Group #23789457 — subgroups

Subgroups — C₄₁×#13256 #23789457

Order 10496.

462 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#362	C₂	1	4	yes	maximal
4	C₄×#108×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₄₁	—	2	1	no	—
6	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₄₁	—	2	1	no	—
8	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#494	—	2	2	no	—
10	C₄₁×#494	C₄	1	4	yes	—
11	C₂×#108×C₄₁	C₂²	1	2	yes	—
12	C₄×M₁₆×C₄₁	—	2	1	no	—
13	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×C₄⋊C₁₆	—	2	2	no	—
15	C₄₁×#362	—	2	8	no	—
16	C₄₁×#341	—	2	1	no	—
17	C₂²×Q₈×C₄₁	—	2	2	no	—
18	C₂×Q₁₆×C₄₁	—	2	4	no	—
19	C₂×Q₁₆×C₄₁	—	2	4	no	—
20	C₂×Q₁₆×C₄₁	—	2	2	no	—
21	C₂×Q₁₆×C₄₁	—	2	4	no	—
22	C₂×Q₁₆×C₄₁	—	4	4	no	—
23	C₂×Q₁₆×C₄₁	C₂×C₄	1	4	yes	—
24	C₂×M₁₆×C₄₁	—	2	2	no	—
25	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₁	—	2	4	no	—
28	C₄⋊Q₈×C₄₁	—	4	1	no	—
29	C₄⋊Q₈×C₄₁	—	4	1	no	—
30	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
31	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
32	M₃₂×C₄₁	—	2	4	no	—
33	C₂×C₁₆×C₄₁	—	2	2	no	—
34	#108×C₄₁	—	2	2	no	—
35	#108×C₄₁	D₄	1	4	yes	—
36	#97×C₄₁	—	4	1	no	—
37	C₄×C₈×C₄₁	—	4	1	no	—
38	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₁	D₄	1	2	yes	—
40	C₂×Q₈×C₄₁	—	2	4	no	—
41	C₂×Q₈×C₄₁	—	4	2	no	—
42	C₂×Q₈×C₄₁	—	4	4	no	—
43	C₂×Q₈×C₄₁	—	4	4	no	—
44	C₂²×C₄×C₄₁	—	4	2	no	—
45	C₂²×C₄×C₄₁	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₄₁	—	2	8	no	—
48	Q₁₆×C₄₁	—	4	8	no	—
49	Q₁₆×C₄₁	—	4	2	no	—
50	Q₁₆×C₄₁	—	4	2	no	—
51	M₁₆×C₄₁	—	2	4	no	—
52	M₁₆×C₄₁	—	4	2	no	—
53	C₂×C₈×C₄₁	—	2	2	no	—
54	C₂×C₈×C₄₁	—	4	2	no	—
55	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₁	#41	1	2	yes	—
60	C₄⋊C₄×C₄₁	—	4	4	no	—
61	C₄⋊C₄×C₄₁	—	4	4	no	—
62	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₄₁	D₈	1	1	yes	characteristic
64	C₄²×C₄₁	D₈	1	1	yes	characteristic
65	C₁₆×C₄₁	—	4	2	no	—
66	C₂³×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₄₁	—	4	4	no	—
68	Q₈×C₄₁	—	4	4	no	—
69	Q₈×C₄₁	—	8	4	no	—
70	C₂×C₄×C₄₁	—	2	4	no	—
71	C₂×C₄×C₄₁	—	4	4	no	—
72	C₂×C₄×C₄₁	—	8	4	no	—
73	C₂×C₄×C₄₁	#102	1	2	yes	—
74	C₂×C₄×C₄₁	#102	1	2	yes	—
75	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₁	#121	1	2	yes	—
77	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₄₁	—	2	2	no	—
80	C₈×C₄₁	—	4	4	no	—
81	C₈×C₄₁	C₄×D₄	1	2	yes	—
82	C₈×C₄₁	#124	1	1	yes	characteristic
83	C₈×C₄₁	#127	1	1	yes	characteristic
84	#13256	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₄₁	—	2	2	no	—
86	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
87	C₂²×C₄₁	#460	1	2	yes	—
88	C₄×C₄₁	—	2	2	no	—
89	C₄×C₄₁	—	8	4	no	—
90	C₄×C₄₁	#362	1	4	yes	complemented
91	C₄×C₄₁	#390	1	1	yes	characteristic
92	C₄×C₄₁	#418	1	1	yes	characteristic
93	C₄×C₄₁	#459	1	2	yes	—
94	C₂×#494	C₂×C₄₁	1	1	yes	characteristic
95	#2150	C₂×C₄₁	1	1	yes	characteristic
96	C₂×#362	C₂×C₄₁	1	4	yes	—
97	C₄×#108	C₂×C₄₁	1	1	yes	characteristic
98	C₂×C₄₁	—	2	2	no	—
99	C₂×C₄₁	#1982	1	1	yes	characteristic
100	C₂×C₄₁	C₂×#362	1	2	yes	—
101	C₂²×Q₁₆	—	2	1	no	—
102	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
105	#494	—	2	2	no	—
106	#494	C₄×C₄₁	1	4	yes	—
107	C₂×#108	C₂²×C₄₁	1	2	yes	—
108	C₄×M₁₆	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#362	—	2	8	no	—
112	#341	—	2	1	no	—
113	C₄₁	#13256	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	4	no	—
119	C₂×Q₁₆	—	4	4	no	—
120	C₂×Q₁₆	C₂×C₄×C₄₁	1	4	yes	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
129	M₃₂	—	2	4	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	2	2	no	—
132	#108	D₄×C₄₁	1	4	yes	—
133	#97	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
136	C₄×C₈	D₄×C₄₁	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	SD₁₆×C₄₁	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	4	8	no	—
146	Q₁₆	—	4	2	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
153	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
154	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
156	C₂×C₈	#41×C₄₁	1	2	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
160	C₄²	D₈×C₄₁	1	1	yes	characteristic
161	C₄²	D₈×C₄₁	1	1	yes	characteristic
162	C₁₆	—	4	2	no	—
163	C₂³	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	Q₈	—	4	4	no	—
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	#102×C₄₁	1	2	yes	—
171	C₂×C₄	#102×C₄₁	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
173	C₂×C₄	#121×C₄₁	1	2	yes	—
174	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
175	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	C₄×D₄×C₄₁	1	2	yes	—
179	C₈	#124×C₄₁	1	1	yes	characteristic
180	C₈	#127×C₄₁	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
183	C₂²	C₄₁×#460	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	C₄₁×#362	1	4	yes	complemented
187	C₄	C₄₁×#390	1	1	yes	characteristic
188	C₄	C₄₁×#418	1	1	yes	characteristic
189	C₄	C₄₁×#459	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	C₄₁×#1982	1	1	yes	characteristic
192	C₂	C₂×C₄₁×#362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄₁×#494

