Subgroups — #65218

Order 320.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	#3963	C₂	1	4	yes	maximal
5	C₂²×C₄×C₅	C₂²	1	2	yes	—
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×Dic₅	C₂²	1	2	yes	—
8	C₂²×Dic₅	C₂²	1	2	yes	—
9	C₂×Dic₁₀	—	2	2	no	—
10	C₂×Dic₁₀	—	2	2	no	—
11	C₂×Dic₁₀	C₄	1	4	yes	complemented
12	C₄²×C₅	—	2	2	no	—
13	C₄⋊Dic₅	—	2	2	no	—
14	#391	—	5	1	no	maximal, Sylow
15	C₂³×C₅	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₅	—	2	2	no	—
17	C₂×C₄×C₅	—	2	2	no	—
18	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅	D₄	1	2	yes	—
20	C₂×C₄×C₅	D₄	1	2	yes	—
21	C₂×C₄×C₅	Q₈	1	4	yes	complemented
22	C₂×Dic₅	—	2	2	no	—
23	C₂×Dic₅	—	2	2	no	—
24	C₂×Dic₅	—	2	2	no	—
25	C₂×Dic₅	—	2	4	no	—
26	C₂×Dic₅	C₂×C₄	1	4	yes	—
27	Dic₁₀	—	2	8	no	—
28	C₂²×Q₈	—	5	1	no	—
29	C₂×C₄⋊C₄	—	5	1	no	—
30	C₂×C₄²	D₅	1	1	yes	characteristic
31	#95	—	5	4	no	—
32	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₅	C₂×Q₈	1	2	yes	—
36	C₂²×C₅	#41	1	2	yes	—
37	C₄×C₅	—	2	4	no	—
38	C₄×C₅	C₂²⋊C₄	1	4	yes	—
39	Dic₅	—	2	4	no	—
40	Dic₅	—	4	2	no	—
41	C₂×Q₈	—	5	4	no	—
42	C₂×Q₈	—	10	2	no	—
43	C₂×Q₈	—	10	2	no	—
44	C₂²×C₄	—	5	2	no	—
45	C₂²×C₄	—	5	2	no	—
46	C₂²×C₄	C₂×D₅	1	2	yes	—
47	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
48	C₄⋊C₄	—	10	2	no	—
49	C₄²	—	2	2	no	—
50	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₅	C₄×Q₈	1	2	yes	—
52	C₂×C₅	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₅	#124	1	1	yes	characteristic
54	C₂×C₅	C₄⋊Q₈	1	1	yes	characteristic
55	C₂³	C₂²×D₅	1	1	yes	characteristic
56	Q₈	—	10	8	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	5	4	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	—	10	2	no	—
62	C₂×C₄	—	10	2	no	—
63	C₂×C₄	—	10	4	no	—
64	C₂×C₄	Dic₁₀	1	4	yes	complemented
65	C₂×C₄	C₄×D₅	1	2	yes	—
66	C₂×C₄	D₂₀	1	2	yes	—
67	C₂×C₄	C₂²⋊D₅	1	2	yes	—
68	C₅	#391	1	1	yes	characteristic, complemented, Sylow
69	C₂²	C₂×Dic₁₀	1	2	yes	—
70	C₂²	C₂×C₄×D₅	1	1	yes	characteristic
71	C₂²	C₂×D₂₀	1	1	yes	characteristic
72	C₂²	#709	1	2	yes	—
73	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
74	C₄	—	2	4	no	—
75	C₄	—	10	4	no	—
76	C₄	—	20	2	no	—
77	C₄	#685	1	4	yes	—
78	C₂	C₄×Dic₁₀	1	2	yes	—
79	C₂	#4015	1	1	yes	characteristic
80	C₂	#4021	1	1	yes	characteristic
81	C₂	#4070	1	2	yes	—
82	C₂	C₂×#685	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₄²×C₅	C₂
socle, prime radical, cosocle	C₂³×C₅	C₂³
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	#391
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₂×Dic₁₀	C₄	semidirect
21. C₂×C₄×C₅	Q₈	semidirect
64. C₂×C₄	Dic₁₀	semidirect
68. C₅	#391	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀	2
C₂×C₄²×C₅	2
C₂×C₄⋊Dic₅	2
#3963	2
#391	5