Group #28379213 — subgroups

Subgroups — #28379213

Order 13952.

300 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20295972	C₂	1	1	yes	maximal, characteristic
3	#20295899	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#89137	—	2	1	no	—
7	C₂×C₄⋊#89137	—	2	1	no	—
8	C₂×C₄⋊#89137	C₂²	1	2	yes	—
9	C₄×#11724786	—	2	1	no	—
10	C₄×#11724786	—	2	1	no	—
11	C₂×Q₈×D₁₀₉	—	2	1	no	—
12	#16812699	C₄	1	1	yes	characteristic, complemented
13	#16812698	—	2	1	no	—
14	#16812698	C₄	1	2	yes	complemented
15	#16812697	—	2	1	no	—
16	C₄⋊C₄×D₁₀₉	—	2	1	no	—
17	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
18	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	C₂²×#89137	—	2	2	no	—
21	D₁₀₉⋊C₈	—	2	2	no	—
22	D₁₀₉⋊C₈	Q₈	1	4	yes	complemented
23	C₂×#11724786	—	2	2	no	—
24	C₄⋊#89137	—	2	4	no	—
25	C₄⋊#89137	—	4	2	no	—
26	C₄⋊#89137	—	4	2	no	—
27	C₂×Dic₂₁₈	—	2	1	no	—
28	C₂×Dic₂₁₈	—	2	2	no	—
29	Q₈×D₁₀₉	—	4	2	no	—
30	Q₈×D₁₀₉	—	4	2	no	—
31	C₂×C₄×D₁₀₉	—	2	2	no	—
32	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₁₀₉	—	2	1	no	—
36	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
37	#13182008	—	2	2	no	—
38	#13182008	—	2	2	no	—
39	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
41	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₀₉	—	2	1	no	—
43	C₄⋊C₄×C₁₀₉	—	2	1	no	—
44	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×#89137	—	4	2	no	—
48	C₂×#89137	—	4	2	no	—
49	C₂×#89137	—	4	2	no	—
50	Q₈×C₁₀₉	—	4	2	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₁₀₉	—	2	2	no	—
56	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₄×D₁₀₉	—	4	2	no	—
60	C₄×D₁₀₉	—	4	2	no	—
61	C₄×D₁₀₉	D₈	1	2	yes	—
62	C₄×D₁₀₉	SD₁₆	1	2	yes	—
63	C₄×D₁₀₉	SD₁₆	1	2	yes	—
64	C₄×D₁₀₉	Q₁₆	1	2	yes	—
65	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
66	C₄×D₁₀₉	#41	1	2	yes	—
67	Dic₂₁₈	—	4	2	no	—
68	Dic₂₁₈	—	4	4	no	—
69	#11724786	—	2	4	no	—
70	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₁₀₉	#124	1	1	yes	characteristic
72	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×D₁₀₉	C₂×D₈	1	1	yes	characteristic
74	C₂×D₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×D₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×D₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
77	#89137	—	8	2	no	—
78	C₄×C₁₀₉	—	4	2	no	—
79	C₄×C₁₀₉	#102	1	2	yes	—
80	C₄×C₁₀₉	#103	1	2	yes	—
81	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
82	Dic₁₀₉	—	4	2	no	—
83	Dic₁₀₉	#102	1	2	yes	—
84	Dic₁₀₉	#103	1	2	yes	—
85	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₁₀₉	#391	1	1	yes	characteristic
87	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
89	D₁₀₉	#486	1	1	yes	characteristic
