Group #41033600 — subgroups

Subgroups — C₁₀₇×#7636 #41033600

Order 27392.

436 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 146 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	C₄×#125×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1767	C₂	1	2	yes	maximal
3	C₁₀₇×#1271	C₂	1	4	yes	maximal
4	C₂×#125×C₁₀₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₁₀₇	—	2	2	no	—
6	C₂×#97×C₁₀₇	—	2	2	no	—
7	#389×C₁₀₇	C₂²	1	2	yes	—
8	#387×C₁₀₇	—	2	2	no	—
9	#382×C₁₀₇	C₄	1	1	yes	characteristic
10	C₄×C₄⋊C₄×C₁₀₇	—	2	2	no	—
11	C₄×C₄⋊C₄×C₁₀₇	C₄	1	2	yes	—
12	C₄³×C₁₀₇	C₂²	1	1	yes	characteristic
13	#336×C₁₀₇	C₂²	1	4	yes	—
14	C₂²×C₈×C₁₀₇	—	2	4	no	—
15	#125×C₁₀₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₀₇	—	2	2	no	—
21	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
24	C₂×C₄²×C₁₀₇	D₄	1	2	yes	—
25	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₁₀₇	—	2	8	no	—
27	#97×C₁₀₇	—	4	4	no	—
28	#95×C₁₀₇	—	2	2	no	—
29	#95×C₁₀₇	—	2	2	no	—
30	#95×C₁₀₇	—	2	2	no	—
31	C₂²×C₄×C₁₀₇	—	2	2	no	—
32	C₂²×C₄×C₁₀₇	—	2	2	no	—
33	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₁₀₇	—	2	8	no	—
39	C₂×C₈×C₁₀₇	—	4	4	no	—
40	C₂×C₈×C₁₀₇	—	4	4	no	—
41	C₄⋊C₄×C₁₀₇	—	2	8	no	—
42	C₄⋊C₄×C₁₀₇	—	2	2	no	—
43	C₄⋊C₄×C₁₀₇	—	4	2	no	—
44	C₄⋊C₄×C₁₀₇	—	4	2	no	—
45	C₄⋊C₄×C₁₀₇	—	4	2	no	—
46	C₄⋊C₄×C₁₀₇	M₁₆	1	4	yes	—
47	C₄²×C₁₀₇	—	2	2	no	—
48	C₄²×C₁₀₇	—	2	4	no	—
49	C₄²×C₁₀₇	—	2	2	no	—
50	C₄²×C₁₀₇	—	2	2	no	—
51	C₄²×C₁₀₇	—	2	2	no	—
52	C₄²×C₁₀₇	—	4	1	no	—
53	C₄²×C₁₀₇	—	4	4	no	—
54	C₂³×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₇	—	2	1	no	—
56	C₂×C₄×C₁₀₇	—	2	2	no	—
57	C₂×C₄×C₁₀₇	—	2	1	no	—
58	C₂×C₄×C₁₀₇	—	2	2	no	—
59	C₂×C₄×C₁₀₇	—	2	2	no	—
60	C₂×C₄×C₁₀₇	—	2	4	no	—
61	C₂×C₄×C₁₀₇	—	2	1	no	—
62	C₂×C₄×C₁₀₇	—	2	2	no	—
63	C₂×C₄×C₁₀₇	—	2	2	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	—	4	2	no	—
69	C₂×C₄×C₁₀₇	—	4	2	no	—
70	C₂×C₄×C₁₀₇	#101	1	2	yes	—
71	C₂×C₄×C₁₀₇	#104	1	4	yes	—
72	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₁₀₇	#137	1	2	yes	—
75	C₈×C₁₀₇	—	4	8	no	—
76	C₂²×C₁₀₇	#407	1	1	yes	characteristic
77	C₂²×C₁₀₇	#410	1	1	yes	characteristic
78	C₂²×C₁₀₇	C₂×#101	1	1	yes	characteristic
79	C₂²×C₁₀₇	#417	1	2	yes	—
80	C₂²×C₁₀₇	C₂×#104	1	2	yes	—
81	C₄×C₁₀₇	—	2	2	no	—
82	C₄×C₁₀₇	—	2	2	no	—
83	C₄×C₁₀₇	—	2	2	no	—
84	C₄×C₁₀₇	—	4	1	no	—
85	C₄×C₁₀₇	—	4	2	no	—
