Group #41037455 — subgroups

Subgroups — C₁₀₇×#11491 #41037455

Order 27392.

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

normal subgroups only

This group has 356 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	C₁₀₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#2163	C₂	1	2	yes	maximal, complemented
3	C₁₀₇×#1315	C₂	1	4	yes	maximal, complemented
4	C₂×#131×C₁₀₇	C₄	1	2	yes	—
5	#532×C₁₀₇	C₄	1	1	yes	characteristic
6	#436×C₁₀₇	—	2	2	no	—
7	C₈×D₄×C₁₀₇	—	2	2	no	—
8	#423×C₁₀₇	C₄	1	1	yes	characteristic
9	#408×C₁₀₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₁₀₇	—	2	2	no	—
12	C₂²⋊C₁₆×C₁₀₇	C₂²	1	4	yes	—
13	#346×C₁₀₇	—	2	2	no	—
14	C₂×#41×C₁₀₇	C₂×C₄	1	2	yes	—
15	#131×C₁₀₇	—	4	4	no	—
16	C₂×M₁₆×C₁₀₇	—	2	2	no	—
17	C₂×M₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₀₇	—	2	2	no	—
19	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₀₇	C₈	1	1	yes	characteristic
22	#127×C₁₀₇	C₈	1	1	yes	characteristic
23	#124×C₁₀₇	C₈	1	2	yes	—
24	#121×C₁₀₇	—	2	2	no	—
25	C₄×D₄×C₁₀₇	—	2	2	no	—
26	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₀₇	—	2	4	no	—
28	C₄⋊C₈×C₁₀₇	—	2	2	no	—
29	C₂²⋊C₈×C₁₀₇	—	2	4	no	—
30	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
31	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
32	#41×C₁₀₇	—	4	2	no	—
33	#41×C₁₀₇	—	4	2	no	—
34	C₂×Q₈×C₁₀₇	C₂×C₈	1	2	yes	—
35	C₂×D₄×C₁₀₇	—	2	2	no	—
36	C₂×D₄×C₁₀₇	—	2	2	no	—
37	C₂×D₄×C₁₀₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₁₀₇	—	2	2	no	—
39	C₂²×C₄×C₁₀₇	M₁₆	1	2	yes	—
40	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
41	M₁₆×C₁₀₇	—	4	4	no	—
42	M₁₆×C₁₀₇	—	4	2	no	—
43	C₂×C₈×C₁₀₇	—	2	2	no	—
44	C₂×C₈×C₁₀₇	—	2	2	no	—
45	C₂×C₈×C₁₀₇	—	4	1	no	—
46	C₂×C₈×C₁₀₇	—	4	4	no	—
47	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
50	C₄⋊C₄×C₁₀₇	—	2	2	no	—
51	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
52	C₄²×C₁₀₇	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₁₀₇	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₁₀₇	C₂×C₈	1	1	yes	characteristic
55	C₁₆×C₁₀₇	—	4	4	no	—
56	C₂³×C₁₀₇	—	2	2	no	—
57	C₂³×C₁₀₇	C₂×M₁₆	1	1	yes	characteristic
58	Q₈×C₁₀₇	—	4	2	no	—
59	D₄×C₁₀₇	—	4	2	no	—
60	D₄×C₁₀₇	—	4	4	no	—
61	D₄×C₁₀₇	—	4	4	no	—
62	C₂×C₄×C₁₀₇	—	2	1	no	—
63	C₂×C₄×C₁₀₇	—	2	1	no	—
64	C₂×C₄×C₁₀₇	—	2	2	no	—
65	C₂×C₄×C₁₀₇	—	4	2	no	—
66	C₂×C₄×C₁₀₇	—	4	2	no	—
67	C₂×C₄×C₁₀₇	—	4	2	no	—
68	C₂×C₄×C₁₀₇	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₁₀₇	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₁₀₇	—	2	2	no	—
73	C₈×C₁₀₇	—	2	1	no	—
74	C₈×C₁₀₇	—	4	2	no	—
75	C₈×C₁₀₇	#100	1	2	yes	—
76	C₂²×C₁₀₇	—	2	2	no	—
77	C₂²×C₁₀₇	—	2	1	no	—
78	C₂²×C₁₀₇	—	4	4	no	—
79	C₂²×C₁₀₇	—	4	2	no	—
