Group #42128289 — subgroups

Subgroups — C₁₀₉×#10295 #42128289

Order 27904.

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

normal subgroups only

This group has 504 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×#3230	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×#1549	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	#500×C₁₀₉	—	2	2	no	—
9	#487×C₁₀₉	—	2	2	no	—
10	#484×C₁₀₉	—	2	4	no	—
11	#477×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	#332×C₁₀₉	—	2	4	no	—
19	C₂²×Q₈×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	—	2	1	no	—
21	C₄⋊Q₈×C₁₀₉	—	2	2	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	D₄	1	2	yes	—
26	#125×C₁₀₉	—	2	1	no	—
27	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
28	C₄×Q₈×C₁₀₉	—	2	2	no	—
29	#117×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
32	C₂×C₄²×C₁₀₉	—	2	2	no	—
33	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
35	#107×C₁₀₉	—	2	8	no	—
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	2	no	—
43	C₂×Q₈×C₁₀₉	—	4	8	no	—
44	C₂×Q₈×C₁₀₉	—	4	2	no	—
45	C₂×Q₈×C₁₀₉	—	4	2	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	M₁₆×C₁₀₉	—	4	2	no	—
51	C₂×C₈×C₁₀₉	—	2	4	no	—
52	C₂×C₈×C₁₀₉	—	4	1	no	—
53	C₄⋊C₄×C₁₀₉	—	2	4	no	—
54	C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄⋊C₄×C₁₀₉	—	2	4	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	4	2	no	—
58	C₄⋊C₄×C₁₀₉	—	4	2	no	—
59	C₄⋊C₄×C₁₀₉	—	4	2	no	—
60	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
61	C₄²×C₁₀₉	—	2	2	no	—
62	C₄²×C₁₀₉	—	4	4	no	—
63	C₄²×C₁₀₉	—	4	1	no	—
64	C₄²×C₁₀₉	Q₁₆	1	4	yes	—
65	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
67	C₄²×C₁₀₉	#41	1	1	yes	characteristic
68	C₂³×C₁₀₉	#120	1	1	yes	characteristic
69	Q₈×C₁₀₉	—	4	4	no	—
70	Q₈×C₁₀₉	—	8	4	no	—
71	Q₈×C₁₀₉	—	8	4	no	—
72	C₂×C₄×C₁₀₉	—	2	4	no	—
73	C₂×C₄×C₁₀₉	—	2	4	no	—
74	C₂×C₄×C₁₀₉	—	4	4	no	—
75	C₂×C₄×C₁₀₉	—	4	8	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	4	no	—
78	C₂×C₄×C₁₀₉	#120	1	2	yes	—
79	C₂×C₄×C₁₀₉	#121	1	2	yes	—
80	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₁₀₉	#137	1	2	yes	—
84	C₈×C₁₀₉	—	4	4	no	—
85	C₈×C₁₀₉	—	8	1	no	—
86	C₂²×C₁₀₉	—	2	2	no	—
87	C₂²×C₁₀₉	#392	1	1	yes	characteristic
88	C₂²×C₁₀₉	#451	1	2	yes	—
89	C₄×C₁₀₉	—	2	2	no	—
90	C₄×C₁₀₉	—	4	4	no	—
91	C₄×C₁₀₉	—	8	4	no	—
92	C₄×C₁₀₉	—	8	2	no	—
93	C₄×C₁₀₉	#454	1	4	yes	—
94	C₄×C₁₀₉	#462	1	2	yes	—
95	C₄×C₁₀₉	#494	1	1	yes	characteristic
96	C₄×C₁₀₉	#497	1	1	yes	characteristic
97	#10295	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₀₉	—	2	2	no	—
99	C₂×C₁₀₉	#2003	1	2	yes	—
100	C₂×C₁₀₉	#2016	1	1	yes	characteristic
101	#3230	C₂×C₁₀₉	1	2	yes	—
102	#2862	C₂×C₁₀₉	1	1	yes	characteristic
103	#1994	C₂×C₁₀₉	1	1	yes	characteristic
104	#1549	C₂×C₁₀₉	1	2	yes	—
105	#1276	C₂×C₁₀₉	1	1	yes	characteristic
106	C₁₀₉	#10295	1	1	yes	characteristic, direct product factor, Sylow
107	#533	C₂²×C₁₀₉	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
109	#500	—	2	2	no	—
110	#487	—	2	2	no	—
111	#484	—	2	4	no	—
112	#477	—	2	4	no	—
113	#431	C₂²×C₁₀₉	1	2	yes	—
114	#423	C₂²×C₁₀₉	1	1	yes	characteristic
115	#391	—	2	4	no	—
116	#382	—	2	2	no	—
117	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
118	#339	—	2	2	no	—
119	#332	—	2	4	no	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
127	#125	—	2	1	no	—
128	C₂²⋊Q₈	—	2	2	no	—
129	C₄×Q₈	—	2	2	no	—
130	#117	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
135	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
136	#107	—	2	8	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
139	#103	—	2	8	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	#95	—	2	8	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	4	8	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
150	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
151	M₁₆	—	4	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	1	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	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	C₂²⋊C₄	—	4	1	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	4	no	—
164	C₄²	—	4	1	no	—
165	C₄²	Q₁₆×C₁₀₉	1	4	yes	—
166	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
167	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
168	C₄²	#41×C₁₀₉	1	1	yes	characteristic
169	C₂³	#120×C₁₀₉	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	8	4	no	—
172	Q₈	—	8	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	#120×C₁₀₉	1	2	yes	—
180	C₂×C₄	#121×C₁₀₉	1	2	yes	—
181	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
184	C₂×C₄	#137×C₁₀₉	1	2	yes	—
185	C₈	—	4	4	no	—
186	C₈	—	8	1	no	—
187	C₂²	—	2	2	no	—
188	C₂²	#392×C₁₀₉	1	1	yes	characteristic
189	C₂²	#451×C₁₀₉	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	4	no	—
193	C₄	—	8	2	no	—
194	C₄	#454×C₁₀₉	1	4	yes	—
195	C₄	#462×C₁₀₉	1	2	yes	—
196	C₄	#494×C₁₀₉	1	1	yes	characteristic
197	C₄	#497×C₁₀₉	1	1	yes	characteristic
198	C₂	—	2	2	no	—
199	C₂	C₁₀₉×#2003	1	2	yes	—
200	C₂	C₁₀₉×#2016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3230

