Group #42131435 — subgroups

Subgroups — C₁₀₉×#13441 #42131435

Order 27904.

378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 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×Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂×#362×C₁₀₉	C₂	1	2	yes	maximal
4	C₁₀₉×#1979	C₂	1	2	yes	maximal
5	C₁₀₉×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
7	C₂×Q₃₂×C₁₀₉	—	2	4	no	—
8	C₂×Q₃₂×C₁₀₉	—	2	2	no	—
9	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
12	#419×C₁₀₉	—	2	2	no	—
13	C₂×#101×C₁₀₉	—	2	2	no	—
14	#405×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₀₉	—	2	2	no	—
16	#362×C₁₀₉	—	2	4	no	—
17	C₂²×Q₈×C₁₀₉	—	2	2	no	—
18	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
19	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
20	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
21	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
22	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
23	C₂×Q₁₆×C₁₀₉	D₄	1	4	yes	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	#117×C₁₀₉	D₄	1	1	yes	characteristic
28	Q₃₂×C₁₀₉	—	2	8	no	—
29	M₃₂×C₁₀₉	—	2	2	no	—
30	C₂×C₁₆×C₁₀₉	—	2	2	no	—
31	C₂×C₁₆×C₁₀₉	—	2	1	no	—
32	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
33	#108×C₁₀₉	—	2	2	no	—
34	#108×C₁₀₉	—	2	2	no	—
35	#103×C₁₀₉	—	4	4	no	—
36	#101×C₁₀₉	—	4	4	no	—
37	#97×C₁₀₉	—	2	1	no	—
38	C₄×C₈×C₁₀₉	—	2	1	no	—
39	C₂×Q₈×C₁₀₉	—	2	4	no	—
40	C₂×Q₈×C₁₀₉	—	4	2	no	—
41	C₂×Q₈×C₁₀₉	—	4	2	no	—
42	C₂×Q₈×C₁₀₉	—	4	2	no	—
43	C₂×Q₈×C₁₀₉	—	4	2	no	—
44	C₂×Q₈×C₁₀₉	—	4	2	no	—
45	C₂²×C₄×C₁₀₉	—	4	2	no	—
46	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₁₀₉	—	2	4	no	—
48	Q₁₆×C₁₀₉	—	2	4	no	—
49	Q₁₆×C₁₀₉	—	4	2	no	—
50	Q₁₆×C₁₀₉	—	4	2	no	—
51	M₁₆×C₁₀₉	—	2	2	no	—
52	M₁₆×C₁₀₉	—	4	2	no	—
53	M₁₆×C₁₀₉	—	4	2	no	—
54	C₂×C₈×C₁₀₉	—	2	1	no	—
55	C₂×C₈×C₁₀₉	—	4	2	no	—
56	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₀₉	#41	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	D₈	1	2	yes	—
61	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
62	C₄²×C₁₀₉	—	2	1	no	—
63	C₁₆×C₁₀₉	—	2	2	no	—
64	C₁₆×C₁₀₉	—	4	1	no	—
65	C₂³×C₁₀₉	#135	1	1	yes	characteristic
66	Q₈×C₁₀₉	—	4	4	no	—
67	Q₈×C₁₀₉	—	4	4	no	—
68	Q₈×C₁₀₉	—	8	2	no	—
69	Q₈×C₁₀₉	—	8	2	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	4	4	no	—
72	C₂×C₄×C₁₀₉	—	8	2	no	—
73	C₂×C₄×C₁₀₉	—	8	2	no	—
74	C₂×C₄×C₁₀₉	#121	1	2	yes	—
75	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₁₀₉	—	4	1	no	—
79	C₈×C₁₀₉	—	8	2	no	—
80	C₈×C₁₀₉	#120	1	1	yes	characteristic
81	C₈×C₁₀₉	#123	1	2	yes	—
82	C₈×C₁₀₉	#124	1	1	yes	characteristic
83	C₂²×C₁₀₉	—	2	2	no	—
84	C₂²×C₁₀₉	#463	1	2	yes	—
85	C₂²×C₁₀₉	#466	1	1	yes	characteristic
86	C₄×C₁₀₉	—	4	2	no	—
87	C₄×C₁₀₉	—	8	4	no	—
88	C₄×C₁₀₉	#394	1	1	yes	characteristic
89	C₄×C₁₀₉	#459	1	2	yes	—
90	C₄×C₁₀₉	#469	1	1	yes	characteristic
91	#13441	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₀₉	—	2	2	no	—
93	C₂×C₁₀₉	#2054	1	1	yes	characteristic
94	C₂×C₁₀₉	#2222	1	2	yes	—
95	C₂²×Q₃₂	C₂×C₁₀₉	1	1	yes	characteristic
96	#2151	C₂×C₁₀₉	1	1	yes	characteristic
97	C₂×#362	C₂×C₁₀₉	1	2	yes	—
98	#1979	C₂×C₁₀₉	1	2	yes	—
99	#1381	C₂×C₁₀₉	1	1	yes	characteristic
100	C₁₀₉	#13441	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
102	C₂×Q₃₂	—	2	4	no	—
103	C₂×Q₃₂	—	2	2	no	—
104	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
106	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
107	#419	—	2	2	no	—
108	C₂×#101	—	2	2	no	—
109	#405	C₂²×C₁₀₉	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#362	—	2	4	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	D₄×C₁₀₉	1	4	yes	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	D₄×C₁₀₉	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
122	#117	D₄×C₁₀₉	1	1	yes	characteristic
123	Q₃₂	—	2	8	no	—
124	M₃₂	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#103	—	4	4	no	—
131	#101	—	4	4	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
152	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
154	C₂×C₈	#41×C₁₀₉	1	2	yes	—
155	C₄⋊C₄	D₈×C₁₀₉	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	4	1	no	—
160	C₂³	#135×C₁₀₉	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	#121×C₁₀₉	1	2	yes	—
170	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
171	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
173	C₈	—	4	1	no	—
174	C₈	—	8	2	no	—
175	C₈	#120×C₁₀₉	1	1	yes	characteristic
176	C₈	#123×C₁₀₉	1	2	yes	—
177	C₈	#124×C₁₀₉	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#463×C₁₀₉	1	2	yes	—
180	C₂²	#466×C₁₀₉	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₄	#394×C₁₀₉	1	1	yes	characteristic
184	C₄	#459×C₁₀₉	1	2	yes	—
185	C₄	#469×C₁₀₉	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	C₁₀₉×#2054	1	1	yes	characteristic
188	C₂	C₁₀₉×#2222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#2151

