Subgroups — #65316

Order 320.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#688	C₂	1	2	yes	maximal, complemented
3	#4020	C₂	1	1	yes	maximal, characteristic
4	#3941	C₂	1	2	yes	maximal, complemented
5	C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×D₂₀	—	2	1	no	—
8	C₂×D₂₀	—	2	1	no	—
9	C₂×D₂₀	C₂²	1	2	yes	—
10	C₅×C₄⋊C₄	—	2	1	no	—
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₄²×C₅	—	2	1	no	—
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	#688	—	2	4	no	—
15	C₂×#168	C₂²	1	2	yes	—
16	#460	—	5	1	no	maximal, Sylow
17	C₂²×D₅	—	2	2	no	—
18	C₅×Q₈	—	2	1	no	—
19	C₅×Q₈	D₄	1	2	yes	complemented
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
25	D₂₀	—	2	2	no	—
26	D₂₀	—	2	2	no	—
27	D₂₀	—	4	2	no	—
28	D₂₀	—	4	2	no	—
29	#168	—	2	2	no	—
30	C₂×SD₁₆	—	5	2	no	—
31	#127	—	5	1	no	—
32	C₄×Q₈	D₅	1	1	yes	characteristic, complemented
33	C₄⋊C₈	—	5	1	no	—
34	#102	—	5	2	no	—
35	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
36	C₂×D₅	—	4	2	no	—
37	C₂×D₅	—	4	2	no	—
38	C₂×D₅	—	4	2	no	—
39	C₄×C₅	—	2	2	no	—
40	C₄×C₅	—	2	1	no	—
41	C₄×C₅	—	4	1	no	—
42	C₄×C₅	SD₁₆	1	2	yes	complemented
43	C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄×C₅	#41	1	1	yes	characteristic
45	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
46	C₂×D₄	—	5	2	no	—
47	C₂×D₄	—	10	1	no	—
48	C₂×D₄	—	10	1	no	—
49	SD₁₆	—	10	4	no	—
50	C₂×C₈	—	5	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₅	1	1	yes	characteristic
55	C₂×C₅	#121	1	1	yes	characteristic
56	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
58	D₅	—	8	2	no	—
59	C₂³	—	10	2	no	—
60	Q₈	—	2	1	no	—
61	Q₈	D₂₀	1	2	yes	complemented
62	D₄	—	10	2	no	—
63	D₄	—	10	2	no	—
64	D₄	—	20	2	no	—
65	D₄	—	20	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
71	C₈	—	10	2	no	—
72	C₅	#460	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	20	2	no	—
74	C₂²	—	20	2	no	—
75	C₂²	—	20	2	no	—
76	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#688	1	2	yes	complemented
81	C₄	C₂×D₂₀	1	1	yes	characteristic
82	C₄	#709	1	1	yes	characteristic
83	C₂	—	40	2	no	—
84	C₂	#4076	1	1	yes	characteristic
85	C₂	C₂×#688	1	1	yes	characteristic
86	C₂	#4095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	#4020	C₂
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	#460
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×C₅×Q₈	C₂	semidirect
2. C₂×#688	C₂	semidirect
4. #3941	C₂	semidirect
5. C₄⋊#168	C₂	semidirect
19. C₅×Q₈	D₄	semidirect
32. C₄×Q₈	D₅	semidirect
42. C₄×C₅	SD₁₆	semidirect
61. Q₈	D₂₀	semidirect
72. C₅	#460	semidirect
80. C₄	#688	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×Q₈	2
C₂×#688	2
#4020	2
#3941	2
C₄⋊#168	2
#460	5