Group #28378335 — subgroups

Subgroups — #28378335

Order 13952.

248 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20296045	C₂	1	1	yes	maximal, characteristic
2	#20296044	C₂	1	1	yes	maximal, characteristic
3	#20295876	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#11724786	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812766	—	2	1	no	—
7	C₄⋊#11724786	C₂²	1	2	yes	—
8	#16812756	—	2	1	no	—
9	#16812756	—	2	1	no	—
10	C₄×#11724786	—	2	1	no	—
11	C₄×#11724786	C₂²	1	2	yes	—
12	C₂×Q₈×D₁₀₉	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×Dic₂₁₈	—	2	1	no	—
15	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
16	#16812664	—	2	1	no	—
17	#16812664	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₁₀₉	—	2	1	no	—
19	C₄⋊C₄×D₁₀₉	C₄	1	1	yes	characteristic
20	C₄²×D₁₀₉	—	2	1	no	—
21	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
23	C₂×#11724786	—	2	2	no	—
24	C₂×#11724786	—	2	2	no	—
25	C₂×Dic₂₁₈	—	2	1	no	—
26	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
27	Q₈×D₁₀₉	—	2	2	no	—
28	Q₈×D₁₀₉	C₈	1	4	yes	complemented
29	C₂×C₄×D₁₀₉	—	2	1	no	—
30	C₂×C₄×D₁₀₉	—	2	1	no	—
31	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
34	C₄⋊Dic₁₀₉	—	2	1	no	—
35	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
36	#13182008	—	2	1	no	—
37	#13182008	—	2	1	no	—
38	#13182008	—	2	1	no	—
39	C₄×Dic₁₀₉	—	2	1	no	—
40	C₄×Dic₁₀₉	—	2	1	no	—
41	C₄×Dic₁₀₉	—	2	1	no	—
42	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
43	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
44	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₁₀₉	—	2	1	no	—
47	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₀₉	—	2	1	no	—
49	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₁₀₉	—	2	1	no	—
52	Q₈×C₁₀₉	C₂×C₈	1	2	yes	—
53	C₂×C₄×C₁₀₉	—	2	1	no	—
54	C₂×C₄×C₁₀₉	—	2	1	no	—
55	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₁₀₉	—	2	1	no	—
59	C₂×Dic₁₀₉	—	2	1	no	—
60	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₄×D₁₀₉	—	2	4	no	—
64	C₄×D₁₀₉	—	2	2	no	—
65	C₄×D₁₀₉	—	4	1	no	—
66	C₄×D₁₀₉	—	4	2	no	—
67	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
68	C₄×D₁₀₉	M₁₆	1	2	yes	—
69	Dic₂₁₈	—	2	1	no	—
70	Dic₂₁₈	—	2	4	no	—
71	Dic₂₁₈	C₂×C₈	1	2	yes	—
72	#11724786	—	4	2	no	—
73	#11724786	—	4	2	no	—
74	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×D₁₀₉	—	2	1	no	—
76	C₂×D₁₀₉	—	2	1	no	—
77	C₂×D₁₀₉	C₂²⋊C₈	1	2	yes	—
78	C₄×C₁₀₉	—	2	2	no	—
79	C₄×C₁₀₉	—	2	1	no	—
80	C₄×C₁₀₉	—	2	1	no	—
81	C₄×C₁₀₉	—	4	1	no	—
82	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
83	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
84	Dic₁₀₉	—	2	2	no	—
85	Dic₁₀₉	—	2	1	no	—
86	Dic₁₀₉	—	4	1	no	—
87	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