C₂

yes

maximal

C₁₀₉×#2862

C₂

yes

maximal, characteristic

C₁₀₉×#1994

C₂

yes

maximal, characteristic

C₁₀₉×#1549

C₂

yes

maximal

C₁₀₉×#1276

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#500×C₁₀₉

—

#487×C₁₀₉

—

#484×C₁₀₉

—

#477×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

characteristic

#391×C₁₀₉

—

#382×C₁₀₉

—

C₄³×C₁₀₉

C₂²

yes

characteristic

#339×C₁₀₉

—

#332×C₁₀₉

—

C₂²×Q₈×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

D₄

yes

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#107×C₁₀₉

—

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

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

—

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

#454

yes

—

C₄×C₁₀₉

#462

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#2003

yes

—

100

C₂×C₁₀₉

#2016

yes

characteristic

101

#3230

C₂×C₁₀₉

yes

—

102

#2862

C₂×C₁₀₉

yes

characteristic

103

#1994

C₂×C₁₀₉

yes

characteristic

104

#1549

C₂×C₁₀₉

yes

—

105

#1276

C₂×C₁₀₉

yes

characteristic

106

C₁₀₉

#10295

yes

characteristic, direct product factor, Sylow

107

#533

C₂²×C₁₀₉

yes

characteristic

108

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

109

#500

—

110

#487

—

111

#484

—

112

#477

—

113

#431

C₂²×C₁₀₉

yes

—

114

#423

C₂²×C₁₀₉

yes

characteristic

115

#391

—

116

#382

—

117

C₄³

C₂²×C₁₀₉

yes

characteristic

118

#339

—

119

#332

—

120

C₂²×Q₈

—

121

C₂×M₁₆

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

D₄×C₁₀₉

yes

—

127

#125

—

128

C₂²⋊Q₈

—

129

C₄×Q₈

—

130

#117

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

133

C₂×C₄²

—

134

C₂×C₄²

D₄×C₁₀₉

yes

—

135

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

D₄×C₁₀₉

yes

—

139

#103

—

140

C₂²⋊C₈

—

141

C₄×C₈

—

142

#95

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

150

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

Q₁₆×C₁₀₉

yes

—

166

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

167

C₄²

C₂×D₄×C₁₀₉

yes

—

168

C₄²

#41×C₁₀₉

yes

characteristic

169

C₂³

#120×C₁₀₉

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#120×C₁₀₉

yes

—

180

C₂×C₄

#121×C₁₀₉

yes

—

181

C₂×C₄

#121×C₁₀₉

yes

characteristic

182

C₂×C₄

#127×C₁₀₉

yes

characteristic

183

C₂×C₄

C₂×Q₁₆×C₁₀₉

yes

—

184

C₂×C₄

#137×C₁₀₉

yes

—

185

C₈

—

186

C₈

—

187

C₂²

—

188

C₂²

#392×C₁₀₉

yes

characteristic

189

C₂²

#451×C₁₀₉

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#454×C₁₀₉

yes

—

195

C₄

#462×C₁₀₉

yes

—

196

C₄

#494×C₁₀₉

yes

characteristic

197

C₄

#497×C₁₀₉

yes

characteristic

198

C₂

—

199

C₂

C₁₀₉×#2003

yes

—

200

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
97. #10295	C₁₀₉	direct (N × H)
106. C₁₀₉	#10295	direct (N × H)

H ≅ G/N

type

97. #10295

C₁₀₉

direct (N × H)

106. C₁₀₉

#10295

direct (N × H)

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

subgroup	index
C₁₀₉×#3230	2
C₁₀₉×#2862	2
C₁₀₉×#1994	2
C₁₀₉×#1549	2
C₁₀₉×#1276	2
#10295	109

subgroup

index

C₁₀₉×#3230

C₁₀₉×#2862

C₁₀₉×#1994

C₁₀₉×#1549

C₁₀₉×#1276

#10295

109

Subgroups — C₁₀₉×#10295 #42128289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10295 #42128289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10295 #42128289