Group #10614835 — subgroups

Subgroups — #10614835

Order 640.

516 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₅×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4022	C₂	1	1	yes	maximal, characteristic, complemented
3	#65279	C₂	1	3	yes	maximal, complemented
4	#65261	C₂	1	3	yes	maximal, complemented
5	#65246	C₂	1	3	yes	maximal
6	#65234	C₂	1	3	yes	maximal, complemented
7	C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₅	C₂²	1	1	yes	characteristic
9	C₅×#126	—	2	4	no	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₅	C₂²	1	3	yes	—
14	C₂×#685	C₂²	1	3	yes	—
15	C₂×#685	C₂²	1	3	yes	—
16	C₂×#683	C₂²	1	3	yes	—
17	C₂×#683	C₂²	1	3	yes	—
18	C₂×C₄×Dic₅	C₂²	1	3	yes	—
19	C₂×C₄×Dic₅	C₂²	1	3	yes	—
20	#4022	—	2	4	no	—
21	#3963	C₂²	1	6	yes	—
22	#2860	—	5	1	no	maximal, Sylow
23	C₂⁴×C₅	C₂³	1	1	yes	characteristic
24	C₂³×D₅	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅	C₂³	1	3	yes	—
26	C₂²×C₄×C₅	C₂³	1	3	yes	—
27	C₂×C₂²⋊D₅	—	2	6	no	—
28	C₂²×Dic₅	C₂³	1	3	yes	—
29	C₂²×Dic₅	C₂³	1	3	yes	—
30	C₂²×Dic₅	C₂³	1	1	yes	characteristic
31	C₅×C₄⋊C₄	—	2	6	no	—
32	C₅×C₂²⋊C₄	—	2	6	no	—
33	C₄²×C₅	D₄	1	4	yes	complemented
34	C₂²⋊Dic₅	—	2	6	no	—
35	#685	—	2	6	no	—
36	#685	—	2	6	no	—
37	#683	—	2	6	no	—
38	#683	—	2	6	no	—
39	C₄×Dic₅	—	2	6	no	—
40	C₄×Dic₅	—	2	6	no	—
41	C₂×#126	—	5	1	no	—
42	C₂×#126	D₅	1	1	yes	characteristic, complemented
43	#390	—	5	3	no	—
44	#389	—	5	3	no	—
45	#388	—	5	3	no	—
46	#388	—	5	3	no	—
47	C₄³	—	5	1	no	—
48	C₂³×C₅	—	2	1	no	—
49	C₂³×C₅	—	2	3	no	—
50	C₂³×C₅	—	2	3	no	—
51	C₂³×C₅	C₂⁴	1	1	yes	characteristic
52	C₂²×D₅	—	2	1	no	—
53	C₂²×D₅	—	2	3	no	—
54	C₂²×D₅	—	2	3	no	—
55	C₂×C₄×C₅	—	2	3	no	—
56	C₂×C₄×C₅	—	2	3	no	—
57	C₂×C₄×C₅	—	2	3	no	—
58	C₂×C₄×C₅	—	2	3	no	—
59	C₂×C₄×C₅	—	2	3	no	—
60	C₂×C₄×C₅	C₂×D₄	1	6	yes	—
61	C₂²⋊D₅	—	4	4	no	—
62	C₂×Dic₅	—	2	3	no	—
63	C₂×Dic₅	—	2	3	no	—
64	C₂×Dic₅	—	2	3	no	—
65	C₂×Dic₅	—	2	3	no	—
66	C₂×Dic₅	—	2	3	no	—
67	C₂×Dic₅	#41	1	6	yes	—
68	C₂×Dic₅	#41	1	6	yes	—
69	C₂²×D₄	—	5	1	no	—
70	#126	—	2	4	no	—
71	#126	—	10	4	no	—
72	C₂×C₄⋊C₄	—	5	3	no	—
73	C₂×C₄⋊C₄	—	5	3	no	—
74	C₂×C₄⋊C₄	C₂×D₅	1	3	yes	—
