Group #28363219 — subgroups

Subgroups — #28363219

Order 13952.

360 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂²×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812632	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic
4	#20294788	C₂	1	2	yes	maximal, complemented
5	#380×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₉	—	2	1	no	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₉	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	#16812632	—	2	4	no	—
12	C₂²⋊#11724784	C₄	1	4	yes	complemented
13	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
17	#95×C₁₀₉	—	2	1	no	—
18	C₂²×Dic₁₀₉	—	2	4	no	—
19	C₂²×Dic₁₀₉	—	2	2	no	—
20	C₂²×Dic₁₀₉	—	4	2	no	—
21	C₂²×Dic₁₀₉	—	4	2	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₂×#11724784	C₂×C₄	1	4	yes	complemented
25	C₄⋊Dic₁₀₉	—	2	4	no	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	D₄	1	4	yes	—
28	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	—	2	2	no	—
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	—	2	1	no	—
32	C₂²×C₄×C₁₀₉	—	2	1	no	—
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₀₉	—	2	2	no	—
38	C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
40	C₂³×C₁₀₉	—	2	1	no	—
41	C₂³×C₁₀₉	—	2	1	no	—
42	C₂³×C₁₀₉	—	2	1	no	—
43	C₂³×C₁₀₉	—	2	1	no	—
44	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₂³×C₁₀₉	D₈	1	2	yes	—
46	C₂³×C₁₀₉	Q₁₆	1	2	yes	—
47	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	—	2	1	no	—
49	C₂×C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	—	2	4	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	—	4	1	no	—
56	C₂×C₄×C₁₀₉	—	4	1	no	—
57	C₂×C₄×C₁₀₉	—	4	1	no	—
58	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	#41	1	2	yes	—
63	C₂×Dic₁₀₉	—	2	4	no	—
64	C₂×Dic₁₀₉	—	4	8	no	—
65	C₂×Dic₁₀₉	—	4	2	no	—
66	C₂×Dic₁₀₉	—	4	2	no	—
67	#11724784	—	2	4	no	—
68	C₂²×C₁₀₉	—	2	2	no	—
69	C₂²×C₁₀₉	—	2	2	no	—
70	C₂²×C₁₀₉	—	2	2	no	—
71	C₂²×C₁₀₉	—	2	2	no	—
72	C₂²×C₁₀₉	—	2	2	no	—
73	C₂²×C₁₀₉	—	2	2	no	—
74	C₂²×C₁₀₉	#107	1	4	yes	complemented
75	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₀₉	#120	1	1	yes	characteristic
77	C₂²×C₁₀₉	#123	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	—	2	2	no	—
83	C₄×C₁₀₉	—	8	1	no	—
84	C₄×C₁₀₉	—	8	1	no	—
85	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
86	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
87	Dic₁₀₉	—	4	4	no	—
88	C₂×C₁₀₉	—	2	4	no	—
89	C₂×C₁₀₉	#385	1	1	yes	characteristic
90	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
91	C₂×C₁₀₉	#428	1	1	yes	characteristic
92	C₂×C₁₀₉	#447	1	1	yes	characteristic
93	C₂×C₁₀₉	#451	1	1	yes	characteristic
94	C₂×C₁₀₉	#480	1	1	yes	characteristic
95	C₂×C₁₀₉	#484	1	1	yes	characteristic
96	#1852	—	109	1	no	maximal, Sylow
97	C₁₀₉	#1852	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₄⋊C₄	—	109	1	no	—
99	C₂×#107	—	109	2	no	—
100	C₂×C₂²⋊C₈	—	109	1	no	—
101	#380	D₁₀₉	1	1	yes	characteristic
102	#340	—	109	2	no	—
103	C₂³×C₄	—	218	1	no	—
104	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
105	C₂²×C₈	—	109	2	no	—
106	C₂×C₄⋊C₄	—	109	2	no	—
107	C₂×C₄⋊C₄	—	218	4	no	—
108	C₂×C₄⋊C₄	—	218	1	no	—
109	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	#107	—	218	4	no	—
113	C₂²⋊C₈	—	109	4	no	—
114	#95	—	2	1	no	—
115	C₂⁴	C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	436	2	no	—
121	C₂²×C₄	—	436	2	no	—
122	C₂²×C₄	—	218	4	no	—
123	C₂²×C₄	—	218	2	no	—
124	C₂²×C₄	Dic₂₁₈	1	2	yes	—
125	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
128	C₂×C₈	—	109	4	no	—
129	C₂×C₈	—	218	2	no	—
130	C₂×C₈	—	218	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	109	4	no	—
134	C₄⋊C₄	—	218	4	no	—
135	C₄⋊C₄	—	218	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	1	no	—
140	C₂³	—	2	1	no	—
141	C₂³	#13182008	1	2	yes	—
142	C₂³	#13182014	1	2	yes	—
143	C₂³	#13182019	1	2	yes	—
144	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	1	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	—	436	8	no	—
156	C₂×C₄	—	436	2	no	—
157	C₂×C₄	—	436	2	no	—
158	C₂×C₄	—	218	4	no	—
159	C₂×C₄	#13182008	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
161	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
162	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
163	C₂×C₄	#13182028	1	2	yes	—
164	C₈	—	218	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	#16812632	1	4	yes	complemented
172	C₂²	C₂×#13182008	1	1	yes	characteristic
173	C₂²	#16812667	1	1	yes	characteristic
174	C₂²	#16812680	1	1	yes	characteristic
175	C₂²	C₂×#13182014	1	1	yes	characteristic
176	C₂²	C₂×#13182019	1	1	yes	characteristic
177	C₂²	#16812724	1	1	yes	characteristic
178	C₂²	#16812729	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	8	1	no	—
181	C₄	—	8	1	no	—
182	C₄	—	436	4	no	—
183	C₄	#16812659	1	2	yes	—
184	C₄	#16812676	1	2	yes	—
185	C₂	—	2	4	no	—
186	C₂	#20294898	1	1	yes	characteristic
187	C₂	C₂×#16812632	1	1	yes	characteristic
188	C₂	#20295069	1	1	yes	characteristic
189	C₂	#20295159	1	1	yes	characteristic
190	C₂	#20295181	1	1	yes	characteristic
191	C₂	#20295337	1	1	yes	characteristic
192	C₂	#20295359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#16812632

