Subgroups — #65355

Order 320.

102 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	C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
3	#3941	C₂	1	1	yes	maximal, characteristic
4	#3941	C₂	1	1	yes	maximal, characteristic
5	#3940	C₂	1	1	yes	maximal, characteristic, complemented
6	#3939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₀	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₅	—	2	1	no	—
10	C₅×C₄⋊C₄	—	2	1	no	—
11	C₅×C₄⋊C₄	—	2	1	no	—
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄²×C₅	C₂²	1	1	yes	characteristic
15	#685	—	2	1	no	—
16	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
17	C₂×#168	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#478	—	5	1	no	maximal, Sylow
20	C₂²×D₅	—	2	1	no	—
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	—	2	1	no	—
23	C₂×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₅	—	2	1	no	—
27	D₂₀	—	2	1	no	—
28	D₂₀	Q₈	1	2	yes	—
29	C₄×D₅	—	4	2	no	—
30	#168	—	2	1	no	—
31	#168	—	2	1	no	—
32	#125	D₅	1	1	yes	characteristic, complemented
33	C₄×D₄	—	5	1	no	—
34	#107	—	5	1	no	—
35	#106	—	5	1	no	—
36	C₄⋊C₈	—	5	1	no	—
37	#102	—	5	1	no	—
38	#102	—	5	1	no	—
39	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×D₅	—	2	2	no	—
41	C₂×D₅	—	4	1	no	—
42	C₂×D₅	—	4	1	no	—
43	C₄×C₅	—	2	1	no	—
44	C₄×C₅	—	2	1	no	—
45	C₄×C₅	—	4	1	no	—
46	C₄×C₅	—	4	1	no	—
47	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₄×C₅	#41	1	1	yes	characteristic
49	Dic₅	—	4	1	no	—
50	C₂×D₄	—	5	1	no	—
51	C₂²×C₄	—	10	1	no	—
52	C₂×C₈	—	5	1	no	—
53	C₂×C₈	—	5	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	5	1	no	—
57	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
58	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
59	C₂²⋊C₄	—	10	1	no	—
60	C₄²	C₂×D₅	1	1	yes	characteristic
61	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₅	#135	1	1	yes	characteristic
63	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
64	D₅	—	4	2	no	—
65	C₂³	—	10	1	no	—
66	D₄	—	5	2	no	—
67	D₄	—	10	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	10	1	no	—
71	C₂×C₄	—	20	2	no	—
72	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
74	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
75	C₈	—	10	1	no	—
76	C₈	—	10	1	no	—
77	C₅	#478	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	10	2	no	—
79	C₂²	—	20	1	no	—
80	C₂²	—	20	1	no	—
81	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	20	1	no	—
87	C₄	Q₈×D₅	1	1	yes	characteristic
88	C₄	#713	1	1	yes	characteristic
89	C₂	—	20	2	no	—
90	C₂	#4092	1	1	yes	characteristic
91	C₂	#4095	1	1	yes	characteristic
92	C₂	#4096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#125	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₂₀	C₂²
hypercenter	#125	D₅
hypocenter	C₅	#478
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
5. #3940	C₂	semidirect
6. #3939	C₂	semidirect
7. C₄⋊#168	C₂	semidirect
32. #125	D₅	semidirect
77. C₅	#478	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
C₄×D₂₀	2
#3941	2
#3940	2
#3939	2
C₄⋊#168	2
#478	5