Group #38793319 — subgroups

Subgroups — C₁₀₁×#11168 #38793319

Order 25856.

802 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 222 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	C₂²×#101×C₁₀₁	C₂	1	2	yes	maximal
2	C₁₀₁×#2310	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₁₀₁	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₀₁	C₂²	1	4	yes	—
6	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
7	C₂×#101×C₁₀₁	—	2	4	no	—
8	C₂×#101×C₁₀₁	—	2	8	no	—
9	#391×C₁₀₁	—	2	4	no	—
10	#381×C₁₀₁	C₄	1	4	yes	—
11	#344×C₁₀₁	—	2	8	no	—
12	C₂²×Q₈×C₁₀₁	—	2	4	no	—
13	C₂²×Q₈×C₁₀₁	—	2	4	no	—
14	C₂²×Q₈×C₁₀₁	—	2	4	no	—
15	C₂²×Q₈×C₁₀₁	D₄	1	4	yes	—
16	C₂³×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₁	—	2	4	no	—
19	C₂×M₁₆×C₁₀₁	—	2	4	no	—
20	C₂×M₁₆×C₁₀₁	—	2	4	no	—
21	C₂×M₁₆×C₁₀₁	—	2	4	no	—
22	C₂×M₁₆×C₁₀₁	—	2	4	no	—
23	C₂×M₁₆×C₁₀₁	D₄	1	8	yes	—
24	C₂²×C₈×C₁₀₁	—	2	4	no	—
25	#117×C₁₀₁	—	2	4	no	—
26	#117×C₁₀₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₀₁	—	2	2	no	—
30	#101×C₁₀₁	—	2	16	no	—
31	#95×C₁₀₁	—	2	8	no	—
32	C₂⁴×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₁₀₁	—	2	4	no	—
34	C₂×Q₈×C₁₀₁	—	2	8	no	—
35	C₂×Q₈×C₁₀₁	—	2	8	no	—
36	C₂×Q₈×C₁₀₁	—	2	4	no	—
37	C₂×Q₈×C₁₀₁	—	4	4	no	—
38	C₂×Q₈×C₁₀₁	—	4	4	no	—
39	C₂×Q₈×C₁₀₁	—	4	4	no	—
40	C₂×Q₈×C₁₀₁	—	4	4	no	—
41	C₂²×C₄×C₁₀₁	—	2	4	no	—
42	C₂²×C₄×C₁₀₁	—	2	8	no	—
43	C₂²×C₄×C₁₀₁	—	2	4	no	—
44	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
48	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
49	M₁₆×C₁₀₁	—	2	8	no	—
50	M₁₆×C₁₀₁	—	2	8	no	—
51	M₁₆×C₁₀₁	—	4	4	no	—
52	M₁₆×C₁₀₁	—	4	4	no	—
53	C₂×C₈×C₁₀₁	—	2	8	no	—
54	C₂×C₈×C₁₀₁	—	4	4	no	—
55	C₂×C₈×C₁₀₁	—	4	4	no	—
56	C₄⋊C₄×C₁₀₁	—	4	4	no	—
57	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
58	C₄²×C₁₀₁	—	4	2	no	—
59	C₄²×C₁₀₁	—	4	2	no	—
60	C₂³×C₁₀₁	—	4	2	no	—
61	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₁₀₁	#117	1	2	yes	—
63	C₂³×C₁₀₁	C₄×D₄	1	4	yes	—
64	Q₈×C₁₀₁	—	4	8	no	—
65	Q₈×C₁₀₁	—	4	8	no	—
66	C₂×C₄×C₁₀₁	—	2	4	no	—
67	C₂×C₄×C₁₀₁	—	2	8	no	—
68	C₂×C₄×C₁₀₁	—	4	8	no	—
69	C₂×C₄×C₁₀₁	—	4	8	no	—
70	C₂×C₄×C₁₀₁	—	4	4	no	—
71	C₂×C₄×C₁₀₁	#120	1	2	yes	—
72	C₂×C₄×C₁₀₁	#121	1	4	yes	—
73	C₂×C₄×C₁₀₁	#121	1	4	yes	—
74	C₂×C₄×C₁₀₁	#123	1	4	yes	—
75	C₂×C₄×C₁₀₁	#123	1	4	yes	—
76	C₂×C₄×C₁₀₁	#123	1	2	yes	—
77	C₂×C₄×C₁₀₁	#124	1	4	yes	—
78	C₂×C₄×C₁₀₁	#126	1	4	yes	—
79	C₈×C₁₀₁	—	4	8	no	—
80	C₂²×C₁₀₁	—	2	2	no	—
81	C₂²×C₁₀₁	—	2	4	no	—
82	C₂²×C₁₀₁	—	4	4	no	—
83	C₂²×C₁₀₁	#381	1	1	yes	characteristic
84	C₂²×C₁₀₁	#386	1	2	yes	—
85	C₂²×C₁₀₁	#388	1	4	yes	—
86	C₄×C₁₀₁	—	4	4	no	—
