Group #41034208 — subgroups

Subgroups — C₁₀₇×#8244 #41034208

Order 27392.

368 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 170 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×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1544	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1526	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#1451	C₂	1	2	yes	maximal
5	C₁₀₇×#1299	C₂	1	2	yes	maximal
6	#533×C₁₀₇	C₂²	1	1	yes	characteristic
7	#446×C₁₀₇	—	2	2	no	—
8	C₈×Q₈×C₁₀₇	—	2	2	no	—
9	#431×C₁₀₇	C₂²	1	2	yes	—
10	#424×C₁₀₇	C₄	1	2	yes	—
11	#423×C₁₀₇	C₂²	1	1	yes	characteristic
12	#405×C₁₀₇	C₄	1	2	yes	—
13	C₄×M₁₆×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	#332×C₁₀₇	C₄	1	4	yes	—
16	#330×C₁₀₇	C₄	1	4	yes	—
17	#322×C₁₀₇	—	2	2	no	—
18	#322×C₁₀₇	—	2	2	no	—
19	C₂×M₁₆×C₁₀₇	—	2	2	no	—
20	C₂×M₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₀₇	C₂×C₄	1	2	yes	—
24	#125×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
26	C₄×Q₈×C₁₀₇	—	2	2	no	—
27	#117×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₇	—	2	4	no	—
30	C₄⋊C₈×C₁₀₇	—	2	2	no	—
31	C₄⋊C₈×C₁₀₇	C₂×C₄	1	4	yes	—
32	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
33	#97×C₁₀₇	—	2	2	no	—
34	#97×C₁₀₇	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₁₀₇	—	2	2	no	—
36	C₄×C₈×C₁₀₇	—	2	2	no	—
37	C₄×C₈×C₁₀₇	—	2	2	no	—
38	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
39	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
40	C₂×Q₈×C₁₀₇	—	2	2	no	—
41	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₇	#41	1	2	yes	—
43	M₁₆×C₁₀₇	—	4	4	no	—
44	M₁₆×C₁₀₇	—	4	2	no	—
45	C₂×C₈×C₁₀₇	—	2	2	no	—
46	C₂×C₈×C₁₀₇	—	2	2	no	—
47	C₂×C₈×C₁₀₇	—	2	4	no	—
48	C₂×C₈×C₁₀₇	—	2	4	no	—
49	C₂×C₈×C₁₀₇	—	4	1	no	—
50	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₄⋊C₄×C₁₀₇	—	2	4	no	—
54	C₄⋊C₄×C₁₀₇	—	2	2	no	—
55	C₄⋊C₄×C₁₀₇	C₄²	1	4	yes	—
56	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
57	C₄²×C₁₀₇	—	2	2	no	—
58	C₄²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
61	C₂³×C₁₀₇	#117	1	1	yes	characteristic
62	Q₈×C₁₀₇	—	4	4	no	—
63	C₂×C₄×C₁₀₇	—	2	1	no	—
64	C₂×C₄×C₁₀₇	—	2	1	no	—
65	C₂×C₄×C₁₀₇	—	2	4	no	—
66	C₂×C₄×C₁₀₇	—	2	4	no	—
67	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₁₀₇	C₄×D₄	1	2	yes	—
71	C₈×C₁₀₇	—	2	1	no	—
72	C₈×C₁₀₇	—	2	2	no	—
73	C₈×C₁₀₇	—	4	4	no	—
74	C₈×C₁₀₇	—	4	2	no	—
75	C₈×C₁₀₇	—	4	4	no	—
76	C₈×C₁₀₇	#101	1	2	yes	—
77	C₂²×C₁₀₇	—	2	1	no	—
78	C₂²×C₁₀₇	—	2	2	no	—
79	C₂²×C₁₀₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₁₀₇	—	2	1	no	—
81	C₄×C₁₀₇	—	2	2	no	—
82	C₄×C₁₀₇	—	4	4	no	—
