Group #42128291 — subgroups

Subgroups — C₁₀₉×#10297 #42128291

Order 27904.

528 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₁₀₉×#3209	C₂	1	2	yes	maximal
2	C₁₀₉×#2862	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1994	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1548	C₂	1	2	yes	maximal
5	C₁₀₉×#1276	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
8	#494×C₁₀₉	—	2	2	no	—
9	#487×C₁₀₉	—	2	2	no	—
10	#462×C₁₀₉	—	2	4	no	—
11	#451×C₁₀₉	—	2	4	no	—
12	#431×C₁₀₉	C₂²	1	2	yes	—
13	#423×C₁₀₉	C₂²	1	1	yes	characteristic
14	#391×C₁₀₉	—	2	4	no	—
15	#382×C₁₀₉	—	2	2	no	—
16	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
17	#339×C₁₀₉	—	2	2	no	—
18	#333×C₁₀₉	—	2	4	no	—
19	C₂²×Q₈×C₁₀₉	D₄	1	4	yes	—
20	C₂×Q₁₆×C₁₀₉	—	2	8	no	—
21	C₂×M₁₆×C₁₀₉	—	2	1	no	—
22	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
23	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
24	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
25	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
27	#125×C₁₀₉	—	2	1	no	—
28	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
29	C₄×Q₈×C₁₀₉	—	2	2	no	—
30	#117×C₁₀₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄²×C₁₀₉	—	2	2	no	—
34	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₉	—	2	2	no	—
37	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
38	#103×C₁₀₉	—	2	8	no	—
39	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
40	C₄×C₈×C₁₀₉	—	2	2	no	—
41	#95×C₁₀₉	—	2	8	no	—
42	C₂×Q₈×C₁₀₉	—	2	4	no	—
43	C₂×Q₈×C₁₀₉	—	2	2	no	—
44	C₂×Q₈×C₁₀₉	—	2	4	no	—
45	C₂×Q₈×C₁₀₉	—	4	8	no	—
46	C₂²×C₄×C₁₀₉	—	2	2	no	—
47	C₂²×C₄×C₁₀₉	—	2	4	no	—
48	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	Q₁₆×C₁₀₉	—	8	8	no	—
51	M₁₆×C₁₀₉	—	4	2	no	—
52	C₂×C₈×C₁₀₉	—	2	4	no	—
53	C₂×C₈×C₁₀₉	—	4	1	no	—
54	C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₄⋊C₄×C₁₀₉	—	4	2	no	—
57	C₄⋊C₄×C₁₀₉	—	4	2	no	—
58	C₄⋊C₄×C₁₀₉	—	4	2	no	—
59	C₄⋊C₄×C₁₀₉	—	4	2	no	—
60	C₄⋊C₄×C₁₀₉	—	4	2	no	—
61	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₄²×C₁₀₉	—	4	4	no	—
64	C₄²×C₁₀₉	—	4	1	no	—
65	C₄²×C₁₀₉	Q₁₆	1	4	yes	complemented
66	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
68	C₄²×C₁₀₉	#41	1	1	yes	characteristic
69	C₂³×C₁₀₉	#120	1	1	yes	characteristic
70	Q₈×C₁₀₉	—	4	4	no	—
71	Q₈×C₁₀₉	—	4	8	no	—
72	Q₈×C₁₀₉	—	8	4	no	—
73	C₂×C₄×C₁₀₉	—	2	4	no	—
74	C₂×C₄×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	—	4	4	no	—
76	C₂×C₄×C₁₀₉	—	4	8	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	—	4	4	no	—
79	C₂×C₄×C₁₀₉	#120	1	2	yes	—
80	C₂×C₄×C₁₀₉	#121	1	2	yes	—
81	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₁₀₉	#137	1	2	yes	—