80	C₂²×C₁₀₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₄×C₁₀₇	—	2	2	no	—
82	C₄×C₁₀₇	—	2	2	no	—
83	C₄×C₁₀₇	—	2	1	no	—
84	C₄×C₁₀₇	—	4	2	no	—
85	C₄×C₁₀₇	#410	1	1	yes	characteristic
86	C₄×C₁₀₇	C₂×#100	1	1	yes	characteristic
87	#11491	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₁₀₇	—	4	1	no	—
89	C₂×C₁₀₇	—	8	2	no	—
90	C₂×C₁₀₇	#1460	1	1	yes	characteristic
91	C₂×C₁₀₇	#2117	1	2	yes	—
92	#2931	C₂×C₁₀₇	1	1	yes	characteristic
93	#2163	C₂×C₁₀₇	1	2	yes	complemented
94	#1315	C₂×C₁₀₇	1	4	yes	complemented
95	C₁₀₇	#11491	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×#131	C₄×C₁₀₇	1	2	yes	—
97	#532	C₄×C₁₀₇	1	1	yes	characteristic
98	#436	—	2	2	no	—
99	C₈×D₄	—	2	2	no	—
100	#423	C₄×C₁₀₇	1	1	yes	characteristic
101	#408	C₂²×C₁₀₇	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
103	C₄⋊C₁₆	—	2	2	no	—
104	C₂²⋊C₁₆	C₂²×C₁₀₇	1	4	yes	—
105	#346	—	2	2	no	—
106	C₂×#41	C₂×C₄×C₁₀₇	1	2	yes	—
107	#131	—	4	4	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
113	C₄⋊Q₈	C₈×C₁₀₇	1	1	yes	characteristic
114	#127	C₈×C₁₀₇	1	1	yes	characteristic
115	#124	C₈×C₁₀₇	1	2	yes	—
116	#121	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
119	C₂×C₁₆	—	2	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	4	no	—
122	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
123	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
124	#41	—	4	2	no	—
125	#41	—	4	2	no	—
126	C₂×Q₈	C₂×C₈×C₁₀₇	1	2	yes	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	C₂×C₈×C₁₀₇	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	M₁₆×C₁₀₇	1	2	yes	—
132	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
133	M₁₆	—	4	4	no	—
134	M₁₆	—	4	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
141	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
142	C₄⋊C₄	—	2	2	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₄²	C₂²⋊C₄×C₁₀₇	1	2	yes	—
145	C₄²	C₂×C₈×C₁₀₇	1	1	yes	characteristic
146	C₄²	C₂×C₈×C₁₀₇	1	1	yes	characteristic
147	C₁₆	—	4	4	no	—
148	C₂³	—	2	2	no	—
149	C₂³	C₂×M₁₆×C₁₀₇	1	1	yes	characteristic
150	Q₈	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	4	no	—
153	D₄	—	4	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂²⋊C₈×C₁₀₇	1	2	yes	—
161	C₂×C₄	C₂²⋊C₈×C₁₀₇	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	2	no	—
167	C₈	#100×C₁₀₇	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×C₂²⋊C₈×C₁₀₇	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	2	no	—
177	C₄	#410×C₁₀₇	1	1	yes	characteristic
178	C₄	C₂×#100×C₁₀₇	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	—	8	2	no	—
181	C₂	C₁₀₇×#1460	1	1	yes	characteristic
182	C₂	C₁₀₇×#2117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#2931

