Group #10600718 — subgroups

Subgroups — #10600718

Order 640.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#65501	C₂	1	1	yes	maximal, characteristic
4	#64771	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂³×Dic₅	—	2	1	no	—
7	C₂²×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₅	—	2	1	no	—
9	C₂×C₄⋊Dic₅	—	2	1	no	—
10	C₂×C₄⋊Dic₅	—	2	2	no	—
11	C₂×C₄⋊Dic₅	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₅×C₂²⋊C₈	—	2	2	no	—
15	#3963	—	2	1	no	—
16	#1824	—	5	1	no	maximal, Sylow
17	C₂⁴×C₅	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅	—	2	4	no	—
19	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅	D₄	1	2	yes	—
22	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅	—	2	4	no	—
24	C₂²×Dic₅	—	2	2	no	—
25	C₂²×Dic₅	—	2	2	no	—
26	C₂²×Dic₅	—	4	4	no	—
27	C₂×C₅×C₈	—	2	2	no	—
28	C₂×C₅×C₈	—	2	4	no	—
29	C₂²⋊Dic₅	—	4	2	no	—
30	C₄⋊Dic₅	—	2	4	no	—
31	C₄⋊Dic₅	—	2	2	no	—
32	C₄⋊Dic₅	—	2	4	no	—
33	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
34	C₂²×C₄⋊C₄	—	5	1	no	—
35	C₂×C₂²⋊C₈	D₅	1	1	yes	characteristic
36	#380	—	5	1	no	—
37	#340	—	5	4	no	—
38	C₂³×C₅	—	2	2	no	—
39	C₂³×C₅	—	2	2	no	—
40	C₂³×C₅	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₅	SD₁₆	1	2	yes	—
42	C₂³×C₅	Q₁₆	1	2	yes	—
43	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅	—	2	2	no	—
45	C₂×C₄×C₅	—	2	4	no	—
46	C₂×C₄×C₅	—	2	4	no	—
47	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅	#41	1	4	yes	—
51	C₂×Dic₅	—	2	4	no	—
52	C₂×Dic₅	—	4	8	no	—
53	C₂×Dic₅	—	4	4	no	—
54	C₂×Dic₅	—	4	2	no	—
55	C₂×Dic₅	—	4	2	no	—
56	C₂×Dic₅	—	4	2	no	—
57	C₅×C₈	—	4	2	no	—
58	C₂³×C₄	—	10	1	no	—
59	C₂³×C₄	C₂×D₅	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₅	1	2	yes	—
61	C₂×C₄⋊C₄	—	5	4	no	—
62	C₂×C₄⋊C₄	—	5	2	no	—
63	C₂×C₄⋊C₄	—	5	2	no	—
64	C₂×C₄⋊C₄	—	10	1	no	—
65	C₂×C₄⋊C₄	—	10	2	no	—
66	C₂×C₂²⋊C₄	—	10	1	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	#95	—	10	1	no	—
69	C₂²×C₅	—	2	2	no	—
70	C₂²×C₅	—	2	2	no	—
71	C₂²×C₅	—	2	4	no	—
72	C₂²×C₅	—	2	4	no	—
73	C₂²×C₅	#103	1	4	yes	—
74	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₅	#123	1	2	yes	—
76	C₂²×C₅	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²×C₅	C₂×Q₁₆	1	1	yes	characteristic
78	C₂²×C₅	AGL(1,ℤ/8)	1	2	yes	—
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	#117	1	2	yes	—
81	C₄×C₅	#123	1	2	yes	—
82	Dic₅	—	4	4	no	—
83	Dic₅	—	8	2	no	—
84	C₂⁴	D₂₀	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	10	4	no	—
87	C₂²×C₄	—	10	2	no	—
88	C₂²×C₄	—	10	2	no	—
89	C₂²×C₄	—	20	4	no	—
90	C₂²×C₄	C₄×D₅	1	2	yes	—
91	C₂²×C₄	D₂₀	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₅	1	2	yes	—
93	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₄⋊C₄	—	5	4	no	—
97	C₄⋊C₄	—	10	4	no	—
98	C₄⋊C₄	—	10	2	no	—
99	C₄⋊C₄	—	10	4	no	—
100	C₂²⋊C₄	—	20	2	no	—
101	C₂×C₅	—	2	4	no	—
102	C₂×C₅	#381	1	1	yes	characteristic
103	C₂×C₅	C₂×#103	1	1	yes	characteristic
104	C₂×C₅	#418	1	1	yes	characteristic
105	C₂×C₅	#481	1	2	yes	—
106	C₂×C₅	#484	1	2	yes	—
107	C₂³	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	#677	1	2	yes	—
110	C₂³	Dic₂₀	1	2	yes	—
111	C₂³	#685	1	2	yes	—
112	C₂³	C₂×D₂₀	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	10	4	no	—
117	C₂×C₄	—	20	8	no	—
118	C₂×C₄	—	20	4	no	—
119	C₂×C₄	—	20	2	no	—
120	C₂×C₄	—	20	2	no	—
121	C₂×C₄	—	20	2	no	—
122	C₂×C₄	#685	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
124	C₂×C₄	#711	1	4	yes	—
125	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₅	#1824	1	1	yes	characteristic, complemented, Sylow
128	C₂²	—	2	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	4	no	—
132	C₂²	#3948	1	4	yes	—
133	C₂²	#4032	1	2	yes	—
134	C₂²	C₂×#677	1	1	yes	characteristic
135	C₂²	C₂×Dic₂₀	1	1	yes	characteristic
136	C₂²	#4054	1	2	yes	—
137	C₂²	C₂×#685	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	20	4	no	—
140	C₄	—	40	2	no	—
141	C₄	#4023	1	2	yes	—
142	C₄	#4081	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#65011	1	2	yes	—
145	C₂	#65024	1	2	yes	—
146	C₂	#65232	1	1	yes	characteristic
147	C₂	C₂×#3948	1	1	yes	characteristic
148	C₂	#65420	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₅

