Subgroups — #65036

Order 320.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4144	C₂	1	1	yes	maximal, characteristic, complemented
2	#4086	C₂	1	1	yes	maximal, characteristic, complemented
3	#4054	C₂	1	2	yes	maximal, complemented
4	C₅×#100	C₂	1	1	yes	maximal, characteristic, complemented
5	#3957	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₅	—	2	2	no	—
8	C₂×C₂²⋊D₅	—	2	2	no	—
9	#711	—	2	2	no	—
10	D₄×D₅	—	2	2	no	—
11	#709	C₂²	1	2	yes	—
12	C₂×D₂₀	C₂²	1	1	yes	characteristic, complemented
13	C₅×M₁₆	C₂²	1	2	yes	complemented
14	C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
15	D₄₀	—	2	2	no	—
16	#677	—	2	2	no	—
17	#453	—	5	1	no	maximal, Sylow
18	C₂³×C₅	D₄	1	2	yes	—
19	C₂²×D₅	—	2	2	no	—
20	C₂²×D₅	—	2	1	no	—
21	C₅×D₄	—	2	2	no	—
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²⋊D₅	—	2	4	no	—
24	C₂²⋊D₅	—	2	2	no	—
25	C₂²⋊D₅	—	4	4	no	—
26	C₂×Dic₅	—	2	2	no	—
27	C₂×Dic₅	—	2	1	no	—
28	D₂₀	—	2	2	no	—
29	D₂₀	D₄	1	2	yes	complemented
30	C₄×D₅	—	2	2	no	—
31	Dic₁₀	D₄	1	2	yes	complemented
32	C₅×C₈	—	2	2	no	—
33	H(5,𝔽₂)	—	5	1	no	—
34	AGL(1,ℤ/8)	—	5	2	no	—
35	#127	—	5	1	no	—
36	#104	—	5	2	no	—
37	#100	D₅	1	1	yes	characteristic, complemented
38	C₂²×C₅	—	2	2	no	—
39	C₂²×C₅	—	4	2	no	—
40	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂×D₅	—	2	2	no	—
42	C₂×D₅	—	4	2	no	—
43	C₂×D₅	—	4	1	no	—
44	C₂×D₅	—	8	2	no	—
45	C₄×C₅	C₂×D₄	1	2	yes	—
46	Dic₅	—	2	2	no	—
47	Dic₅	—	2	2	no	—
48	#41	—	5	2	no	—
49	#41	—	10	2	no	—
50	C₂×D₄	—	5	1	no	—
51	C₂×D₄	—	10	2	no	—
52	C₂×D₄	—	10	2	no	—
53	C₂×D₄	—	10	2	no	—
54	C₂×D₄	C₂×D₅	1	1	yes	characteristic
55	SD₁₆	—	10	2	no	—
56	D₈	—	10	2	no	—
57	M₁₆	C₂×D₅	1	2	yes	complemented
58	C₄²	—	5	1	no	—
59	C₂×C₅	—	2	1	no	—
60	C₂×C₅	—	4	2	no	—
61	C₂×C₅	#120	1	1	yes	characteristic
62	D₅	—	4	2	no	—
63	D₅	—	8	1	no	—
64	C₂³	—	10	2	no	—
65	C₂³	—	10	1	no	—
66	C₂³	D₂₀	1	2	yes	—
67	Q₈	—	5	2	no	—
68	D₄	—	2	2	no	—
69	D₄	—	5	2	no	—
70	D₄	—	10	4	no	—
71	D₄	—	10	2	no	—
72	D₄	—	10	2	no	—
73	D₄	—	20	4	no	—
74	C₂×C₄	—	10	2	no	—
75	C₂×C₄	—	10	1	no	—
76	C₂×C₄	—	10	2	no	—
77	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₅	#453	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	10	2	no	—
83	C₂²	—	20	2	no	—
84	C₂²	—	20	1	no	—
85	C₂²	—	40	2	no	—
86	C₂²	C₂×D₂₀	1	1	yes	characteristic
87	C₄	—	10	2	no	—
88	C₄	—	10	2	no	—
89	C₄	D₄×D₅	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	20	2	no	—
93	C₂	—	40	1	no	—
94	C₂	#4028	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4028
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#100	C₂
socle	C₂×C₅	#120
hypercenter	#100	D₅
hypocenter	C₅	#453
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
1. #4144	C₂	semidirect
2. #4086	C₂	semidirect
3. #4054	C₂	semidirect
4. C₅×#100	C₂	semidirect
5. #3957	C₂	semidirect
12. C₂×D₂₀	C₂²	semidirect
13. C₅×M₁₆	C₂²	semidirect
14. C₄×Dic₅	C₂²	semidirect
29. D₂₀	D₄	semidirect
31. Dic₁₀	D₄	semidirect
37. #100	D₅	semidirect
57. M₁₆	C₂×D₅	semidirect
79. C₅	#453	semidirect

Maximal subgroups

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

subgroup	index
#4144	2
#4086	2
#4054	2
C₅×#100	2
#3957	2
#453	5