Subgroups — #65827

Order 320.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
2	C₂×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4076	C₂	1	4	yes	maximal, complemented
4	#4032	C₂	1	4	yes	maximal, complemented
5	#4028	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₅	C₂²	1	1	yes	characteristic
7	C₂³×D₅	C₂²	1	2	yes	complemented
8	C₂²×C₄×C₅	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₅	—	2	8	no	—
10	C₂×C₂²⋊D₅	C₂²	1	4	yes	—
11	C₂×C₂²⋊D₅	C₂²	1	4	yes	complemented
12	C₂²×Dic₅	C₂²	1	2	yes	complemented
13	C₂×D₂₀	C₂²	1	4	yes	complemented
14	C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
15	#685	C₂²	1	8	yes	complemented
16	C₄⋊Dic₅	C₂²	1	4	yes	complemented
17	#534	—	5	1	no	maximal, Sylow
18	C₂³×C₅	—	4	2	no	—
19	C₂³×C₅	D₄	1	4	yes	complemented
20	C₂³×C₅	C₂³	1	2	yes	—
21	C₂³×C₅	C₂³	1	1	yes	characteristic
22	C₂²×D₅	—	2	4	no	—
23	C₂²×D₅	—	4	4	no	—
24	C₂²×D₅	C₂³	1	4	yes	—
25	C₂×C₄×C₅	—	4	2	no	—
26	C₂×C₄×C₅	C₂³	1	4	yes	—
27	C₂²⋊D₅	—	2	8	no	—
28	C₂²⋊D₅	—	2	8	no	—
29	C₂×Dic₅	—	2	4	no	—
30	C₂×Dic₅	C₂³	1	4	yes	—
31	D₂₀	—	4	4	no	—
32	C₂²×D₄	—	5	2	no	—
33	#123	—	5	4	no	—
34	#121	—	5	4	no	—
35	#120	—	5	4	no	—
36	C₂×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
37	C₂²×C₅	—	2	2	no	—
38	C₂²×C₅	—	2	4	no	—
39	C₂²×C₅	—	4	4	no	—
40	C₂²×C₅	C₂×D₄	1	4	yes	—
41	C₂²×C₅	C₂×D₄	1	2	yes	—
42	C₂²×C₅	C₂⁴	1	1	yes	characteristic
43	C₂×D₅	—	2	4	no	—
44	C₂×D₅	—	2	8	no	—
45	C₂×D₅	—	4	8	no	—
46	C₄×C₅	—	2	4	no	—
47	Dic₅	—	2	4	no	—
48	C₂⁴	—	5	2	no	—
49	C₂⁴	C₂×D₅	1	1	yes	characteristic
50	C₂×D₄	—	5	4	no	—
51	C₂×D₄	—	5	4	no	—
52	C₂×D₄	—	5	4	no	—
53	C₂×D₄	—	10	8	no	—
54	C₂²×C₄	—	5	2	no	—
55	C₂²×C₄	C₂×D₅	1	2	yes	complemented
56	C₄⋊C₄	—	5	4	no	—
57	C₂²⋊C₄	—	5	8	no	—
58	C₂²⋊C₄	C₂×D₅	1	4	yes	complemented
59	C₂×C₅	—	2	4	no	—
60	C₂×C₅	—	2	2	no	—
61	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₅	H(5,𝔽₂)	1	2	yes	—
63	D₅	—	4	4	no	—
64	C₂³	—	4	2	no	—
65	C₂³	—	5	4	no	—
66	C₂³	—	10	4	no	—
67	C₂³	—	20	4	no	—
68	C₂³	D₂₀	1	4	yes	complemented
69	C₂³	C₂²×D₅	1	2	yes	—
70	C₂³	C₂²×D₅	1	1	yes	characteristic
71	D₄	—	10	8	no	—
72	D₄	—	10	8	no	—
73	D₄	—	20	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	5	4	no	—
76	C₂×C₄	—	10	4	no	—
77	C₂×C₄	C₂²×D₅	1	4	yes	—
78	C₅	#534	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	4	no	—
82	C₂²	—	10	4	no	—
83	C₂²	—	10	8	no	—
84	C₂²	—	20	8	no	—
85	C₂²	C₂×D₂₀	1	4	yes	—
86	C₂²	C₂×D₂₀	1	2	yes	—
87	C₂²	C₂³×D₅	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	10	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	20	4	no	—
93	C₂	C₂²×D₂₀	1	1	yes	characteristic
94	C₂	#4144	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₂×C₅×C₂²⋊C₄	C₂
hypercenter	C₂×C₂²⋊C₄	D₅
hypocenter	C₅	#534
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₅	C₂	semidirect
2. C₂×C₅×C₂²⋊C₄	C₂	semidirect
3. #4076	C₂	semidirect
4. #4032	C₂	semidirect
5. #4028	C₂	semidirect
7. C₂³×D₅	C₂²	semidirect
8. C₂²×C₄×C₅	C₂²	semidirect
11. C₂×C₂²⋊D₅	C₂²	semidirect
12. C₂²×Dic₅	C₂²	semidirect
13. C₂×D₂₀	C₂²	semidirect
14. C₅×C₂²⋊C₄	C₂²	semidirect
15. #685	C₂²	semidirect
16. C₄⋊Dic₅	C₂²	semidirect
19. C₂³×C₅	D₄	semidirect
36. C₂×C₂²⋊C₄	D₅	semidirect
55. C₂²×C₄	C₂×D₅	semidirect
58. C₂²⋊C₄	C₂×D₅	semidirect
68. C₂³	D₂₀	semidirect
78. C₅	#534	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊D₅	2
C₂×C₅×C₂²⋊C₄	2
#4076	2
#4032	2
#4028	2
#534	5