88	Dic₁₀₉	#137	1	2	yes	—
89	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₁₀₉	#419	1	1	yes	characteristic
91	C₂×C₁₀₉	#421	1	1	yes	characteristic
92	D₁₀₉	—	2	2	no	—
93	#1479	—	109	1	no	maximal, Sylow
94	C₁₀₉	#1479	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄×Q₈	—	109	1	no	—
96	#431	—	109	1	no	—
97	#431	—	109	1	no	—
98	Q₈⋊C₈	—	109	2	no	—
99	Q₈⋊C₈	—	109	2	no	—
100	C₂²×Q₈	—	109	1	no	—
101	C₄×Q₈	—	109	1	no	—
102	C₄×Q₈	—	109	1	no	—
103	C₄×Q₈	—	109	1	no	—
104	C₄×Q₈	—	218	1	no	—
105	C₄×Q₈	—	218	1	no	—
106	C₄×Q₈	#89137	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	109	1	no	—
108	C₂×C₄⋊C₄	—	218	1	no	—
109	C₂×C₄²	—	109	1	no	—
110	C₂×C₄²	—	218	1	no	—
111	C₄⋊C₈	—	109	2	no	—
112	C₄⋊C₈	—	218	1	no	—
113	C₂²⋊C₈	—	218	1	no	—
114	C₂²⋊C₈	—	218	1	no	—
115	C₄×C₈	—	109	2	no	—
116	C₄×C₈	—	218	1	no	—
117	C₂×Q₈	—	109	4	no	—
118	C₂×Q₈	—	109	1	no	—
119	C₂×Q₈	—	218	1	no	—
120	C₂×Q₈	—	218	2	no	—
121	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
122	C₂²×C₄	—	109	1	no	—
123	C₂²×C₄	—	109	1	no	—
124	C₂²×C₄	—	109	1	no	—
125	C₂²×C₄	—	218	1	no	—
126	C₂²×C₄	—	218	1	no	—
127	C₂×C₈	—	218	2	no	—
128	C₂×C₈	—	218	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	109	1	no	—
131	C₄⋊C₄	—	218	1	no	—
132	C₄⋊C₄	—	218	1	no	—
133	C₄⋊C₄	—	218	1	no	—
134	C₄⋊C₄	—	218	1	no	—
135	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	—	109	1	no	—
138	C₄²	—	109	1	no	—
139	C₄²	—	109	1	no	—
140	C₄²	—	218	1	no	—
141	C₄²	—	218	1	no	—
142	C₄²	—	218	1	no	—
143	C₄²	C₂×#89137	1	1	yes	characteristic
144	C₂³	—	109	1	no	—
145	Q₈	—	2	1	no	—
146	Q₈	—	109	2	no	—
147	Q₈	—	218	4	no	—
148	Q₈	—	218	1	no	—
149	Q₈	D₁₀₉⋊C₈	1	2	yes	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	109	1	no	—
153	C₂×C₄	—	109	1	no	—
154	C₂×C₄	—	109	2	no	—
155	C₂×C₄	—	109	2	no	—
156	C₂×C₄	—	109	1	no	—
157	C₂×C₄	—	436	1	no	—
158	C₂×C₄	—	436	2	no	—
159	C₂×C₄	—	218	2	no	—
160	C₂×C₄	—	218	4	no	—
161	C₂×C₄	—	218	1	no	—
162	C₂×C₄	—	218	1	no	—
163	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
164	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
165	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
166	C₈	—	436	2	no	—
167	C₈	—	436	2	no	—
168	C₂²	—	109	2	no	—
169	C₂²	—	218	1	no	—
170	C₂²	—	218	1	no	—
171	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	109	2	no	—
177	C₄	—	109	2	no	—
178	C₄	—	436	1	no	—
179	C₄	—	218	2	no	—
180	C₄	—	218	1	no	—
181	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
182	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
183	C₂	—	218	2	no	—
184	C₂	D₁₀₉⋊(C₂²⋊C₈)	1	1	yes	characteristic
185	C₂	#20296019	1	1	yes	characteristic
186	C₂	#20296023	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296045

