Group #26143443 — subgroups

Subgroups — #26143443

Order 12032.

4390 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 4390 conjugacy classes of subgroups — too many to draw. Only its 651 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×#15688983	C₂	1	6	yes	maximal, complemented
2	#20241659	C₂	1	2	yes	maximal, complemented
3	#20241629	C₂	1	6	yes	maximal, complemented
4	C₂³×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₂²⋊Dic₄₇	C₂²	1	12	yes	—
7	#15688983	—	2	24	no	—
8	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
9	C₂²×#41×C₄₇	—	2	24	no	—
10	C₂²×#41×C₄₇	C₄	1	8	yes	complemented
11	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₄₇	C₂²	1	3	yes	—
13	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
14	C₂³×Dic₄₇	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₄₇	—	2	24	no	—
16	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
17	C₂×C₂²⋊Dic₄₇	—	4	24	no	—
18	C₄₇⋊#95	—	2	24	no	—
19	C₂⁵×C₄₇	C₂³	1	3	yes	—
20	C₂×#41×C₄₇	—	2	128	no	—
21	C₂×#41×C₄₇	—	2	96	no	—
22	C₂²×Q₈×C₄₇	—	2	12	no	—
23	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×D₄×C₄₇	—	2	24	no	—
25	C₂²×D₄×C₄₇	—	2	12	no	—
26	C₂²×D₄×C₄₇	C₂×C₄	1	12	yes	—
27	C₂³×C₄×C₄₇	—	2	24	no	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	12	yes	—
29	C₂³×C₄×C₄₇	D₄	1	24	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂⁴×C₄₇	—	2	12	no	—
33	C₂⁴×C₄₇	—	2	12	no	—
34	C₂⁴×C₄₇	—	2	12	no	—
35	C₂⁴×C₄₇	C₂²×C₄	1	6	yes	—
36	C₂⁴×C₄₇	C₂×D₄	1	12	yes	—
37	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
38	#41×C₄₇	—	2	256	no	—
39	C₂×Q₈×C₄₇	—	2	32	no	—
40	C₂×Q₈×C₄₇	—	2	24	no	—
41	C₂×D₄×C₄₇	—	2	96	no	—
42	C₂×D₄×C₄₇	—	2	48	no	—
43	C₂×D₄×C₄₇	—	2	24	no	—
44	C₂²×C₄×C₄₇	—	2	12	no	—
45	C₂²×C₄×C₄₇	—	2	48	no	—
46	C₂²×C₄×C₄₇	—	2	96	no	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	48	yes	complemented
48	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
52	C₂²×Dic₄₇	—	2	24	no	—
53	C₂²×Dic₄₇	—	2	24	no	—
54	C₂²×Dic₄₇	—	2	8	no	—
55	C₂²⋊Dic₄₇	—	4	48	no	—
56	C₄⋊Dic₄₇	—	4	16	no	—
57	C₂³×C₄₇	—	2	12	no	—
58	C₂³×C₄₇	—	2	24	no	—
59	C₂³×C₄₇	—	2	24	no	—
60	C₂³×C₄₇	—	2	12	no	—
61	C₂³×C₄₇	—	2	12	no	—
62	C₂³×C₄₇	—	2	24	no	—
63	C₂³×C₄₇	—	4	12	no	—
64	C₂³×C₄₇	—	4	12	no	—
65	C₂³×C₄₇	—	4	24	no	—
66	C₂³×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
67	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
68	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
69	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
70	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
71	Q₈×C₄₇	—	2	32	no	—
72	D₄×C₄₇	—	2	96	no	—
73	C₂×C₄×C₄₇	—	2	12	no	—
74	C₂×C₄×C₄₇	—	2	24	no	—
75	C₂×C₄×C₄₇	—	2	96	no	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
77	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
78	C₂×C₄×C₄₇	#120	1	16	yes	—
79	C₂×Dic₄₇	—	4	8	no	—
80	C₂×Dic₄₇	—	4	24	no	—
81	C₂×Dic₄₇	—	4	24	no	—
82	#14337	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	24	no	—
84	C₂²×C₄₇	—	2	12	no	—
85	C₂²×C₄₇	—	4	24	no	—
86	C₂²×C₄₇	—	4	24	no	—
87	C₂²×C₄₇	—	4	24	no	—
88	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	3	yes	—
89	C₂²×C₄₇	#520	1	4	yes	—
90	C₂²×C₄₇	C₂×#120	1	4	yes	—
91	C₂²×C₄₇	#535	1	6	yes	—
92	C₂²×C₄₇	#536	1	6	yes	—
