Group #28372802 — subgroups

Subgroups — C₂×#20295320 #28372802

Order 13952.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16812698	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812664	C₂	1	1	yes	maximal, characteristic
3	#20295320	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812632	C₂	1	1	yes	maximal, characteristic
5	C₂×#16812628	C₂	1	1	yes	maximal, characteristic
6	C₂×#16812624	C₂	1	1	yes	maximal, characteristic
7	C₂×#16812622	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₈	—	2	1	no	—
10	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	1	yes	characteristic
12	#16812698	—	2	2	no	—
13	#16812698	C₂²	1	4	yes	—
14	#16812664	—	2	2	no	—
15	#16812664	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182008	—	2	1	no	—
18	C₂×#13182008	—	2	1	no	—
19	C₂×C₄×Dic₁₀₉	—	2	1	no	—
20	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
21	#16812632	C₂²	1	4	yes	—
22	#16812628	C₂²	1	4	yes	—
23	#16812624	C₂²	1	4	yes	—
24	#16812622	C₂²	1	4	yes	—
25	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	#107×C₁₀₉	C₂²	1	4	yes	—
29	C₂×Dic₂₁₈	—	2	1	no	—
30	C₂×Dic₂₁₈	—	2	2	no	—
31	C₂×Dic₂₁₈	—	2	4	no	—
32	C₂×Dic₂₁₈	—	4	4	no	—
33	C₂×Dic₂₁₈	Q₈	1	4	yes	complemented
34	C₂×Dic₂₁₈	C₂³	1	2	yes	—
35	C₂²×Dic₁₀₉	—	2	1	no	—
36	C₂²×Dic₁₀₉	—	2	1	no	—
37	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
38	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
39	C₂×#11724784	—	2	2	no	—
40	C₂×#11724784	C₂³	1	2	yes	—
41	C₄⋊Dic₁₀₉	—	2	1	no	—
42	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
43	#13182008	—	2	4	no	—
44	#13182008	—	2	4	no	—
45	C₄×Dic₁₀₉	—	2	4	no	—
46	C₄×Dic₁₀₉	D₄	1	2	yes	—
47	C₄×Dic₁₀₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	—	2	2	no	—
52	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	—	2	1	no	—
54	C₄⋊C₄×C₁₀₉	—	2	1	no	—
55	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
57	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	4	2	no	—
61	C₂×C₄×C₁₀₉	—	4	2	no	—
62	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₀₉	#41	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₁₀₉	—	2	4	no	—
67	C₂×Dic₁₀₉	—	2	2	no	—
68	C₂×Dic₁₀₉	—	2	2	no	—
69	C₂×Dic₁₀₉	—	2	2	no	—
70	C₂×Dic₁₀₉	—	4	2	no	—
71	C₂×Dic₁₀₉	Q₁₆	1	4	yes	—
72	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
73	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
74	Dic₂₁₈	—	2	4	no	—
75	Dic₂₁₈	—	2	2	no	—
76	Dic₂₁₈	—	4	8	no	—
77	Dic₂₁₈	C₂×Q₈	1	4	yes	complemented
78	C₈×C₁₀₉	—	2	2	no	—
79	#11724784	—	2	2	no	—
80	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
82	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₁₀₉	—	4	2	no	—
85	C₄×C₁₀₉	—	4	2	no	—
86	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
89	Dic₁₀₉	—	2	4	no	—
90	Dic₁₀₉	—	2	2	no	—
91	Dic₁₀₉	—	4	2	no	—