87	C₄×C₁₀₁	—	8	4	no	—
88	C₄×C₁₀₁	#394	1	4	yes	—
89	C₄×C₁₀₁	#397	1	4	yes	—
90	#11168	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₁₀₁	—	2	2	no	—
92	C₂×C₁₀₁	—	2	4	no	—
93	C₂×C₁₀₁	#1440	1	1	yes	characteristic
94	C₂×C₁₀₁	#1830	1	2	yes	—
95	C₂²×#101	C₂×C₁₀₁	1	2	yes	—
96	#2310	C₂×C₁₀₁	1	1	yes	characteristic
97	C₂×#344	C₂×C₁₀₁	1	4	yes	—
98	C₁₀₁	#11168	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
100	C₂²×M₁₆	C₂²×C₁₀₁	1	4	yes	—
101	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
102	C₂×#101	—	2	4	no	—
103	C₂×#101	—	2	8	no	—
104	#391	—	2	4	no	—
105	#381	C₄×C₁₀₁	1	4	yes	—
106	#344	—	2	8	no	—
107	C₂²×Q₈	—	2	4	no	—
108	C₂²×Q₈	—	2	4	no	—
109	C₂²×Q₈	—	2	4	no	—
110	C₂²×Q₈	D₄×C₁₀₁	1	4	yes	—
111	C₂³×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
112	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	—	2	4	no	—
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	2	4	no	—
117	C₂×M₁₆	—	2	4	no	—
118	C₂×M₁₆	D₄×C₁₀₁	1	8	yes	—
119	C₂²×C₈	—	2	4	no	—
120	#117	—	2	4	no	—
121	#117	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
124	C₂×C₄²	—	2	2	no	—
125	#101	—	2	16	no	—
126	#95	—	2	8	no	—
127	C₂⁴	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	8	no	—
130	C₂×Q₈	—	2	8	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	8	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
140	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
141	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
142	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
143	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
144	M₁₆	—	2	8	no	—
145	M₁₆	—	2	8	no	—
146	M₁₆	—	4	4	no	—
147	M₁₆	—	4	4	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	2	8	no	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
157	C₂³	#117×C₁₀₁	1	2	yes	—
158	C₂³	C₄×D₄×C₁₀₁	1	4	yes	—
159	Q₈	—	4	8	no	—
160	Q₈	—	4	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	#120×C₁₀₁	1	2	yes	—
167	C₂×C₄	#121×C₁₀₁	1	4	yes	—
168	C₂×C₄	#121×C₁₀₁	1	4	yes	—
169	C₂×C₄	#123×C₁₀₁	1	2	yes	—
170	C₂×C₄	#123×C₁₀₁	1	4	yes	—
171	C₂×C₄	#123×C₁₀₁	1	4	yes	—
172	C₂×C₄	#124×C₁₀₁	1	4	yes	—
173	C₂×C₄	#126×C₁₀₁	1	4	yes	—
174	C₈	—	4	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	4	4	no	—
178	C₂²	#381×C₁₀₁	1	1	yes	characteristic
179	C₂²	#386×C₁₀₁	1	2	yes	—
180	C₂²	#388×C₁₀₁	1	4	yes	—
181	C₄	—	4	4	no	—
182	C₄	—	8	4	no	—
183	C₄	#394×C₁₀₁	1	4	yes	—
184	C₄	#397×C₁₀₁	1	4	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	C₁₀₁×#1440	1	1	yes	characteristic
188	C₂	C₁₀₁×#1830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#101×C₁₀₁

