Group #41036480 — subgroups

Subgroups — C₁₀₇×#10516 #41036480

Order 27392.

364 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₁₀₇×#3249	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1843	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1552	C₂	1	2	yes	maximal
4	C₁₀₇×#1506	C₂	1	2	yes	maximal
5	C₁₀₇×#1290	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₇	C₂²	1	2	yes	—
7	#477×C₁₀₇	—	2	2	no	—
8	#467×C₁₀₇	—	2	2	no	—
9	#462×C₁₀₇	—	2	2	no	—
10	#432×C₁₀₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₀₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	#376×C₁₀₇	C₂²	1	1	yes	characteristic
15	#332×C₁₀₇	—	2	2	no	—
16	#332×C₁₀₇	—	2	2	no	—
17	#328×C₁₀₇	—	2	4	no	—
18	C₂×Q₁₆×C₁₀₇	—	2	2	no	—
19	C₂²×C₈×C₁₀₇	—	2	1	no	—
20	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
21	C₄⋊Q₈×C₁₀₇	D₄	1	2	yes	—
22	C₄⋊Q₈×C₁₀₇	D₄	1	2	yes	—
23	#125×C₁₀₇	—	2	2	no	—
24	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
25	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
26	C₄×Q₈×C₁₀₇	—	2	2	no	—
27	C₄×Q₈×C₁₀₇	—	2	2	no	—
28	#117×C₁₀₇	—	2	2	no	—
29	C₂×C₄²×C₁₀₇	—	2	1	no	—
30	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
31	#107×C₁₀₇	—	2	2	no	—
32	C₄⋊C₈×C₁₀₇	—	2	2	no	—
33	C₄⋊C₈×C₁₀₇	—	2	2	no	—
34	C₄⋊C₈×C₁₀₇	—	2	2	no	—
35	C₄⋊C₈×C₁₀₇	—	4	4	no	—
36	#103×C₁₀₇	—	2	4	no	—
37	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
38	#97×C₁₀₇	—	2	2	no	—
39	#95×C₁₀₇	—	2	1	no	—
40	#95×C₁₀₇	D₄	1	2	yes	—
41	C₂×Q₈×C₁₀₇	—	2	2	no	—
42	C₂×Q₈×C₁₀₇	—	2	2	no	—
43	C₂²×C₄×C₁₀₇	—	2	2	no	—
44	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₇	#41	1	1	yes	characteristic
47	Q₁₆×C₁₀₇	—	8	2	no	—
48	C₂×C₈×C₁₀₇	—	2	2	no	—
49	C₂×C₈×C₁₀₇	—	4	2	no	—
50	C₂×C₈×C₁₀₇	—	4	2	no	—
51	C₂×C₈×C₁₀₇	—	4	1	no	—
52	C₂×C₈×C₁₀₇	—	4	2	no	—
53	C₄⋊C₄×C₁₀₇	—	2	2	no	—
54	C₄⋊C₄×C₁₀₇	—	2	2	no	—
55	C₄⋊C₄×C₁₀₇	—	2	2	no	—
56	C₄⋊C₄×C₁₀₇	—	2	2	no	—
57	C₄⋊C₄×C₁₀₇	—	4	2	no	—
58	C₄⋊C₄×C₁₀₇	—	4	2	no	—
59	C₂²⋊C₄×C₁₀₇	—	4	2	no	—
60	C₄²×C₁₀₇	—	2	2	no	—
61	C₄²×C₁₀₇	—	2	2	no	—
62	C₄²×C₁₀₇	—	4	2	no	—
63	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
64	C₄²×C₁₀₇	#41	1	2	yes	—
65	C₂³×C₁₀₇	#121	1	1	yes	characteristic
66	Q₈×C₁₀₇	—	4	2	no	—
67	Q₈×C₁₀₇	—	4	2	no	—
68	Q₈×C₁₀₇	—	8	2	no	—
69	C₂×C₄×C₁₀₇	—	2	2	no	—
70	C₂×C₄×C₁₀₇	—	2	1	no	—
71	C₂×C₄×C₁₀₇	—	2	1	no	—
72	C₂×C₄×C₁₀₇	—	2	4	no	—
73	C₂×C₄×C₁₀₇	—	4	1	no	—
74	C₂×C₄×C₁₀₇	—	4	2	no	—
75	C₂×C₄×C₁₀₇	—	4	2	no	—
76	C₂×C₄×C₁₀₇	—	4	2	no	—
77	C₂×C₄×C₁₀₇	—	4	2	no	—
78	C₂×C₄×C₁₀₇	#120	1	1	yes	characteristic
79	C₂×C₄×C₁₀₇	#121	1	2	yes	—
80	C₂×C₄×C₁₀₇	#123	1	2	yes	—
81	C₂×C₄×C₁₀₇	#124	1	1	yes	characteristic
