Subgroups — #66051

Order 320.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₅	C₂	1	4	yes	maximal, complemented
2	C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#4084	C₂	1	4	yes	maximal, complemented
4	#4083	C₂	1	4	yes	maximal, complemented
5	C₄×D₂₀	C₂	1	2	yes	maximal, complemented
6	C₂³×D₅	C₂²	1	4	yes	complemented
7	C₂×C₅×D₄	C₂²	1	4	yes	complemented
8	C₂×C₅×D₄	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₅	C₂²	1	8	yes	complemented
10	D₄×D₅	—	2	16	no	—
11	C₂×D₂₀	C₂²	1	2	yes	—
12	C₂×C₄×D₅	C₂²	1	4	yes	complemented
13	C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
14	C₂²⋊Dic₅	C₂²	1	4	yes	complemented
15	#685	C₂²	1	4	yes	complemented
16	C₄⋊Dic₅	C₂²	1	2	yes	complemented
17	D₄²	—	5	1	no	maximal, Sylow
18	C₂³×C₅	C₂³	1	4	yes	—
19	C₂²×D₅	—	2	16	no	—
20	C₂²×D₅	—	4	4	no	—
21	C₂²×D₅	C₂³	1	4	yes	—
22	C₅×D₄	—	2	8	no	—
23	C₅×D₄	—	4	2	no	—
24	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₅	C₂³	1	2	yes	—
26	C₂²⋊D₅	—	2	16	no	—
27	C₂×Dic₅	C₂³	1	4	yes	—
28	D₂₀	D₄	1	8	yes	complemented
29	C₄×D₅	—	2	8	no	—
30	C₂²×D₄	—	5	4	no	—
31	#127	D₅	1	1	yes	characteristic, complemented
32	#121	—	5	4	no	—
33	#120	—	5	4	no	—
34	C₄×D₄	—	5	2	no	—
35	C₂²×C₅	—	2	8	no	—
36	C₂²×C₅	—	2	4	no	—
37	C₂²×C₅	C₂⁴	1	1	yes	characteristic
38	C₂×D₅	—	2	4	no	—
39	C₂×D₅	—	2	4	no	—
40	C₂×D₅	—	4	16	no	—
41	C₂×D₅	C₂×D₄	1	8	yes	complemented
42	C₄×C₅	—	2	1	no	—
43	C₄×C₅	C₂×D₄	1	4	yes	complemented
44	Dic₅	—	2	4	no	—
45	C₂⁴	—	5	4	no	—
46	C₂×D₄	—	5	8	no	—
47	C₂×D₄	—	5	2	no	—
48	C₂×D₄	—	10	16	no	—
49	C₂×D₄	C₂×D₅	1	4	yes	complemented
50	C₂×D₄	C₂×D₅	1	2	yes	complemented
51	C₂²×C₄	—	5	4	no	—
52	C₄⋊C₄	—	5	2	no	—
53	C₂²⋊C₄	—	5	4	no	—
54	C₂²⋊C₄	—	5	4	no	—
55	C₄²	C₂×D₅	1	1	yes	characteristic, complemented
56	C₂×C₅	—	4	4	no	—
57	C₂×C₅	C₂²×D₄	1	2	yes	—
58	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
59	D₅	—	2	8	no	—
60	C₂³	—	5	4	no	—
61	C₂³	—	10	16	no	—
62	C₂³	—	20	4	no	—
63	C₂³	C₂²×D₅	1	4	yes	—
64	D₄	—	2	8	no	—
65	D₄	—	4	2	no	—
66	D₄	—	5	8	no	—
67	D₄	—	10	16	no	—
68	C₂×C₄	—	5	4	no	—
69	C₂×C₄	—	10	8	no	—
70	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₅	1	2	yes	—
72	C₅	D₄²	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	5	8	no	—
76	C₂²	—	10	4	no	—
77	C₂²	—	10	4	no	—
78	C₂²	—	20	16	no	—
79	C₂²	C₂³×D₅	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	10	4	no	—
82	C₄	D₄×D₅	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	10	8	no	—
85	C₂	C₂×D₄×D₅	1	2	yes	—
86	C₂	#4144	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#127	C₂
hypercenter	#127	D₅
hypocenter	C₅	D₄²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₅	C₂	semidirect
2. C₅×#127	C₂	semidirect
3. #4084	C₂	semidirect
4. #4083	C₂	semidirect
5. C₄×D₂₀	C₂	semidirect
6. C₂³×D₅	C₂²	semidirect
7. C₂×C₅×D₄	C₂²	semidirect
8. C₂×C₅×D₄	C₂²	semidirect
9. C₂×C₂²⋊D₅	C₂²	semidirect
12. C₂×C₄×D₅	C₂²	semidirect
13. C₄²×C₅	C₂²	semidirect
14. C₂²⋊Dic₅	C₂²	semidirect
15. #685	C₂²	semidirect
16. C₄⋊Dic₅	C₂²	semidirect
28. D₂₀	D₄	semidirect
31. #127	D₅	semidirect
41. C₂×D₅	C₂×D₄	semidirect
43. C₄×C₅	C₂×D₄	semidirect
49. C₂×D₄	C₂×D₅	semidirect
50. C₂×D₄	C₂×D₅	semidirect
55. C₄²	C₂×D₅	semidirect
72. C₅	D₄²	semidirect
82. C₄	D₄×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₅	2
C₅×#127	2
#4084	2
#4083	2
C₄×D₂₀	2
D₄²	5