C₂

yes

maximal, characteristic

C₂×#362×C₁₀₉

C₂

yes

maximal

C₁₀₉×#1979

C₂

yes

maximal

C₁₀₉×#1381

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×Q₃₂×C₁₀₉

—

C₂×Q₃₂×C₁₀₉

—

C₂×M₃₂×C₁₀₉

C₂²

yes

characteristic

C₂²×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×#108×C₁₀₉

C₂²

yes

—

#419×C₁₀₉

—

C₂×#101×C₁₀₉

—

#405×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₉

—

#362×C₁₀₉

—

C₂²×Q₈×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

D₄

yes

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

D₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

#117×C₁₀₉

D₄

yes

characteristic

Q₃₂×C₁₀₉

—

M₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#108×C₁₀₉

—

#108×C₁₀₉

—

#103×C₁₀₉

—

#101×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

C₂×C₈×C₁₀₉

yes

C₂×C₈×C₁₀₉

yes

—

C₄⋊C₄×C₁₀₉

D₈

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

#135

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#121

yes

—

C₂×C₄×C₁₀₉

#121

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#120

yes

characteristic

C₈×C₁₀₉

#123

yes

—

C₈×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#463

yes

—

C₂²×C₁₀₉

#466

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#394

yes

characteristic

C₄×C₁₀₉

#459

yes

—

C₄×C₁₀₉

#469

yes

characteristic

#13441

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#2054

yes

characteristic

C₂×C₁₀₉

#2222

yes

—

C₂²×Q₃₂

C₂×C₁₀₉

yes

characteristic

#2151

C₂×C₁₀₉

yes

characteristic

C₂×#362

C₂×C₁₀₉

yes

—

#1979

C₂×C₁₀₉

yes

—

#1381

C₂×C₁₀₉

yes

characteristic

100

C₁₀₉

#13441

yes

characteristic, direct product factor, Sylow

101

C₂²×Q₁₆

C₂²×C₁₀₉

yes

—

102

C₂×Q₃₂

—

103

C₂×Q₃₂

—

104

C₂×M₃₂

C₂²×C₁₀₉

yes

characteristic

105

C₂²×C₁₆

C₂²×C₁₀₉

yes

characteristic

106

C₂×#108

C₂²×C₁₀₉

yes

—

107

#419

—

108

C₂×#101

—

109

#405

C₂²×C₁₀₉

yes

characteristic

110

C₄⋊C₁₆

—

111

#362

—

112

C₂²×Q₈

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

D₄×C₁₀₉

yes

—

119

C₂×M₁₆

—

120

C₂×M₁₆

D₄×C₁₀₉

yes

characteristic

121

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

122

#117

D₄×C₁₀₉

yes

characteristic

123

Q₃₂

—

124

M₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

D₄×C₁₀₉

yes

—

128

#108

—

129

#108

—

130

#103

—

131

#101

—

132

#97

—

133

C₄×C₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

152

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

153

C₂×C₈

#41×C₁₀₉

yes

characteristic

154

C₂×C₈

#41×C₁₀₉

yes

—

155

C₄⋊C₄

D₈×C₁₀₉

yes

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

#135×C₁₀₉

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

#121×C₁₀₉

yes

—

170

C₂×C₄

#121×C₁₀₉

yes

characteristic

171

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

172

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

—

173

C₈

—

174

C₈

—

175

C₈

#120×C₁₀₉

yes

characteristic

176

C₈

#123×C₁₀₉

yes

—

177

C₈

#124×C₁₀₉

yes

characteristic

178

C₂²

—

179

C₂²

#463×C₁₀₉

yes

—

180

C₂²

#466×C₁₀₉

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

#394×C₁₀₉

yes

characteristic

184

C₄

#459×C₁₀₉

yes

—

185

C₄

#469×C₁₀₉

yes

characteristic

186

C₂

—

187

C₂

C₁₀₉×#2054

yes

characteristic

188

C₂

C₁₀₉×#2222

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #405×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #405×C₁₀₉ ◁ G

N	H ≅ G/N	type
91. #13441	C₁₀₉	direct (N × H)
100. C₁₀₉	#13441	direct (N × H)

H ≅ G/N

type

91. #13441

C₁₀₉

direct (N × H)

100. C₁₀₉

#13441

direct (N × H)

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

subgroup	index
C₂²×Q₃₂×C₁₀₉	2
C₁₀₉×#2151	2
C₂×#362×C₁₀₉	2
C₁₀₉×#1979	2
C₁₀₉×#1381	2
#13441	109

subgroup

index

C₂²×Q₃₂×C₁₀₉

C₁₀₉×#2151

C₂×#362×C₁₀₉

C₁₀₉×#1979

C₁₀₉×#1381

#13441

109

Subgroups — C₁₀₉×#13441 #42131435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#13441 #42131435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#13441 #42131435