Subgroups — #65852

Order 320.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#117	C₂	1	1	yes	maximal, characteristic
2	#4070	C₂	1	2	yes	maximal
3	#4035	C₂	1	4	yes	maximal, complemented
4	#4025	C₂	1	4	yes	maximal
5	#4016	C₂	1	1	yes	maximal, characteristic, complemented
6	#4015	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₁₀	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀	C₂²	1	2	yes	—
10	C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
12	C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂²⋊Dic₅	C₂²	1	4	yes	—
15	C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₄⋊Dic₅	C₂²	1	4	yes	—
17	#683	C₂²	1	4	yes	—
18	#683	C₂²	1	4	yes	—
19	#683	C₂²	1	4	yes	—
20	C₄×Dic₅	C₂²	1	4	yes	—
21	#541	—	5	1	no	maximal, Sylow
22	C₂³×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₅	—	2	2	no	—
24	C₂×C₄×C₅	C₂³	1	2	yes	—
25	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₅	C₂³	1	2	yes	—
27	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×Dic₅	C₂³	1	4	yes	—
29	C₂×Dic₅	C₂³	1	4	yes	—
30	Dic₁₀	—	2	4	no	—
31	C₄⋊Q₈	—	5	1	no	—
32	#126	—	5	4	no	—
33	#125	—	5	4	no	—
34	#125	—	5	1	no	—
35	C₂²⋊Q₈	—	5	2	no	—
36	C₄×Q₈	—	5	2	no	—
37	#117	D₅	1	1	yes	characteristic
38	C₂²×C₅	—	2	1	no	—
39	C₂²×C₅	—	2	2	no	—
40	C₂²×C₅	C₂⁴	1	1	yes	characteristic
41	C₄×C₅	—	2	1	no	—
42	C₄×C₅	—	2	2	no	—
43	C₄×C₅	—	2	2	no	—
44	C₄×C₅	#41	1	2	yes	—
45	Dic₅	—	2	4	no	—
46	Dic₅	—	2	4	no	—
47	C₂×Q₈	—	5	2	no	—
48	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
49	C₄⋊C₄	—	5	2	no	—
50	C₄⋊C₄	—	5	4	no	—
51	C₄⋊C₄	—	5	4	no	—
52	C₄⋊C₄	—	5	4	no	—
53	C₄⋊C₄	—	5	4	no	—
54	C₄⋊C₄	C₂×D₅	1	2	yes	—
55	C₂²⋊C₄	—	5	4	no	—
56	C₂²⋊C₄	C₂×D₅	1	2	yes	—
57	C₄²	—	5	4	no	—
58	C₄²	C₂×D₅	1	1	yes	characteristic
59	C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₅	—	4	1	no	—
61	C₂×C₅	C₂×#41	1	1	yes	characteristic
62	C₂×C₅	2₋¹⁺⁴	1	2	yes	—
63	C₂³	C₂²×D₅	1	1	yes	characteristic
64	Q₈	—	10	4	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	5	4	no	—
67	C₂×C₄	—	5	4	no	—
68	C₂×C₄	C₂²×D₅	1	2	yes	—
69	C₂×C₄	C₂²×D₅	1	2	yes	—
70	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
72	C₅	#541	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	1	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂³×D₅	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	10	4	no	—
80	C₄	—	10	4	no	—
81	C₄	#709	1	2	yes	—
82	C₂	—	4	1	no	—
83	C₂	C₂×#709	1	1	yes	characteristic
84	C₂	#4150	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#117	C₂
prime radical	C₂³×C₅	C₂³
hypercenter	#117	D₅
hypocenter	C₅	#541
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
3. #4035	C₂	semidirect
5. #4016	C₂	semidirect
6. #4015	C₂	semidirect
7. C₄×Dic₁₀	C₂	semidirect
72. C₅	#541	semidirect

Maximal subgroups

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

subgroup	index
C₅×#117	2
#4070	2
#4035	2
#4025	2
#4016	2
#4015	2
C₄×Dic₁₀	2
#541	5