75	C₂×C₂²⋊C₄	—	5	3	no	—
76	C₂×C₂²⋊C₄	—	5	3	no	—
77	C₂×C₂²⋊C₄	—	5	3	no	—
78	C₂×C₂²⋊C₄	C₂×D₅	1	3	yes	—
79	C₂×C₄²	—	5	3	no	—
80	C₂×C₄²	—	5	3	no	—
81	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
82	#95	—	5	6	no	—
83	C₂²×C₅	—	4	1	no	—
84	C₂²×C₅	—	4	3	no	—
85	C₂²×C₅	—	4	3	no	—
86	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₅	C₂×#41	1	3	yes	—
88	C₂²×C₅	C₂×#41	1	3	yes	—
89	C₂×D₅	—	4	1	no	—
90	C₂×D₅	—	4	3	no	—
91	C₂×D₅	—	4	3	no	—
92	C₄×C₅	—	2	6	no	—
93	C₄×C₅	—	4	3	no	—
94	Dic₅	—	2	6	no	—
95	Dic₅	—	4	3	no	—
96	Dic₅	#126	1	4	yes	complemented
97	C₂⁴	—	5	1	no	—
98	C₂⁴	C₂²×D₅	1	1	yes	characteristic
99	C₂×D₄	—	10	6	no	—
100	C₂²×C₄	—	5	1	no	—
101	C₂²×C₄	—	5	3	no	—
102	C₂²×C₄	—	5	3	no	—
103	C₂²×C₄	C₂²×D₅	1	3	yes	—
104	C₂²×C₄	C₂²×D₅	1	3	yes	—
105	C₄⋊C₄	—	2	6	no	—
106	C₄⋊C₄	—	10	6	no	—
107	C₄⋊C₄	—	10	6	no	—
108	C₂²⋊C₄	—	2	6	no	—
109	C₂²⋊C₄	—	10	6	no	—
110	C₂²⋊C₄	—	10	6	no	—
111	C₂²⋊C₄	—	10	6	no	—
112	C₄²	—	10	6	no	—
113	C₄²	—	10	6	no	—
114	C₄²	C₂²⋊D₅	1	4	yes	complemented
115	C₂×C₅	—	8	1	no	—
116	C₂×C₅	C₂×#126	1	1	yes	characteristic
117	C₂×C₅	#529	1	3	yes	—
118	C₂×C₅	#532	1	3	yes	—
119	D₅	—	8	1	no	—
120	C₂³	—	2	3	no	—
121	C₂³	—	2	3	no	—
122	C₂³	—	2	1	no	—
123	C₂³	—	10	3	no	—
124	C₂³	—	10	3	no	—
125	C₂³	—	10	1	no	—
126	C₂³	C₂³×D₅	1	1	yes	characteristic
127	D₄	—	20	4	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	5	6	no	—
134	C₂×C₄	—	5	6	no	—
135	C₂×C₄	—	10	3	no	—
136	C₂×C₄	—	10	3	no	—
137	C₂×C₄	—	10	3	no	—
138	C₂×C₄	—	10	3	no	—
139	C₂×C₄	—	10	3	no	—
140	C₂×C₄	C₂×C₂²⋊D₅	1	6	yes	—
141	C₅	#2860	1	1	yes	characteristic, complemented, Sylow
142	C₂²	—	4	3	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	3	no	—
145	C₂²	—	20	3	no	—
146	C₂²	—	20	1	no	—
147	C₂²	—	20	3	no	—
148	C₂²	D₅×#41	1	3	yes	—
149	C₂²	D₅×#41	1	3	yes	—
150	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
151	C₄	—	2	6	no	—
152	C₄	—	4	3	no	—
153	C₄	—	5	4	no	—
154	C₄	—	10	6	no	—
155	C₄	—	20	3	no	—
156	C₂	—	8	1	no	—
157	C₂	—	40	1	no	—
158	C₂	D₅×#126	1	1	yes	characteristic
159	C₂	#66040	1	3	yes	—
160	C₂	#66166	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#126

