Subgroups — #65048

Order 320.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
2	#4034	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#102	C₂	1	1	yes	maximal, characteristic
4	#3964	C₂	1	1	yes	maximal, characteristic
5	#3950	C₂	1	1	yes	maximal, characteristic, complemented
6	#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	#3939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅	—	2	1	no	—
10	C₂×Dic₁₀	—	2	1	no	—
11	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅	—	2	1	no	—
14	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	#683	—	2	1	no	—
17	C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#474	—	5	1	no	maximal, Sylow
20	C₂³×C₅	—	2	1	no	—
21	C₅×D₄	—	2	1	no	—
22	C₅×D₄	Q₈	1	2	yes	complemented
23	C₂×C₄×C₅	—	2	1	no	—
24	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×Dic₅	—	2	1	no	—
26	C₂×Dic₅	—	2	1	no	—
27	C₂×Dic₅	—	4	2	no	—
28	C₂×Dic₅	D₄	1	1	yes	characteristic
29	C₂×Dic₅	D₄	1	1	yes	characteristic
30	Dic₁₀	—	4	2	no	—
31	C₅×C₈	—	2	1	no	—
32	#168	—	2	1	no	—
33	C₄⋊Q₈	—	5	1	no	—
34	C₄×D₄	—	5	1	no	—
35	#107	—	5	1	no	—
36	#107	—	5	1	no	—
37	C₄⋊C₈	—	5	1	no	—
38	#102	—	5	1	no	—
39	#102	D₅	1	1	yes	characteristic
40	C₂²×C₅	—	2	2	no	—
41	C₂²×C₅	—	4	1	no	—
42	C₂²×C₅	—	4	1	no	—
43	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄×C₅	—	4	1	no	—
45	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
46	C₄×C₅	#41	1	1	yes	characteristic
47	Dic₅	—	2	1	no	—
48	Dic₅	—	4	1	no	—
49	Dic₅	—	4	1	no	—
50	Dic₅	D₈	1	2	yes	—
51	C₂×Q₈	—	10	1	no	—
52	C₂×D₄	C₂×D₅	1	1	yes	characteristic
53	C₂²×C₄	—	10	1	no	—
54	C₂×C₈	—	5	1	no	—
55	C₂×C₈	C₂×D₅	1	1	yes	characteristic
56	C₄⋊C₄	—	5	1	no	—
57	C₄⋊C₄	—	5	1	no	—
58	C₄⋊C₄	—	10	1	no	—
59	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
60	C₂²⋊C₄	—	10	1	no	—
61	C₄²	—	5	1	no	—
62	C₂×C₅	—	4	2	no	—
63	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₅	C₂×D₈	1	1	yes	characteristic
65	C₂×C₅	#137	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	Q₈	—	20	2	no	—
68	D₄	—	2	1	no	—
69	D₄	Dic₁₀	1	2	yes	complemented
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	5	1	no	—
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	10	1	no	—
74	C₂×C₄	—	10	1	no	—
75	C₂×C₄	—	20	2	no	—
76	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	10	1	no	—
79	C₅	#474	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₅	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	5	2	no	—
86	C₄	—	10	1	no	—
87	C₄	—	20	1	no	—
88	C₄	—	20	1	no	—
89	C₄	C₂×Dic₁₀	1	1	yes	characteristic
90	C₄	#711	1	1	yes	characteristic
91	C₂	—	4	2	no	—
92	C₂	#4024	1	1	yes	characteristic
93	C₂	D₅×D₈	1	1	yes	characteristic
94	C₂	#4061	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#102	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×C₅×D₄	C₂²
hypercenter	#102	D₅
hypocenter	C₅	#474
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
2. #4034	C₂	semidirect
5. #3950	C₂	semidirect
6. #3946	C₂	semidirect
7. #3939	C₂	semidirect
22. C₅×D₄	Q₈	semidirect
69. D₄	Dic₁₀	semidirect
79. C₅	#474	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₅	2
#4034	2
C₅×#102	2
#3964	2
#3950	2
#3946	2
#3939	2
#474	5