C₂

yes

maximal, characteristic

C₁₀₇×#2163

C₂

yes

maximal, complemented

C₁₀₇×#1315

C₂

yes

maximal, complemented

C₂×#131×C₁₀₇

C₄

yes

—

#532×C₁₀₇

C₄

yes

characteristic

#436×C₁₀₇

—

C₈×D₄×C₁₀₇

—

#423×C₁₀₇

C₄

yes

characteristic

#408×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₇

—

C₂²⋊C₁₆×C₁₀₇

C₂²

yes

—

#346×C₁₀₇

—

C₂×#41×C₁₀₇

C₂×C₄

yes

—

#131×C₁₀₇

—

C₂×M₁₆×C₁₀₇

—

C₂×M₁₆×C₁₀₇

C₂×C₄

yes

—

C₂²×C₈×C₁₀₇

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₈

yes

characteristic

#127×C₁₀₇

C₈

yes

characteristic

#124×C₁₀₇

C₈

yes

—

#121×C₁₀₇

—

C₄×D₄×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₂²⋊C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

D₄

yes

—

C₄×C₈×C₁₀₇

D₄

yes

—

#41×C₁₀₇

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂×C₈

yes

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

C₂×C₈

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

M₁₆

yes

—

C₂²×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂²⋊C₄

yes

—

C₄²×C₁₀₇

C₂×C₈

yes

characteristic

C₄²×C₁₀₇

C₂×C₈

yes

characteristic

C₁₆×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂×M₁₆

yes

characteristic

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₈

yes

characteristic

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

#100

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₈

yes

characteristic

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

—

C₂×C₁₀₇

#1460

yes

characteristic

C₂×C₁₀₇

#2117

yes

—

#2931

C₂×C₁₀₇

yes

characteristic

#2163

C₂×C₁₀₇

yes

complemented

#1315

C₂×C₁₀₇

yes

complemented

C₁₀₇

#11491

yes

characteristic, direct product factor, Sylow

C₂×#131

C₄×C₁₀₇

yes

—

#532

C₄×C₁₀₇

yes

characteristic

#436

—

C₈×D₄

—

100

#423

C₄×C₁₀₇

yes

characteristic

101

#408

C₂²×C₁₀₇

yes

—

102

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

103

C₄⋊C₁₆

—

104

C₂²⋊C₁₆

C₂²×C₁₀₇

yes

—

105

#346

—

106

C₂×#41

C₂×C₄×C₁₀₇

yes

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

C₂×C₄×C₁₀₇

yes

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×C₄×C₁₀₇

yes

characteristic

112

C₂²×C₈

C₂³×C₁₀₇

yes

characteristic

113

C₄⋊Q₈

C₈×C₁₀₇

yes

characteristic

114

#127

C₈×C₁₀₇

yes

characteristic

115

#124

C₈×C₁₀₇

yes

—

116

#121

—

117

C₄×D₄

—

118

C₂×C₄²

C₂×C₄×C₁₀₇

yes

characteristic

119

C₂×C₁₆

—

120

C₄⋊C₈

—

121

C₂²⋊C₈

—

122

C₄×C₈

D₄×C₁₀₇

yes

—

123

C₄×C₈

D₄×C₁₀₇

yes

—

124

#41

—

125

#41

—

126

C₂×Q₈

C₂×C₈×C₁₀₇

yes

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

C₂×C₈×C₁₀₇

yes

—

130

C₂²×C₄

—

131

C₂²×C₄

M₁₆×C₁₀₇

yes

—

132

C₂²×C₄

C₂²×C₄×C₁₀₇

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

characteristic

140

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

characteristic

141

C₂×C₈

C₂×D₄×C₁₀₇

yes

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

C₂²⋊C₄×C₁₀₇

yes

—

145

C₄²

C₂×C₈×C₁₀₇

yes

characteristic

146

C₄²

C₂×C₈×C₁₀₇

yes

characteristic

147

C₁₆

—

148

C₂³

—

149

C₂³

C₂×M₁₆×C₁₀₇

yes

characteristic

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₂²⋊C₈×C₁₀₇

yes

—

161

C₂×C₄

C₂²⋊C₈×C₁₀₇

yes

—

162

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₇

yes

characteristic

163

C₂×C₄

C₂²×C₈×C₁₀₇

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

#100×C₁₀₇

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂×C₂²⋊C₈×C₁₀₇

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#410×C₁₀₇

yes

characteristic

178

C₄

C₂×#100×C₁₀₇

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

C₁₀₇×#1460

yes

characteristic

182

C₂

C₁₀₇×#2117

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₈
prime radical	#532×C₁₀₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₇

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₈×C₁₀₇

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₇

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₈

prime radical

#532×C₁₀₇

C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
2. C₁₀₇×#2163	C₂	semidirect
3. C₁₀₇×#1315	C₂	semidirect
87. #11491	C₁₀₇	direct (N × H)
93. #2163	C₂×C₁₀₇	semidirect
94. #1315	C₂×C₁₀₇	semidirect
95. C₁₀₇	#11491	direct (N × H)

H ≅ G/N

type

2. C₁₀₇×#2163

C₂

semidirect

3. C₁₀₇×#1315

C₂

semidirect

87. #11491

C₁₀₇

direct (N × H)

93. #2163

C₂×C₁₀₇

semidirect

94. #1315

C₂×C₁₀₇

semidirect

95. C₁₀₇

#11491

direct (N × H)

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

subgroup	index
C₁₀₇×#2931	2
C₁₀₇×#2163	2
C₁₀₇×#1315	2
#11491	107

subgroup

index

C₁₀₇×#2931

C₁₀₇×#2163

C₁₀₇×#1315

#11491

107

Subgroups — C₁₀₇×#11491 #41037455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#11491 #41037455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#11491 #41037455