82	C₈×C₁₀₇	—	4	2	no	—
83	C₈×C₁₀₇	—	8	2	no	—
84	C₈×C₁₀₇	—	8	1	no	—
85	C₂²×C₁₀₇	—	2	1	no	—
86	C₂²×C₁₀₇	—	2	1	no	—
87	C₂²×C₁₀₇	#394	1	1	yes	characteristic
88	C₂²×C₁₀₇	#473	1	2	yes	—
89	C₄×C₁₀₇	—	2	2	no	—
90	C₄×C₁₀₇	—	2	1	no	—
91	C₄×C₁₀₇	—	2	2	no	—
92	C₄×C₁₀₇	—	4	4	no	—
93	C₄×C₁₀₇	—	8	2	no	—
94	C₄×C₁₀₇	—	8	2	no	—
95	C₄×C₁₀₇	#453	1	1	yes	characteristic
96	C₄×C₁₀₇	#456	1	1	yes	characteristic
97	#10516	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₁₀₇	—	2	2	no	—
99	C₂×C₁₀₇	#2039	1	1	yes	characteristic
100	C₂×C₁₀₇	#2047	1	1	yes	characteristic
101	C₂×C₁₀₇	#2053	1	1	yes	characteristic
102	#3249	C₂×C₁₀₇	1	1	yes	characteristic
103	#1843	C₂×C₁₀₇	1	1	yes	characteristic
104	#1552	C₂×C₁₀₇	1	2	yes	—
105	#1506	C₂×C₁₀₇	1	2	yes	—
106	#1290	C₂×C₁₀₇	1	1	yes	characteristic
107	C₁₀₇	#10516	1	1	yes	characteristic, direct product factor, Sylow
108	#533	C₂²×C₁₀₇	1	2	yes	—
109	#477	—	2	2	no	—
110	#467	—	2	2	no	—
111	#462	—	2	2	no	—
112	#432	C₂²×C₁₀₇	1	2	yes	—
113	C₂×C₄⋊C₈	—	2	2	no	—
114	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₁₀₇	1	1	yes	characteristic
116	#376	C₂²×C₁₀₇	1	1	yes	characteristic
117	#332	—	2	2	no	—
118	#332	—	2	2	no	—
119	#328	—	2	4	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
123	C₄⋊Q₈	D₄×C₁₀₇	1	2	yes	—
124	C₄⋊Q₈	D₄×C₁₀₇	1	2	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₄×Q₈	—	2	2	no	—
130	#117	—	2	2	no	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
133	#107	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	4	4	no	—
138	#103	—	2	4	no	—
139	C₂²⋊C₈	—	2	2	no	—
140	#97	—	2	2	no	—
141	#95	—	2	1	no	—
142	#95	D₄×C₁₀₇	1	2	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
148	C₂²×C₄	#41×C₁₀₇	1	1	yes	characteristic
149	Q₁₆	—	8	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
166	C₄²	#41×C₁₀₇	1	2	yes	—
167	C₂³	#121×C₁₀₇	1	1	yes	characteristic
168	Q₈	—	4	2	no	—
169	Q₈	—	4	2	no	—
170	Q₈	—	8	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	#120×C₁₀₇	1	1	yes	characteristic
181	C₂×C₄	#121×C₁₀₇	1	2	yes	—
182	C₂×C₄	#123×C₁₀₇	1	2	yes	—
183	C₂×C₄	#124×C₁₀₇	1	1	yes	characteristic
184	C₈	—	4	2	no	—
185	C₈	—	8	2	no	—
186	C₈	—	8	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#394×C₁₀₇	1	1	yes	characteristic
190	C₂²	#473×C₁₀₇	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	4	no	—
195	C₄	—	8	2	no	—
196	C₄	—	8	2	no	—
197	C₄	#453×C₁₀₇	1	1	yes	characteristic
198	C₄	#456×C₁₀₇	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	C₁₀₇×#2039	1	1	yes	characteristic
201	C₂	C₁₀₇×#2047	1	1	yes	characteristic
202	C₂	C₁₀₇×#2053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#3249