85	C₈×C₁₀₉	—	4	4	no	—
86	C₈×C₁₀₉	—	8	1	no	—
87	C₂²×C₁₀₉	—	2	2	no	—
88	C₂²×C₁₀₉	#392	1	1	yes	characteristic
89	C₂²×C₁₀₉	#451	1	2	yes	—
90	C₄×C₁₀₉	—	2	2	no	—
91	C₄×C₁₀₉	—	4	4	no	—
92	C₄×C₁₀₉	—	8	4	no	—
93	C₄×C₁₀₉	—	8	2	no	—
94	C₄×C₁₀₉	#455	1	4	yes	—
95	C₄×C₁₀₉	#462	1	2	yes	—
96	C₄×C₁₀₉	#494	1	1	yes	characteristic
97	C₄×C₁₀₉	#497	1	1	yes	characteristic
98	#10297	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₁₀₉	—	2	2	no	—
100	C₂×C₁₀₉	#2004	1	2	yes	—
101	C₂×C₁₀₉	#2016	1	1	yes	characteristic
102	#3209	C₂×C₁₀₉	1	2	yes	—
103	#2862	C₂×C₁₀₉	1	1	yes	characteristic
104	#1994	C₂×C₁₀₉	1	1	yes	characteristic
105	#1548	C₂×C₁₀₉	1	2	yes	—
106	#1276	C₂×C₁₀₉	1	1	yes	characteristic
107	C₁₀₉	#10297	1	1	yes	characteristic, direct product factor, Sylow
108	#533	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
110	#494	—	2	2	no	—
111	#487	—	2	2	no	—
112	#462	—	2	4	no	—
113	#451	—	2	4	no	—
114	#431	C₂²×C₁₀₉	1	2	yes	—
115	#423	C₂²×C₁₀₉	1	1	yes	characteristic
116	#391	—	2	4	no	—
117	#382	—	2	2	no	—
118	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
119	#339	—	2	2	no	—
120	#333	—	2	4	no	—
121	C₂²×Q₈	D₄×C₁₀₉	1	4	yes	—
122	C₂×Q₁₆	—	2	8	no	—
123	C₂×M₁₆	—	2	1	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
129	#125	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	#117	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
137	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
140	#103	—	2	8	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	#95	—	2	8	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
152	Q₁₆	—	8	8	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	1	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	4	no	—
166	C₄²	—	4	1	no	—
167	C₄²	Q₁₆×C₁₀₉	1	4	yes	complemented
168	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
170	C₄²	#41×C₁₀₉	1	1	yes	characteristic
171	C₂³	#120×C₁₀₉	1	1	yes	characteristic
172	Q₈	—	4	4	no	—
173	Q₈	—	4	8	no	—
174	Q₈	—	8	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	#120×C₁₀₉	1	2	yes	—
182	C₂×C₄	#121×C₁₀₉	1	2	yes	—
183	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
186	C₂×C₄	#137×C₁₀₉	1	2	yes	—
187	C₈	—	4	4	no	—
188	C₈	—	8	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#392×C₁₀₉	1	1	yes	characteristic
191	C₂²	#451×C₁₀₉	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	4	no	—
195	C₄	—	8	2	no	—
196	C₄	#455×C₁₀₉	1	4	yes	—
197	C₄	#462×C₁₀₉	1	2	yes	—
198	C₄	#494×C₁₀₉	1	1	yes	characteristic
199	C₄	#497×C₁₀₉	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	C₁₀₉×#2004	1	2	yes	—
202	C₂	C₁₀₉×#2016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3209

