Group #31816696 — subgroups

Subgroups — #31816696

Order 17792.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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	#22581558	C₂	1	1	yes	maximal, characteristic
2	#22581430	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581278	C₂	1	2	yes	maximal, complemented
4	#22581261	C₂	1	2	yes	maximal, complemented
5	#22581154	C₂	1	2	yes	maximal, complemented
6	#22581094	C₂	1	2	yes	maximal, complemented
7	#22581055	C₂	1	2	yes	maximal, complemented
8	C₄×#14555774	C₂	1	2	yes	maximal, complemented
9	#433×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
11	C₂×#14555774	C₂²	1	2	yes	—
12	C₂×#14555774	C₂²	1	2	yes	—
13	#17984839	C₂²	1	1	yes	characteristic
14	#17984834	C₂²	1	1	yes	characteristic
15	#17984827	—	2	2	no	—
16	#17984816	C₂²	1	2	yes	—
17	#17984815	—	2	2	no	—
18	C₄×Dic₂₇₈	C₂²	1	2	yes	—
19	C₄×Dic₂₇₈	C₂²	1	2	yes	—
20	#17984807	—	2	2	no	—
21	C₄×D₅₅₆	C₂²	1	2	yes	—
22	#17984801	C₂²	1	1	yes	characteristic
23	#17984798	—	2	2	no	—
24	#17984776	C₂²	1	2	yes	—
25	#17984774	C₂²	1	2	yes	—
26	#17984767	C₂²	1	2	yes	—
27	#17984767	C₂²	1	2	yes	—
28	#17984764	C₂²	1	2	yes	—
29	#17984764	C₂²	1	2	yes	—
30	#117×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₁₃₉	C₂²	1	2	yes	—
33	#97×C₁₃₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₂₇₈	—	2	4	no	—
36	C₂×Dic₂₇₈	—	2	1	no	—
37	C₂×Dic₂₇₈	C₂³	1	2	yes	—
38	C₂×D₅₅₆	—	2	2	no	—
39	C₂×D₅₅₆	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
41	C₂²×Dic₁₃₉	—	2	1	no	—
42	C₂×C₄×D₁₃₉	—	2	2	no	—
43	#14555774	—	2	4	no	—
44	#14555774	—	2	4	no	—
45	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
46	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
47	Dic₁₃₉⋊C₄	—	2	2	no	—
48	Dic₁₃₉⋊C₄	—	2	2	no	—
49	Dic₁₃₉⋊C₄	—	2	2	no	—
50	C₂²⋊Dic₁₃₉	—	2	1	no	—
51	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
52	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
53	C₄×Dic₁₃₉	—	2	2	no	—
54	C₄×Dic₁₃₉	—	2	2	no	—
55	C₄×Dic₁₃₉	—	2	1	no	—
56	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
58	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
59	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
60	C₂²⋊C₄×C₁₃₉	D₄	1	2	yes	—
61	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
63	C₂²×D₁₃₉	—	2	2	no	—
64	Dic₂₇₈	—	2	4	no	—
65	Dic₂₇₈	—	2	2	no	—
66	Dic₂₇₈	#41	1	4	yes	—
67	D₅₅₆	—	2	2	no	—
68	D₅₅₆	—	2	2	no	—
69	D₅₅₆	—	8	2	no	—
70	C₂²⋊D₁₃₉	—	8	2	no	—
71	C₂×Dic₁₃₉	—	2	4	no	—
72	C₂×Dic₁₃₉	—	2	2	no	—
73	C₂×Dic₁₃₉	—	2	2	no	—
74	C₂×Dic₁₃₉	—	2	2	no	—
75	C₄×D₁₃₉	—	8	2	no	—
76	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	—	2	2	no	—
78	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₃₉	—	2	2	no	—
83	C₈×C₁₃₉	—	2	2	no	—
84	C₂²×C₁₃₉	—	2	1	no	—
85	C₂²×C₁₃₉	—	2	2	no	—
86	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
87	C₂×D₁₃₉	—	4	2	no	—
88	C₂×D₁₃₉	—	4	2	no	—
89	C₂×D₁₃₉	—	4	2	no	—
90	C₄×C₁₃₉	—	2	1	no	—
91	C₄×C₁₃₉	—	2	2	no	—
92	C₄×C₁₃₉	—	2	2	no	—
93	C₄×C₁₃₉	C₂×#41	1	2	yes	—
94	Dic₁₃₉	—	2	4	no	—