90	D₁₀₉	#487	1	1	yes	characteristic
91	D₁₀₉	#498	1	1	yes	characteristic
92	D₁₀₉	#500	1	1	yes	characteristic
93	#1990	—	109	1	no	maximal, Sylow
94	C₁₀₉	#1990	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄⋊Q₈	—	109	1	no	—
96	C₂×C₄×C₈	—	109	1	no	—
97	#384	—	109	1	no	—
98	#340	—	109	4	no	—
99	C₂²×Q₈	—	218	1	no	—
100	C₂²×C₈	—	109	2	no	—
101	C₄⋊Q₈	—	109	2	no	—
102	C₄⋊Q₈	—	109	1	no	—
103	C₄⋊Q₈	—	218	1	no	—
104	C₄⋊Q₈	—	218	1	no	—
105	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
106	C₂×C₄⋊C₄	—	109	2	no	—
107	C₂×C₄⋊C₄	—	109	2	no	—
108	C₂×C₄⋊C₄	—	218	1	no	—
109	C₂×C₄⋊C₄	—	218	1	no	—
110	C₂×C₄⋊C₄	—	218	1	no	—
111	C₂×C₄²	—	109	1	no	—
112	C₄×C₈	—	218	1	no	—
113	C₄×C₈	—	218	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	436	2	no	—
116	C₂×Q₈	—	436	2	no	—
117	C₂×Q₈	—	218	2	no	—
118	C₂×Q₈	—	218	1	no	—
119	C₂²×C₄	—	109	1	no	—
120	C₂²×C₄	—	109	1	no	—
121	C₂²×C₄	—	109	1	no	—
122	C₂²×C₄	—	218	2	no	—
123	C₂²×C₄	—	218	2	no	—
124	C₂×C₈	—	109	4	no	—
125	C₂×C₈	—	218	2	no	—
126	C₂×C₈	—	218	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	109	2	no	—
129	C₄⋊C₄	—	436	2	no	—
130	C₄⋊C₄	—	436	2	no	—
131	C₄⋊C₄	—	218	2	no	—
132	C₄⋊C₄	—	218	2	no	—
133	C₄⋊C₄	—	218	4	no	—
134	C₄⋊C₄	—	218	1	no	—
135	C₄⋊C₄	C₂×#89137	1	2	yes	—
136	C₄²	—	109	1	no	—
137	C₄²	—	109	1	no	—
138	C₄²	—	109	1	no	—
139	C₄²	C₂×#89137	1	1	yes	characteristic
140	C₂³	—	109	1	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	436	4	no	—
143	Q₈	—	436	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	109	1	no	—
146	C₂×C₄	—	109	1	no	—
147	C₂×C₄	—	109	1	no	—
148	C₂×C₄	—	109	2	no	—
149	C₂×C₄	—	109	2	no	—
150	C₂×C₄	—	109	2	no	—
151	C₂×C₄	—	109	2	no	—
152	C₂×C₄	—	109	2	no	—
153	C₂×C₄	—	109	2	no	—
154	C₂×C₄	—	436	2	no	—
155	C₂×C₄	—	436	2	no	—
156	C₂×C₄	—	436	2	no	—
157	C₂×C₄	—	436	2	no	—
158	C₂×C₄	—	436	2	no	—
159	C₂×C₄	—	218	2	no	—
160	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
162	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
163	C₈	—	218	4	no	—
164	C₂²	—	109	1	no	—
165	C₂²	—	109	1	no	—
166	C₂²	—	109	1	no	—
167	C₂²	—	109	1	no	—
168	C₂²	—	109	1	no	—
169	C₂²	—	109	1	no	—
170	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	872	2	no	—
173	C₄	—	109	2	no	—
174	C₄	—	109	2	no	—
175	C₄	—	109	2	no	—
176	C₄	—	436	2	no	—
177	C₄	#16812761	1	2	yes	—
178	C₄	#16812762	1	2	yes	—
179	C₄	Q₈×#89137	1	2	yes	—
180	C₂	—	109	1	no	—
181	C₂	—	109	1	no	—
182	C₂	—	109	1	no	—
183	C₂	—	109	1	no	—
184	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
185	C₂	C₂×#16812761	1	1	yes	characteristic
186	C₂	C₂×#16812762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

