Subgroups — #64688

Order 320.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#685	C₂	1	1	yes	maximal, characteristic
2	C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂²×Dic₅	C₄	1	1	yes	characteristic
7	C₂×C₅×C₈	—	2	1	no	—
8	#685	—	2	1	no	—
9	#685	—	2	1	no	—
10	C₂×#168	—	2	1	no	—
11	#323	—	5	1	no	maximal, Sylow
12	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×D₅	—	2	1	no	—
14	C₂²×D₅	—	2	1	no	—
15	C₂²×D₅	—	4	1	no	—
16	C₂²×D₅	—	4	1	no	—
17	C₂²×D₅	C₈	1	2	yes	complemented
18	C₂×C₄×C₅	—	4	1	no	—
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂×Dic₅	—	2	1	no	—
22	C₂×Dic₅	—	4	1	no	—
23	C₅×C₈	—	4	1	no	—
24	#168	—	4	1	no	—
25	C₂×C₂²⋊C₄	—	5	1	no	—
26	C₂²⋊C₈	—	5	1	no	—
27	C₂²⋊C₈	D₅	1	1	yes	characteristic, complemented
28	C₂²×C₅	—	2	1	no	—
29	C₂²×C₅	—	2	1	no	—
30	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₅	M₁₆	1	1	yes	characteristic
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	—	4	2	no	—
35	C₂×D₅	—	4	1	no	—
36	C₂×D₅	—	4	1	no	—
37	C₂×D₅	—	4	2	no	—
38	C₄×C₅	—	2	1	no	—
39	C₄×C₅	—	2	1	no	—
40	Dic₅	—	4	1	no	—
41	C₂⁴	—	5	1	no	—
42	C₂²×C₄	—	5	1	no	—
43	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	10	1	no	—
46	C₂²⋊C₄	—	10	1	no	—
47	C₂²⋊C₄	—	10	1	no	—
48	C₂×C₅	—	2	1	no	—
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₅	#99	1	1	yes	characteristic
52	C₂×C₅	#100	1	1	yes	characteristic
53	D₅	—	4	2	no	—
54	C₂³	—	5	2	no	—
55	C₂³	—	10	1	no	—
56	C₂³	—	10	1	no	—
57	C₂³	—	20	1	no	—
58	C₂³	—	20	1	no	—
59	C₂³	C₄×D₅	1	1	yes	characteristic
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	10	1	no	—
62	C₂×C₄	—	20	1	no	—
63	C₂×C₄	D₂₀	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
65	C₈	—	4	1	no	—
66	C₈	—	20	1	no	—
67	C₅	#323	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	10	2	no	—
71	C₂²	—	20	2	no	—
72	C₂²	—	20	2	no	—
73	C₂²	—	20	1	no	—
74	C₂²	—	20	1	no	—
75	C₂²	C₈×D₅	1	1	yes	characteristic
76	C₂²	#676	1	1	yes	characteristic
77	C₂²	#685	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	20	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	20	2	no	—
84	C₂	#3938	1	1	yes	characteristic
85	C₂	#3952	1	1	yes	characteristic
86	C₂	#3955	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#685
commutator	C₂²×C₅	C₂×C₈
Frattini	C₂²×C₄	C₂×D₅
Fitting	C₅×C₂²⋊C₈	C₂
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂³×D₅	C₄
cosocle	C₂²×C₄×C₅	C₂²
hypercenter	C₂²⋊C₈	D₅
hypocenter	C₅	#323
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₅×C₂²⋊C₈	C₂	semidirect
3. C₂²⋊#168	C₂	semidirect
17. C₂²×D₅	C₈	semidirect
27. C₂²⋊C₈	D₅	semidirect
67. C₅	#323	semidirect

Maximal subgroups

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

subgroup	index
C₂×#685	2
C₅×C₂²⋊C₈	2
C₂²⋊#168	2
#323	5