Subgroups — #65377

Order 320.

118 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	#4020	C₂	1	1	yes	maximal, characteristic
3	#3941	C₂	1	4	yes	maximal, complemented
4	C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×Q₈	—	2	1	no	—
6	C₂×D₂₀	—	2	1	no	—
7	C₂×D₂₀	—	2	1	no	—
8	C₂×D₂₀	C₂²	1	2	yes	—
9	C₅×C₄⋊C₄	—	2	1	no	—
10	C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×#168	C₂²	1	2	yes	—
13	#486	—	5	1	no	maximal, Sylow
14	C₂²×D₅	—	2	2	no	—
15	C₅×Q₈	—	4	2	no	—
16	C₂×C₄×C₅	—	2	2	no	—
17	C₂×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	D₂₀	—	2	4	no	—
21	D₂₀	—	4	2	no	—
22	D₂₀	—	4	2	no	—
23	#168	—	2	2	no	—
24	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
25	#127	—	5	1	no	—
26	#102	—	5	4	no	—
27	C₄×C₈	—	5	1	no	—
28	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
29	C₂×D₅	—	4	2	no	—
30	C₂×D₅	—	4	2	no	—
31	C₂×D₅	—	4	2	no	—
32	C₄×C₅	—	4	2	no	—
33	C₄×C₅	D₈	1	2	yes	—
34	C₄×C₅	SD₁₆	1	2	yes	—
35	C₄×C₅	#41	1	2	yes	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×D₄	—	5	2	no	—
38	C₂×D₄	—	10	1	no	—
39	C₂×D₄	—	10	1	no	—
40	C₂×C₈	—	5	2	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	C₂×D₅	1	2	yes	—
43	C₄²	C₂×D₅	1	1	yes	characteristic
44	C₂×C₅	#124	1	1	yes	characteristic
45	C₂×C₅	C₂×D₈	1	1	yes	characteristic
46	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
47	D₅	—	8	2	no	—
48	C₂³	—	10	2	no	—
49	Q₈	—	4	2	no	—
50	D₄	—	10	4	no	—
51	D₄	—	20	2	no	—
52	D₄	—	20	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
57	C₈	—	10	2	no	—
58	C₅	#486	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	20	2	no	—
60	C₂²	—	20	2	no	—
61	C₂²	—	20	2	no	—
62	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#686	1	2	yes	—
65	C₄	#688	1	2	yes	—
66	C₄	#713	1	2	yes	—
67	C₂	—	40	2	no	—
68	C₂	C₂×#686	1	1	yes	characteristic
69	C₂	C₂×#688	1	1	yes	characteristic
70	C₂	#4093	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×C₄⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	#4020	C₂
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#486
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
3. #3941	C₂	semidirect
4. C₄×#168	C₂	semidirect
24. C₄⋊Q₈	D₅	semidirect
58. C₅	#486	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
#4020	2
#3941	2
C₄×#168	2
#486	5