Subgroups — #65352

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
3	#3942	C₂	1	1	yes	maximal, characteristic
4	#3942	C₂	1	1	yes	maximal, characteristic
5	#3940	C₂	1	1	yes	maximal, characteristic
6	#3939	C₂	1	1	yes	maximal, characteristic
7	C₄⋊#168	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊C₄	—	2	1	no	—
10	C₅×C₄⋊C₄	—	2	1	no	—
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	#683	—	2	1	no	—
16	C₄×Dic₅	—	2	1	no	—
17	C₂×#168	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#479	—	5	1	no	maximal, Sylow
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	C₂×Dic₅	—	2	1	no	—
26	C₂×Dic₅	—	2	1	no	—
27	Dic₁₀	—	2	1	no	—
28	Dic₁₀	Q₈	1	2	yes	—
29	#168	—	2	1	no	—
30	#168	—	2	1	no	—
31	#125	D₅	1	1	yes	characteristic
32	C₄×Q₈	—	5	1	no	—
33	#107	—	5	1	no	—
34	#106	—	5	1	no	—
35	C₄⋊C₈	—	5	1	no	—
36	#103	—	5	1	no	—
37	#103	—	5	1	no	—
38	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅	—	2	1	no	—
40	C₄×C₅	—	2	1	no	—
41	C₄×C₅	—	4	1	no	—
42	C₄×C₅	—	4	1	no	—
43	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
44	C₄×C₅	#41	1	1	yes	characteristic
45	Dic₅	—	2	2	no	—
46	Dic₅	—	4	1	no	—
47	C₂×Q₈	—	5	1	no	—
48	C₂×C₈	—	5	1	no	—
49	C₂×C₈	—	5	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	5	1	no	—
53	C₄⋊C₄	—	10	1	no	—
54	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
56	C₄²	—	10	1	no	—
57	C₄²	C₂×D₅	1	1	yes	characteristic
58	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₅	#135	1	1	yes	characteristic
60	C₂×C₅	#137	1	1	yes	characteristic
61	Q₈	—	5	2	no	—
62	Q₈	—	10	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	10	1	no	—
66	C₂×C₄	—	10	1	no	—
67	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
70	C₈	—	10	1	no	—
71	C₈	—	10	1	no	—
72	C₅	#479	1	1	yes	characteristic, complemented, Sylow
73	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	10	2	no	—
79	C₄	—	20	1	no	—
80	C₄	Q₈×D₅	1	1	yes	characteristic
81	C₄	#713	1	1	yes	characteristic
82	C₂	#4092	1	1	yes	characteristic
83	C₂	#4096	1	1	yes	characteristic
84	C₂	#4097	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×#125	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#125	D₅
hypocenter	C₅	#479
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
72. C₅	#479	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
C₄×Dic₁₀	2
#3942	2
#3940	2
#3939	2
C₄⋊#168	2
#479	5