C₂

yes

maximal, characteristic

C₂×C₅×C₂²⋊C₈

C₂

yes

maximal, characteristic

#65501

C₂

yes

maximal, characteristic

#64771

C₂

yes

maximal, complemented

C₂³×C₄×C₅

C₂²

yes

characteristic

C₂³×Dic₅

—

C₂²×C₅×C₈

C₂²

yes

—

C₂×C₂²⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

C₄

yes

complemented

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₅×C₂²⋊C₈

—

#3963

—

#1824

—

maximal, Sylow

C₂⁴×C₅

D₄

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×C₄×C₅

D₄

yes

characteristic

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₂²×C₄⋊C₄

—

C₂×C₂²⋊C₈

D₅

yes

characteristic

#380

—

#340

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂²⋊C₄

yes

—

C₂³×C₅

SD₁₆

yes

—

C₂³×C₅

Q₁₆

yes

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅

yes

characteristic

C₂²×C₈

C₂×D₅

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#103

yes

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

#123

yes

—

C₂²×C₅

C₂×SD₁₆

yes

characteristic

C₂²×C₅

C₂×Q₁₆

yes

characteristic

C₂²×C₅

AGL(1,ℤ/8)

yes

—

C₄×C₅

—

C₄×C₅

#117

yes

—

C₄×C₅

#123

yes

—

Dic₅

—

Dic₅

—

C₂⁴

D₂₀

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅

yes

—

C₂²×C₄

D₂₀

yes

characteristic

C₂²×C₄

C₂²⋊D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂×C₅

—

102

C₂×C₅

#381

yes

characteristic

103

C₂×C₅

C₂×#103

yes

characteristic

104

C₂×C₅

#418

yes

characteristic

105

C₂×C₅

#481

yes

—

106

C₂×C₅

#484

yes

—

107

C₂³

—

108

C₂³

—

109

C₂³

#677

yes

—

110

C₂³

Dic₂₀

yes

—

111

C₂³

#685

yes

—

112

C₂³

C₂×D₂₀

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#685

yes

—

123

C₂×C₄

C₂×C₄×D₅

yes

characteristic

124

C₂×C₄

#711

yes

—

125

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

126

C₈

—

127

C₅

#1824

yes

characteristic, complemented, Sylow

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#3948

yes

—

133

C₂²

#4032

yes

—

134

C₂²

C₂×#677

yes

characteristic

135

C₂²

C₂×Dic₂₀

yes

characteristic

136

C₂²

#4054

yes

—

137

C₂²

C₂×#685

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#4023

yes

—

142

C₄

#4081

yes

—

143

C₂

—

144

C₂

#65011

yes

—

145

C₂

#65024

yes

—

146

C₂

#65232

yes

characteristic

147

C₂

C₂×#3948

yes

characteristic

148

C₂

#65420

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂×C₅×C₂²⋊C₈	C₂
socle	C₂³×C₅	C₂×D₄
prime radical	C₂⁴×C₅	D₄
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	C₂×C₂²⋊C₈	D₅
hypocenter	C₅	#1824
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₀

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₂×C₅×C₂²⋊C₈

C₂

socle

C₂³×C₅

C₂×D₄

prime radical

C₂⁴×C₅

D₄

cosocle

C₂²×C₄×C₅

C₂³

hypercenter

C₂×C₂²⋊C₈

D₅

hypocenter

C₅

#1824

LCS

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
4. #64771	C₂	semidirect
11. C₂×C₄⋊Dic₅	C₄	semidirect
127. C₅	#1824	semidirect

H ≅ G/N

type

4. #64771

C₂

semidirect

11. C₂×C₄⋊Dic₅

C₄

semidirect

127. C₅

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₅	2
C₂×C₅×C₂²⋊C₈	2
#65501	2
#64771	2
#1824	5

subgroup

index

C₂²×C₄⋊Dic₅

C₂×C₅×C₂²⋊C₈

#65501

#64771

#1824

Subgroups — #10600718

Conjugacy classes of subgroups

Complements

Maximal subgroups