Group #11990311 — subgroups

Subgroups — #11990311

Order 1280.

384 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 133 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	#10615059	C₂	1	1	yes	maximal, characteristic
2	C₅×#1544	C₂	1	1	yes	maximal, characteristic
3	#10602614	C₂	1	1	yes	maximal, characteristic
4	#10601532	C₂	1	2	yes	maximal
5	#10598234	C₂	1	2	yes	maximal
6	C₅×#533	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	2	yes	—
8	#65313	—	2	2	no	—
9	Q₈×#168	—	2	2	no	—
10	#65288	C₄	1	2	yes	—
11	#65286	C₄	1	2	yes	—
12	#65213	C₂²	1	1	yes	characteristic
13	C₄×#681	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	C₅×#332	C₄	1	4	yes	—
16	#64763	C₄	1	4	yes	—
17	#64674	—	2	2	no	—
18	C₈×#168	—	2	2	no	—
19	#8242	—	5	1	no	maximal, Sylow
20	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
21	C₅×#125	C₂×C₄	1	2	yes	—
22	C₅×C₂²⋊Q₈	—	2	2	no	—
23	C₄×C₅×Q₈	—	2	2	no	—
24	C₅×#117	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂×#681	—	2	2	no	—
27	C₂×#681	C₂×C₄	1	2	yes	—
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₅×C₂²⋊C₈	—	2	2	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₄⋊#168	—	2	2	no	—
34	C₄⋊#168	—	2	4	no	—
35	#3935	—	2	2	no	—
36	#3935	C₂×C₄	1	4	yes	—
37	C₄×#168	—	2	2	no	—
38	C₄×#168	—	2	2	no	—
39	C₄×#168	D₄	1	2	yes	—
40	C₄×#168	D₄	1	2	yes	—
41	#2960	—	5	1	no	—
42	#1544	D₅	1	1	yes	characteristic
43	#1526	—	5	1	no	—
44	C₈×M₁₆	—	5	2	no	—
45	#1299	—	5	2	no	—
46	C₂×C₅×Q₈	—	2	2	no	—
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	2	yes	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₅×C₄⋊C₄	—	2	4	no	—
51	C₅×C₄⋊C₄	—	2	2	no	—
52	C₅×C₄⋊C₄	C₄²	1	4	yes	—
53	C₅×C₂²⋊C₄	—	2	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅	C₂×D₄	1	2	yes	—
58	#681	—	2	8	no	—
59	#681	—	4	2	no	—
60	C₂×#168	—	2	4	no	—
61	C₂×#168	—	2	2	no	—
62	C₂×#168	—	2	2	no	—
63	C₂×#168	—	4	1	no	—
64	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×#168	C₂²⋊C₄	1	2	yes	—
66	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
67	#533	C₂×D₅	1	1	yes	characteristic
68	#446	—	10	2	no	—
69	C₈×Q₈	—	10	2	no	—
70	#431	C₂×D₅	1	2	yes	—
71	#424	—	5	2	no	—
72	#423	—	5	1	no	—
73	#405	—	5	2	no	—
74	C₄×M₁₆	—	5	2	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	#332	Dic₅	1	4	yes	—
77	#330	—	5	4	no	—
78	#322	—	10	2	no	—
79	C₈²	—	10	2	no	—
80	C₂³×C₅	#117	1	1	yes	characteristic
81	C₅×Q₈	—	4	4	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
90	C₅×C₈	—	4	4	no	—
91	#168	—	2	8	no	—
92	#168	—	2	1	no	—
93	#168	—	2	2	no	—
94	#168	—	4	2	no	—
95	#168	#101	1	2	yes	—
96	C₂×M₁₆	—	5	2	no	—
97	C₂×M₁₆	—	10	2	no	—
98	C₂²×C₈	—	5	1	no	—
99	C₂²×C₈	—	5	1	no	—
100	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
101	#125	C₄×D₅	1	2	yes	—
102	C₂²⋊Q₈	—	2	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	#117	C₄×D₅	1	2	yes	—
105	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
106	C₄⋊C₈	—	10	4	no	—
107	C₄⋊C₈	—	10	2	no	—
108	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
109	C₂²⋊C₈	—	2	2	no	—
110	#97	—	5	4	no	—
111	#97	—	10	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	5	2	no	—
114	C₄×C₈	—	5	2	no	—
115	C₄×C₈	—	10	2	no	—
116	C₄×C₈	—	10	2	no	—
117	C₂²×C₅	—	2	1	no	—
118	C₂²×C₅	—	2	2	no	—
119	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₅	—	2	1	no	—
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	4	4	no	—
123	C₄×C₅	C₂×#101	1	1	yes	characteristic
124	C₄×C₅	#413	1	1	yes	characteristic
125	C₄×C₅	#443	1	4	yes	—
126	C₂×Q₈	—	2	2	no	—
127	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
128	C₂²×C₄	#711	1	2	yes	—
129	M₁₆	—	10	8	no	—
130	M₁₆	—	20	2	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	5	1	no	—
133	C₂×C₈	—	5	1	no	—
134	C₂×C₈	—	5	2	no	—
135	C₂×C₈	—	10	2	no	—
136	C₂×C₈	—	10	4	no	—
137	C₂×C₈	—	10	2	no	—
138	C₂×C₈	—	20	1	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
142	C₂²⋊C₄	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
145	C₄²	D₄×D₅	1	2	yes	—
146	C₄²	C₂²×Dic₅	1	1	yes	characteristic
147	C₂×C₅	—	4	1	no	—
148	C₂×C₅	C₄×#101	1	1	yes	characteristic
149	C₂×C₅	#1764	1	2	yes	—
150	C₂³	#4023	1	1	yes	characteristic
151	Q₈	—	4	4	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#4027	1	2	yes	—
158	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
159	C₂×C₄	D₄×Dic₅	1	2	yes	—
160	C₈	—	4	4	no	—
161	C₈	—	5	2	no	—
162	C₈	—	10	8	no	—
163	C₈	—	10	1	no	—
164	C₈	—	10	2	no	—
165	C₈	—	20	2	no	—
166	C₅	#8242	1	1	yes	characteristic, complemented, Sylow
167	C₂²	—	2	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	4	no	—
173	C₄	D₅×#101	1	1	yes	characteristic
174	C₄	#65040	1	1	yes	characteristic
175	C₄	#65485	1	4	yes	—
176	C₂	—	4	1	no	—
177	C₂	Dic₅×#101	1	1	yes	characteristic
178	C₂	#10601082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10615059

