Group #12002410 — subgroups

Subgroups — #12002410

Order 1280.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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×#2864	C₂	1	1	yes	maximal, characteristic
2	#10600339	C₂	1	2	yes	maximal
3	#10600189	C₂	1	1	yes	maximal, characteristic
4	#10598122	C₂	1	2	yes	maximal
5	#10598102	C₂	1	1	yes	maximal, characteristic
6	#65804	C₂²	1	1	yes	characteristic
7	C₅×#389	—	2	2	no	—
8	C₅×#387	—	2	4	no	—
9	C₅×#382	C₂²	1	2	yes	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	#65214	C₂²	1	2	yes	—
12	#65213	C₂²	1	1	yes	characteristic
13	#64743	—	2	2	no	—
14	#64672	—	2	4	no	—
15	#10367	—	5	1	no	maximal, Sylow
16	C₂×C₅×C₄⋊C₄	—	2	4	no	—
17	C₂×C₅×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄²×C₅	—	2	2	no	—
19	C₂×C₄²×C₅	D₄	1	2	yes	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	#4072	—	2	1	no	—
22	#4070	—	2	2	no	—
23	C₂×#681	—	2	1	no	—
24	#4016	—	2	1	no	—
25	#4015	Q₈	1	2	yes	—
26	C₄×Dic₁₀	—	2	2	no	—
27	C₅×#95	—	2	4	no	—
28	C₅×#95	—	2	2	no	—
29	C₅×#95	D₄	1	4	yes	—
30	C₂²⋊#168	—	2	2	no	—
31	C₄⋊#168	—	2	2	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄×#168	—	2	2	no	—
34	#2864	D₅	1	1	yes	characteristic
35	#1994	—	5	1	no	—
36	#1518	—	5	2	no	—
37	#1286	—	5	2	no	—
38	#1276	—	5	1	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	—	2	4	no	—
41	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₁₀	—	2	2	no	—
44	C₅×C₄⋊C₄	—	4	8	no	—
45	C₅×C₄⋊C₄	—	4	2	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₄²×C₅	—	2	2	no	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	—	4	4	no	—
50	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₅	#41	1	1	yes	characteristic
52	C₄²×C₅	#41	1	2	yes	—
53	C₂²⋊Dic₅	—	4	1	no	—
54	C₄⋊Dic₅	—	2	2	no	—
55	C₄⋊Dic₅	—	2	2	no	—
56	#683	—	4	2	no	—
57	C₄×Dic₅	—	4	1	no	—
58	#681	—	4	2	no	—
59	C₂×#168	—	4	2	no	—
60	C₂×#168	—	4	1	no	—
61	#533	—	5	1	no	—
62	#431	—	5	2	no	—
63	#423	—	5	1	no	—
64	#389	—	2	2	no	—
65	#387	—	2	4	no	—
66	#382	C₂×D₅	1	2	yes	—
67	C₄³	C₂×D₅	1	1	yes	characteristic
68	#339	—	10	2	no	—
69	#328	—	10	4	no	—
70	C₂³×C₅	#120	1	1	yes	characteristic
71	C₂×C₄×C₅	—	2	4	no	—
72	C₂×C₄×C₅	—	4	2	no	—
73	C₂×C₄×C₅	—	4	2	no	—
74	C₂×C₄×C₅	—	4	2	no	—
75	C₂×C₄×C₅	—	4	8	no	—
76	C₂×C₄×C₅	—	4	4	no	—
77	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅	#124	1	2	yes	—
80	C₂×C₄×C₅	#124	1	1	yes	characteristic
81	C₂×C₄×C₅	#135	1	2	yes	—
82	C₂×C₄×C₅	#137	1	2	yes	—
83	C₂×Dic₅	—	4	2	no	—
84	Dic₁₀	—	8	2	no	—
85	#168	—	8	2	no	—
86	#168	—	8	1	no	—
87	C₂×M₁₆	—	10	1	no	—
88	C₄⋊Q₈	—	5	2	no	—
89	#125	—	10	1	no	—
90	C₂²⋊Q₈	—	10	2	no	—
91	C₄×Q₈	—	10	2	no	—
92	#117	—	10	1	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
97	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
98	C₄⋊C₈	—	10	2	no	—
99	C₄⋊C₈	—	10	2	no	—
100	C₂²⋊C₈	—	10	2	no	—
101	C₄×C₈	—	10	2	no	—
102	#95	—	2	4	no	—
103	#95	—	2	2	no	—
104	#95	C₂²⋊D₅	1	4	yes	—
105	C₂²×C₅	—	2	2	no	—
106	C₂²×C₅	#393	1	1	yes	characteristic
107	C₂²×C₅	#452	1	2	yes	—
108	C₄×C₅	—	2	2	no	—
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	—	4	4	no	—
111	C₄×C₅	—	8	4	no	—
112	C₄×C₅	#479	1	2	yes	—
113	C₄×C₅	#488	1	1	yes	characteristic
114	C₄×C₅	#491	1	1	yes	characteristic
115	Dic₅	—	8	2	no	—
116	C₂×Q₈	—	10	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
121	M₁₆	—	20	2	no	—
122	C₂×C₈	—	20	2	no	—
123	C₂×C₈	—	20	1	no	—
124	C₄⋊C₄	—	4	8	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	10	2	no	—
128	C₄⋊C₄	—	10	2	no	—
129	C₄⋊C₄	—	20	2	no	—
130	C₂²⋊C₄	—	20	1	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	4	4	no	—
134	C₄²	—	20	1	no	—
135	C₄²	Q₈×D₅	1	1	yes	characteristic
136	C₄²	#713	1	1	yes	characteristic
137	C₄²	#713	1	2	yes	—
138	C₂×C₅	—	2	2	no	—
139	C₂×C₅	#2026	1	1	yes	characteristic
140	C₂×C₅	#2031	1	2	yes	—
141	C₂³	#4099	1	1	yes	characteristic
142	Q₈	—	40	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	20	2	no	—
150	C₂×C₄	#4092	1	2	yes	—
151	C₂×C₄	#4092	1	1	yes	characteristic
152	C₂×C₄	#4093	1	2	yes	—
153	C₂×C₄	#4093	1	1	yes	characteristic
154	C₂×C₄	#4096	1	2	yes	—
155	C₂×C₄	#4097	1	2	yes	—
156	C₈	—	40	2	no	—
157	C₈	—	40	1	no	—
158	C₅	#10367	1	1	yes	characteristic, complemented, Sylow
159	C₂²	—	2	2	no	—
160	C₂²	#65520	1	1	yes	characteristic
161	C₂²	#65528	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	4	no	—
166	C₄	—	40	2	no	—
167	C₄	#65352	1	2	yes	—
168	C₄	#65357	1	1	yes	characteristic
169	C₄	#65358	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#10602537	1	1	yes	characteristic
172	C₂	#10602635	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2864

