Subgroups — #65287

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#117	C₂	1	1	yes	maximal, characteristic
2	#4072	C₂	1	1	yes	maximal, characteristic
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic
4	#3940	C₂	1	2	yes	maximal, complemented
5	#3939	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₅×C₂²⋊C₄	—	2	1	no	—
9	C₄²×C₅	—	2	1	no	—
10	C₂²⋊Dic₅	—	2	1	no	—
11	C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₄×Dic₅	—	2	1	no	—
13	C₂×#168	C₄	1	4	yes	complemented
14	C₂×#168	C₂²	1	1	yes	characteristic
15	C₂×#168	C₂²	1	1	yes	characteristic
16	#427	—	5	1	no	maximal, Sylow
17	C₂³×C₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	—	2	2	no	—
19	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅	Q₈	1	2	yes	—
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×Dic₅	—	2	2	no	—
24	#168	C₂×C₄	1	2	yes	—
25	#168	C₂×C₄	1	2	yes	—
26	C₂²×C₈	—	5	1	no	—
27	#117	—	5	1	no	—
28	#117	D₅	1	1	yes	characteristic
29	#107	—	5	2	no	—
30	#106	—	5	2	no	—
31	C₂²×C₅	—	2	2	no	—
32	C₂²×C₅	C₄⋊C₄	1	2	yes	—
33	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅	—	4	2	no	—
35	C₄×C₅	C₄⋊C₄	1	2	yes	—
36	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
38	Dic₅	—	4	2	no	—
39	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
40	C₂×C₈	—	5	4	no	—
41	C₂×C₈	—	5	1	no	—
42	C₂×C₈	—	5	1	no	—
43	C₄⋊C₄	—	5	2	no	—
44	C₄⋊C₄	C₂×D₅	1	2	yes	—
45	C₂²⋊C₄	—	2	1	no	—
46	C₂²⋊C₄	—	10	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	10	1	no	—
49	C₂×C₅	—	2	2	no	—
50	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅	#135	1	2	yes	—
52	C₂³	C₂²⋊D₅	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	10	2	no	—
55	C₂×C₄	Dic₁₀	1	2	yes	—
56	C₂×C₄	C₄×D₅	1	2	yes	—
57	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
58	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
59	C₈	—	5	2	no	—
60	C₈	—	5	2	no	—
61	C₅	#427	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	2	no	—
63	C₂²	#683	1	2	yes	—
64	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
65	C₄	—	4	2	no	—
66	C₄	—	20	2	no	—
67	C₄	#683	1	2	yes	—
68	C₄	C₂×Dic₁₀	1	1	yes	characteristic
69	C₄	C₂×C₄×D₅	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×#683	1	1	yes	characteristic
72	C₂	#4096	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#117	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#117	D₅
hypocenter	C₅	#427
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #3940	C₂	semidirect
5. #3939	C₂	semidirect
13. C₂×#168	C₄	semidirect
61. C₅	#427	semidirect

Maximal subgroups

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

subgroup	index
C₅×#117	2
#4072	2
C₂²×#168	2
#3940	2
#3939	2
#427	5