C₂

yes

maximal

C₁₀₁×#2310

C₂

yes

maximal, characteristic

C₂×#344×C₁₀₁

C₂

yes

maximal

C₂³×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂²×M₁₆×C₁₀₁

C₂²

yes

—

C₂×#117×C₁₀₁

C₂²

yes

—

C₂×#101×C₁₀₁

—

C₂×#101×C₁₀₁

—

#391×C₁₀₁

—

#381×C₁₀₁

C₄

yes

—

#344×C₁₀₁

—

C₂²×Q₈×C₁₀₁

—

C₂²×Q₈×C₁₀₁

—

C₂²×Q₈×C₁₀₁

—

C₂²×Q₈×C₁₀₁

D₄

yes

—

C₂³×C₄×C₁₀₁

C₂×C₄

yes

—

C₂³×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

D₄

yes

—

C₂²×C₈×C₁₀₁

—

#117×C₁₀₁

—

#117×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₂²⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂×C₄²×C₁₀₁

—

#101×C₁₀₁

—

#95×C₁₀₁

—

C₂⁴×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×D₄

yes

—

C₂²×C₄×C₁₀₁

C₂×D₄

yes

—

C₂²×C₄×C₁₀₁

#41

yes

—

C₂²×C₄×C₁₀₁

#41

yes

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₁₀₁

#117

yes

—

C₂³×C₁₀₁

C₄×D₄

yes

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

#120

yes

—

C₂×C₄×C₁₀₁

#121

yes

—

C₂×C₄×C₁₀₁

#121

yes

—

C₂×C₄×C₁₀₁

#123

yes

—

C₂×C₄×C₁₀₁

#123

yes

—

C₂×C₄×C₁₀₁

#123

yes

—

C₂×C₄×C₁₀₁

#124

yes

—

C₂×C₄×C₁₀₁

#126

yes

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#381

yes

characteristic

C₂²×C₁₀₁

#386

yes

—

C₂²×C₁₀₁

#388

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#394

yes

—

C₄×C₁₀₁

#397

yes

—

#11168

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#1440

yes

characteristic

C₂×C₁₀₁

#1830

yes

—

C₂²×#101

C₂×C₁₀₁

yes

—

#2310

C₂×C₁₀₁

yes

characteristic

C₂×#344

C₂×C₁₀₁

yes

—

C₁₀₁

#11168

yes

characteristic, direct product factor, Sylow

C₂³×Q₈

C₂²×C₁₀₁

yes

characteristic

100

C₂²×M₁₆

C₂²×C₁₀₁

yes

—

101

C₂×#117

C₂²×C₁₀₁

yes

—

102

C₂×#101

—

103

C₂×#101

—

104

#391

—

105

#381

C₄×C₁₀₁

yes

—

106

#344

—

107

C₂²×Q₈

—

108

C₂²×Q₈

—

109

C₂²×Q₈

—

110

C₂²×Q₈

D₄×C₁₀₁

yes

—

111

C₂³×C₄

C₂×C₄×C₁₀₁

yes

—

112

C₂³×C₄

C₂³×C₁₀₁

yes

characteristic

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

D₄×C₁₀₁

yes

—

119

C₂²×C₈

—

120

#117

—

121

#117

—

122

C₂×C₄⋊C₄

—

123

C₂×C₂²⋊C₄

C₂×C₄×C₁₀₁

yes

—

124

C₂×C₄²

—

125

#101

—

126

#95

—

127

C₂⁴

C₂²×C₄×C₁₀₁

yes

characteristic

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²⋊C₄×C₁₀₁

yes

—

140

C₂²×C₄

C₂×D₄×C₁₀₁

yes

—

141

C₂²×C₄

C₂×D₄×C₁₀₁

yes

—

142

C₂²×C₄

#41×C₁₀₁

yes

—

143

C₂²×C₄

#41×C₁₀₁

yes

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₂³

—

156

C₂³

C₂×C₂²⋊C₄×C₁₀₁

yes

characteristic

157

C₂³

#117×C₁₀₁

yes

—

158

C₂³

C₄×D₄×C₁₀₁

yes

—

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#120×C₁₀₁

yes

—

167

C₂×C₄

#121×C₁₀₁

yes

—

168

C₂×C₄

#121×C₁₀₁

yes

—

169

C₂×C₄

#123×C₁₀₁

yes

—

170

C₂×C₄

#123×C₁₀₁

yes

—

171

C₂×C₄

#123×C₁₀₁

yes

—

172

C₂×C₄

#124×C₁₀₁

yes

—

173

C₂×C₄

#126×C₁₀₁

yes

—

174

C₈

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#381×C₁₀₁

yes

characteristic

179

C₂²

#386×C₁₀₁

yes

—

180

C₂²

#388×C₁₀₁

yes

—

181

C₄

—

182

C₄

—

183

C₄

#394×C₁₀₁

yes

—

184

C₄

#397×C₁₀₁

yes

—

185

C₂

—

186

C₂

—

187

C₂

C₁₀₁×#1440

yes

characteristic

188

C₂

C₁₀₁×#1830

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#381
commutator	C₂⁴	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₁

#381

commutator

C₂⁴

C₂²×C₄×C₁₀₁

Frattini, cosocle

C₂³×C₄

C₂³×C₁₀₁

prime radical

C₂⁴×C₁₀₁

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

N	H ≅ G/N	type
90. #11168	C₁₀₁	direct (N × H)
98. C₁₀₁	#11168	direct (N × H)

H ≅ G/N

type

90. #11168

C₁₀₁

direct (N × H)

98. C₁₀₁

#11168

direct (N × H)

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

subgroup	index
C₂²×#101×C₁₀₁	2
C₁₀₁×#2310	2
C₂×#344×C₁₀₁	2
#11168	101

subgroup

index

C₂²×#101×C₁₀₁

C₁₀₁×#2310

C₂×#344×C₁₀₁

#11168

101

Subgroups — C₁₀₁×#11168 #38793319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#11168 #38793319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#11168 #38793319