Subgroups — #64734

Order 320.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#676	C₂	1	1	yes	maximal, characteristic
2	C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₅	C₄	1	1	yes	characteristic
5	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#168	C₄	1	1	yes	characteristic
7	#676	—	2	1	no	—
8	#676	—	2	1	no	—
9	C₅×C₁₆	—	2	1	no	—
10	#672	—	2	1	no	—
11	#349	—	5	1	no	maximal, Sylow
12	C₂²×D₅	C₈	1	1	yes	characteristic
13	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₅	C₈	1	1	yes	characteristic
15	C₄×D₅	—	2	1	no	—
16	C₄×D₅	—	2	1	no	—
17	C₅×C₈	D₄	1	1	yes	characteristic
18	C₅×C₈	D₄	1	1	yes	characteristic
19	#168	—	2	1	no	—
20	C₂×M₁₆	—	5	1	no	—
21	M₃₂	—	5	1	no	—
22	M₃₂	D₅	1	1	yes	characteristic, complemented
23	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
24	C₂×D₅	—	2	1	no	—
25	C₂×D₅	—	4	1	no	—
26	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₅	M₁₆	1	1	yes	characteristic
28	Dic₅	—	2	1	no	—
29	C₂²×C₄	—	5	1	no	—
30	M₁₆	—	10	1	no	—
31	M₁₆	—	10	1	no	—
32	C₂×C₈	—	5	1	no	—
33	C₂×C₈	C₂×D₅	1	1	yes	characteristic
34	C₁₆	—	2	1	no	—
35	C₁₆	—	10	1	no	—
36	C₂×C₅	—	2	1	no	—
37	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
38	D₅	—	4	1	no	—
39	C₂³	—	5	1	no	—
40	C₂×C₄	—	5	1	no	—
41	C₂×C₄	—	10	1	no	—
42	C₂×C₄	—	10	1	no	—
43	C₂×C₄	C₄×D₅	1	1	yes	characteristic
44	C₈	—	10	1	no	—
45	C₈	D₂₀	1	1	yes	characteristic
46	C₈	C₂²⋊D₅	1	1	yes	characteristic
47	C₅	#349	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	10	1	no	—
49	C₂²	—	20	1	no	—
50	C₂²	C₈×D₅	1	1	yes	characteristic
51	C₄	—	10	1	no	—
52	C₄	#676	1	1	yes	characteristic
53	C₄	#685	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	20	1	no	—
56	C₂	#3952	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#685
commutator	C₂²×C₅	C₂×C₈
Frattini	C₂×C₈	C₂×D₅
Fitting	C₅×M₃₂	C₂
socle	C₂×C₅	C₂²⋊C₈
prime radical	C₂²×D₅	C₈
cosocle	C₂×C₅×C₈	C₂²
hypercenter	M₃₂	D₅
hypocenter	C₅	#349
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₅×M₃₂	C₂	semidirect
3. #3944	C₂	semidirect
22. M₃₂	D₅	semidirect
47. C₅	#349	semidirect

Maximal subgroups

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

subgroup	index
C₂×#676	2
C₅×M₃₂	2
#3944	2
#349	5