Subgroups — #66034

Order 320.

220 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	#4042	C₂	1	4	yes	maximal, complemented
3	#4041	C₂	1	4	yes	maximal, complemented
4	#4040	C₂	1	4	yes	maximal, complemented
5	#4021	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₀	C₂²	1	2	yes	—
7	C₂×D₂₀	C₂²	1	4	yes	—
8	C₂×C₄×D₅	C₂²	1	4	yes	—
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	4	yes	—
14	#685	C₂²	1	4	yes	—
15	#685	C₂²	1	4	yes	—
16	#683	C₂²	1	4	yes	—
17	#562	—	5	1	no	maximal, Sylow
18	C₂²×D₅	C₂³	1	4	yes	—
19	C₂×C₄×C₅	C₂³	1	4	yes	—
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅	C₂³	1	2	yes	—
22	C₂×Dic₅	C₂³	1	4	yes	—
23	D₂₀	—	4	2	no	—
24	D₂₀	—	4	4	no	—
25	C₄×D₅	—	4	4	no	—
26	Dic₁₀	—	4	2	no	—
27	#125	D₅	1	1	yes	characteristic, complemented
28	#124	—	5	2	no	—
29	#123	—	5	4	no	—
30	C₂²⋊Q₈	—	5	4	no	—
31	#121	—	5	4	no	—
32	C₂²×C₅	C₂⁴	1	1	yes	characteristic
33	C₂×D₅	—	2	4	no	—
34	C₂×D₅	—	2	4	no	—
35	C₂×D₅	—	2	4	no	—
36	C₄×C₅	—	2	2	no	—
37	C₄×C₅	—	2	4	no	—
38	C₄×C₅	—	2	1	no	—
39	Dic₅	—	2	4	no	—
40	C₂×Q₈	—	5	2	no	—
41	C₂×D₄	—	5	4	no	—
42	C₂×D₄	—	5	2	no	—
43	C₂²×C₄	—	5	4	no	—
44	C₄⋊C₄	—	5	4	no	—
45	C₄⋊C₄	C₂×D₅	1	4	yes	—
46	C₄⋊C₄	C₂×D₅	1	2	yes	—
47	C₂²⋊C₄	—	5	4	no	—
48	C₂²⋊C₄	—	5	4	no	—
49	C₂²⋊C₄	—	5	4	no	—
50	C₄²	C₂×D₅	1	1	yes	characteristic
51	C₂×C₅	H(5,𝔽₂)	1	2	yes	—
52	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
53	D₅	—	4	4	no	—
54	C₂³	—	5	4	no	—
55	Q₈	—	20	2	no	—
56	D₄	—	20	4	no	—
57	D₄	—	20	2	no	—
58	C₂×C₄	—	5	4	no	—
59	C₂×C₄	—	20	4	no	—
60	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₅	1	2	yes	—
62	C₂×C₄	C₂²×D₅	1	4	yes	—
63	C₅	#562	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	10	4	no	—
65	C₂²	—	10	4	no	—
66	C₂²	—	10	4	no	—
67	C₂²	C₂³×D₅	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	4	no	—
70	C₄	—	2	2	no	—
71	C₄	—	10	4	no	—
72	C₂	—	20	4	no	—
73	C₂	#4147	1	1	yes	characteristic
74	C₂	#4149	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#125	C₂
hypercenter	#125	D₅
hypocenter	C₅	#562
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
2. #4042	C₂	semidirect
3. #4041	C₂	semidirect
4. #4040	C₂	semidirect
5. #4021	C₂	semidirect
27. #125	D₅	semidirect
63. C₅	#562	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
#4042	2
#4041	2
#4040	2
#4021	2
#562	5