92	Dic₁₀₉	C₂×Q₁₆	1	4	yes	—
93	C₂×C₁₀₉	#477	1	4	yes	complemented
94	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×#477	—	109	1	no	maximal, Sylow
98	C₁₀₉	C₂×#477	1	1	yes	characteristic, complemented, Sylow
99	C₂×C₄⋊Q₈	—	109	1	no	—
100	C₂×C₄×Q₈	—	109	1	no	—
101	#477	—	109	8	no	—
102	C₂×#107	—	109	1	no	—
103	C₂×#107	D₁₀₉	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	109	1	no	—
105	C₂×#103	—	109	1	no	—
106	C₂×#103	—	109	1	no	—
107	C₂²×Q₈	—	109	1	no	—
108	C₂²×Q₈	—	218	1	no	—
109	C₂²×C₈	—	109	1	no	—
110	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
111	C₄⋊Q₈	—	109	4	no	—
112	C₄⋊Q₈	—	218	2	no	—
113	C₄×Q₈	—	109	4	no	—
114	C₄×Q₈	—	218	2	no	—
115	C₂×C₄⋊C₄	—	109	1	no	—
116	C₂×C₄⋊C₄	—	218	1	no	—
117	C₂×C₄⋊C₄	—	218	1	no	—
118	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
120	C₂×C₄²	—	109	1	no	—
121	C₂×C₄²	—	218	1	no	—
122	#107	—	109	4	no	—
123	#107	C₂×D₁₀₉	1	4	yes	—
124	C₄⋊C₈	—	109	4	no	—
125	#103	—	109	4	no	—
126	#103	—	109	4	no	—
127	C₂×Q₈	—	109	4	no	—
128	C₂×Q₈	—	109	2	no	—
129	C₂×Q₈	—	436	4	no	—
130	C₂×Q₈	—	218	4	no	—
131	C₂×Q₈	—	218	1	no	—
132	C₂×Q₈	—	218	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	109	1	no	—
136	C₂²×C₄	—	109	1	no	—
137	C₂²×C₄	—	218	1	no	—
138	C₂²×C₄	—	218	1	no	—
139	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	109	2	no	—
142	C₂×C₈	—	218	2	no	—
143	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	109	2	no	—
147	C₄⋊C₄	—	218	4	no	—
148	C₄⋊C₄	—	218	4	no	—
149	C₄⋊C₄	—	218	1	no	—
150	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
151	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
152	C₄²	—	109	2	no	—
153	C₄²	—	109	2	no	—
154	C₄²	—	218	4	no	—
155	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
156	Q₈	—	109	4	no	—
157	Q₈	—	436	8	no	—
158	Q₈	—	218	4	no	—
159	Q₈	—	218	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	109	4	no	—
165	C₂×C₄	—	109	2	no	—
166	C₂×C₄	—	109	2	no	—
167	C₂×C₄	—	436	2	no	—
168	C₂×C₄	—	218	2	no	—
169	C₂×C₄	—	218	4	no	—
170	C₂×C₄	—	218	2	no	—
171	C₂×C₄	—	218	2	no	—
172	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
174	C₂×C₄	#13182029	1	2	yes	—
175	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	218	2	no	—
178	C₂²	#16812677	1	2	yes	—
179	C₂²	Q₁₆×D₁₀₉	1	2	yes	—
180	C₂²	#16812723	1	2	yes	—
181	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	109	4	no	—
185	C₄	—	436	2	no	—
186	C₄	—	218	4	no	—
187	C₄	—	218	2	no	—
188	C₄	#16812677	1	2	yes	—
189	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
190	C₄	C₂×#13182029	1	1	yes	characteristic
191	C₂	#20295320	1	4	yes	direct product factor
192	C₂	C₂×#16812677	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₁₀₉	1	1	yes	characteristic
194	C₂	C₂×#16812723	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812698