95	Dic₁₃₉	—	4	2	no	—
96	Dic₁₃₉	—	4	2	no	—
97	C₂×C₁₃₉	—	4	1	no	—
98	C₂×C₁₃₉	#525	1	1	yes	characteristic
99	C₂×C₁₃₉	#577	1	2	yes	—
100	D₁₃₉	—	8	2	no	—
101	C₁₃₉	#3126	1	1	yes	characteristic, complemented, Sylow
102	#3126	—	139	1	no	maximal, Sylow
103	#540	—	139	1	no	—
104	#519	—	139	1	no	—
105	#482	—	139	2	no	—
106	#479	—	139	2	no	—
107	#460	—	139	2	no	—
108	#448	—	139	2	no	—
109	#440	—	139	2	no	—
110	C₄×SD₁₆	—	139	2	no	—
111	#433	D₁₃₉	1	1	yes	characteristic, complemented
112	C₂²×Q₈	—	139	1	no	—
113	C₂×SD₁₆	—	139	2	no	—
114	C₂×SD₁₆	—	139	2	no	—
115	#127	—	139	1	no	—
116	#125	—	139	1	no	—
117	#123	—	278	2	no	—
118	#121	—	278	2	no	—
119	#121	—	139	2	no	—
120	C₄×Q₈	—	278	2	no	—
121	C₄×Q₈	—	139	2	no	—
122	C₄×Q₈	—	139	2	no	—
123	C₄×D₄	—	139	2	no	—
124	#117	—	278	2	no	—
125	#117	—	139	1	no	—
126	#117	C₂×D₁₃₉	1	1	yes	characteristic
127	#107	—	139	2	no	—
128	#106	—	139	2	no	—
129	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
130	#103	—	139	2	no	—
131	#103	—	139	2	no	—
132	#102	—	139	2	no	—
133	#102	—	139	2	no	—
134	C₂²⋊C₈	C₂×D₁₃₉	1	2	yes	—
135	#97	C₂×D₁₃₉	1	1	yes	characteristic
136	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
137	C₂×Q₈	—	278	4	no	—
138	C₂×Q₈	—	278	1	no	—
139	C₂×Q₈	—	139	2	no	—
140	C₂×D₄	—	278	2	no	—
141	C₂×D₄	—	278	2	no	—
142	C₂×D₄	—	139	2	no	—
143	C₂²×C₄	—	278	1	no	—
144	C₂²×C₄	—	278	2	no	—
145	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
146	SD₁₆	—	278	4	no	—
147	SD₁₆	—	278	4	no	—
148	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
149	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
150	C₄⋊C₄	—	278	2	no	—
151	C₄⋊C₄	—	278	2	no	—
152	C₄⋊C₄	—	278	2	no	—
153	C₄⋊C₄	—	139	2	no	—
154	C₄⋊C₄	—	139	2	no	—
155	C₄⋊C₄	D₅₅₆	1	2	yes	—
156	C₂²⋊C₄	—	278	1	no	—
157	C₂²⋊C₄	—	278	2	no	—
158	C₂²⋊C₄	—	278	2	no	—
159	C₂²⋊C₄	D₅₅₆	1	2	yes	—
160	C₄²	—	278	2	no	—
161	C₄²	—	278	2	no	—
162	C₄²	—	278	1	no	—
163	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
164	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
165	C₂³	—	278	2	no	—
166	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
167	Q₈	—	278	4	no	—
168	Q₈	—	278	2	no	—
169	Q₈	—	139	4	no	—
170	D₄	—	278	2	no	—
171	D₄	—	278	2	no	—
172	D₄	—	1112	2	no	—
173	D₄	—	1112	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	278	4	no	—
176	C₂×C₄	—	278	2	no	—
177	C₂×C₄	—	278	2	no	—
178	C₂×C₄	—	278	2	no	—
179	C₂×C₄	—	1112	2	no	—
180	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
181	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
182	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
183	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	556	2	no	—
189	C₂²	—	556	2	no	—
190	C₂²	—	556	2	no	—
191	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	278	4	no	—
196	C₄	—	556	2	no	—
197	C₄	—	556	2	no	—
198	C₄	#41×D₁₃₉	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	1112	2	no	—
201	C₂	#22581465	1	1	yes	characteristic
202	C₂	#22581765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581558

