Group #66023 — subgroups

Subgroups — #66023

Order 320.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4043	C₂	1	2	yes	maximal
3	#4042	C₂	1	2	yes	maximal
4	#4038	C₂	1	1	yes	maximal, characteristic
5	D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#4036	C₂	1	1	yes	maximal, characteristic, complemented
7	#4035	C₂	1	2	yes	maximal, complemented
8	#4034	C₂	1	1	yes	maximal, characteristic, complemented
9	#4034	C₂	1	2	yes	maximal, complemented
10	#4018	C₂	1	1	yes	maximal, characteristic
11	#4015	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₅	C₂²	1	2	yes	—
13	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀	C₂²	1	2	yes	—
15	C₂×Dic₁₀	C₂²	1	2	yes	—
16	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₄²×C₅	C₂²	1	1	yes	characteristic
21	#685	C₂²	1	1	yes	characteristic
22	#685	C₂²	1	1	yes	characteristic
23	#685	C₂²	1	2	yes	—
24	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₅	C₂²	1	2	yes	—
28	#683	C₂²	1	1	yes	characteristic
29	#683	C₂²	1	1	yes	characteristic
30	#683	C₂²	1	2	yes	—
31	#683	C₂²	1	2	yes	—
32	#683	C₂²	1	2	yes	—
33	C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₄×Dic₅	C₂²	1	2	yes	—
35	#544	—	5	1	no	maximal, Sylow
36	C₂²×D₅	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂³	1	2	yes	—
38	C₂×C₄×C₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×Dic₅	C₂³	1	1	yes	characteristic
43	C₂×Dic₅	C₂³	1	1	yes	characteristic
44	C₂×Dic₅	C₂³	1	2	yes	—
45	C₂×Dic₅	C₂³	1	2	yes	—
46	C₂×Dic₅	C₂³	1	1	yes	characteristic
47	C₄×D₅	—	4	2	no	—
48	C₄×D₅	Q₈	1	4	yes	complemented
49	Dic₁₀	—	4	2	no	—
50	Dic₁₀	—	4	2	no	—
51	C₄⋊Q₈	—	5	2	no	—
52	C₄⋊Q₈	—	5	1	no	—
53	C₄⋊Q₈	—	5	1	no	—
54	#125	—	5	2	no	—
55	#125	—	5	1	no	—
56	#125	D₅	1	1	yes	characteristic, complemented
57	C₂²⋊Q₈	—	5	2	no	—
58	C₂²⋊Q₈	—	5	2	no	—
59	#117	—	5	1	no	—
60	#117	—	5	1	no	—
61	C₂×C₄⋊C₄	—	5	1	no	—
62	C₂²×C₅	C₂⁴	1	1	yes	characteristic
63	C₂×D₅	—	2	1	no	—
64	C₂×D₅	—	2	1	no	—
65	C₂×D₅	C₂×Q₈	1	2	yes	—
66	C₄×C₅	—	2	1	no	—
67	C₄×C₅	—	2	2	no	—
68	C₄×C₅	—	2	2	no	—
69	C₄×C₅	—	2	1	no	—
70	C₄×C₅	C₂×Q₈	1	2	yes	—
71	Dic₅	—	2	1	no	—
72	Dic₅	—	2	2	no	—
73	Dic₅	—	2	2	no	—
74	Dic₅	—	2	1	no	—
75	Dic₅	C₂×Q₈	1	2	yes	—
76	C₂×Q₈	—	5	2	no	—
77	C₂×Q₈	—	5	2	no	—
78	C₂²×C₄	—	5	1	no	—
79	C₂²×C₄	—	5	2	no	—
80	C₄⋊C₄	—	5	2	no	—
81	C₄⋊C₄	—	5	1	no	—
82	C₄⋊C₄	—	5	1	no	—
83	C₄⋊C₄	—	5	1	no	—
84	C₄⋊C₄	—	5	1	no	—
85	C₄⋊C₄	—	5	2	no	—
86	C₄⋊C₄	—	5	2	no	—
87	C₄⋊C₄	—	5	2	no	—
88	C₄⋊C₄	—	5	2	no	—
89	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
90	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
91	C₄⋊C₄	C₂×D₅	1	2	yes	—
92	C₄⋊C₄	C₂×D₅	1	2	yes	—
93	C₂²⋊C₄	—	5	1	no	—
94	C₂²⋊C₄	—	5	2	no	—
95	C₂²⋊C₄	—	5	1	no	—
96	C₄²	—	5	2	no	—
97	C₄²	—	5	1	no	—
98	C₄²	C₂×D₅	1	1	yes	characteristic
99	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
102	D₅	—	2	2	no	—
103	C₂³	—	5	1	no	—
104	Q₈	—	20	2	no	—
105	Q₈	—	20	2	no	—
106	C₂×C₄	—	5	1	no	—
107	C₂×C₄	—	5	4	no	—
108	C₂×C₄	—	5	1	no	—
109	C₂×C₄	—	5	1	no	—
110	C₂×C₄	—	5	2	no	—
111	C₂×C₄	—	5	2	no	—
112	C₂×C₄	—	20	2	no	—
113	C₂×C₄	C₂²×D₅	1	2	yes	—
114	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₅	1	2	yes	—
117	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
118	C₅	#544	1	1	yes	characteristic, complemented, Sylow
119	C₂²	—	5	2	no	—
120	C₂²	—	10	1	no	—
121	C₂²	—	10	1	no	—
122	C₂²	C₂³×D₅	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	5	2	no	—
128	C₄	—	10	2	no	—
129	C₄	—	10	1	no	—
130	C₄	—	10	2	no	—
131	C₄	—	10	1	no	—
132	C₄	Q₈×D₅	1	2	yes	—
133	C₂	—	10	2	no	—
134	C₂	C₂×Q₈×D₅	1	1	yes	characteristic
135	C₂	#4149	1	1	yes	characteristic
136	C₂	#4150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#125

