Subgroups — #65374

Order 320.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#688	C₂	1	4	yes	maximal, complemented
3	#4020	C₂	1	1	yes	maximal, characteristic
4	C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×Q₈	C₂²	1	2	yes	—
6	C₂×D₂₀	—	2	2	no	—
7	C₂×D₂₀	C₂²	1	2	yes	—
8	C₅×C₄⋊C₄	—	2	2	no	—
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	#688	—	2	8	no	—
11	C₂×#168	C₂²	1	2	yes	—
12	#492	—	5	1	no	maximal, Sylow
13	C₂²×D₅	—	2	2	no	—
14	C₅×Q₈	—	2	4	no	—
15	C₂×C₄×C₅	—	2	2	no	—
16	C₂×C₄×C₅	D₄	1	2	yes	—
17	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
18	D₂₀	—	2	4	no	—
19	D₂₀	—	4	4	no	—
20	#168	D₄	1	4	yes	complemented
21	C₂×SD₁₆	—	5	4	no	—
22	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
23	#127	—	5	1	no	—
24	C₄×C₈	—	5	1	no	—
25	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
26	C₂×D₅	—	4	2	no	—
27	C₂×D₅	—	4	4	no	—
28	C₄×C₅	—	4	2	no	—
29	C₄×C₅	SD₁₆	1	4	yes	complemented
30	C₄×C₅	C₂×D₄	1	2	yes	—
31	C₂×Q₈	C₂×D₅	1	2	yes	—
32	C₂×D₄	—	5	2	no	—
33	C₂×D₄	—	10	2	no	—
34	SD₁₆	—	10	8	no	—
35	C₂×C₈	—	5	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄²	C₂×D₅	1	1	yes	characteristic
38	C₂×C₅	#127	1	1	yes	characteristic
39	C₂×C₅	C₂×SD₁₆	1	2	yes	—
40	D₅	—	8	2	no	—
41	C₂³	—	10	2	no	—
42	Q₈	—	2	4	no	—
43	D₄	—	10	4	no	—
44	D₄	—	20	4	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₂²⋊D₅	1	2	yes	—
47	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
48	C₈	—	5	4	no	—
49	C₅	#492	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	20	2	no	—
51	C₂²	—	20	4	no	—
52	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#688	1	4	yes	complemented
55	C₄	D₄×D₅	1	2	yes	—
56	C₂	—	40	2	no	—
57	C₂	#4086	1	1	yes	characteristic
58	C₂	C₂×#688	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	#4020	C₂
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#492
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. C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×#688	C₂	semidirect
4. C₄×#168	C₂	semidirect
20. #168	D₄	semidirect
22. C₄⋊Q₈	D₅	semidirect
29. C₄×C₅	SD₁₆	semidirect
49. C₅	#492	semidirect
54. C₄	#688	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄⋊Q₈	2
C₂×#688	2
#4020	2
C₄×#168	2
#492	5