C₂

yes

maximal, characteristic

#10600339

C₂

yes

maximal

#10600189

C₂

yes

maximal, characteristic

#10598122

C₂

yes

maximal

#10598102

C₂

yes

maximal, characteristic

#65804

C₂²

yes

—

—

C₂²

yes

—

C₄³×C₅

C₂²

yes

characteristic

#65214

C₂²

yes

—

#65213

C₂²

yes

—

—

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

—

Q₈

yes

—

C₄×Dic₁₀

—

C₅×#95

—

C₅×#95

—

C₅×#95

D₄

yes

—

C₂²⋊#168

—

—

—

—

D₅

yes

—

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×Q₈

yes

C₄²×C₅

yes

C₄²×C₅

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

—

—

—

—

—

—

C₂×D₅

yes

—

C₄³

C₂×D₅

yes

characteristic

#339

—

#328

—

C₂³×C₅

#120

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅

#124

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

#168

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

#95

—

103

#95

—

104

#95

C₂²⋊D₅

yes

—

105

C₂²×C₅

—

106

C₂²×C₅

#393

yes

characteristic

107

C₂²×C₅

#452

yes

—

108

C₄×C₅

—

109

C₄×C₅

—

110

C₄×C₅

—

111

C₄×C₅

—

112

C₄×C₅

#479

yes

—

113

C₄×C₅

#488

yes

characteristic

114

C₄×C₅

#491

yes

characteristic

115

Dic₅

—

116

C₂×Q₈

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

120

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

Q₈×D₅

yes

characteristic

136

C₄²

#713

yes

characteristic

137

C₄²

#713

yes

—

138

C₂×C₅

—

139

C₂×C₅

#2026

yes

characteristic

140

C₂×C₅

#2031

yes

—

141

C₂³

#4099

yes

characteristic

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#4092

yes

—

151

C₂×C₄

#4092

yes

characteristic

152

C₂×C₄

#4093

yes

—

153

C₂×C₄

#4093

yes

characteristic

154

C₂×C₄

#4096

yes

—

155

C₂×C₄

#4097

yes

—

156

C₈

—

157

C₈

—

158

C₅

#10367

yes

characteristic, complemented, Sylow

159

C₂²

—

160

C₂²

#65520

yes

characteristic

161

C₂²

#65528

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#65352

yes

—

168

C₄

#65357

yes

characteristic

169

C₄

#65358

yes

characteristic

170

C₂

—

171

C₂

#10602537

yes

characteristic

172

C₂

#10602635

yes

—

	subgroup	G/H
center	C₂²	#65520
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2864	C₂
socle	C₂²×C₅	#393
prime radical	C₂³×C₅	#120
hypercenter	#2864	D₅
hypocenter	C₅	#10367
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

subgroup

G/H

center

C₂²

#65520

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

N	H ≅ G/N	type
158. C₅	#10367	semidirect

H ≅ G/N

type

158. C₅

#10367

semidirect

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

subgroup	index
C₅×#2864	2
#10600339	2
#10600189	2
#10598122	2
#10598102	2
#10367	5

subgroup

index

Subgroups — #12002410

Conjugacy classes of subgroups

Complements

Maximal subgroups