C₂

yes

maximal, characteristic

#20296044

C₂

yes

maximal, characteristic

#20295876

C₂

yes

maximal, complemented

Q₈⋊#11724786

C₂

yes

maximal, complemented

C₄×Q₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812766

—

C₄⋊#11724786

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×Q₈×D₁₀₉

C₄

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

C₄

yes

characteristic

#16812664

—

#16812664

C₂²

yes

characteristic

C₄⋊C₄×D₁₀₉

—

C₄⋊C₄×D₁₀₉

C₄

yes

characteristic

C₄²×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

C₂×#11724786

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂×C₄

yes

characteristic

Q₈×D₁₀₉

—

Q₈×D₁₀₉

C₈

yes

complemented

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

—

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×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

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₈

yes

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₈

yes

—

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

C₂×C₂²⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1479

—

109

maximal, Sylow

C₁₀₉

#1479

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

109

—

#431

—

109

—

#431

—

109

—

Q₈⋊C₈

—

109

—

Q₈⋊C₈

—

109

—

100

C₂²×Q₈

—

109

—

101

C₄×Q₈

—

109

—

102

C₄×Q₈

—

109

—

103

C₄×Q₈

—

109

—

104

C₄×Q₈

—

218

—

105

C₄×Q₈

—

218

—

106

C₄×Q₈

#89137

yes

characteristic

107

C₂×C₄⋊C₄

—

109

—

108

C₂×C₄⋊C₄

—

218

—

109

C₂×C₄²

—

109

—

110

C₂×C₄²

—

218

—

111

C₄⋊C₈

—

109

—

112

C₄⋊C₈

—

218

—

113

C₂²⋊C₈

—

218

—

114

C₂²⋊C₈

—

218

—

115

C₄×C₈

—

109

—

116

C₄×C₈

—

218

—

117

C₂×Q₈

—

109

—

118

C₂×Q₈

—

109

—

119

C₂×Q₈

—

218

—

120

C₂×Q₈

—

218

—

121

C₂×Q₈

C₂×#89137

yes

characteristic

122

C₂²×C₄

—

109

—

123

C₂²×C₄

—

109

—

124

C₂²×C₄

—

109

—

125

C₂²×C₄

—

218

—

126

C₂²×C₄

—

218

—

127

C₂×C₈

—

218

—

128

C₂×C₈

—

218

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

109

—

131

C₄⋊C₄

—

218

—

132

C₄⋊C₄

—

218

—

133

C₄⋊C₄

—

218

—

134

C₄⋊C₄

—

218

—

135

C₄⋊C₄

C₂×#89137

yes

characteristic

136

C₄²

—

137

C₄²

—

109

—

138

C₄²

—

109

—

139

C₄²

—

109

—

140

C₄²

—

218

—

141

C₄²

—

218

—

142

C₄²

—

218

—

143

C₄²

C₂×#89137

yes

characteristic

144

C₂³

—

109

—

145

Q₈

—

146

Q₈

—

109

—

147

Q₈

—

218

—

148

Q₈

—

218

—

149

Q₈

D₁₀₉⋊C₈

yes

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

109

—

153

C₂×C₄

—

109

—

154

C₂×C₄

—

109

—

155

C₂×C₄

—

109

—

156

C₂×C₄

—

109

—

157

C₂×C₄

—

436

—

158

C₂×C₄

—

436

—

159

C₂×C₄

—

218

—

160

C₂×C₄

—

218

—

161

C₂×C₄

—

218

—

162

C₂×C₄

—

218

—

163

C₂×C₄

C₂²⋊#89137

yes

characteristic

164

C₂×C₄

C₂²⋊#89137

yes

characteristic

165

C₂×C₄

C₂²×#89137

yes

characteristic

166

C₈

—

436

—

167

C₈

—

436

—

168

C₂²

—

109

—

169

C₂²

—

218

—

170

C₂²

—

218

—

171

C₂²

C₂×C₂²⋊#89137

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

109

—

177

C₄

—

109

—

178

C₄

—

436

—

179

C₄

—

218

—

180

C₄

—

218

—

181

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

182

C₄

D₁₀₉⋊M₁₆

yes

characteristic

183

C₂

—

218

—

184

C₂

D₁₀₉⋊(C₂²⋊C₈)

yes

characteristic

185

C₂

#20296019

yes

characteristic

186

C₂

#20296023

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×Q₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄×Q₈	#89137
hypocenter	C₁₀₉	#1479
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×Q₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄×Q₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
3. #20295876	C₂	semidirect
4. Q₈⋊#11724786	C₂	semidirect
28. Q₈×D₁₀₉	C₈	semidirect
94. C₁₀₉	#1479	semidirect

H ≅ G/N

type

3. #20295876

C₂

semidirect

4. Q₈⋊#11724786

C₂

semidirect

28. Q₈×D₁₀₉

C₈

semidirect

94. C₁₀₉

#1479

semidirect

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

subgroup	index
#20296045	2
#20296044	2
#20295876	2
Q₈⋊#11724786	2
C₄×Q₈×D₁₀₉	2
#1479	109

subgroup

index

C₄×Q₈×D₁₀₉

109

Subgroups — #28378335

Conjugacy classes of subgroups

Complements

Maximal subgroups