Subgroups — #65804

Order 320.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	#4072	C₂	1	2	yes	maximal
3	#4070	C₂	1	4	yes	maximal
4	#4016	C₂	1	2	yes	maximal, complemented
5	#4015	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₁₀	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×C₅	C₂²	1	2	yes	—
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀	C₂²	1	4	yes	—
10	C₄²×C₅	C₂²	1	2	yes	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅	C₂²	1	4	yes	—
14	C₄⋊Dic₅	C₂²	1	4	yes	—
15	C₄⋊Dic₅	C₂²	1	4	yes	—
16	#683	C₂²	1	8	yes	—
17	C₄×Dic₅	C₂²	1	4	yes	—
18	#533	—	5	1	no	maximal, Sylow
19	C₂³×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅	—	2	4	no	—
21	C₂×C₄×C₅	Q₈	1	4	yes	complemented
22	C₂×C₄×C₅	C₂³	1	2	yes	—
23	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₅	C₂³	1	2	yes	—
25	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₅	C₂³	1	8	yes	—
27	Dic₁₀	—	2	8	no	—
28	C₄⋊Q₈	—	5	2	no	—
29	#125	—	5	2	no	—
30	C₂²⋊Q₈	—	5	4	no	—
31	C₄×Q₈	—	5	4	no	—
32	#117	—	5	2	no	—
33	C₂×C₄²	D₅	1	1	yes	characteristic
34	C₂²×C₅	—	2	2	no	—
35	C₂²×C₅	C₂×Q₈	1	2	yes	—
36	C₂²×C₅	C₂⁴	1	1	yes	characteristic
37	C₄×C₅	—	2	2	no	—
38	C₄×C₅	C₂×Q₈	1	2	yes	—
39	C₄×C₅	C₂×Q₈	1	2	yes	—
40	C₄×C₅	#41	1	4	yes	—
41	Dic₅	—	2	8	no	—
42	C₂×Q₈	—	5	4	no	—
43	C₂²×C₄	C₂×D₅	1	2	yes	—
44	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
45	C₄⋊C₄	—	5	4	no	—
46	C₄⋊C₄	—	5	8	no	—
47	C₄⋊C₄	—	5	4	no	—
48	C₂²⋊C₄	—	5	4	no	—
49	C₄²	—	5	4	no	—
50	C₄²	C₂×D₅	1	1	yes	characteristic
51	C₄²	C₂×D₅	1	1	yes	characteristic
52	C₄²	C₂×D₅	1	2	yes	—
53	C₂×C₅	—	2	2	no	—
54	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
55	C₂×C₅	C₂×#41	1	2	yes	—
56	C₂³	C₂²×D₅	1	1	yes	characteristic
57	Q₈	—	10	8	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	5	8	no	—
60	C₂×C₄	Dic₁₀	1	4	yes	complemented
61	C₂×C₄	C₂²×D₅	1	2	yes	—
62	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
63	C₂×C₄	C₂²×D₅	1	2	yes	—
64	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
65	C₅	#533	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Dic₁₀	1	2	yes	—
68	C₂²	C₂³×D₅	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	10	8	no	—
71	C₄	C₂×Dic₁₀	1	2	yes	—
72	C₄	C₂×Dic₁₀	1	2	yes	—
73	C₄	#709	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₁₀	1	1	yes	characteristic
76	C₂	C₂×#709	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Frattini	C₂²	C₂³×D₅
Fitting	C₂×C₄²×C₅	C₂
prime radical	C₂³×C₅	C₂³
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	#533
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #4016	C₂	semidirect
5. #4015	C₂	semidirect
6. C₄×Dic₁₀	C₂	semidirect
21. C₂×C₄×C₅	Q₈	semidirect
60. C₂×C₄	Dic₁₀	semidirect
65. C₅	#533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅	2
#4072	2
#4070	2
#4016	2
#4015	2
C₄×Dic₁₀	2
#533	5