C₂

yes

maximal

C₁₀₉×#2862

C₂

yes

maximal, characteristic

C₁₀₉×#1994

C₂

yes

maximal, characteristic

C₁₀₉×#1548

C₂

yes

maximal

C₁₀₉×#1276

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#494×C₁₀₉

—

#487×C₁₀₉

—

#462×C₁₀₉

—

#451×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

characteristic

#391×C₁₀₉

—

#382×C₁₀₉

—

C₄³×C₁₀₉

C₂²

yes

characteristic

#339×C₁₀₉

—

#333×C₁₀₉

—

C₂²×Q₈×C₁₀₉

D₄

yes

—

C₂×Q₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

#125×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

D₄

yes

—

#103×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

#95×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

Q₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

Q₁₆

yes

complemented

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

yes

C₂³×C₁₀₉

yes

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#120

yes

—

C₂×C₄×C₁₀₉

#121

yes

—

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₀₉

#137

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#392

yes

characteristic

C₂²×C₁₀₉

#451

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#455

yes

—

C₄×C₁₀₉

#462

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

100

C₂×C₁₀₉

#2004

yes

—

101

C₂×C₁₀₉

#2016

yes

characteristic

102

#3209

C₂×C₁₀₉

yes

—

103

#2862

C₂×C₁₀₉

yes

characteristic

104

#1994

C₂×C₁₀₉

yes

characteristic

105

#1548

C₂×C₁₀₉

yes

—

106

#1276

C₂×C₁₀₉

yes

characteristic

107

C₁₀₉

#10297

yes

characteristic, direct product factor, Sylow

108

#533

C₂²×C₁₀₉

yes

characteristic

109

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

110

#494

—

111

#487

—

112

#462

—

113

#451

—

114

#431

C₂²×C₁₀₉

yes

—

115

#423

C₂²×C₁₀₉

yes

characteristic

116

#391

—

117

#382

—

118

C₄³

C₂²×C₁₀₉

yes

characteristic

119

#339

—

120

#333

—

121

C₂²×Q₈

D₄×C₁₀₉

yes

—

122

C₂×Q₁₆

—

123

C₂×M₁₆

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

D₄×C₁₀₉

yes

—

129

#125

—

130

C₂²⋊Q₈

—

131

C₄×Q₈

—

132

#117

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

D₄×C₁₀₉

yes

—

137

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

D₄×C₁₀₉

yes

—

140

#103

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

#95

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

151

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

152

Q₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

Q₁₆×C₁₀₉

yes

complemented

168

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

169

C₄²

C₂×D₄×C₁₀₉

yes

—

170

C₄²

#41×C₁₀₉

yes

characteristic

171

C₂³

#120×C₁₀₉

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#120×C₁₀₉

yes

—

182

C₂×C₄

#121×C₁₀₉

yes

—

183

C₂×C₄

#121×C₁₀₉

yes

characteristic

184

C₂×C₄

#127×C₁₀₉

yes

characteristic

185

C₂×C₄

C₂×Q₁₆×C₁₀₉

yes

—

186

C₂×C₄

#137×C₁₀₉

yes

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

#392×C₁₀₉

yes

characteristic

191

C₂²

#451×C₁₀₉

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#455×C₁₀₉

yes

—

197

C₄

#462×C₁₀₉

yes

—

198

C₄

#494×C₁₀₉

yes

characteristic

199

C₄

#497×C₁₀₉

yes

characteristic

200

C₂

—

201

C₂

C₁₀₉×#2004

yes

—

202

C₂

C₁₀₉×#2016

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
65. C₄²×C₁₀₉	Q₁₆	semidirect
98. #10297	C₁₀₉	direct (N × H)
107. C₁₀₉	#10297	direct (N × H)
167. C₄²	Q₁₆×C₁₀₉	semidirect

H ≅ G/N

type

65. C₄²×C₁₀₉

Q₁₆

semidirect

98. #10297

C₁₀₉

direct (N × H)

107. C₁₀₉

#10297

direct (N × H)

167. C₄²

Q₁₆×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#3209	2
C₁₀₉×#2862	2
C₁₀₉×#1994	2
C₁₀₉×#1548	2
C₁₀₉×#1276	2
#10297	109

subgroup

index

C₁₀₉×#3209

C₁₀₉×#2862

C₁₀₉×#1994

C₁₀₉×#1548

C₁₀₉×#1276

#10297

109

Subgroups — C₁₀₉×#10297 #42128291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10297 #42128291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10297 #42128291