Subgroups — #4085

Order 160.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#685	C₂	1	1	yes	maximal, characteristic, complemented
7	#683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅	C₂²	1	1	yes	characteristic
9	C₂³×C₅	C₂²	1	1	yes	characteristic
10	C₂²×D₅	C₂²	1	1	yes	characteristic
11	C₅×D₄	—	2	2	no	—
12	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²⋊D₅	—	2	2	no	—
14	C₂²⋊D₅	—	2	2	no	—
15	C₂×Dic₅	—	2	2	no	—
16	C₂×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×Dic₅	C₂²	1	1	yes	characteristic
19	#121	—	5	1	no	maximal, Sylow
20	C₂²×C₅	—	2	1	no	—
21	C₂²×C₅	—	2	1	no	—
22	C₂²×C₅	—	2	1	no	—
23	C₂²×C₅	—	2	1	no	—
24	C₂²×C₅	—	2	1	no	—
25	C₂²×C₅	D₄	1	2	yes	complemented
26	C₂²×C₅	C₂³	1	1	yes	characteristic
27	C₂×D₅	—	2	1	no	—
28	C₂×D₅	—	2	1	no	—
29	C₂×D₅	—	2	1	no	—
30	C₄×C₅	—	2	1	no	—
31	Dic₅	—	2	1	no	—
32	Dic₅	—	2	1	no	—
33	Dic₅	D₄	1	2	yes	complemented
34	C₂×D₄	—	5	1	no	—
35	C₂×D₄	—	5	1	no	—
36	C₂×D₄	D₅	1	1	yes	characteristic, complemented
37	C₂²×C₄	—	5	1	no	—
38	C₄⋊C₄	—	5	1	no	—
39	C₂²⋊C₄	—	5	1	no	—
40	C₂²⋊C₄	—	5	1	no	—
41	C₂×C₅	—	2	2	no	—
42	C₂×C₅	—	4	1	no	—
43	C₂×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅	#41	1	1	yes	characteristic
46	D₅	—	4	1	no	—
47	C₂³	—	5	1	no	—
48	C₂³	C₂×D₅	1	1	yes	characteristic
49	C₂³	C₂×D₅	1	1	yes	characteristic
50	D₄	—	2	2	no	—
51	D₄	—	10	2	no	—
52	D₄	—	10	2	no	—
53	C₂×C₄	—	5	1	no	—
54	C₂×C₄	—	5	1	no	—
55	C₂×C₄	—	5	1	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	C₂×D₅	1	1	yes	characteristic
58	C₅	#121	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	10	1	no	—
65	C₂²	—	10	1	no	—
66	C₂²	—	10	1	no	—
67	C₂²	C₂²⋊D₅	1	2	yes	complemented
68	C₂²	C₂²×D₅	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	5	2	no	—
71	C₄	—	10	1	no	—
72	C₄	—	10	1	no	—
73	C₂	—	2	2	no	—
74	C₂	—	4	1	no	—
75	C₂	—	20	1	no	—
76	C₂	D₄×D₅	1	1	yes	characteristic
77	C₂	#711	1	1	yes	characteristic
78	C₂	C₂×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅
commutator, socle, cosocle	C₂²×C₅	C₂³
Fitting	C₂×C₅×D₄	C₂
hypercenter	C₂×D₄	D₅
hypocenter	C₅	#121
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₄	C₂	semidirect
2. C₂×C₂²⋊D₅	C₂	semidirect
3. C₂×C₂²⋊D₅	C₂	semidirect
4. C₂²×Dic₅	C₂	semidirect
5. C₂²⋊Dic₅	C₂	semidirect
6. #685	C₂	semidirect
7. #683	C₂	semidirect
25. C₂²×C₅	D₄	semidirect
33. Dic₅	D₄	semidirect
36. C₂×D₄	D₅	semidirect
58. C₅	#121	semidirect
67. C₂²	C₂²⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₄	2
C₂×C₂²⋊D₅	2
C₂²×Dic₅	2
C₂²⋊Dic₅	2
#685	2
#683	2
#121	5