C₂

yes

maximal, complemented

C₂×C₂²⋊#11724784

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal, complemented

#380×C₁₀₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₉

—

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812632

—

C₂²⋊#11724784

C₄

yes

complemented

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₉

—

#95×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

complemented

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂⁴×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

D₈

yes

—

C₂³×C₁₀₉

Q₁₆

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

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#107

yes

complemented

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

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

#1852

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

109

—

C₂×#107

—

109

—

100

C₂×C₂²⋊C₈

—

109

—

101

#380

D₁₀₉

yes

characteristic

102

#340

—

109

—

103

C₂³×C₄

—

218

—

104

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

105

C₂²×C₈

—

109

—

106

C₂×C₄⋊C₄

—

109

—

107

C₂×C₄⋊C₄

—

218

—

108

C₂×C₄⋊C₄

—

218

—

109

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

110

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

#107

—

218

—

113

C₂²⋊C₈

—

109

—

114

#95

—

115

C₂⁴

C₂²⋊D₁₀₉

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

436

—

121

C₂²×C₄

—

436

—

122

C₂²×C₄

—

218

—

123

C₂²×C₄

—

218

—

124

C₂²×C₄

Dic₂₁₈

yes

—

125

C₂²×C₄

C₄×D₁₀₉

yes

—

126

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

127

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

128

C₂×C₈

—

109

—

129

C₂×C₈

—

218

—

130

C₂×C₈

—

218

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

109

—

134

C₄⋊C₄

—

218

—

135

C₄⋊C₄

—

218

—

136

C₂²⋊C₄

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

#13182008

yes

—

142

C₂³

#13182014

yes

—

143

C₂³

#13182019

yes

—

144

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

436

—

156

C₂×C₄

—

436

—

157

C₂×C₄

—

436

—

158

C₂×C₄

—

218

—

159

C₂×C₄

#13182008

yes

—

160

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

161

C₂×C₄

D₄×D₁₀₉

yes

—

162

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

163

C₂×C₄

#13182028

yes

—

164

C₈

—

218

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#16812632

yes

complemented

172

C₂²

C₂×#13182008

yes

characteristic

173

C₂²

#16812667

yes

characteristic

174

C₂²

#16812680

yes

characteristic

175

C₂²

C₂×#13182014

yes

characteristic

176

C₂²

C₂×#13182019

yes

characteristic

177

C₂²

#16812724

yes

characteristic

178

C₂²

#16812729

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

436

—

183

C₄

#16812659

yes

—

184

C₄

#16812676

yes

—

185

C₂

—

186

C₂

#20294898

yes

characteristic

187

C₂

C₂×#16812632

yes

characteristic

188

C₂

#20295069

yes

characteristic

189

C₂

#20295159

yes

characteristic

190

C₂

#20295181

yes

characteristic

191

C₂

#20295337

yes

characteristic

192

C₂

#20295359

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#380×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#380	D₁₀₉
hypocenter	C₁₀₉	#1852
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#380×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂⁴×C₁₀₉

D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
2. C₂×#16812632	C₂	semidirect
4. #20294788	C₂	semidirect
12. C₂²⋊#11724784	C₄	semidirect
24. C₂×#11724784	C₂×C₄	semidirect
74. C₂²×C₁₀₉	#107	semidirect
97. C₁₀₉	#1852	semidirect
171. C₂²	#16812632	semidirect

H ≅ G/N

type

2. C₂×#16812632

C₂

semidirect

4. #20294788

C₂

semidirect

12. C₂²⋊#11724784

C₄

semidirect

24. C₂×#11724784

C₂×C₄

semidirect

74. C₂²×C₁₀₉

#107

semidirect

97. C₁₀₉

#1852

semidirect

171. C₂²

#16812632

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₀₉	2
C₂×#16812632	2
C₂×C₂²⋊#11724784	2
#20294788	2
#380×C₁₀₉	2
#1852	109

subgroup

index

C₂²×C₄⋊Dic₁₀₉

C₂×#16812632

C₂×C₂²⋊#11724784

#20294788

#380×C₁₀₉

#1852

109

Subgroups — #28363219

Conjugacy classes of subgroups

Complements

Maximal subgroups