Group #28363034 — subgroups

Subgroups — #28363034

Order 13952.

308 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812622	C₂	1	2	yes	maximal
4	#20294788	C₂	1	2	yes	maximal
5	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₄×Dic₂₁₈	—	2	2	no	—
9	C₄×Dic₂₁₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#13182008	—	2	1	no	—
12	C₂×C₄×Dic₁₀₉	—	2	1	no	—
13	C₄⋊#11724784	—	2	2	no	—
14	#16812622	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
16	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₈	—	2	1	no	—
20	C₂×Dic₂₁₈	—	2	1	no	—
21	C₂×Dic₂₁₈	—	2	1	no	—
22	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
23	C₂×Dic₂₁₈	D₄	1	2	yes	—
24	C₂×Dic₂₁₈	Q₈	1	2	yes	—
25	C₂²×Dic₁₀₉	—	2	1	no	—
26	C₂²×Dic₁₀₉	—	2	1	no	—
27	C₂×#11724784	—	2	2	no	—
28	C₂×#11724784	—	2	2	no	—
29	C₂×#11724784	—	2	2	no	—
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
32	#13182008	—	2	4	no	—
33	C₄×Dic₁₀₉	—	2	4	no	—
34	C₂²×C₄×C₁₀₉	—	2	2	no	—
35	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	—	4	2	no	—
41	C₄⋊C₄×C₁₀₉	—	4	2	no	—
42	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄²×C₁₀₉	D₄	1	2	yes	—
44	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄×C₁₀₉	—	2	1	no	—
47	C₂×C₄×C₁₀₉	—	4	2	no	—
48	C₂×C₄×C₁₀₉	—	4	2	no	—
49	C₂×C₄×C₁₀₉	—	4	2	no	—
50	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
60	C₂×Dic₁₀₉	—	2	2	no	—
61	C₂×Dic₁₀₉	—	2	2	no	—
62	C₂×Dic₁₀₉	—	2	2	no	—
63	C₂×Dic₁₀₉	—	2	2	no	—
64	C₂×Dic₁₀₉	—	4	1	no	—
65	C₂×Dic₁₀₉	—	4	1	no	—
66	Dic₂₁₈	—	2	2	no	—
67	Dic₂₁₈	—	2	2	no	—
68	Dic₂₁₈	—	2	2	no	—
69	Dic₂₁₈	C₄⋊C₄	1	4	yes	complemented
70	#11724784	—	4	2	no	—
71	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₀₉	#121	1	1	yes	characteristic
73	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²×C₁₀₉	#137	1	1	yes	characteristic
78	C₄×C₁₀₉	—	2	2	no	—
79	C₄×C₁₀₉	—	8	2	no	—
80	C₄×C₁₀₉	#103	1	4	yes	complemented
81	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₁₀₉	#117	1	1	yes	characteristic
83	C₄×C₁₀₉	#121	1	1	yes	characteristic
84	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
85	Dic₁₀₉	—	2	4	no	—
86	Dic₁₀₉	—	4	2	no	—
87	C₂×C₁₀₉	#380	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
89	C₂×C₁₀₉	#417	1	1	yes	characteristic
90	C₂×C₁₀₉	#460	1	1	yes	characteristic
91	C₂×C₁₀₉	#462	1	1	yes	characteristic
92	C₂×C₁₀₉	#475	1	1	yes	characteristic
93	C₂×C₁₀₉	#477	1	1	yes	characteristic
94	#1803	—	109	1	no	maximal, Sylow
95	C₁₀₉	#1803	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	—	109	1	no	—
97	C₂×C₄⋊C₈	—	109	1	no	—
98	C₂×#103	—	109	2	no	—
99	#384	D₁₀₉	1	1	yes	characteristic
100	#340	—	109	2	no	—
101	C₂²×Q₈	—	109	1	no	—
102	C₂²×C₈	—	109	2	no	—
103	C₄×Q₈	—	109	4	no	—
104	C₄×Q₈	—	218	2	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	109	1	no	—
108	C₂×C₄⋊C₄	—	218	1	no	—
109	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
110	C₂×C₄²	—	218	1	no	—
111	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
112	C₄⋊C₈	—	218	2	no	—
113	#103	—	218	4	no	—
114	C₂×Q₈	—	109	2	no	—
115	C₂×Q₈	—	109	2	no	—
116	C₂×Q₈	—	109	2	no	—
117	C₂×Q₈	—	218	1	no	—
118	C₂×Q₈	—	218	1	no	—
119	C₂×Q₈	—	218	1	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	218	1	no	—
122	C₂²×C₄	—	218	1	no	—
123	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
126	C₂×C₈	—	218	2	no	—
127	C₂×C₈	—	218	2	no	—
128	C₂×C₈	—	218	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	109	2	no	—
134	C₄⋊C₄	—	218	4	no	—
135	C₄⋊C₄	—	218	1	no	—
136	C₄²	—	218	4	no	—
137	C₄²	C₄×D₁₀₉	1	2	yes	—
138	C₄²	D₄₃₆	1	2	yes	—
139	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
140	Q₈	—	109	4	no	—
141	Q₈	—	218	2	no	—
142	Q₈	—	218	2	no	—
143	Q₈	—	218	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	436	1	no	—
150	C₂×C₄	—	436	1	no	—
151	C₂×C₄	—	218	2	no	—
152	C₂×C₄	—	218	2	no	—
153	C₂×C₄	—	218	2	no	—
154	C₂×C₄	—	218	2	no	—
155	C₂×C₄	#13182006	1	2	yes	—
156	C₂×C₄	#13182006	1	2	yes	—
157	C₂×C₄	#13182017	1	2	yes	—
158	C₂×C₄	#13182019	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
160	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
163	C₂×C₄	#13182028	1	1	yes	characteristic
164	C₂×C₄	#13182030	1	1	yes	characteristic
165	C₈	—	436	2	no	—
166	C₂²	C₂×#13182006	1	1	yes	characteristic
167	C₂²	#16812670	1	1	yes	characteristic
168	C₂²	#16812675	1	1	yes	characteristic
169	C₂²	C₂×#13182017	1	1	yes	characteristic
170	C₂²	C₂×#13182019	1	1	yes	characteristic
171	C₂²	#16812725	1	1	yes	characteristic
172	C₂²	#16812730	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	8	2	no	—
175	C₄	—	436	2	no	—
176	C₄	—	218	4	no	—
177	C₄	#16812622	1	4	yes	complemented
178	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
179	C₄	#16812657	1	1	yes	characteristic
180	C₄	#16812674	1	1	yes	characteristic
181	C₄	#16812679	1	1	yes	characteristic
182	C₂	#20294883	1	1	yes	characteristic
183	C₂	C₂×#16812622	1	1	yes	characteristic
184	C₂	#20295025	1	1	yes	characteristic
185	C₂	#20295223	1	1	yes	characteristic
186	C₂	#20295233	1	1	yes	characteristic
187	C₂	#20295308	1	1	yes	characteristic
188	C₂	#20295318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₁₈