86	C₄×C₁₀₇	—	4	4	no	—
87	C₄×C₁₀₇	—	8	2	no	—
88	C₄×C₁₀₇	#330	1	4	yes	—
89	#7636	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₁₀₇	#1467	1	1	yes	characteristic
91	C₂×C₁₀₇	#1489	1	2	yes	—
92	C₂×C₁₀₇	#1509	1	2	yes	—
93	C₂×C₁₀₇	C₂×#330	1	1	yes	characteristic
94	C₂×C₁₀₇	#1561	1	1	yes	characteristic
95	C₄×#125	C₂×C₁₀₇	1	1	yes	characteristic
96	#1767	C₂×C₁₀₇	1	2	yes	—
97	#1271	C₂×C₁₀₇	1	4	yes	—
98	C₁₀₇	#7636	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×#125	C₄×C₁₀₇	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×#97	—	2	2	no	—
102	#389	C₂²×C₁₀₇	1	2	yes	—
103	#387	—	2	2	no	—
104	#382	C₄×C₁₀₇	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	2	no	—
106	C₄×C₄⋊C₄	C₄×C₁₀₇	1	2	yes	—
107	C₄³	C₂²×C₁₀₇	1	1	yes	characteristic
108	#336	C₂²×C₁₀₇	1	4	yes	—
109	C₂²×C₈	—	2	4	no	—
110	#125	—	2	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×C₄×C₁₀₇	1	2	yes	—
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
118	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
119	C₂×C₄²	D₄×C₁₀₇	1	2	yes	—
120	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
121	C₄⋊C₈	—	2	8	no	—
122	#97	—	4	4	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
130	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
131	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
133	C₂×C₈	—	2	8	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	4	4	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	M₁₆×C₁₀₇	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	4	no	—
149	C₂³	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	#101×C₁₀₇	1	2	yes	—
166	C₂×C₄	#104×C₁₀₇	1	4	yes	—
167	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
168	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
169	C₂×C₄	#137×C₁₀₇	1	2	yes	—
170	C₈	—	4	8	no	—
171	C₂²	#407×C₁₀₇	1	1	yes	characteristic
172	C₂²	#410×C₁₀₇	1	1	yes	characteristic
173	C₂²	C₂×#101×C₁₀₇	1	1	yes	characteristic
174	C₂²	#417×C₁₀₇	1	2	yes	—
175	C₂²	C₂×#104×C₁₀₇	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	4	no	—
181	C₄	—	4	1	no	—
182	C₄	—	8	2	no	—
183	C₄	#330×C₁₀₇	1	4	yes	—
184	C₂	C₁₀₇×#1467	1	1	yes	characteristic
185	C₂	C₁₀₇×#1489	1	2	yes	—
186	C₂	C₁₀₇×#1509	1	2	yes	—
187	C₂	C₂×#330×C₁₀₇	1	1	yes	characteristic
188	C₂	C₁₀₇×#1561	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#125×C₁₀₇