C₂

yes

maximal, characteristic, complemented

C₂×#4022

C₂

yes

maximal, characteristic, complemented

#65279

C₂

yes

maximal, complemented

#65261

C₂

yes

maximal, complemented

#65246

C₂

yes

maximal

#65234

C₂

yes

maximal, complemented

C₄²×Dic₅

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₅

C₂²

yes

characteristic

C₅×#126

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₂²⋊Dic₅

C₂²

yes

—

C₂×#685

C₂²

yes

—

C₂×#685

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

#4022

—

#3963

C₂²

yes

—

#2860

—

maximal, Sylow

C₂⁴×C₅

C₂³

yes

characteristic

C₂³×D₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂×C₂²⋊D₅

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

D₄

yes

complemented

C₂²⋊Dic₅

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#126

—

C₂×#126

D₅

yes

characteristic, complemented

—

—

—

—

C₄³

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#41

yes

—

C₂×Dic₅

#41

yes

—

C₂²×D₄

—

#126

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

characteristic

#95

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂×#41

yes

—

C₂²×C₅

C₂×#41

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

#126

yes

complemented

C₂⁴

—

C₂⁴

C₂²×D₅

yes

characteristic

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₅

yes

—

104

C₂²×C₄

C₂²×D₅

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²⋊D₅

yes

complemented

115

C₂×C₅

—

116

C₂×C₅

C₂×#126

yes

characteristic

117

C₂×C₅

#529

yes

—

118

C₂×C₅

#532

yes

—

119

D₅

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

C₂³×D₅

yes

characteristic

127

D₄

—

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

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₂²⋊D₅

yes

—

141

C₅

#2860

yes

characteristic, complemented, Sylow

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

D₅×#41

yes

—

149

C₂²

D₅×#41

yes

—

150

C₂²

C₂²×C₂²⋊D₅

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₂

—

157

C₂

—

158

C₂

D₅×#126

yes

characteristic

159

C₂

#66040

yes

—

160

C₂

#66166

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₅×#126	C₂
prime radical	C₂²×C₂²⋊D₅	C₂²
hypercenter	C₂×#126	D₅
hypocenter	C₅	#2860
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, cosocle

C₂³×C₅

C₂⁴

Fitting

C₂×C₅×#126

C₂

prime radical

C₂²×C₂²⋊D₅

C₂²

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×C₅×#126	C₂	semidirect
2. C₂×#4022	C₂	semidirect
3. #65279	C₂	semidirect
4. #65261	C₂	semidirect
6. #65234	C₂	semidirect
7. C₄²×Dic₅	C₂	semidirect
33. C₄²×C₅	D₄	semidirect
42. C₂×#126	D₅	semidirect
96. Dic₅	#126	semidirect
114. C₄²	C₂²⋊D₅	semidirect
141. C₅	#2860	semidirect

H ≅ G/N

type

1. C₂×C₅×#126

C₂

semidirect

2. C₂×#4022

C₂

semidirect

3. #65279

C₂

semidirect

4. #65261

C₂

semidirect

6. #65234

C₂

semidirect

7. C₄²×Dic₅

C₂

semidirect

33. C₄²×C₅

D₄

semidirect

42. C₂×#126

D₅

semidirect

96. Dic₅

#126

semidirect

114. C₄²

C₂²⋊D₅

semidirect

141. C₅

#2860

semidirect

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

subgroup	index
C₂×C₅×#126	2
C₂×#4022	2
#65279	2
#65261	2
#65246	2
#65234	2
C₄²×Dic₅	2
#2860	5

subgroup

index

C₂×C₅×#126

C₄²×Dic₅

Subgroups — #10614835

Conjugacy classes of subgroups

Complements

Maximal subgroups