93	C₂²×C₄₇	#537	1	12	yes	—
94	C₄×C₄₇	—	2	12	no	—
95	C₄×C₄₇	#379	1	8	yes	—
96	Dic₄₇	—	8	8	no	—
97	C₂³×#41	D₄₇	1	1	yes	characteristic
98	C₂×#380	—	47	6	no	—
99	#1446	—	47	2	no	—
100	#1438	—	47	6	no	—
101	C₂×C₄₇	—	4	12	no	—
102	C₂×C₄₇	C₂×#379	1	1	yes	characteristic
103	C₂×C₄₇	#2311	1	6	yes	—
104	C₂×C₄₇	#2615	1	4	yes	—
105	C₂×C₄₇	#2616	1	4	yes	—
106	C₂²×#41	—	2	24	no	—
107	C₂²×#41	Dic₄₇	1	8	yes	complemented
108	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
109	C₂³×D₄	C₂×D₄₇	1	3	yes	—
110	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
111	C₂²×C₄⋊C₄	—	47	4	no	—
112	C₂²×C₂²⋊C₄	—	47	12	no	—
113	#380	—	94	24	no	—
114	C₂×#95	—	47	12	no	—
115	C₄₇	#14337	1	1	yes	characteristic, complemented, Sylow
116	C₂⁵	C₂²×D₄₇	1	3	yes	—
117	C₂×#41	—	2	128	no	—
118	C₂×#41	—	2	96	no	—
119	C₂²×Q₈	—	2	12	no	—
120	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
121	C₂²×D₄	—	2	24	no	—
122	C₂²×D₄	—	2	12	no	—
123	C₂²×D₄	C₂×Dic₄₇	1	12	yes	—
124	C₂³×C₄	—	2	24	no	—
125	C₂³×C₄	—	47	8	no	—
126	C₂³×C₄	C₂×Dic₄₇	1	12	yes	—
127	C₂³×C₄	C₂²⋊D₄₇	1	24	yes	—
128	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
129	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	94	24	no	—
131	C₂×C₂²⋊C₄	—	94	24	no	—
132	C₂×C₂²⋊C₄	—	188	24	no	—
133	#95	—	94	24	no	—
134	C₂⁴	—	2	12	no	—
135	C₂⁴	—	2	12	no	—
136	C₂⁴	—	2	12	no	—
137	C₂⁴	C₂²×Dic₄₇	1	6	yes	—
138	C₂⁴	C₂×C₂²⋊D₄₇	1	12	yes	—
139	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
140	#41	—	2	256	no	—
141	C₂×Q₈	—	2	32	no	—
142	C₂×Q₈	—	2	24	no	—
143	C₂×D₄	—	2	96	no	—
144	C₂×D₄	—	2	48	no	—
145	C₂×D₄	—	2	24	no	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	2	48	no	—
148	C₂²×C₄	—	2	96	no	—
149	C₂²×C₄	—	94	24	no	—
150	C₂²×C₄	—	94	24	no	—
151	C₂²×C₄	—	94	8	no	—
152	C₂²×C₄	C₂²⋊Dic₄₇	1	48	yes	complemented
153	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
154	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
157	C₄⋊C₄	—	188	16	no	—
158	C₂²⋊C₄	—	188	48	no	—
159	C₂³	—	2	12	no	—
160	C₂³	—	2	24	no	—
161	C₂³	—	2	24	no	—
162	C₂³	—	2	12	no	—
163	C₂³	—	2	12	no	—
164	C₂³	—	2	24	no	—
165	C₂³	—	4	12	no	—
166	C₂³	—	4	12	no	—
167	C₂³	—	4	24	no	—
168	C₂³	C₂×C₂²⋊Dic₄₇	1	12	yes	—
169	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
170	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
171	C₂³	#13094070	1	4	yes	—
172	C₂³	#13094071	1	4	yes	—
173	Q₈	—	2	32	no	—
174	D₄	—	2	96	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	24	no	—
177	C₂×C₄	—	2	96	no	—
178	C₂×C₄	—	188	8	no	—
179	C₂×C₄	—	188	24	no	—
180	C₂×C₄	—	188	24	no	—
181	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
183	C₂×C₄	#13093993	1	16	yes	—
184	C₂²	—	2	24	no	—
185	C₂²	—	2	12	no	—
186	C₂²	—	4	24	no	—
187	C₂²	—	4	24	no	—
188	C₂²	—	4	24	no	—
189	C₂²	C₂²×C₂²⋊Dic₄₇	1	3	yes	—
190	C₂²	#15689606	1	4	yes	—
191	C₂²	C₂×#13093993	1	4	yes	—
192	C₂²	#15689687	1	6	yes	—
193	C₂²	#15689694	1	6	yes	—
194	C₂²	#15689701	1	12	yes	—
195	C₄	—	2	12	no	—
196	C₄	—	376	8	no	—
197	C₄	#15688979	1	8	yes	—
198	C₂	—	4	12	no	—
199	C₂	C₂×#15688979	1	1	yes	characteristic
200	C₂	#20245980	1	6	yes	—
201	C₂	#20249112	1	4	yes	—
202	C₂	#20249117	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688983

