Group #28363787 — subgroups

Subgroups — #28363787

Order 13952.

256 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16812632	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812628	C₂	1	1	yes	maximal, characteristic
3	#20294895	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#20294789	C₂	1	1	yes	maximal, characteristic
6	#20294788	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#13182008	—	2	1	no	—
12	C₂×#13182008	—	2	1	no	—
13	C₂×C₄×Dic₁₀₉	—	2	1	no	—
14	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#16812632	—	2	2	no	—
16	#16812628	C₄	1	4	yes	complemented
17	#16812606	—	2	1	no	—
18	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	#107×C₁₀₉	—	2	2	no	—
22	C₂²×Dic₁₀₉	—	2	1	no	—
23	C₂²×Dic₁₀₉	—	2	1	no	—
24	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
25	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
26	C₂×#11724784	—	2	1	no	—
27	C₂×#11724784	—	2	1	no	—
28	C₂×#11724784	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	Q₈	1	2	yes	—
33	#13182008	—	4	2	no	—
34	#13182008	—	4	2	no	—
35	C₄×Dic₁₀₉	—	4	2	no	—
36	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₀₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₀₉	—	2	1	no	—
39	C₂²×C₄×C₁₀₉	—	2	1	no	—
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	—	2	1	no	—
42	C₂×C₈×C₁₀₉	—	2	1	no	—
43	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	—	2	1	no	—
45	C₄⋊C₄×C₁₀₉	—	2	1	no	—
46	C₄⋊C₄×C₁₀₉	—	2	1	no	—
47	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
48	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	—	4	1	no	—
51	C₂×C₄×C₁₀₉	—	4	1	no	—
52	C₂×C₄×C₁₀₉	—	4	1	no	—
53	C₂×C₄×C₁₀₉	—	4	1	no	—
54	C₂×C₄×C₁₀₉	—	4	1	no	—
55	C₂×C₄×C₁₀₉	C₂²×C₄	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	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
61	C₂×Dic₁₀₉	—	2	1	no	—
62	C₂×Dic₁₀₉	—	2	1	no	—
63	C₂×Dic₁₀₉	—	2	2	no	—
64	C₂×Dic₁₀₉	—	4	1	no	—
65	C₂×Dic₁₀₉	—	4	1	no	—
66	C₂×Dic₁₀₉	—	4	1	no	—
67	C₂×Dic₁₀₉	—	4	1	no	—
68	C₂×Dic₁₀₉	—	4	1	no	—
69	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
70	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
71	C₂×Dic₁₀₉	D₈	1	2	yes	—
72	C₂×Dic₁₀₉	Q₁₆	1	2	yes	—
73	C₈×C₁₀₉	—	2	2	no	—
74	#11724784	—	2	2	no	—
75	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₀₉	#121	1	1	yes	characteristic
77	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
80	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₁₀₉	#137	1	1	yes	characteristic
82	C₄×C₁₀₉	—	4	2	no	—
83	C₄×C₁₀₉	—	8	1	no	—
84	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
85	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
86	C₄×C₁₀₉	#125	1	1	yes	characteristic
87	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
88	Dic₁₀₉	—	2	2	no	—
89	Dic₁₀₉	—	4	2	no	—
90	Dic₁₀₉	—	8	1	no	—
91	Dic₁₀₉	#107	1	4	yes	complemented
92	C₂×C₁₀₉	#389	1	1	yes	characteristic
93	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
