
28	C₂×Dic₅	C₂³	1	1	yes	characteristic
29	C₄×D₅	—	2	2	no	—
30	Dic₁₀	—	2	4	no	—
31	C₄⋊Q₈	—	5	1	no	—
32	#126	—	5	4	no	—
33	#125	—	5	4	no	—
34	#125	D₅	1	1	yes	characteristic, complemented
35	C₂²⋊Q₈	—	5	2	no	—
36	C₄×Q₈	—	5	2	no	—
37	#117	—	5	1	no	—
38	C₂²×C₅	C₂⁴	1	1	yes	characteristic
39	C₂×D₅	—	2	1	no	—
40	C₂×D₅	—	2	2	no	—
41	C₄×C₅	—	2	4	no	—
42	C₄×C₅	—	2	2	no	—
43	C₄×C₅	#41	1	2	yes	—
44	Dic₅	—	2	4	no	—
45	Dic₅	—	2	1	no	—
46	Dic₅	—	2	2	no	—
47	C₂×Q₈	—	5	2	no	—
48	C₂²×C₄	—	5	1	no	—
49	C₄⋊C₄	—	5	2	no	—
50	C₄⋊C₄	—	5	4	no	—
51	C₄⋊C₄	—	5	4	no	—
52	C₄⋊C₄	—	5	4	no	—
53	C₄⋊C₄	C₂×D₅	1	2	yes	—
54	C₄⋊C₄	C₂×D₅	1	4	yes	—
55	C₂²⋊C₄	—	5	2	no	—
56	C₂²⋊C₄	—	5	4	no	—
57	C₄²	—	5	1	no	—
58	C₄²	—	5	4	no	—
59	C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₅	C₂×#41	1	1	yes	characteristic
61	C₂×C₅	2₋¹⁺⁴	1	2	yes	—
62	D₅	—	4	1	no	—
63	C₂³	—	5	1	no	—
64	Q₈	—	10	4	no	—
65	C₂×C₄	—	5	2	no	—
66	C₂×C₄	—	5	4	no	—
67	C₂×C₄	—	5	1	no	—
68	C₂×C₄	—	10	2	no	—
69	C₂×C₄	C₂²×D₅	1	4	yes	—
70	C₂×C₄	C₂²×D₅	1	2	yes	—
71	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
72	C₅	#541	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	10	1	no	—
74	C₂²	—	10	2	no	—
75	C₂²	C₂³×D₅	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	10	1	no	—
79	C₄	—	10	4	no	—
80	C₄	—	10	2	no	—
81	C₄	#713	1	2	yes	—
82	C₂	—	20	1	no	—
83	C₂	C₂×#713	1	1	yes	characteristic
84	C₂	#4150	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#125	C₂
prime radical	C₂²×D₅	C₂³
hypercenter	#125	D₅
hypocenter	C₅	#541
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
4. #4036	C₂	semidirect
5. #4033	C₂	semidirect
7. #4015	C₂	semidirect
34. #125	D₅	semidirect
72. C₅	#541	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
#4044	2
#4043	2
#4036	2
#4033	2
#4018	2
#4015	2
#541	5