Group #11998734 — subgroups

Subgroups — #11998734

Order 1280.

414 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 155 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×#64969	C₂	1	1	yes	maximal, characteristic
2	C₅×#1450	C₂	1	1	yes	maximal, characteristic
3	#10601573	C₂	1	2	yes	maximal
4	#10601572	C₂	1	2	yes	maximal
5	#10600585	C₂	1	1	yes	maximal, characteristic
6	C₂×#4015	C₂²	1	2	yes	—
7	C₅×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3951	—	2	2	no	—
10	C₂×#3949	—	2	2	no	—
11	#65213	C₂²	1	1	yes	characteristic
12	#65213	C₂²	1	1	yes	characteristic
13	#64969	—	2	1	no	—
14	#64969	—	2	1	no	—
15	#64969	C₄	1	4	yes	—
16	C₅×#322	—	2	1	no	—
17	C₅×C₈²	—	2	1	no	—
18	#64702	—	2	4	no	—
19	#64701	—	2	4	no	—
20	#9767	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×#681	—	2	1	no	—
27	C₂×#681	—	2	1	no	—
28	#4015	—	2	2	no	—
29	#4015	—	2	2	no	—
30	#4015	—	2	2	no	—
31	#4015	C₂×C₄	1	4	yes	—
32	C₅×C₄⋊C₈	D₄	1	4	yes	—
33	C₅×C₂²⋊C₈	—	2	2	no	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₅×C₈	D₄	1	1	yes	characteristic
37	C₄×C₅×C₈	D₄	1	1	yes	characteristic
38	#3951	—	4	4	no	—
39	#3949	—	2	4	no	—
40	#3949	—	2	4	no	—
41	C₄⋊#168	—	2	2	no	—
42	C₄⋊#168	—	2	2	no	—
43	C₂×#498	—	5	1	no	—
44	#1848	—	5	1	no	—
45	#1549	—	5	2	no	—
46	#1548	—	5	2	no	—
47	#1450	D₅	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	#41	1	2	yes	—
50	C₂²×Dic₅	—	4	2	no	—
51	C₂×Dic₁₀	—	2	4	no	—
52	C₂×Dic₁₀	—	4	2	no	—
53	C₂×Dic₁₀	—	8	4	no	—
54	C₂×C₅×C₈	—	2	4	no	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₅×C₈	SD₁₆	1	4	yes	—
58	C₂×C₅×C₈	Q₁₆	1	4	yes	—
59	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅	C₂×D₄	1	2	yes	—
61	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Dic₅	—	2	4	no	—
63	C₄⋊Dic₅	—	2	4	no	—
64	C₄⋊Dic₅	—	4	2	no	—
65	C₄⋊Dic₅	—	4	2	no	—
66	#681	—	4	2	no	—
67	#681	—	4	2	no	—
68	C₂×#168	—	4	1	no	—
69	C₂×#168	—	4	1	no	—
70	C₂×C₄⋊Q₈	—	5	2	no	—
71	#498	—	5	4	no	—
72	#498	—	10	1	no	—
73	#498	—	10	1	no	—
74	#431	C₂×D₅	1	2	yes	—
75	C₂×#108	—	10	2	no	—
76	C₂×#106	—	10	2	no	—
77	#423	—	5	1	no	—
78	#423	—	5	1	no	—
79	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
80	#333	—	10	4	no	—
81	#332	—	10	4	no	—
82	#322	—	2	1	no	—
83	C₈²	—	2	1	no	—
84	C₂³×C₅	#124	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₅	#121	1	2	yes	—
89	C₂×C₄×C₅	#127	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
92	C₂×Dic₅	—	4	4	no	—
93	C₂×Dic₅	—	8	4	no	—
94	Dic₁₀	—	8	4	no	—
95	Dic₁₀	—	8	4	no	—
96	C₅×C₈	—	4	4	no	—
97	C₅×C₈	#103	1	4	yes	—
98	C₅×C₈	#103	1	4	yes	—
99	#168	—	8	1	no	—
100	#168	—	8	1	no	—
101	C₂²×Q₈	—	10	2	no	—
102	C₂×M₁₆	—	10	1	no	—
103	C₂×M₁₆	—	10	1	no	—
104	C₂²×C₈	D₂₀	1	2	yes	—
105	C₄⋊Q₈	—	5	4	no	—
106	C₄⋊Q₈	—	10	2	no	—
107	C₄⋊Q₈	—	10	2	no	—
108	C₄⋊Q₈	—	10	2	no	—
109	C₂×C₄⋊C₄	—	10	2	no	—
110	C₂×C₄⋊C₄	—	10	2	no	—
111	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
112	#108	—	20	4	no	—
113	#106	—	10	4	no	—
114	#106	—	10	4	no	—
115	C₄⋊C₈	—	10	2	no	—
116	C₄⋊C₈	—	10	2	no	—
117	C₄⋊C₈	D₂₀	1	4	yes	—
118	C₂²⋊C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₄×D₅	1	2	yes	—
121	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	#390	1	1	yes	characteristic
125	C₂²×C₅	#487	1	2	yes	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	C₂×#103	1	2	yes	—
128	C₄×C₅	#471	1	4	yes	—
129	C₄×C₅	#492	1	1	yes	characteristic
130	C₄×C₅	#494	1	1	yes	characteristic
131	Dic₅	—	8	4	no	—
132	C₂×Q₈	—	10	4	no	—
133	C₂×Q₈	—	20	2	no	—
134	C₂×Q₈	—	40	4	no	—
135	C₂²×C₄	—	20	2	no	—
136	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
137	C₂²×C₄	#709	1	2	yes	—
138	M₁₆	—	20	2	no	—
139	M₁₆	—	20	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	20	1	no	—
142	C₂×C₈	—	20	1	no	—
143	C₂×C₈	#677	1	4	yes	—
144	C₂×C₈	Dic₂₀	1	4	yes	—
145	C₂×C₈	#685	1	2	yes	—
146	C₂×C₈	#685	1	2	yes	—
147	C₄⋊C₄	—	10	4	no	—
148	C₄⋊C₄	—	10	4	no	—
149	C₄⋊C₄	—	20	2	no	—
150	C₄⋊C₄	—	20	2	no	—
151	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
152	C₄²	C₂×D₂₀	1	2	yes	—
153	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₅	—	2	2	no	—
155	C₂×C₅	#1956	1	1	yes	characteristic
156	C₂×C₅	#1977	1	2	yes	—
157	C₂³	#4021	1	1	yes	characteristic
158	Q₈	—	40	4	no	—
159	Q₈	—	40	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	20	4	no	—
162	C₂×C₄	—	40	4	no	—
163	C₂×C₄	C₄×D₂₀	1	2	yes	—
164	C₂×C₄	#4020	1	1	yes	characteristic
165	C₂×C₄	C₂×#677	1	2	yes	—
166	C₂×C₄	C₂×Dic₂₀	1	2	yes	—
167	C₂×C₄	C₂×#685	1	1	yes	characteristic
168	C₂×C₄	#4076	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	40	1	no	—
171	C₈	—	40	1	no	—
172	C₈	#3948	1	4	yes	—
173	C₈	#3948	1	4	yes	—
174	C₅	#9767	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	#64970	1	2	yes	—
177	C₂²	#65228	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	40	4	no	—
180	C₄	#64982	1	1	yes	characteristic
181	C₄	#64988	1	1	yes	characteristic
182	C₄	C₂×#3948	1	2	yes	—
183	C₄	#65430	1	4	yes	—
184	C₂	—	2	2	no	—
185	C₂	#10600564	1	1	yes	characteristic
186	C₂	#10600695	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64969