94	C₂×C₁₀₉	#428	1	1	yes	characteristic
95	C₂×C₁₀₉	#459	1	1	yes	characteristic
96	C₂×C₁₀₉	#462	1	1	yes	characteristic
97	C₂×C₁₀₉	#474	1	1	yes	characteristic
98	C₂×C₁₀₉	#477	1	1	yes	characteristic
99	#1947	—	109	1	no	maximal, Sylow
100	C₁₀₉	#1947	1	1	yes	characteristic, complemented, Sylow
101	C₂×#107	—	109	1	no	—
102	C₂×#107	D₁₀₉	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	109	1	no	—
104	#384	—	109	1	no	—
105	C₄×C₄⋊C₄	—	109	1	no	—
106	#340	—	109	1	no	—
107	#340	—	109	1	no	—
108	C₂²×C₈	—	109	1	no	—
109	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	109	1	no	—
111	C₂×C₄⋊C₄	—	109	1	no	—
112	C₂×C₄⋊C₄	—	218	1	no	—
113	C₂×C₄⋊C₄	—	218	1	no	—
114	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄²	—	109	1	no	—
117	C₂×C₄²	—	218	1	no	—
118	#107	—	2	2	no	—
119	#107	—	218	2	no	—
120	C₄⋊C₈	—	109	4	no	—
121	#95	—	218	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	109	1	no	—
125	C₂²×C₄	—	109	1	no	—
126	C₂²×C₄	—	218	1	no	—
127	C₂²×C₄	—	218	1	no	—
128	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	109	2	no	—
132	C₂×C₈	—	218	1	no	—
133	C₂×C₈	—	218	1	no	—
134	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	109	2	no	—
139	C₄⋊C₄	—	436	2	no	—
140	C₄⋊C₄	—	436	2	no	—
141	C₄⋊C₄	—	218	1	no	—
142	C₄⋊C₄	—	218	1	no	—
143	C₄⋊C₄	—	218	1	no	—
144	C₄⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
145	C₄²	—	109	2	no	—
146	C₄²	—	109	2	no	—
147	C₄²	—	436	2	no	—
148	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	109	2	no	—
156	C₂×C₄	—	109	2	no	—
157	C₂×C₄	—	109	2	no	—
158	C₂×C₄	—	109	2	no	—
159	C₂×C₄	—	436	1	no	—
160	C₂×C₄	—	436	1	no	—
161	C₂×C₄	—	436	1	no	—
162	C₂×C₄	—	436	1	no	—
163	C₂×C₄	—	436	1	no	—
164	C₂×C₄	—	218	2	no	—
165	C₂×C₄	—	218	1	no	—
166	C₂×C₄	—	218	1	no	—
167	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
170	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
171	C₂×C₄	#13182028	1	1	yes	characteristic
172	C₂×C₄	#13182029	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	218	2	no	—
175	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
176	C₂²	#16812671	1	1	yes	characteristic
177	C₂²	#16812677	1	1	yes	characteristic
178	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
179	C₂²	Q₁₆×D₁₀₉	1	1	yes	characteristic
180	C₂²	#16812723	1	1	yes	characteristic
181	C₂²	#16812731	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	872	1	no	—
185	C₄	—	109	4	no	—
186	C₄	—	436	2	no	—
187	C₄	—	218	2	no	—
188	C₄	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
189	C₄	#16812664	1	1	yes	characteristic
190	C₄	#16812689	1	1	yes	characteristic
191	C₄	#16812697	1	1	yes	characteristic
192	C₂	#20294921	1	1	yes	characteristic
193	C₂	#107×D₁₀₉	1	1	yes	characteristic
194	C₂	#20295067	1	1	yes	characteristic
195	C₂	#20295219	1	1	yes	characteristic
196	C₂	#20295234	1	1	yes	characteristic
197	C₂	#20295305	1	1	yes	characteristic
198	C₂	#20295320	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812632