C₂

yes

maximal, characteristic

C₅×#1544

C₂

yes

maximal, characteristic

#10602614

C₂

yes

maximal, characteristic

#10601532

C₂

yes

maximal

#10598234

C₂

yes

maximal

C₅×#533

C₂²

yes

characteristic

C₅×#431

C₂²

yes

—

#65313

—

Q₈×#168

—

#65288

C₄

yes

—

#65286

C₄

yes

—

#65213

C₂²

yes

characteristic

C₄×#681

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

characteristic

C₅×#332

C₄

yes

—

#64763

C₄

yes

—

—

—

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₅×#125

C₂×C₄

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×#681

—

C₂×#681

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

characteristic

C₅×C₄⋊C₈

C₂×C₄

yes

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

—

—

—

C₂×C₄

yes

—

C₄×#168

—

C₄×#168

—

C₄×#168

D₄

yes

—

C₄×#168

D₄

yes

—

#2960

—

#1544

D₅

yes

characteristic

#1526

—

C₈×M₁₆

—

#1299

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₄²

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#168

C₂²⋊C₄

yes

—

C₂×#168

C₂²⋊C₄

yes

characteristic

#533

C₂×D₅

yes

characteristic

#446

—

C₈×Q₈

—

#431

C₂×D₅

yes

—

#424

—

#423

—

#405

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#332

Dic₅

yes

—

#330

—

#322

—

C₈²

—

C₂³×C₅

#117

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

C₄×D₄

yes

—

C₅×C₈

—

—

—

—

—

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₄⋊Q₈

C₂×Dic₅

yes

—

101

#125

C₄×D₅

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

C₄×D₅

yes

—

105

C₂×C₄²

C₂²×D₅

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

C₂×Dic₅

yes

—

109

C₂²⋊C₈

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₂²×C₅

—

118

C₂²×C₅

—

119

C₂²×C₅

C₄×C₂²⋊C₄

yes

characteristic

120

C₄×C₅

—

121

C₄×C₅

—

122

C₄×C₅

—

123

C₄×C₅

C₂×#101

yes

characteristic

124

C₄×C₅

#413

yes

characteristic

125

C₄×C₅

#443

yes

—

126

C₂×Q₈

—

127

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

128

C₂²×C₄

#711

yes

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

C₄×Dic₅

yes

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

C₂×C₄×D₅

yes

characteristic

145

C₄²

D₄×D₅

yes

—

146

C₄²

C₂²×Dic₅

yes

characteristic

147

C₂×C₅

—

148

C₂×C₅

C₄×#101

yes

characteristic

149

C₂×C₅

#1764

yes

—

150

C₂³

#4023

yes

characteristic

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

D₅×C₂²⋊C₄

yes

characteristic

157

C₂×C₄

#4027

yes

—

158

C₂×C₄

C₂×C₄×Dic₅

yes

characteristic

159

C₂×C₄

D₄×Dic₅

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₅

#8242

yes

characteristic, complemented, Sylow

167

C₂²

—

168

C₂²

—

169

C₂²

C₂²⋊C₄×Dic₅

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

D₅×#101

yes

characteristic

174

C₄

#65040

yes

characteristic

175

C₄

#65485

yes

—

176

C₂

—

177

C₂

Dic₅×#101

yes

characteristic

178

C₂

#10601082

yes

—

	subgroup	G/H
center	C₂×C₄	D₅×C₂²⋊C₄
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1544	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂³×C₅	#117
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1544	D₅
hypocenter	C₅	#8242
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

subgroup

G/H

center

C₂×C₄

D₅×C₂²⋊C₄

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1544

C₂

socle

C₂²×C₅

C₄×C₂²⋊C₄

prime radical

C₂³×C₅

#117

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

N	H ≅ G/N	type
166. C₅	#8242	semidirect

H ≅ G/N

type

166. C₅

#8242

semidirect

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

subgroup	index
#10615059	2
C₅×#1544	2
#10602614	2
#10601532	2
#10598234	2
#8242	5

subgroup

index

Subgroups — #11990311

Conjugacy classes of subgroups

Complements

Maximal subgroups