Group #10614210 — subgroups

Subgroups — #10614210

Order 640.

468 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4015	C₂	1	1	yes	maximal, characteristic
3	#65519	C₂	1	2	yes	maximal
4	#65275	C₂	1	4	yes	maximal
5	#65266	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
7	#65218	C₂	1	4	yes	maximal
8	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀	C₂²	1	2	yes	—
10	C₄×C₅×Q₈	—	2	4	no	—
11	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
18	C₂×C₄×Dic₅	C₂²	1	2	yes	—
19	#4015	—	2	4	no	—
20	#3963	C₂²	1	8	yes	—
21	#3963	C₂²	1	8	yes	—
22	#3963	C₂²	1	2	yes	—
23	#2568	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₈	—	2	2	no	—
25	C₂×C₅×Q₈	—	2	2	no	—
26	C₂×C₅×Q₈	Q₈	1	4	yes	complemented
27	C₂²×C₄×C₅	C₂³	1	2	yes	—
28	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅	C₂³	1	2	yes	—
30	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅	C₂³	1	4	yes	—
33	C₂²×Dic₅	C₂³	1	4	yes	—
34	C₂×Dic₁₀	—	2	4	no	—
35	C₂×Dic₁₀	—	2	8	no	—
36	C₅×C₄⋊C₄	—	2	4	no	—
37	C₅×C₄⋊C₄	—	2	2	no	—
38	C₄²×C₅	—	2	4	no	—
39	C₄²×C₅	D₄	1	4	yes	—
40	C₄⋊Dic₅	—	2	4	no	—
41	C₄⋊Dic₅	—	2	4	no	—
42	C₄⋊Dic₅	—	2	4	no	—
43	C₄×Dic₅	—	2	4	no	—
44	C₂×C₄⋊Q₈	—	5	1	no	—
45	C₂×C₄×Q₈	D₅	1	1	yes	characteristic
46	#400	—	5	4	no	—
47	#391	—	5	2	no	—
48	#391	—	5	4	no	—
49	#382	—	5	2	no	—
50	C₄×C₄⋊C₄	—	5	1	no	—
51	C₂³×C₅	C₂⁴	1	1	yes	characteristic
52	C₅×Q₈	—	2	8	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	4	no	—
58	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₅	#41	1	2	yes	—
64	C₂×C₄×C₅	#41	1	4	yes	—
65	C₂×Dic₅	—	2	4	no	—
66	C₂×Dic₅	—	2	4	no	—
67	C₂×Dic₅	—	2	4	no	—
68	C₂×Dic₅	—	2	8	no	—
69	C₂×Dic₅	—	2	4	no	—
70	Dic₁₀	—	4	8	no	—
71	C₂²×Q₈	—	5	2	no	—
72	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic
73	C₄⋊Q₈	—	10	4	no	—
74	C₄×Q₈	—	2	4	no	—
75	C₂×C₄⋊C₄	—	5	2	no	—
76	C₂×C₄⋊C₄	—	5	2	no	—
77	C₂×C₄⋊C₄	—	5	2	no	—
78	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
80	C₂×C₄²	—	5	2	no	—
81	C₂×C₄²	C₂×D₅	1	2	yes	—
82	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
83	#95	—	5	2	no	—
84	#95	—	5	8	no	—
85	#95	—	5	8	no	—
86	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
88	C₂²×C₅	C₂×#41	1	1	yes	characteristic
89	C₂²×C₅	C₂×#41	1	2	yes	—
90	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
91	C₄×C₅	—	2	4	no	—
92	C₄×C₅	—	2	2	no	—
93	C₄×C₅	—	4	2	no	—
94	C₄×C₅	C₂²⋊Q₈	1	4	yes	—
95	C₄×C₅	#124	1	4	yes	—
96	Dic₅	—	4	4	no	—
97	Dic₅	—	4	4	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	10	8	no	—
101	C₂×Q₈	—	10	4	no	—
102	C₂×Q₈	Dic₁₀	1	4	yes	complemented
103	C₂²×C₄	—	5	4	no	—
104	C₂²×C₄	—	5	4	no	—
105	C₂²×C₄	C₂²×D₅	1	2	yes	—
106	C₂²×C₄	C₂²×D₅	1	2	yes	—
107	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	10	4	no	—
113	C₄⋊C₄	—	10	4	no	—
114	C₄⋊C₄	—	10	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	—	10	4	no	—
117	C₄²	C₂²⋊D₅	1	4	yes	—
118	C₂×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₅	C₂×#124	1	1	yes	characteristic
120	C₂×C₅	#536	1	1	yes	characteristic
121	C₂×C₅	#556	1	2	yes	—
122	C₂×C₅	#557	1	2	yes	—
123	C₂³	C₂³×D₅	1	1	yes	characteristic
124	Q₈	—	2	8	no	—
125	Q₈	—	20	8	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	10	4	no	—
132	C₂×C₄	—	10	4	no	—
133	C₂×C₄	—	10	8	no	—
134	C₂×C₄	—	10	4	no	—
135	C₂×C₄	—	10	4	no	—
136	C₂×C₄	C₂×Dic₁₀	1	4	yes	—
137	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
138	C₂×C₄	#709	1	2	yes	—
139	C₂×C₄	#713	1	4	yes	—
140	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
143	C₅	#2568	1	1	yes	characteristic, complemented, Sylow
144	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
145	C₂²	C₂×#709	1	1	yes	characteristic
146	C₂²	C₂×#713	1	2	yes	—
147	C₂²	#4150	1	2	yes	—
148	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	20	4	no	—
153	C₄	—	20	4	no	—
154	C₄	#4070	1	4	yes	—
155	C₄	#4093	1	4	yes	—
156	C₂	#65904	1	2	yes	—
157	C₂	#65918	1	2	yes	—
158	C₂	C₂×#4070	1	1	yes	characteristic
159	C₂	C₂×#4093	1	1	yes	characteristic
160	C₂	#66159	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₅×Q₈