C₂

yes

maximal, characteristic

C₁₀₇×#1767

C₂

yes

maximal

C₁₀₇×#1271

C₂

yes

maximal

C₂×#125×C₁₀₇

C₄

yes

characteristic

C₂×C₄⋊C₈×C₁₀₇

—

C₂×#97×C₁₀₇

—

#389×C₁₀₇

C₂²

yes

—

#387×C₁₀₇

—

#382×C₁₀₇

C₄

yes

characteristic

C₄×C₄⋊C₄×C₁₀₇

—

C₄×C₄⋊C₄×C₁₀₇

C₄

yes

—

C₄³×C₁₀₇

C₂²

yes

characteristic

#336×C₁₀₇

C₂²

yes

—

C₂²×C₈×C₁₀₇

—

#125×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₂×C₄²×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₀₇

D₄

yes

—

C₂×C₄²×C₁₀₇

D₄

yes

—

C₂×C₄²×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₇

—

#97×C₁₀₇

—

#95×C₁₀₇

—

#95×C₁₀₇

—

#95×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂×D₄

yes

—

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

M₁₆

yes

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

#101

yes

—

C₂×C₄×C₁₀₇

#104

yes

—

C₂×C₄×C₁₀₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₀₇

#137

yes

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

—

C₂²×C₁₀₇

C₂×#104

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#330

yes

—

#7636

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

#1467

yes

characteristic

C₂×C₁₀₇

#1489

yes

—

C₂×C₁₀₇

#1509

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

characteristic

#1767

C₂×C₁₀₇

yes

—

#1271

C₂×C₁₀₇

yes

—

C₁₀₇

#7636

yes

characteristic, direct product factor, Sylow

C₂×#125

C₄×C₁₀₇

yes

characteristic

100

C₂×C₄⋊C₈

—

101

C₂×#97

—

102

#389

C₂²×C₁₀₇

yes

—

103

#387

—

104

#382

C₄×C₁₀₇

yes

characteristic

105

C₄×C₄⋊C₄

—

106

C₄×C₄⋊C₄

C₄×C₁₀₇

yes

—

107

C₄³

C₂²×C₁₀₇

yes

characteristic

108

#336

C₂²×C₁₀₇

yes

—

109

C₂²×C₈

—

110

#125

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×C₄×C₁₀₇

yes

—

114

C₂×C₄⋊C₄

C₂×C₄×C₁₀₇

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×C₄×C₁₀₇

yes

characteristic

117

C₂×C₄²

C₂×C₄×C₁₀₇

yes

characteristic

118

C₂×C₄²

D₄×C₁₀₇

yes

—

119

C₂×C₄²

D₄×C₁₀₇

yes

—

120

C₂×C₄²

C₂³×C₁₀₇

yes

characteristic

121

C₄⋊C₈

—

122

#97

—

123

#95

—

124

#95

—

125

#95

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂²⋊C₄×C₁₀₇

yes

characteristic

129

C₂²×C₄

C₂²⋊C₄×C₁₀₇

yes

—

130

C₂²×C₄

C₂²⋊C₄×C₁₀₇

yes

characteristic

131

C₂²×C₄

C₂²×C₄×C₁₀₇

yes

characteristic

132

C₂²×C₄

C₂×D₄×C₁₀₇

yes

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

M₁₆×C₁₀₇

yes

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₂³

C₂×C₂²⋊C₄×C₁₀₇

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#101×C₁₀₇

yes

—

166

C₂×C₄

#104×C₁₀₇

yes

—

167

C₂×C₄

C₂×M₁₆×C₁₀₇

yes

—

168

C₂×C₄

AGL(1,ℤ/8)×C₁₀₇

yes

—

169

C₂×C₄

#137×C₁₀₇

yes

—

170

C₈

—

171

C₂²

#407×C₁₀₇

yes

characteristic

172

C₂²

#410×C₁₀₇

yes

characteristic

173

C₂²

C₂×#101×C₁₀₇

yes

characteristic

174

C₂²

#417×C₁₀₇

yes

—

175

C₂²

C₂×#104×C₁₀₇

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#330×C₁₀₇

yes

—

184

C₂

C₁₀₇×#1467

yes

characteristic

185

C₂

C₁₀₇×#1489

yes

—

186

C₂

C₁₀₇×#1509

yes

—

187

C₂

C₂×#330×C₁₀₇

yes

characteristic

188

C₂

C₁₀₇×#1561

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₇

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₁₀₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₇

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₇ ◁ C₄³×C₁₀₇ ◁ G

N	H ≅ G/N	type
89. #7636	C₁₀₇	direct (N × H)
98. C₁₀₇	#7636	direct (N × H)

H ≅ G/N

type

89. #7636

C₁₀₇

direct (N × H)

98. C₁₀₇

#7636

direct (N × H)

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

subgroup	index
C₄×#125×C₁₀₇	2
C₁₀₇×#1767	2
C₁₀₇×#1271	2
#7636	107

subgroup

index

C₄×#125×C₁₀₇

C₁₀₇×#1767

C₁₀₇×#1271

#7636

107

Subgroups — C₁₀₇×#7636 #41033600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#7636 #41033600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7636 #41033600