Subgroups — #66022

Order 320.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4042	C₂	1	4	yes	maximal
3	#4038	C₂	1	2	yes	maximal
4	#4034	C₂	1	2	yes	maximal, complemented
5	#4033	C₂	1	4	yes	maximal, complemented
6	C₄²×D₅	C₂	1	1	yes	maximal, characteristic
7	#4016	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₅	C₂²	1	2	yes	—
9	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₀	C₂²	1	4	yes	—
11	C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	#685	C₂²	1	4	yes	—
15	C₄⋊Dic₅	C₂²	1	2	yes	—
16	#683	C₂²	1	4	yes	—
17	#683	C₂²	1	4	yes	—
18	C₄×Dic₅	C₂²	1	2	yes	—
19	C₄×Dic₅	C₂²	1	1	yes	characteristic
20	C₄×Dic₅	C₂²	1	4	yes	—
21	#533	—	5	1	no	maximal, Sylow
22	C₂²×D₅	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂³	1	2	yes	—
24	C₂×C₄×C₅	C₂³	1	4	yes	—
25	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₅	C₂³	1	4	yes	—
27	C₂×Dic₅	C₂³	1	1	yes	characteristic
28	C₂×Dic₅	C₂³	1	2	yes	—
29	C₄×D₅	—	2	4	no	—
30	C₄×D₅	Q₈	1	4	yes	complemented
31	Dic₁₀	—	2	8	no	—
32	C₄⋊Q₈	—	5	2	no	—
33	#125	—	5	1	no	—
34	#125	D₅	1	1	yes	characteristic, complemented
35	C₂²⋊Q₈	—	5	4	no	—
36	C₄×Q₈	—	5	4	no	—
37	#117	—	5	2	no	—
38	C₂×C₄²	—	5	1	no	—
39	C₂²×C₅	C₂⁴	1	1	yes	characteristic
40	C₂×D₅	—	2	2	no	—
41	C₂×D₅	C₂×Q₈	1	2	yes	—
42	C₄×C₅	—	2	4	no	—
43	C₄×C₅	—	2	2	no	—
44	C₄×C₅	C₂×Q₈	1	2	yes	—
45	Dic₅	—	2	4	no	—
46	Dic₅	C₂×Q₈	1	2	yes	—
47	Dic₅	#41	1	4	yes	—
48	C₂×Q₈	—	5	4	no	—
49	C₂²×C₄	—	5	1	no	—
50	C₂²×C₄	—	5	2	no	—
51	C₄⋊C₄	—	5	2	no	—
52	C₄⋊C₄	—	5	4	no	—
53	C₄⋊C₄	—	5	4	no	—
54	C₄⋊C₄	C₂×D₅	1	2	yes	—
55	C₄⋊C₄	C₂×D₅	1	4	yes	—
56	C₂²⋊C₄	—	5	4	no	—
57	C₄²	—	5	2	no	—
58	C₄²	—	5	1	no	—
59	C₄²	—	5	4	no	—
60	C₄²	C₂×D₅	1	1	yes	characteristic
61	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₅	C₂×#41	1	2	yes	—
63	D₅	—	2	2	no	—
64	C₂³	—	5	1	no	—
65	Q₈	—	10	8	no	—
66	C₂×C₄	—	5	1	no	—
67	C₂×C₄	—	5	2	no	—
68	C₂×C₄	—	5	4	no	—
69	C₂×C₄	—	5	4	no	—
70	C₂×C₄	—	10	4	no	—
71	C₂×C₄	C₂²×D₅	1	2	yes	—
72	C₂×C₄	C₂²×D₅	1	4	yes	—
73	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
74	C₅	#533	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	5	2	no	—
76	C₂²	—	10	2	no	—
77	C₂²	C₂³×D₅	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	4	no	—
80	C₄	—	5	2	no	—
81	C₄	—	5	4	no	—
82	C₄	—	10	4	no	—
83	C₄	Q₈×D₅	1	2	yes	—
84	C₂	—	10	2	no	—
85	C₂	C₂×Q₈×D₅	1	1	yes	characteristic
86	C₂	D₅×#41	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#125	C₂
prime radical	C₂²×D₅	C₂³
hypercenter	#125	D₅
hypocenter	C₅	#533
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
4. #4034	C₂	semidirect
5. #4033	C₂	semidirect
7. #4016	C₂	semidirect
30. C₄×D₅	Q₈	semidirect
34. #125	D₅	semidirect
74. C₅	#533	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
#4042	2
#4038	2
#4034	2
#4033	2
C₄²×D₅	2
#4016	2
#533	5