C₂

yes

maximal, characteristic

C₁₀₇×#1843

C₂

yes

maximal, characteristic

C₁₀₇×#1552

C₂

yes

maximal

C₁₀₇×#1506

C₂

yes

maximal

C₁₀₇×#1290

C₂

yes

maximal, characteristic

#533×C₁₀₇

C₂²

yes

—

#477×C₁₀₇

—

#467×C₁₀₇

—

#462×C₁₀₇

—

#432×C₁₀₇

C₂²

yes

—

C₂×C₄⋊C₈×C₁₀₇

—

C₂×C₄⋊C₈×C₁₀₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₁₀₇

C₂²

yes

characteristic

#376×C₁₀₇

C₂²

yes

characteristic

#332×C₁₀₇

—

#332×C₁₀₇

—

#328×C₁₀₇

—

C₂×Q₁₆×C₁₀₇

—

C₂²×C₈×C₁₀₇

—

C₂²×C₈×C₁₀₇

D₄

yes

—

C₄⋊Q₈×C₁₀₇

D₄

yes

—

C₄⋊Q₈×C₁₀₇

D₄

yes

—

#125×C₁₀₇

—

C₂²⋊Q₈×C₁₀₇

—

C₂²⋊Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

#117×C₁₀₇

—

C₂×C₄²×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂³

yes

characteristic

#107×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

#103×C₁₀₇

—

C₂²⋊C₈×C₁₀₇

—

#97×C₁₀₇

—

#95×C₁₀₇

—

#95×C₁₀₇

D₄

yes

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₀₇

#41

yes

characteristic

Q₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂×D₄

yes

—

C₄²×C₁₀₇

#41

yes

—

C₂³×C₁₀₇

#121

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

#120

yes

characteristic

C₂×C₄×C₁₀₇

#121

yes

—

C₂×C₄×C₁₀₇

#123

yes

—

C₂×C₄×C₁₀₇

#124

yes

characteristic

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#394

yes

characteristic

C₂²×C₁₀₇

#473

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

#2039

yes

characteristic

100

C₂×C₁₀₇

#2047

yes

characteristic

101

C₂×C₁₀₇

#2053

yes

characteristic

102

#3249

C₂×C₁₀₇

yes

characteristic

103

#1843

C₂×C₁₀₇

yes

characteristic

104

#1552

C₂×C₁₀₇

yes

—

105

#1506

C₂×C₁₀₇

yes

—

106

#1290

C₂×C₁₀₇

yes

characteristic

107

C₁₀₇

#10516

yes

characteristic, direct product factor, Sylow

108

#533

C₂²×C₁₀₇

yes

—

109

#477

—

110

#467

—

111

#462

—

112

#432

C₂²×C₁₀₇

yes

—

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

C₂²×C₁₀₇

yes

characteristic

115

C₂×C₄⋊C₈

C₂²×C₁₀₇

yes

characteristic

116

#376

C₂²×C₁₀₇

yes

characteristic

117

#332

—

118

#332

—

119

#328

—

120

C₂×Q₁₆

—

121

C₂²×C₈

—

122

C₂²×C₈

D₄×C₁₀₇

yes

—

123

C₄⋊Q₈

D₄×C₁₀₇

yes

—

124

C₄⋊Q₈

D₄×C₁₀₇

yes

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

128

C₄×Q₈

—

129

C₄×Q₈

—

130

#117

—

131

C₂×C₄²

—

132

C₂×C₄²

C₂³×C₁₀₇

yes

characteristic

133

#107

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

C₂²⋊C₈

—

140

#97

—

141

#95

—

142

#95

D₄×C₁₀₇

yes

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₁₀₇

yes

characteristic

147

C₂²×C₄

C₂×D₄×C₁₀₇

yes

characteristic

148

C₂²×C₄

#41×C₁₀₇

yes

characteristic

149

Q₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

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

C₂×D₄×C₁₀₇

yes

—

166

C₄²

#41×C₁₀₇

yes

—

167

C₂³

#121×C₁₀₇

yes

characteristic

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#120×C₁₀₇

yes

characteristic

181

C₂×C₄

#121×C₁₀₇

yes

—

182

C₂×C₄

#123×C₁₀₇

yes

—

183

C₂×C₄

#124×C₁₀₇

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

—

187

C₂²

—

188

C₂²

—

189

C₂²

#394×C₁₀₇

yes

characteristic

190

C₂²

#473×C₁₀₇

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#453×C₁₀₇

yes

characteristic

198

C₄

#456×C₁₀₇

yes

characteristic

199

C₂

—

200

C₂

C₁₀₇×#2039

yes

characteristic

201

C₂

C₁₀₇×#2047

yes

characteristic

202

C₂

C₁₀₇×#2053

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
prime radical	C₂³×C₁₀₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₇

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₇

prime radical

C₂³×C₁₀₇

#121

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

N	H ≅ G/N	type
97. #10516	C₁₀₇	direct (N × H)
107. C₁₀₇	#10516	direct (N × H)

H ≅ G/N

type

97. #10516

C₁₀₇

direct (N × H)

107. C₁₀₇

#10516

direct (N × H)

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

subgroup	index
C₁₀₇×#3249	2
C₁₀₇×#1843	2
C₁₀₇×#1552	2
C₁₀₇×#1506	2
C₁₀₇×#1290	2
#10516	107

subgroup

index

C₁₀₇×#3249

C₁₀₇×#1843

C₁₀₇×#1552

C₁₀₇×#1506

C₁₀₇×#1290

#10516

107

Subgroups — C₁₀₇×#10516 #41036480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#10516 #41036480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#10516 #41036480