C₂

yes

maximal, characteristic

#22581430

C₂

yes

maximal, characteristic, complemented

#22581278

C₂

yes

maximal, complemented

#22581261

C₂

yes

maximal, complemented

#22581154

C₂

yes

maximal, complemented

#22581094

C₂

yes

maximal, complemented

#22581055

C₂

yes

maximal, complemented

C₄×#14555774

C₂

yes

maximal, complemented

#433×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₇₈

C₂²

yes

characteristic

C₂×#14555774

C₂²

yes

—

C₂×#14555774

C₂²

yes

—

#17984839

C₂²

yes

characteristic

#17984834

C₂²

yes

characteristic

#17984827

—

#17984816

C₂²

yes

—

#17984815

—

C₄×Dic₂₇₈

C₂²

yes

—

C₄×Dic₂₇₈

C₂²

yes

—

#17984807

—

C₄×D₅₅₆

C₂²

yes

—

#17984801

C₂²

yes

characteristic

#17984798

—

#17984776

C₂²

yes

—

#17984774

C₂²

yes

—

#17984767

C₂²

yes

—

#17984767

C₂²

yes

—

#17984764

C₂²

yes

—

#17984764

C₂²

yes

—

#117×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

C₂²⋊C₈×C₁₃₉

C₂²

yes

—

#97×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂³

yes

—

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂³

yes

—

C₂×C₂²⋊D₁₃₉

—

C₂²×Dic₁₃₉

—

C₂×C₄×D₁₃₉

—

#14555774

—

#14555774

—

C₄⋊Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

#41

yes

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×#41

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#525

yes

characteristic

C₂×C₁₃₉

#577

yes

—

100

D₁₃₉

—

101

C₁₃₉

#3126

yes

characteristic, complemented, Sylow

102

#3126

—

139

maximal, Sylow

103

#540

—

139

—

104

#519

—

139

—

105

#482

—

139

—

106

#479

—

139

—

107

#460

—

139

—

108

#448

—

139

—

109

#440

—

139

—

110

C₄×SD₁₆

—

139

—

111

#433

D₁₃₉

yes

characteristic, complemented

112

C₂²×Q₈

—

139

—

113

C₂×SD₁₆

—

139

—

114

C₂×SD₁₆

—

139

—

115

#127

—

139

—

116

#125

—

139

—

117

#123

—

278

—

118

#121

—

278

—

119

#121

—

139

—

120

C₄×Q₈

—

278

—

121

C₄×Q₈

—

139

—

122

C₄×Q₈

—

139

—

123

C₄×D₄

—

139

—

124

#117

—

278

—

125

#117

—

139

—

126

#117

C₂×D₁₃₉

yes

characteristic

127

#107

—

139

—

128

#106

—

139

—

129

C₄⋊C₈

C₂×D₁₃₉

yes

—

130

#103

—

139

—

131

#103

—

139

—

132

#102

—

139

—

133

#102

—

139

—

134

C₂²⋊C₈

C₂×D₁₃₉

yes

—

135

#97

C₂×D₁₃₉

yes

characteristic

136

C₄×C₈

C₂×D₁₃₉

yes

characteristic

137

C₂×Q₈

—

278

—

138

C₂×Q₈

—

278

—

139

C₂×Q₈

—

139

—

140

C₂×D₄

—

278

—

141

C₂×D₄

—

278

—

142

C₂×D₄

—

139

—

143

C₂²×C₄

—

278

—

144

C₂²×C₄

—

278

—

145

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

146

SD₁₆

—

278

—

147

SD₁₆

—