C₂

yes

maximal, characteristic, complemented

#4043

C₂

yes

maximal

#4042

C₂

yes

maximal

#4038

C₂

yes

maximal, characteristic

D₅×C₄⋊C₄

C₂

yes

maximal, characteristic

#4036

C₂

yes

maximal, characteristic, complemented

#4035

C₂

yes

maximal, complemented

#4034

C₂

yes

maximal, characteristic, complemented

#4034

C₂

yes

maximal, complemented

#4018

C₂

yes

maximal, characteristic

#4015

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₅

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

—

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₅×C₄⋊C₄

C₂²

yes

—

C₅×C₄⋊C₄

C₂²

yes

—

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₄²×C₅

C₂²

yes

characteristic

#685

C₂²

yes

characteristic

#685

C₂²

yes

characteristic

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

C₄⋊Dic₅

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

C₄⋊Dic₅

C₂²

yes

—

#683

C₂²

yes

characteristic

#683

C₂²

yes

characteristic

#683

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

—

#544

—

maximal, Sylow

C₂²×D₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₄×D₅

—

C₄×D₅

Q₈

yes

complemented

Dic₁₀

—

Dic₁₀

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

#125

D₅

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

C₂×Q₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×Q₈

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

C₂×D₅

yes

—

C₄⋊C₄

C₂×D₅

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₅

yes

characteristic

C₂×C₅

C₂²×Q₈

yes

characteristic

100

C₂×C₅

H(5,𝔽₂)

yes

characteristic

101

C₂×C₅

2₋¹⁺⁴

yes

characteristic

102

D₅

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₅

yes

—

114

C₂×C₄

C₂²×D₅

yes

characteristic

115

C₂×C₄

C₂²×D₅

yes

characteristic

116

C₂×C₄

C₂²×D₅

yes

—

117

C₂×C₄

C₂²×D₅

yes

characteristic

118

C₅

#544

yes

characteristic, complemented, Sylow

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₅

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

Q₈×D₅

yes

—

133

C₂

—

134

C₂

C₂×Q₈×D₅

yes

characteristic

135

C₂

#4149

yes

characteristic

136

C₂

#4150

yes

characteristic

	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	#544
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

Fitting

C₅×#125

C₂

prime radical

C₂²×D₅

C₂³

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
6. #4036	C₂	semidirect
7. #4035	C₂	semidirect
8. #4034	C₂	semidirect
9. #4034	C₂	semidirect
11. #4015	C₂	semidirect
48. C₄×D₅	Q₈	semidirect
56. #125	D₅	semidirect
118. C₅	#544	semidirect

H ≅ G/N

type

1. C₅×#125

C₂

semidirect

6. #4036

C₂

semidirect

7. #4035

C₂

semidirect

8. #4034

C₂

semidirect

9. #4034

C₂

semidirect

11. #4015

C₂

semidirect

48. C₄×D₅

Q₈

semidirect

56. #125

D₅

semidirect

118. C₅

#544

semidirect

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

subgroup	index
C₅×#125	2
#4043	2
#4042	2
#4038	2
D₅×C₄⋊C₄	2
#4036	2
#4035	2
#4034	2
#4018	2
#4015	2
#544	5

subgroup

index

D₅×C₄⋊C₄

Subgroups — #66023

Conjugacy classes of subgroups

Complements

Maximal subgroups