Group #10602824 — subgroups

Subgroups — #10602824

Order 640.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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×#568	C₂	1	1	yes	maximal, characteristic
2	#66159	C₂	1	1	yes	maximal, characteristic
3	#65526	C₂	1	1	yes	maximal, characteristic
4	C₂×#3956	C₂	1	2	yes	maximal, complemented
5	#64779	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#41	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₀	C₂²	1	1	yes	characteristic
8	C₅×#131	—	2	4	no	—
9	C₂×C₅×M₁₆	—	2	2	no	—
10	C₂×C₅×M₁₆	C₂²	1	2	yes	—
11	#4090	—	2	1	no	—
12	#4082	—	2	1	no	—
13	#4081	C₄	1	4	yes	—
14	#4072	C₂²	1	1	yes	characteristic
15	#4070	—	2	2	no	—
16	C₂×#681	C₂²	1	2	yes	—
17	#3968	—	2	1	no	—
18	#3965	—	2	1	no	—
19	#3956	—	2	4	no	—
20	#1915	—	5	1	no	maximal, Sylow
21	C₅×#41	—	2	2	no	—
22	C₅×#41	—	2	2	no	—
23	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
24	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
25	C₂×C₅×D₄	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₀	—	2	2	no	—
30	C₂×Dic₁₀	—	2	1	no	—
31	C₂×Dic₁₀	—	2	1	no	—
32	C₂×Dic₁₀	—	2	2	no	—
33	C₅×M₁₆	—	2	4	no	—
34	C₅×M₁₆	—	2	2	no	—
35	C₅×M₁₆	D₄	1	4	yes	—
36	C₂×C₅×C₈	—	2	4	no	—
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂²⋊Dic₅	—	2	4	no	—
39	C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₅	—	2	1	no	—
41	#683	—	2	4	no	—
42	C₄×Dic₅	—	2	1	no	—
43	#681	—	2	2	no	—
44	#681	—	2	2	no	—
45	C₂×#168	—	2	2	no	—
46	#568	D₅	1	1	yes	characteristic
47	#536	—	5	1	no	—
48	#413	—	5	1	no	—
49	C₂×#101	—	5	2	no	—
50	#344	—	5	2	no	—
51	C₂³×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₅×Q₈	—	2	2	no	—
54	C₅×D₄	—	2	4	no	—
55	C₅×D₄	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅	#41	1	2	yes	—
63	C₂×Dic₅	—	2	2	no	—
64	C₂×Dic₅	—	2	2	no	—
65	C₂×Dic₅	—	2	4	no	—
66	Dic₁₀	—	4	1	no	—
67	Dic₁₀	—	4	2	no	—
68	Dic₁₀	—	4	1	no	—
69	C₅×C₈	—	2	4	no	—
70	#168	—	4	2	no	—
71	C₂×#41	C₂×D₅	1	1	yes	characteristic
72	C₂²×Q₈	—	5	1	no	—
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	5	2	no	—
76	C₂×M₁₆	C₂×D₅	1	2	yes	—
77	C₄⋊Q₈	—	10	1	no	—
78	#124	—	10	1	no	—
79	#123	—	5	4	no	—
80	C₂²⋊Q₈	—	10	2	no	—
81	#117	—	5	1	no	—
82	#101	—	10	1	no	—
83	#101	—	10	4	no	—
84	#100	—	10	1	no	—
85	C₂²×C₅	—	2	2	no	—
86	C₂²×C₅	—	4	1	no	—
87	C₂²×C₅	—	4	2	no	—
88	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₅	C₄×D₄	1	2	yes	—
90	C₄×C₅	—	2	2	no	—
91	C₄×C₅	#120	1	1	yes	characteristic
92	C₄×C₅	#121	1	2	yes	—
93	C₄×C₅	#123	1	1	yes	characteristic
94	Dic₅	—	4	2	no	—
95	Dic₅	—	4	2	no	—
96	#41	—	2	2	no	—
97	#41	—	2	2	no	—
98	C₂×Q₈	—	10	1	no	—
99	C₂×Q₈	—	10	2	no	—
100	C₂×Q₈	—	10	2	no	—
101	C₂×Q₈	—	10	1	no	—
102	C₂×Q₈	D₂₀	1	1	yes	characteristic
103	C₂×D₄	C₄×D₅	1	2	yes	—
104	C₂×D₄	D₂₀	1	1	yes	characteristic
105	C₂²×C₄	—	5	2	no	—
106	C₂²×C₄	C₂²⋊D₅	1	2	yes	—
107	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	10	2	no	—
111	M₁₆	—	10	2	no	—
112	M₁₆	C₂²⋊D₅	1	4	yes	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	10	2	no	—
116	C₄⋊C₄	—	10	4	no	—
117	C₄⋊C₄	—	10	1	no	—
118	C₂²⋊C₄	—	5	2	no	—
119	C₂²⋊C₄	—	10	4	no	—
120	C₄²	—	10	1	no	—
121	C₂×C₅	—	2	1	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	—	4	2	no	—
124	C₂×C₅	#386	1	1	yes	characteristic
125	C₂³	#685	1	2	yes	—
126	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	20	1	no	—
129	Q₈	—	20	1	no	—
130	Q₈	—	20	2	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	10	2	no	—
136	C₂×C₄	—	10	2	no	—
137	C₂×C₄	—	10	4	no	—
138	C₂×C₄	#685	1	2	yes	—
139	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
140	C₂×C₄	#709	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	20	2	no	—
145	C₅	#1915	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×#685	1	1	yes	characteristic
150	C₂²	C₄×C₂²⋊D₅	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	20	2	no	—
153	C₄	—	20	2	no	—
154	C₄	#4075	1	1	yes	characteristic
155	C₄	#4076	1	2	yes	—
156	C₄	#4099	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#65502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#568