C₂

yes

maximal, characteristic

C₅×#1450

C₂

yes

maximal, characteristic

#10601573

C₂

yes

maximal

#10601572

C₂

yes

maximal

#10600585

C₂

yes

maximal, characteristic

C₂×#4015

C₂²

yes

—

C₅×#431

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

—

C₂²

yes

characteristic

#65213

C₂²

yes

—

—

C₄

yes

—

C₅×#322

—

C₅×C₈²

—

—

—

—

C₂²×Dic₁₀

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

—

—

—

—

—

C₂×C₄

yes

—

C₅×C₄⋊C₈

D₄

yes

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₂×C₅×C₈

Q₁₆

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

C₂×C₄⋊Q₈

—

—

—

—

C₂×D₅

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₈²

—

C₂³×C₅

#124

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#103

yes

—

C₅×C₈

#103

yes

—

#168

—

100

#168

—

101

C₂²×Q₈

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

D₂₀

yes

—

105

C₄⋊Q₈

—

106

C₄⋊Q₈

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄²

C₂²×D₅

yes

characteristic

112

#108

—

113

#106

—

114

#106

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

D₂₀

yes

—

118

C₂²⋊C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₄×D₅

yes

—

121

C₄×C₈

C₂²⋊D₅

yes

characteristic

122

C₄×C₈

C₂²⋊D₅

yes

characteristic

123

C₂²×C₅

—

124

C₂²×C₅

#390

yes

characteristic

125

C₂²×C₅

#487

yes

—

126

C₄×C₅

—

127

C₄×C₅

C₂×#103

yes

—

128

C₄×C₅

#471

yes

—

129

C₄×C₅

#492

yes

characteristic

130

C₄×C₅

#494

yes

characteristic

131

Dic₅

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₂₀

yes

characteristic

137

C₂²×C₄

#709

yes

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#677

yes

—

144

C₂×C₈

Dic₂₀

yes

—

145

C₂×C₈

#685

yes

—

146

C₂×C₈

#685

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₄×D₅

yes

characteristic

152

C₄²

C₂×D₂₀

yes

—

153

C₄²

C₂×C₂²⋊D₅

yes

characteristic

154

C₂×C₅

—

155

C₂×C₅

#1956

yes

characteristic

156

C₂×C₅

#1977

yes

—

157

C₂³

#4021

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₄×D₂₀

yes

—

164

C₂×C₄

#4020

yes

characteristic

165

C₂×C₄

C₂×#677

yes

—

166

C₂×C₄

C₂×Dic₂₀

yes

—

167

C₂×C₄

C₂×#685

yes

characteristic

168

C₂×C₄

#4076

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

#3948

yes

—

173

C₈

#3948

yes

—

174

C₅

#9767

yes

characteristic, complemented, Sylow

175

C₂²

—

176

C₂²

#64970

yes

—

177

C₂²

#65228

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

#64982

yes

characteristic

181

C₄

#64988

yes

characteristic

182

C₄

C₂×#3948

yes

—

183

C₄

#65430

yes

—

184

C₂

—

185

C₂

#10600564

yes

characteristic

186

C₂

#10600695

yes

—

	subgroup	G/H
center	C₂²	#65228
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1450	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1450	D₅
hypocenter	C₅	#9767
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#65228

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
174. C₅	#9767	semidirect

H ≅ G/N

type

174. C₅

#9767

semidirect

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

subgroup	index
C₂×#64969	2
C₅×#1450	2
#10601573	2
#10601572	2
#10600585	2
#9767	5

subgroup

index

Subgroups — #11998734

Conjugacy classes of subgroups

Complements

Maximal subgroups