Subgroups — #65855

Order 320.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	#4072	C₂	1	1	yes	maximal, characteristic, complemented
4	#4039	C₂	1	4	yes	maximal, complemented
5	D₅×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₄×D₂₀	C₂	1	4	yes	maximal, complemented
7	C₂³×D₅	C₂²	1	2	yes	—
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×D₂₀	C₄	1	8	yes	complemented
10	C₂×D₂₀	C₂²	1	2	yes	—
11	C₂×D₂₀	C₂²	1	2	yes	—
12	C₂×C₄×D₅	C₂²	1	8	yes	complemented
13	C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₄²×C₅	C₂²	1	2	yes	complemented
16	C₂²⋊Dic₅	C₂²	1	2	yes	—
17	#685	C₂²	1	8	yes	complemented
18	C₄⋊Dic₅	C₂²	1	2	yes	complemented
19	C₄×Dic₅	C₂²	1	2	yes	complemented
20	#518	—	5	1	no	maximal, Sylow
21	C₂³×C₅	C₂³	1	1	yes	characteristic
22	C₂²×D₅	—	4	4	no	—
23	C₂²×D₅	C₂×C₄	1	8	yes	complemented
24	C₂²×D₅	C₂³	1	4	yes	—
25	C₂×C₄×C₅	C₂×C₄	1	4	yes	complemented
26	C₂×C₄×C₅	C₂³	1	4	yes	—
27	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×Dic₅	C₂³	1	4	yes	—
30	D₂₀	C₂×C₄	1	8	yes	complemented
31	D₂₀	C₂×C₄	1	8	yes	complemented
32	C₄×D₅	—	4	8	no	—
33	C₂²×D₄	—	5	1	no	—
34	C₄×D₄	—	5	4	no	—
35	C₄×D₄	—	5	4	no	—
36	#117	—	5	1	no	—
37	#117	D₅	1	1	yes	characteristic, complemented
38	C₂×C₂²⋊C₄	—	5	4	no	—
39	C₂²×C₅	—	2	2	no	—
40	C₂²×C₅	C₂²×C₄	1	2	yes	—
41	C₂²×C₅	C₂⁴	1	1	yes	characteristic
42	C₂×D₅	—	2	8	no	—
43	C₂×D₅	—	4	8	no	—
44	C₂×D₅	C₂²×C₄	1	8	yes	—
45	C₄×C₅	—	2	4	no	—
46	C₄×C₅	C₂²×C₄	1	2	yes	—
47	C₄×C₅	C₂²×C₄	1	2	yes	—
48	Dic₅	—	2	4	no	—
49	C₂⁴	—	5	2	no	—
50	C₂×D₄	—	5	8	no	—
51	C₂×D₄	—	5	2	no	—
52	C₂×D₄	—	5	2	no	—
53	C₂²×C₄	—	5	8	no	—
54	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
55	C₄⋊C₄	—	5	2	no	—
56	C₄⋊C₄	C₂×D₅	1	2	yes	complemented
57	C₂²⋊C₄	—	5	2	no	—
58	C₂²⋊C₄	—	5	8	no	—
59	C₂²⋊C₄	C₂×D₅	1	2	yes	—
60	C₄²	—	5	2	no	—
61	C₄²	C₂×D₅	1	2	yes	complemented
62	C₂×C₅	—	2	2	no	—
63	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₅	H(5,𝔽₂)	1	2	yes	—
65	D₅	—	2	8	no	—
66	C₂³	—	5	4	no	—
67	C₂³	—	5	8	no	—
68	C₂³	—	20	4	no	—
69	C₂³	C₂²×D₅	1	1	yes	characteristic
70	D₄	—	5	8	no	—
71	D₄	—	5	8	no	—
72	C₂×C₄	—	5	4	no	—
73	C₂×C₄	—	20	8	no	—
74	C₂×C₄	C₄×D₅	1	4	yes	complemented
75	C₂×C₄	C₂²×D₅	1	4	yes	—
76	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
78	C₅	#518	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	—	5	8	no	—
81	C₂²	—	10	8	no	—
82	C₂²	—	20	8	no	—
83	C₂²	C₂×C₄×D₅	1	2	yes	—
84	C₂²	C₂³×D₅	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	10	4	no	—
87	C₄	C₂×C₄×D₅	1	2	yes	—
88	C₄	C₂×C₄×D₅	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	10	8	no	—
91	C₂	C₂²×C₄×D₅	1	1	yes	characteristic
92	C₂	#4149	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator	C₂×C₅	C₂³×C₄
Fitting	C₅×#117	C₂
socle, cosocle	C₂²×C₅	C₂⁴
prime radical	C₂²×D₂₀	C₂
hypercenter	#117	D₅
hypocenter	C₅	#518
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₅×#117	C₂	semidirect
3. #4072	C₂	semidirect
4. #4039	C₂	semidirect
5. D₅×C₂²⋊C₄	C₂	semidirect
6. C₄×D₂₀	C₂	semidirect
9. C₂×D₂₀	C₄	semidirect
12. C₂×C₄×D₅	C₂²	semidirect
13. C₅×C₄⋊C₄	C₂²	semidirect
15. C₄²×C₅	C₂²	semidirect
17. #685	C₂²	semidirect
18. C₄⋊Dic₅	C₂²	semidirect
19. C₄×Dic₅	C₂²	semidirect
23. C₂²×D₅	C₂×C₄	semidirect
25. C₂×C₄×C₅	C₂×C₄	semidirect
30. D₂₀	C₂×C₄	semidirect
31. D₂₀	C₂×C₄	semidirect
37. #117	D₅	semidirect
56. C₄⋊C₄	C₂×D₅	semidirect
61. C₄²	C₂×D₅	semidirect
74. C₂×C₄	C₄×D₅	semidirect
78. C₅	#518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₀	2
C₅×#117	2
#4072	2
#4039	2
D₅×C₂²⋊C₄	2
C₄×D₂₀	2
#518	5