C₂

yes

maximal, characteristic

C₂×#16812628

C₂

yes

maximal, characteristic

#20294895

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294789

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal, characteristic

C₂×#107×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×#13182008

—

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812632

—

#16812628

C₄

yes

complemented

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

D₄

yes

—

—

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

Q₈

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×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

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

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

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

D₈

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

#125

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

100

C₁₀₉

#1947

yes

characteristic, complemented, Sylow

101

C₂×#107

—

109

—

102

C₂×#107

D₁₀₉

yes

characteristic

103

C₂×C₄⋊C₈

—

109

—

104

#384

—

109

—

105

C₄×C₄⋊C₄

—

109

—

106

#340

—

109

—

107

#340

—

109

—

108

C₂²×C₈

—

109

—

109

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

110

C₂×C₄⋊C₄

—

109

—

111

C₂×C₄⋊C₄

—

109

—

112

C₂×C₄⋊C₄

—

218

—

113

C₂×C₄⋊C₄

—

218

—

114

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

116

C₂×C₄²

—

109

—

117

C₂×C₄²

—

218

—

118

#107

—

119

#107

—

218

—

120

C₄⋊C₈

—

109

—

121

#95

—

218

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

109

—

125

C₂²×C₄

—

109

—

126

C₂²×C₄

—

218

—

127

C₂²×C₄

—

218

—

128

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

109

—

132

C₂×C₈

—

218

—

133

C₂×C₈

—

218

—

134

C₂×C₈

C₄×D₁₀₉

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

109

—

139

C₄⋊C₄

—

436

—

140

C₄⋊C₄

—

436

—

141

C₄⋊C₄

—

218

—

142

C₄⋊C₄

—

218

—

143

C₄⋊C₄

—

218

—

144

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

145

C₄²

—

109

—

146

C₄²

—

109

—

147

C₄²

—

436

—

148

C₂³

D₄×D₁₀₉

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

109

—

156

C₂×C₄

—

109

—

157

C₂×C₄

—

109

—

158

C₂×C₄

—

109

—

159

C₂×C₄

—

436

—

160

C₂×C₄

—

436

—

161

C₂×C₄

—

436

—

162

C₂×C₄

—

436

—

163

C₂×C₄

—

436

—

164

C₂×C₄

—

218

—

165

C₂×C₄

—

218

—

166

C₂×C₄

—

218

—

167

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

168

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

169

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

170

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

171

C₂×C₄

#13182028

yes

characteristic

172

C₂×C₄

#13182029

yes

characteristic

173

C₈

—

174

C₈

—

218

—

175

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

176

C₂²

#16812671

yes

characteristic

177

C₂²

#16812677

yes

characteristic

178

C₂²

D₈×D₁₀₉

yes

characteristic

179

C₂²

Q₁₆×D₁₀₉

yes

characteristic

180

C₂²

#16812723

yes

characteristic

181

C₂²

#16812731

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

872

—

185

C₄

—

109

—

186

C₄

—

436

—

187

C₄

—

218

—

188

C₄

C₄×C₂²⋊D₁₀₉

yes

characteristic

189

C₄

#16812664

yes

characteristic

190

C₄

#16812689

yes

characteristic

191

C₄

#16812697

yes

characteristic

192

C₂

#20294921

yes

characteristic

193

C₂

#107×D₁₀₉

yes

characteristic

194

C₂

#20295067

yes

characteristic

195

C₂

#20295219

yes

characteristic

196

C₂

#20295234

yes

characteristic

197

C₂

#20295305

yes

characteristic

198

C₂

#20295320

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#107×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#107	D₁₀₉
hypocenter	C₁₀₉	#1947
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#107×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 ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
16. #16812628	C₄	semidirect
91. Dic₁₀₉	#107	semidirect
100. C₁₀₉	#1947	semidirect

H ≅ G/N

type

16. #16812628

C₄

semidirect

91. Dic₁₀₉

#107

semidirect

100. C₁₀₉

#1947

semidirect

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

subgroup	index
C₂×#16812632	2
C₂×#16812628	2
#20294895	2
C₄⋊C₄×Dic₁₀₉	2
#20294789	2
#20294788	2
C₂×#107×C₁₀₉	2
#1947	109

subgroup

index

C₄⋊C₄×Dic₁₀₉

C₂×#107×C₁₀₉

109

Subgroups — #28363787

Conjugacy classes of subgroups

Complements

Maximal subgroups