83	C₄×C₁₀₇	C₂×#101	1	1	yes	characteristic
84	C₄×C₁₀₇	#413	1	1	yes	characteristic
85	C₄×C₁₀₇	#444	1	4	yes	—
86	#8244	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₁₀₇	—	4	1	no	—
88	C₂×C₁₀₇	C₄×#101	1	1	yes	characteristic
89	C₂×C₁₀₇	#1765	1	2	yes	—
90	#2960	C₂×C₁₀₇	1	1	yes	characteristic
91	#1544	C₂×C₁₀₇	1	1	yes	characteristic
92	#1526	C₂×C₁₀₇	1	1	yes	characteristic
93	#1451	C₂×C₁₀₇	1	2	yes	—
94	#1299	C₂×C₁₀₇	1	2	yes	—
95	C₁₀₇	#8244	1	1	yes	characteristic, direct product factor, Sylow
96	#533	C₂²×C₁₀₇	1	1	yes	characteristic
97	#446	—	2	2	no	—
98	C₈×Q₈	—	2	2	no	—
99	#431	C₂²×C₁₀₇	1	2	yes	—
100	#424	C₄×C₁₀₇	1	2	yes	—
101	#423	C₂²×C₁₀₇	1	1	yes	characteristic
102	#405	C₄×C₁₀₇	1	2	yes	—
103	C₄×M₁₆	C₂²×C₁₀₇	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
105	#332	C₄×C₁₀₇	1	4	yes	—
106	#330	C₄×C₁₀₇	1	4	yes	—
107	#322	—	2	2	no	—
108	#322	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
111	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
113	C₄⋊Q₈	C₂×C₄×C₁₀₇	1	2	yes	—
114	#125	C₂×C₄×C₁₀₇	1	2	yes	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	#117	C₂×C₄×C₁₀₇	1	2	yes	—
118	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	C₂×C₄×C₁₀₇	1	4	yes	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	C₂×C₄×C₁₀₇	1	4	yes	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
129	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
130	C₂×Q₈	—	2	2	no	—
131	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
132	C₂²×C₄	#41×C₁₀₇	1	2	yes	—
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	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	2	yes	—
142	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	C₄²×C₁₀₇	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
149	C₄²	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
150	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
151	C₂³	#117×C₁₀₇	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
159	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
160	C₂×C₄	C₄×D₄×C₁₀₇	1	2	yes	—
161	C₈	—	2	1	no	—
162	C₈	—	2	2	no	—
163	C₈	—	4	4	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	4	no	—
166	C₈	#101×C₁₀₇	1	2	yes	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	C₄×C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	2	no	—
172	C₄	—	4	4	no	—
173	C₄	C₂×#101×C₁₀₇	1	1	yes	characteristic
174	C₄	#413×C₁₀₇	1	1	yes	characteristic
175	C₄	#444×C₁₀₇	1	4	yes	—
176	C₂	—	4	1	no	—
177	C₂	C₄×#101×C₁₀₇	1	1	yes	characteristic
178	C₂	C₁₀₇×#1765	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#2960