C₂

yes

maximal, characteristic

#66159

C₂

yes

maximal, characteristic

#65526

C₂

yes

maximal, characteristic

C₂×#3956

C₂

yes

maximal, complemented

#64779

C₂

yes

maximal, complemented

C₂×C₅×#41

C₂²

yes

characteristic

C₂²×Dic₁₀

C₂²

yes

characteristic

C₅×#131

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

C₂²

yes

—

#4090

—

#4082

—

#4081

C₄

yes

—

#4072

C₂²

yes

characteristic

#4070

—

C₂×#681

C₂²

yes

—

—

—

—

—

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

D₄

yes

characteristic

C₂×C₅×D₄

C₂×C₄

yes

—

C₂×C₅×D₄

D₄

yes

characteristic

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂×C₄

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

D₄

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

—

D₅

yes

—

—

—

—

C₂³×C₅

C₂²⋊C₄

yes

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂×#41

C₂×D₅

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅

yes

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

C₄×D₄

yes

—

C₄×C₅

—

C₄×C₅

#120

yes

characteristic

C₄×C₅

#121

yes

—

C₄×C₅

#123

yes

characteristic

Dic₅

—

Dic₅

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

D₂₀

yes

characteristic

103

C₂×D₄

C₄×D₅

yes

—

104

C₂×D₄

D₂₀

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊D₅

yes

—

107

C₂²×C₄

C₂²×D₅

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

C₂²⋊D₅

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

#386

yes

characteristic

125

C₂³

#685

yes

—

126

C₂³

C₂×C₄×D₅

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#685

yes

—

139

C₂×C₄

C₂×D₂₀

yes

characteristic

140

C₂×C₄

#709

yes

—

141

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₅

yes

—

143

C₈

—

144

C₈

—

145

C₅

#1915

yes

characteristic, complemented, Sylow

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×#685

yes

characteristic

150

C₂²

C₄×C₂²⋊D₅

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#4075

yes

characteristic

155

C₄

#4076

yes

—

156

C₄

#4099

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#65502

yes

characteristic

	subgroup	G/H
center	C₂	#65502
commutator	C₂³×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#568	C₂
socle	C₂×C₅	#386
prime radical	C₂×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#568	D₅
hypocenter	C₅	#1915
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#65502

commutator

C₂³×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

C₂

C₂×C₅

C₂×C₅×#41

C₂²

cosocle

C₂²×C₄×C₅

C₂³

D₅

C₅

G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
4. C₂×#3956	C₂	semidirect
5. #64779	C₂	semidirect
145. C₅	#1915	semidirect

H ≅ G/N

type

4. C₂×#3956

C₂

semidirect

5. #64779

C₂

semidirect

145. C₅

#1915

semidirect

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

subgroup	index
C₅×#568	2
#66159	2
#65526	2
C₂×#3956	2
#64779	2
#1915	5

subgroup

index

Subgroups — #10602824

Conjugacy classes of subgroups

Complements

Maximal subgroups