C₂

yes

maximal, characteristic

C₄₁×#2150

C₂

yes

maximal, characteristic

C₂×C₄₁×#362

C₂

yes

maximal

C₄×#108×C₄₁

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₄₁

—

C₂²×Q₁₆×C₄₁

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₁

—

C₂×M₃₂×C₄₁

C₂²

yes

—

C₄₁×#494

—

C₄₁×#494

C₄

yes

—

C₂×#108×C₄₁

C₂²

yes

—

C₄×M₁₆×C₄₁

—

C₂×C₄×C₈×C₄₁

C₂²

yes

characteristic

C₄₁×C₄⋊C₁₆

—

C₄₁×#362

—

C₄₁×#341

—

C₂²×Q₈×C₄₁

—

C₂×Q₁₆×C₄₁

—

C₂×Q₁₆×C₄₁

—

C₂×Q₁₆×C₄₁

—

C₂×Q₁₆×C₄₁

—

C₂×Q₁₆×C₄₁

—

C₂×Q₁₆×C₄₁

C₂×C₄

yes

—

C₂×M₁₆×C₄₁

—

C₂²×C₈×C₄₁

D₄

yes

characteristic

C₂²×C₈×C₄₁

C₂³

yes

characteristic

C₄⋊Q₈×C₄₁

—

C₄⋊Q₈×C₄₁

—

C₄⋊Q₈×C₄₁

—

C₂×C₄⋊C₄×C₄₁

—

C₂×C₄²×C₄₁

D₄

yes

characteristic

M₃₂×C₄₁

—

C₂×C₁₆×C₄₁

—

#108×C₄₁

—

#108×C₄₁

D₄

yes

—

#97×C₄₁

—

C₄×C₈×C₄₁

—

C₄×C₈×C₄₁

C₂×C₄

yes

—

C₄×C₈×C₄₁

D₄

yes

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂²×C₄×C₄₁

—

C₂²×C₄×C₄₁

SD₁₆

yes

—

C₂²×C₄×C₄₁

C₂×D₄

yes

characteristic

Q₁₆×C₄₁

—

Q₁₆×C₄₁

—

Q₁₆×C₄₁

—

Q₁₆×C₄₁

—

M₁₆×C₄₁

—

M₁₆×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

C₂²⋊C₄

yes

—

C₂×C₈×C₄₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₁

C₂×D₄

yes

—

C₂×C₈×C₄₁

C₂×D₄

yes

characteristic

C₂×C₈×C₄₁

#41

yes

—

C₄⋊C₄×C₄₁

—

C₄⋊C₄×C₄₁

—

C₄²×C₄₁

C₂²⋊C₄

yes

—

C₄²×C₄₁

D₈

yes

characteristic

C₄²×C₄₁

D₈

yes

characteristic

C₁₆×C₄₁

—

C₂³×C₄₁

C₂×SD₁₆

yes

characteristic

Q₈×C₄₁

—

Q₈×C₄₁

—

Q₈×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

#102

yes

—

C₂×C₄×C₄₁

#102

yes

—

C₂×C₄×C₄₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₁

#121

yes

—

C₂×C₄×C₄₁

C₂×D₈

yes

characteristic

C₂×C₄×C₄₁

AGL(1,ℤ/8)