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

C₂×#16812622

C₂

yes

maximal

#20294788

C₂

yes

maximal

#384×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

—

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

—

C₄⋊#11724784

—

#16812622

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂×Dic₂₁₈

D₄

yes

—

C₂×Dic₂₁₈

Q₈

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₄⋊C₄

yes

complemented

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#103

yes

complemented

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1803

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×#103

—

109

—

#384

D₁₀₉

yes

characteristic

100

#340

—

109

—

101

C₂²×Q₈

—

109

—

102

C₂²×C₈

—

109

—

103

C₄×Q₈

—

109

—

104

C₄×Q₈

—

218

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

109

—

108

C₂×C₄⋊C₄

—

218

—

109

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

110

C₂×C₄²

—

218

—

111

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

112

C₄⋊C₈

—

218

—

113

#103

—

218

—

114

C₂×Q₈

—

109

—

115

C₂×Q₈

—

109

—

116

C₂×Q₈

—

109

—

117

C₂×Q₈

—

218

—

118

C₂×Q₈

—

218

—

119

C₂×Q₈

—

218

—

120

C₂²×C₄

—

121

C₂²×C₄

—

218

—

122

C₂²×C₄

—

218

—

123

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

124

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

125

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

126

C₂×C₈

—

218

—

127

C₂×C₈

—

218

—

128

C₂×C₈

—

218

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

109

—

134

C₄⋊C₄

—

218

—

135

C₄⋊C₄

—

218

—

136

C₄²

—

218

—

137

C₄²

C₄×D₁₀₉

yes

—

138

C₄²

D₄₃₆

yes

—

139

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

140

Q₈

—

109

—

141

Q₈

—

218

—

142

Q₈

—

218

—

143

Q₈

—

218

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

436

—

150

C₂×C₄

—

436

—

151

C₂×C₄

—

218

—

152

C₂×C₄

—

218

—

153

C₂×C₄

—

218

—

154

C₂×C₄

—

218

—

155

C₂×C₄

#13182006

yes

—

156

C₂×C₄

#13182006

yes

—

157

C₂×C₄

#13182017

yes

—

158

C₂×C₄

#13182019

yes

—

159

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

160

C₂×C₄

D₄×D₁₀₉

yes

characteristic

161

C₂×C₄

C₂×D₄₃₆

yes

characteristic

162

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

163

C₂×C₄

#13182028

yes

characteristic

164

C₂×C₄

#13182030

yes

characteristic

165

C₈

—

436

—

166

C₂²

C₂×#13182006

yes

characteristic

167

C₂²

#16812670

yes

characteristic

168

C₂²

#16812675

yes

characteristic

169

C₂²

C₂×#13182017

yes

characteristic

170

C₂²

C₂×#13182019

yes

characteristic

171

C₂²

#16812725

yes

characteristic

172

C₂²

#16812730

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

436

—

176

C₄

—

218

—

177

C₄

#16812622

yes

complemented

178

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

179

C₄

#16812657

yes

characteristic

180

C₄

#16812674

yes

characteristic

181

C₄

#16812679

yes

characteristic

182

C₂

#20294883

yes

characteristic

183

C₂

C₂×#16812622

yes

characteristic

184

C₂

#20295025

yes

characteristic

185

C₂

#20295223

yes

characteristic

186

C₂

#20295233

yes

characteristic

187

C₂

#20295308

yes

characteristic

188

C₂

#20295318

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#384×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#1803
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#384×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×Dic₂₁₈	C₄	semidirect
69. Dic₂₁₈	C₄⋊C₄	semidirect
80. C₄×C₁₀₉	#103	semidirect
95. C₁₀₉	#1803	semidirect
177. C₄	#16812622	semidirect

H ≅ G/N

type

9. C₄×Dic₂₁₈

C₄

semidirect

69. Dic₂₁₈

C₄⋊C₄

semidirect

80. C₄×C₁₀₉

#103

semidirect

95. C₁₀₉

#1803

semidirect

177. C₄

#16812622

semidirect

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

subgroup	index
C₂×C₄×Dic₂₁₈	2
C₂×C₄⋊#11724784	2
C₂×#16812622	2
#20294788	2
#384×C₁₀₉	2
#1803	109

subgroup

index

C₂×C₄×Dic₂₁₈

#384×C₁₀₉

109

Subgroups — #28363034

Conjugacy classes of subgroups

Complements

Maximal subgroups