C₂

yes

maximal, characteristic

C₂×#16812664

C₂

yes

maximal, characteristic

#20295320

C₂

yes

maximal, direct product factor

C₂×#16812632

C₂

yes

maximal, characteristic

C₂×#16812628

C₂

yes

maximal, characteristic

C₂×#16812624

C₂

yes

maximal, characteristic

C₂×#16812622

C₂

yes

maximal, characteristic

C₂×#107×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×Dic₂₁₈

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812698

—

#16812698

C₂²

yes

—

#16812664

—

#16812664

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×#13182008

—

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812632

C₂²

yes

—

#16812628

C₂²

yes

—

#16812624

C₂²

yes

—

#16812622

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

Q₈

yes

complemented

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

Q₁₆

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

C₂×Q₈

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

C₂×Q₁₆

yes

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₁₆

yes

—

C₂×C₁₀₉

#477

yes

complemented

C₂×C₁₀₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²×Q₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#477

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

100

C₂×C₄×Q₈

—

109

—

101

#477

—

109

—

102

C₂×#107

—

109

—

103

C₂×#107

D₁₀₉

yes

characteristic

104

C₂×C₄⋊C₈

—

109

—

105

C₂×#103

—

109

—

106

C₂×#103

—

109

—

107

C₂²×Q₈

—

109

—

108

C₂²×Q₈

—

218

—

109

C₂²×C₈

—

109

—

110

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

111

C₄⋊Q₈

—

109

—

112

C₄⋊Q₈

—

218

—

113

C₄×Q₈

—

109

—

114

C₄×Q₈

—

218

—

115

C₂×C₄⋊C₄

—

109

—

116

C₂×C₄⋊C₄

—

218

—

117

C₂×C₄⋊C₄

—

218

—

118

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

120

C₂×C₄²

—

109

—

121

C₂×C₄²

—

218

—

122

#107

—

109

—

123

#107

C₂×D₁₀₉

yes

—

124

C₄⋊C₈

—

109

—

125

#103

—

109

—

126

#103

—

109

—

127

C₂×Q₈

—

109

—

128

C₂×Q₈

—

109

—

129

C₂×Q₈

—

436

—

130

C₂×Q₈

—

218

—

131

C₂×Q₈

—

218

—

132

C₂×Q₈

—

218

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

109

—

136

C₂²×C₄

—

109

—

137

C₂²×C₄

—

218

—

138

C₂²×C₄

—

218

—

139

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

109

—

142

C₂×C₈

—

218

—

143

C₂×C₈

C₂²×D₁₀₉

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

109

—

147

C₄⋊C₄

—

218

—

148

C₄⋊C₄

—

218

—

149

C₄⋊C₄

—

218

—

150

C₄⋊C₄

C₂²×D₁₀₉

yes

—

151

C₄⋊C₄

C₂²×D₁₀₉

yes

—

152

C₄²

—

109

—

153

C₄²

—

109

—

154

C₄²

—

218

—

155

C₂³

D₄×D₁₀₉

yes

characteristic

156

Q₈

—

109

—

157

Q₈

—

436

—

158

Q₈

—

218

—

159

Q₈

—

218

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

109

—

165

C₂×C₄

—

109

—

166

C₂×C₄

—

109

—

167

C₂×C₄

—

436

—

168

C₂×C₄

—

218

—

169

C₂×C₄

—

218

—

170

C₂×C₄

—

218

—

171

C₂×C₄

—

218

—

172

C₂×C₄

D₄×D₁₀₉

yes

characteristic

173

C₂×C₄

Q₈×D₁₀₉

yes

—

174

C₂×C₄

#13182029

yes

—

175

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

176

C₈

—

177

C₈

—

218

—

178

C₂²

#16812677

yes

—

179

C₂²

Q₁₆×D₁₀₉

yes

—

180

C₂²

#16812723

yes

—

181

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

109

—

185

C₄

—

436

—

186

C₄

—

218

—

187

C₄

—

218

—

188

C₄

#16812677

yes

—

189

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

190

C₄

C₂×#13182029

yes

characteristic

191

C₂

#20295320

yes

direct product factor

192

C₂

C₂×#16812677

yes

characteristic

193

C₂

C₂×Q₁₆×D₁₀₉

yes

characteristic

194

C₂

C₂×#16812723

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#107×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×#107	D₁₀₉
hypocenter	C₁₀₉	C₂×#477
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#107×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #20295320	C₂	direct (N × H)
33. C₂×Dic₂₁₈	Q₈	semidirect
77. Dic₂₁₈	C₂×Q₈	semidirect
93. C₂×C₁₀₉	#477	semidirect
98. C₁₀₉	C₂×#477	semidirect
191. C₂	#20295320	direct (N × H)

H ≅ G/N

type

3. #20295320

C₂

direct (N × H)

33. C₂×Dic₂₁₈

Q₈

semidirect

77. Dic₂₁₈

C₂×Q₈

semidirect

93. C₂×C₁₀₉

#477

semidirect

98. C₁₀₉

C₂×#477

semidirect

191. C₂

#20295320

direct (N × H)

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

subgroup	index
C₂×#16812698	2
C₂×#16812664	2
#20295320	2
C₂×#16812632	2
C₂×#16812628	2
C₂×#16812624	2
C₂×#16812622	2
C₂×#107×C₁₀₉	2
C₂×#477	109

subgroup

index

C₂×#107×C₁₀₉

109

Subgroups — C₂×#20295320 #28372802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295320 #28372802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295320 #28372802