yes

—

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

C₄×D₄

yes

—

C₈×C₄₁

yes

C₈×C₄₁

yes

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₁

—

C₂²×C₄₁

C₂×#102

yes

characteristic

C₂²×C₄₁

#460

yes

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

yes

C₄×C₄₁

yes

C₄×C₄₁

yes

C₄×C₄₁

yes

—

C₂×#494

C₂×C₄₁

yes

characteristic

#2150

C₂×C₄₁

yes

characteristic

C₂×#362

C₂×C₄₁

yes

—

C₄×#108

C₂×C₄₁

yes

characteristic

C₂×C₄₁

—

C₂×C₄₁

#1982

yes

characteristic

100

C₂×C₄₁

C₂×#362

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

C₂²×C₄₁

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×M₃₂

C₂²×C₄₁

yes

—

105

#494

—

106

#494

C₄×C₄₁

yes

—

107

C₂×#108

C₂²×C₄₁

yes

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

C₂²×C₄₁

yes

characteristic

110

C₄⋊C₁₆

—

111

#362

—

112

#341

—

113

C₄₁

#13256

yes

characteristic, direct product factor, Sylow

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

C₂×C₄×C₄₁

yes

—

121

C₂×M₁₆

—

122

C₂²×C₈

D₄×C₄₁

yes

characteristic

123

C₂²×C₈

C₂³×C₄₁

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

D₄×C₄₁

yes

characteristic

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

D₄×C₄₁

yes

—

133

#97

—

134

C₄×C₈

—

135

C₄×C₈

C₂×C₄×C₄₁

yes

—

136

C₄×C₈

D₄×C₄₁

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

SD₁₆×C₄₁

yes

—

143

C₂²×C₄

C₂×D₄×C₄₁

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂²⋊C₄×C₄₁

yes

—

153

C₂×C₈

C₂²×C₄×C₄₁

yes

characteristic

154

C₂×C₈

C₂×D₄×C₄₁

yes

—

155

C₂×C₈

C₂×D₄×C₄₁

yes

characteristic

156

C₂×C₈

#41×C₄₁

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

C₂²⋊C₄×C₄₁

yes

—

160

C₄²

D₈×C₄₁

yes

characteristic

161

C₄²

D₈×C₄₁

yes

characteristic

162

C₁₆

—

163

C₂³

C₂×SD₁₆×C₄₁

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#102×C₄₁

yes

—

171

C₂×C₄

#102×C₄₁

yes

—

172

C₂×C₄

C₂×C₂²⋊C₄×C₄₁

yes

characteristic

173

C₂×C₄

#121×C₄₁

yes

—

174

C₂×C₄

C₂×D₈×C₄₁

yes

characteristic

175

C₂×C₄

AGL(1,ℤ/8)×C₄₁

yes

—

176

C₈

—

177

C₈

—

178

C₈

C₄×D₄×C₄₁

yes

—

179

C₈

#124×C₄₁

yes

characteristic

180

C₈

#127×C₄₁

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#102×C₄₁

yes

characteristic

183

C₂²

C₄₁×#460

yes

—

184

C₄

—

185

C₄

—

186

C₄

C₄₁×#362

yes

complemented

187

C₄

C₄₁×#390

yes

characteristic

188

C₄

C₄₁×#418

yes

characteristic

189

C₄

C₄₁×#459

yes

—

190

C₂

—

191

C₂

C₄₁×#1982

yes

characteristic

192

C₂

C₂×C₄₁×#362

yes

—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂³×C₄₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₁

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₄₁

Frattini, cosocle

C₂²×C₈

C₂³×C₄₁

prime radical

C₂³×C₄₁

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

N	H ≅ G/N	type
84. #13256	C₄₁	direct (N × H)
90. C₄×C₄₁	#362	semidirect
113. C₄₁	#13256	direct (N × H)
186. C₄	C₄₁×#362	semidirect

H ≅ G/N

type

84. #13256

C₄₁

direct (N × H)

90. C₄×C₄₁

#362

semidirect

113. C₄₁

#13256

direct (N × H)

186. C₄

C₄₁×#362

semidirect

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

subgroup	index
C₂×C₄₁×#494	2
C₄₁×#2150	2
C₂×C₄₁×#362	2
C₄×#108×C₄₁	2
#13256	41

subgroup

index

C₂×C₄₁×#494

C₄₁×#2150

C₂×C₄₁×#362

C₄×#108×C₄₁

#13256

Subgroups — C₄₁×#13256 #23789457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#13256 #23789457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13256 #23789457