C₂

yes

maximal, characteristic

C₁₀₇×#1544

C₂

yes

maximal, characteristic

C₁₀₇×#1526

C₂

yes

maximal, characteristic

C₁₀₇×#1451

C₂

yes

maximal

C₁₀₇×#1299

C₂

yes

maximal

#533×C₁₀₇

C₂²

yes

characteristic

#446×C₁₀₇

—

C₈×Q₈×C₁₀₇

—

#431×C₁₀₇

C₂²

yes

—

#424×C₁₀₇

C₄

yes

—

#423×C₁₀₇

C₂²

yes

characteristic

#405×C₁₀₇

C₄

yes

—

C₄×M₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

#332×C₁₀₇

C₄

yes

—

#330×C₁₀₇

C₄

yes

—

#322×C₁₀₇

—

#322×C₁₀₇

—

C₂×M₁₆×C₁₀₇

—

C₂×M₁₆×C₁₀₇

C₂×C₄

yes

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂×C₄

yes

—

#125×C₁₀₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

#117×C₁₀₇

C₂×C₄

yes

—

C₂×C₄²×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

C₂×C₄

yes

—

C₂²⋊C₈×C₁₀₇

—

#97×C₁₀₇

—

#97×C₁₀₇

C₂×C₄

yes

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

D₄

yes

—

C₄×C₈×C₁₀₇

D₄

yes

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

#41

yes

—

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

—

C₂×C₈×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

—

C₂×C₈×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

C₄²

yes

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂²×C₄

yes

characteristic

C₄²×C₁₀₇

C₂²×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×D₄

yes

—

C₂³×C₁₀₇

#117

yes

characteristic

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₄×D₄

yes

—

C₂×C₄×C₁₀₇

C₄×D₄

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

#101

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

#8244

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

C₄×#101

yes

characteristic

C₂×C₁₀₇

#1765

yes

—

#2960

C₂×C₁₀₇

yes

characteristic

#1544

C₂×C₁₀₇

yes

characteristic

#1526

C₂×C₁₀₇

yes

characteristic

#1451

C₂×C₁₀₇

yes

—

#1299

C₂×C₁₀₇

yes

—

C₁₀₇

#8244

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₁₀₇

yes

characteristic

#446

—

C₈×Q₈

—

#431

C₂²×C₁₀₇

yes

—

100

#424

C₄×C₁₀₇

yes

—

101

#423

C₂²×C₁₀₇

yes

characteristic

102

#405

C₄×C₁₀₇

yes

—

103

C₄×M₁₆

C₂²×C₁₀₇

yes

—

104

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

105

#332

C₄×C₁₀₇

yes

—

106

#330

C₄×C₁₀₇

yes

—

107

#322

—

108

#322

—

109

C₂×M₁₆

—

110

C₂×M₁₆

C₂×C₄×C₁₀₇

yes

—

111

C₂²×C₈

C₂×C₄×C₁₀₇

yes

characteristic

112

C₂²×C₈

C₂×C₄×C₁₀₇

yes

characteristic

113

C₄⋊Q₈

C₂×C₄×C₁₀₇

yes

—

114

#125

C₂×C₄×C₁₀₇

yes

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

#117

C₂×C₄×C₁₀₇

yes

—

118

C₂×C₄²

C₂³×C₁₀₇

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

C₂×C₄×C₁₀₇

yes

—

122

C₂²⋊C₈

—

123

#97

—

124

#97

C₂×C₄×C₁₀₇

yes

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

D₄×C₁₀₇

yes

—

129

C₄×C₈

D₄×C₁₀₇

yes

—

130

C₂×Q₈

—

131

C₂²×C₄

C₂²×C₄×C₁₀₇

yes

characteristic

132

C₂²×C₄

#41×C₁₀₇

yes

—

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

characteristic

141

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

—

142

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄²×C₁₀₇

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂²×C₄×C₁₀₇

yes

characteristic

149

C₄²

C₂²×C₄×C₁₀₇

yes

characteristic

150

C₄²

C₂×D₄×C₁₀₇

yes

—

151

C₂³

#117×C₁₀₇

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄²×C₁₀₇

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₇

yes

characteristic

159

C₂×C₄

C₄×D₄×C₁₀₇

yes

—

160

C₂×C₄

C₄×D₄×C₁₀₇

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

#101×C₁₀₇

yes

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₄×C₂²⋊C₄×C₁₀₇

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

C₂×#101×C₁₀₇

yes

characteristic

174

C₄

#413×C₁₀₇

yes

characteristic

175

C₄

#444×C₁₀₇

yes

—

176

C₂

—

177

C₂

C₄×#101×C₁₀₇

yes

characteristic

178

C₂

C₁₀₇×#1765

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	C₄×C₂²⋊C₄
prime radical	C₂³×C₁₀₇	#117
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

C₂×C₄²×C₁₀₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₇

socle

C₂²×C₁₀₇

C₄×C₂²⋊C₄

C₂³×C₁₀₇

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
86. #8244	C₁₀₇	direct (N × H)
95. C₁₀₇	#8244	direct (N × H)

H ≅ G/N

type

86. #8244

C₁₀₇

direct (N × H)

95. C₁₀₇

#8244

direct (N × H)

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

subgroup	index
C₁₀₇×#2960	2
C₁₀₇×#1544	2
C₁₀₇×#1526	2
C₁₀₇×#1451	2
C₁₀₇×#1299	2
#8244	107

subgroup

index

C₁₀₇×#2960

C₁₀₇×#1544

C₁₀₇×#1526

C₁₀₇×#1451

C₁₀₇×#1299

#8244

107

Subgroups — C₁₀₇×#8244 #41034208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#8244 #41034208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#8244 #41034208