Subgroups — #66006

Order 320.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#124	C₂	1	1	yes	maximal, characteristic
2	#4090	C₂	1	2	yes	maximal, complemented
3	#4081	C₂	1	2	yes	maximal
4	#4025	C₂	1	4	yes	maximal, complemented
5	#4024	C₂	1	4	yes	maximal, complemented
6	#4016	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅	C₂²	1	2	yes	—
10	C₂×Dic₁₀	C₂²	1	2	yes	—
11	C₅×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅	C₂²	1	4	yes	—
14	C₂²⋊Dic₅	C₂²	1	2	yes	—
15	C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₄⋊Dic₅	C₂²	1	2	yes	—
17	#683	C₂²	1	4	yes	—
18	#683	C₂²	1	4	yes	—
19	#683	C₂²	1	4	yes	—
20	C₄×Dic₅	C₂²	1	2	yes	—
21	#563	—	5	1	no	maximal, Sylow
22	C₂³×C₅	C₂³	1	2	yes	—
23	C₅×Q₈	—	4	1	no	—
24	C₅×D₄	—	4	1	no	—
25	C₂×C₄×C₅	C₂³	1	2	yes	—
26	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂³	1	2	yes	—
28	C₂×Dic₅	—	4	2	no	—
29	C₂×Dic₅	C₂³	1	4	yes	—
30	C₂×Dic₅	C₂³	1	4	yes	—
31	Dic₁₀	—	4	2	no	—
32	C₄⋊Q₈	—	5	2	no	—
33	#126	—	5	4	no	—
34	#125	—	5	2	no	—
35	#124	D₅	1	1	yes	characteristic
36	#123	—	5	2	no	—
37	C₂²⋊Q₈	—	5	4	no	—
38	C₂²×C₅	—	2	2	no	—
39	C₂²×C₅	—	2	4	no	—
40	C₂²×C₅	C₂⁴	1	1	yes	characteristic
41	C₄×C₅	—	2	2	no	—
42	C₄×C₅	—	2	1	no	—
43	C₄×C₅	—	2	2	no	—
44	Dic₅	—	2	4	no	—
45	Dic₅	—	2	4	no	—
46	C₂×Q₈	—	5	2	no	—
47	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
48	C₂×D₄	C₂×D₅	1	1	yes	characteristic
49	C₂²×C₄	—	5	2	no	—
50	C₄⋊C₄	—	5	2	no	—
51	C₄⋊C₄	—	5	4	no	—
52	C₄⋊C₄	—	5	4	no	—
53	C₄⋊C₄	—	5	4	no	—
54	C₄⋊C₄	—	5	2	no	—
55	C₂²⋊C₄	—	5	4	no	—
56	C₂²⋊C₄	—	5	2	no	—
57	C₂²⋊C₄	C₂×D₅	1	4	yes	—
58	C₄²	—	5	2	no	—
59	C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×C₅	—	4	2	no	—
61	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂×C₅	2₋¹⁺⁴	1	2	yes	—
63	C₂³	C₂²×D₅	1	2	yes	—
64	Q₈	—	4	1	no	—
65	Q₈	—	20	2	no	—
66	D₄	—	4	1	no	—
67	C₂×C₄	—	5	4	no	—
68	C₂×C₄	—	5	4	no	—
69	C₂×C₄	—	20	2	no	—
70	C₂×C₄	C₂²×D₅	1	2	yes	—
71	C₂×C₄	C₂²×D₅	1	2	yes	—
72	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
73	C₅	#563	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₅	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	10	4	no	—
81	C₄	—	10	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#4144	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×#124	C₂
prime radical	C₂×C₅×D₄	C₂²
hypercenter	#124	D₅
hypocenter	C₅	#563
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #4090	C₂	semidirect
4. #4025	C₂	semidirect
5. #4024	C₂	semidirect
6. #4016	C₂	semidirect
73. C₅	#563	semidirect

Maximal subgroups

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

subgroup	index
C₅×#124	2
#4090	2
#4081	2
#4025	2
#4024	2
#4016	2
#563	5