278

—

148

C₂×C₈

C₂²×D₁₃₉

yes

—

149

C₂×C₈

C₂²×D₁₃₉

yes

—

150

C₄⋊C₄

—

278

—

151

C₄⋊C₄

—

278

—

152

C₄⋊C₄

—

278

—

153

C₄⋊C₄

—

139

—

154

C₄⋊C₄

—

139

—

155

C₄⋊C₄

D₅₅₆

yes

—

156

C₂²⋊C₄

—

278

—

157

C₂²⋊C₄

—

278

—

158

C₂²⋊C₄

—

278

—

159

C₂²⋊C₄

D₅₅₆

yes

—

160

C₄²

—

278

—

161

C₄²

—

278

—

162

C₄²

—

278

—

163

C₄²

C₂²×D₁₃₉

yes

characteristic

164

C₄²

C₂²×D₁₃₉

yes

characteristic

165

C₂³

—

278

—

166

C₂³

C₂×D₅₅₆

yes

characteristic

167

Q₈

—

278

—

168

Q₈

—

278

—

169

Q₈

—

139

—

170

D₄

—

278

—

171

D₄

—

278

—

172

D₄

—

1112

—

173

D₄

—

1112

—

174

C₂×C₄

—

175

C₂×C₄

—

278

—

176

C₂×C₄

—

278

—

177

C₂×C₄

—

278

—

178

C₂×C₄

—

278

—

179

C₂×C₄

—

1112

—

180

C₂×C₄

C₂×D₅₅₆

yes

—

181

C₂×C₄

C₂×D₅₅₆

yes

—

182

C₂×C₄

C₂×D₅₅₆

yes

characteristic

183

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

184

C₈

—

185

C₈

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

556

—

189

C₂²

—

556

—

190

C₂²

—

556

—

191

C₂²

C₂²×D₅₅₆

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

278

—

196

C₄

—

556

—

197

C₄

—

556

—

198

C₄

#41×D₁₃₉

yes

—

199

C₂

—

200

C₂

—

1112

—

201

C₂

#22581465

yes

characteristic

202

C₂

#22581765

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#433×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581558	C₂
hypercenter	#433	D₁₃₉
hypocenter	C₁₃₉	#3126
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

subgroup

G/H

center

C₂²

C₂²×D₅₅₆

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#433×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

C₂

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #433

N	H ≅ G/N	type
2. #22581430	C₂	semidirect
3. #22581278	C₂	semidirect
4. #22581261	C₂	semidirect
5. #22581154	C₂	semidirect
6. #22581094	C₂	semidirect
7. #22581055	C₂	semidirect
8. C₄×#14555774	C₂	semidirect
9. #433×C₁₃₉	C₂	semidirect
101. C₁₃₉	#3126	semidirect
111. #433	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581430

C₂

semidirect

3. #22581278

C₂

semidirect

4. #22581261

C₂

semidirect

5. #22581154

C₂

semidirect

6. #22581094

C₂

semidirect

7. #22581055

C₂

semidirect

8. C₄×#14555774

C₂

semidirect

9. #433×C₁₃₉

C₂

semidirect

101. C₁₃₉

#3126

semidirect

111. #433

D₁₃₉

semidirect

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

subgroup	index
#22581558	2
#22581430	2
#22581278	2
#22581261	2
#22581154	2
#22581094	2
#22581055	2
C₄×#14555774	2
#433×C₁₃₉	2
#3126	139

subgroup

index

#433×C₁₃₉

139

Subgroups — #31816696

Conjugacy classes of subgroups

Complements

Maximal subgroups