#20295972

C₂

yes

maximal, characteristic

#20295899

C₂

yes

maximal

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₂²

yes

—

C₂×C₄⋊#89137

—

C₂×C₄⋊#89137

—

C₂×C₄⋊#89137

C₂²

yes

—

C₄×#11724786

—

C₄×#11724786

—

C₂×Q₈×D₁₀₉

—

#16812699

C₄

yes

characteristic, complemented

#16812698

—

#16812698

C₄

yes

complemented

#16812697

—

C₄⋊C₄×D₁₀₉

—

C₄⋊C₄×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

Q₈

yes

—

—

—

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

—

—

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₈

yes

—

C₄×D₁₀₉

SD₁₆

yes

—

C₄×D₁₀₉

SD₁₆

yes

—

C₄×D₁₀₉

Q₁₆

yes

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₄×D₁₀₉

#41

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

#124

yes

characteristic

C₂×D₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×D₁₀₉

C₂×D₈

yes

characteristic

C₂×D₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×D₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×D₁₀₉

C₂×Q₁₆

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#102

yes

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

#102

yes

—

Dic₁₀₉

#103

yes

—

Dic₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1990

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄×C₈

—

109

—

#384

—

109

—

#340

—

109

—

C₂²×Q₈

—

218

—

100

C₂²×C₈

—

109

—

101

C₄⋊Q₈

—

109

—

102

C₄⋊Q₈

—

109

—

103

C₄⋊Q₈

—

218

—

104

C₄⋊Q₈

—

218

—

105

C₄⋊Q₈

#89137

yes

characteristic, complemented

106

C₂×C₄⋊C₄

—

109

—

107

C₂×C₄⋊C₄

—

109

—

108

C₂×C₄⋊C₄

—

218

—

109

C₂×C₄⋊C₄

—

218

—

110

C₂×C₄⋊C₄

—

218

—

111

C₂×C₄²

—

109

—

112

C₄×C₈

—

218

—

113

C₄×C₈

—

218

—

114

C₂×Q₈

—

115

C₂×Q₈

—

436

—

116

C₂×Q₈

—

436

—

117

C₂×Q₈

—

218

—

118

C₂×Q₈

—

218

—

119

C₂²×C₄

—

109

—

120

C₂²×C₄

—

109

—

121

C₂²×C₄

—

109

—

122

C₂²×C₄

—

218

—

123

C₂²×C₄

—

218

—

124

C₂×C₈

—

109

—

125

C₂×C₈

—

218

—

126

C₂×C₈

—

218

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

109

—

129

C₄⋊C₄

—

436

—

130

C₄⋊C₄

—

436

—

131

C₄⋊C₄

—

218

—

132

C₄⋊C₄

—

218

—

133

C₄⋊C₄

—

218

—

134

C₄⋊C₄

—

218

—

135

C₄⋊C₄

C₂×#89137

yes

—

136

C₄²

—

109

—

137

C₄²

—

109

—

138

C₄²

—

109

—

139

C₄²

C₂×#89137

yes

characteristic

140

C₂³

—

109

—

141

Q₈

—

142

Q₈

—

436

—

143

Q₈

—

436

—

144

C₂×C₄

—

145

C₂×C₄

—

109

—

146

C₂×C₄

—

109

—

147

C₂×C₄

—

109

—

148

C₂×C₄

—

109

—

149

C₂×C₄

—

109

—

150

C₂×C₄

—

109

—

151

C₂×C₄

—

109

—

152

C₂×C₄

—

109

—

153

C₂×C₄

—

109

—

154

C₂×C₄

—

436

—

155

C₂×C₄

—

436

—

156

C₂×C₄

—

436

—

157

C₂×C₄

—

436

—

158

C₂×C₄

—

436

—

159

C₂×C₄

—

218

—

160

C₂×C₄

C₂²⋊#89137

yes

characteristic

161

C₂×C₄

C₂²⋊#89137

yes

characteristic

162

C₂×C₄

C₂²×#89137

yes

characteristic

163

C₈

—

218

—

164

C₂²

—

109

—

165

C₂²

—

109

—

166

C₂²

—

109

—

167

C₂²

—

109

—

168

C₂²

—

109

—

169

C₂²

—

109

—

170

C₂²

C₂×C₂²⋊#89137

yes

characteristic

171

C₄

—

172

C₄

—

872

—

173

C₄

—

109

—

174

C₄

—

109

—

175

C₄

—

109

—

176

C₄

—

436

—

177

C₄

#16812761

yes

—

178

C₄

#16812762

yes

—

179

C₄

Q₈×#89137

yes

—

180

C₂

—

109

—

181

C₂

—

109

—

182

C₂

—

109

—

183

C₂

—

109

—

184

C₂

(C₂×#89137)⋊Q₈

yes

characteristic

185

C₂

C₂×#16812761

yes

characteristic

186

C₂

C₂×#16812762

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂×D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1990
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄⋊Q₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂×D₄

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #16812699	C₄	semidirect
14. #16812698	C₄	semidirect
19. C₄⋊Q₈×C₁₀₉	C₄	semidirect
22. D₁₀₉⋊C₈	Q₈	semidirect
94. C₁₀₉	#1990	semidirect
105. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

12. #16812699

C₄

semidirect

14. #16812698

C₄

semidirect

19. C₄⋊Q₈×C₁₀₉

C₄

semidirect

22. D₁₀₉⋊C₈

Q₈

semidirect

94. C₁₀₉

#1990

semidirect

105. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
C₄×D₁₀₉⋊C₈	2
#20295972	2
#20295899	2
C₄⋊Q₈×D₁₀₉	2
#1990	109

subgroup

index

C₄×D₁₀₉⋊C₈

#20295972

#20295899

C₄⋊Q₈×D₁₀₉

#1990

109

Subgroups — #28379213

Conjugacy classes of subgroups

Complements

Maximal subgroups