C₂

yes

maximal, characteristic

C₂×#4015

C₂

yes

maximal, characteristic

#65519

C₂

yes

maximal

#65275

C₂

yes

maximal

#65266

C₂

yes

maximal

C₄×C₄⋊Dic₅

C₂

yes

maximal, characteristic

#65218

C₂

yes

maximal

C₂²×C₅×Q₈

C₂²

yes

characteristic

C₂²×Dic₁₀

C₂²

yes

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

#4015

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2568

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

Q₈

yes

complemented

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₅

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

—

C₂×C₄²

C₂×D₅

yes

—

—

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

characteristic

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

C₂×#41

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂²⋊Q₈

yes

—

C₄×C₅

#124

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

Dic₁₀

yes

complemented

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₅

yes

—

106

C₂²×C₄

C₂²×D₅

yes

—

107

C₂²×C₄

C₂²×D₅

yes

characteristic

108

C₂²×C₄

C₂²×D₅

yes

characteristic

109

C₂²×C₄

C₂²×D₅

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂²⋊D₅

yes

—

118

C₂×C₅

C₂×C₂²⋊Q₈

yes

characteristic

119

C₂×C₅

C₂×#124

yes

characteristic

120

C₂×C₅

#536

yes

characteristic

121

C₂×C₅

#556

yes

—

122

C₂×C₅

#557

yes

—

123

C₂³

C₂³×D₅

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂×Dic₁₀

yes

—

137

C₂×C₄

C₂×Dic₁₀

yes

—

138

C₂×C₄

#709

yes

—

139

C₂×C₄

#713

yes

—

140

C₂×C₄

C₂×C₂²⋊D₅

yes

—

141

C₂×C₄

C₂×C₂²⋊D₅

yes

—

142

C₂×C₄

C₂×C₂²⋊D₅

yes

—

143

C₅

#2568

yes

characteristic, complemented, Sylow

144

C₂²

C₂²×Dic₁₀

yes

characteristic

145

C₂²

C₂×#709

yes

characteristic

146

C₂²

C₂×#713

yes

—

147

C₂²

#4150

yes

—

148

C₂²

C₂²×C₂²⋊D₅

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#4070

yes

—

155

C₄

#4093

yes

—

156

C₂

#65904

yes

—

157

C₂

#65918

yes

—

158

C₂

C₂×#4070

yes

characteristic

159

C₂

C₂×#4093

yes

characteristic

160

C₂

#66159

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₄×C₅×Q₈	C₂
hypercenter	C₂×C₄×Q₈	D₅
hypocenter	C₅	#2568
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, prime radical, cosocle

C₂³×C₅

C₂⁴

Fitting

C₂×C₄×C₅×Q₈

C₂

hypercenter

C₂×C₄×Q₈

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
26. C₂×C₅×Q₈	Q₈	semidirect
102. C₂×Q₈	Dic₁₀	semidirect
143. C₅	#2568	semidirect

H ≅ G/N

type

26. C₂×C₅×Q₈

Q₈

semidirect

102. C₂×Q₈

Dic₁₀

semidirect

143. C₅

#2568

semidirect

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

subgroup	index
C₂×C₄×C₅×Q₈	2
C₂×#4015	2
#65519	2
#65275	2
#65266	2
C₄×C₄⋊Dic₅	2
#65218	2
#2568	5

subgroup

index

C₂×C₄×C₅×Q₈

C₄×C₄⋊Dic₅

Subgroups — #10614210

Conjugacy classes of subgroups

Complements

Maximal subgroups