Subgroups — #64671

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4076	C₂	1	1	yes	maximal, characteristic
2	C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×C₅	—	2	1	no	—
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊D₅	—	2	1	no	—
7	C₂×D₂₀	C₄	1	1	yes	characteristic, complemented
8	#685	—	2	1	no	—
9	C₄⋊Dic₅	C₄	1	1	yes	characteristic, complemented
10	C₂×#168	—	2	1	no	—
11	#327	—	5	1	no	maximal, Sylow
12	C₂³×C₅	D₄	1	1	yes	characteristic
13	C₂²×D₅	—	2	1	no	—
14	C₂×C₄×C₅	—	2	2	no	—
15	C₂×C₄×C₅	—	4	1	no	—
16	C₂×C₄×C₅	—	4	1	no	—
17	C₂×C₄×C₅	—	4	1	no	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂²⋊D₅	—	4	1	no	—
21	C₂²⋊D₅	—	4	1	no	—
22	C₂×Dic₅	—	2	1	no	—
23	D₂₀	—	4	1	no	—
24	#168	—	4	1	no	—
25	#121	—	5	1	no	—
26	C₂²⋊C₈	—	5	1	no	—
27	#95	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	D₈	1	1	yes	characteristic
32	C₂²×C₅	SD₁₆	1	1	yes	characteristic
33	C₂×D₅	—	4	1	no	—
34	C₂×D₅	—	4	1	no	—
35	C₂×D₅	—	4	1	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	—	2	1	no	—
38	C₄×C₅	—	4	2	no	—
39	Dic₅	—	4	1	no	—
40	C₂×D₄	—	5	1	no	—
41	C₂×D₄	—	10	1	no	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
44	C₂×C₈	—	10	1	no	—
45	C₄⋊C₄	—	5	1	no	—
46	C₂²⋊C₄	—	10	1	no	—
47	C₂×C₅	—	2	1	no	—
48	C₂×C₅	—	2	1	no	—
49	C₂×C₅	#99	1	1	yes	characteristic
50	C₂×C₅	#102	1	1	yes	characteristic
51	C₂×C₅	#104	1	1	yes	characteristic
52	D₅	—	8	1	no	—
53	C₂³	—	10	1	no	—
54	C₂³	C₂²⋊D₅	1	1	yes	characteristic
55	D₄	—	20	1	no	—
56	D₄	—	20	1	no	—
57	D₄	—	20	1	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	10	1	no	—
63	C₂×C₄	C₄×D₅	1	1	yes	characteristic
64	C₂×C₄	D₂₀	1	1	yes	characteristic
65	C₈	—	20	1	no	—
66	C₅	#327	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	20	1	no	—
70	C₂²	—	20	1	no	—
71	C₂²	—	20	1	no	—
72	C₂²	#685	1	1	yes	characteristic
73	C₂²	#686	1	1	yes	characteristic
74	C₂²	#688	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	20	1	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	40	1	no	—
82	C₂	#3937	1	1	yes	characteristic
83	C₂	#3938	1	1	yes	characteristic
84	C₂	#3941	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#685
commutator	C₂×C₄×C₅	C₂×C₄
Frattini	C₂²×C₄	C₂×D₅
Fitting	C₅×#95	C₂
socle	C₂²×C₅	C₂²⋊C₄
prime radical	#4076	C₂
cosocle	C₂²×C₄×C₅	C₂²
hypercenter	#95	D₅
hypocenter	C₅	#327
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #95

Complements

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

N	H ≅ G/N	type
2. C₅×#95	C₂	semidirect
3. C₂²⋊#168	C₂	semidirect
7. C₂×D₂₀	C₄	semidirect
9. C₄⋊Dic₅	C₄	semidirect
27. #95	D₅	semidirect
66. C₅	#327	semidirect

Maximal subgroups

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

subgroup	index
#4076	2
C₅×#95	2
C₂²⋊#168	2
#327	5