C₂

yes

maximal, complemented

#20241659

C₂

yes

maximal, complemented

#20241629

C₂

yes

maximal, complemented

C₂³×#41×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₇

C₂²

yes

—

C₂²×C₂²⋊Dic₄₇

C₂²

yes

—

#15688983

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂²×#41×C₄₇

—

C₂²×#41×C₄₇

C₄

yes

complemented

C₂³×Q₈×C₄₇

C₂²

yes

characteristic

C₂³×D₄×C₄₇

C₂²

yes

—

C₂⁴×C₄×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₄₇⋊#95

—

C₂⁵×C₄₇

C₂³

yes

—

C₂×#41×C₄₇

—

128

—

C₂×#41×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

—

C₂⁴×C₄₇

C₂×D₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

#41×C₄₇

—

256

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

complemented

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

H(5,𝔽₂)

yes

—

C₂³×C₄₇

2₋¹⁺⁴

yes

—

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#14337

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₂²⋊C₄

yes

—

C₂²×C₄₇

#520

yes

—

C₂²×C₄₇

C₂×#120

yes

—

C₂²×C₄₇

#535

yes

—

C₂²×C₄₇

#536

yes

—

C₂²×C₄₇

#537

yes

—

C₄×C₄₇

—

C₄×C₄₇

#379

yes

—

Dic₄₇

—

C₂³×#41

D₄₇

yes

characteristic

C₂×#380

—

#1446

—

100

#1438

—

101

C₂×C₄₇

—

102

C₂×C₄₇

C₂×#379

yes

characteristic

103

C₂×C₄₇

#2311

yes

—

104

C₂×C₄₇

#2615

yes

—

105

C₂×C₄₇

#2616

yes

—

106

C₂²×#41

—

107

C₂²×#41

Dic₄₇

yes

complemented

108

C₂³×Q₈

C₂×D₄₇

yes

characteristic

109

C₂³×D₄

C₂×D₄₇

yes

—

110

C₂⁴×C₄

C₂×D₄₇

yes

—

111

C₂²×C₄⋊C₄

—

112

C₂²×C₂²⋊C₄

—

113

#380

—

114

C₂×#95

—

115

C₄₇

#14337

yes

characteristic, complemented, Sylow

116

C₂⁵

C₂²×D₄₇

yes

—

117

C₂×#41

—

128

—

118

C₂×#41

—

119

C₂²×Q₈

—

120

C₂²×Q₈

C₂×Dic₄₇

yes

—

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂²×D₄

C₂×Dic₄₇

yes

—

124

C₂³×C₄

—

125

C₂³×C₄

—

126

C₂³×C₄

C₂×Dic₄₇

yes

—

127

C₂³×C₄

C₂²⋊D₄₇

yes

—

128

C₂³×C₄

C₂²×D₄₇

yes

—

129

C₂³×C₄

C₂²×D₄₇

yes

characteristic

130

C₂×C₄⋊C₄

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₂²⋊C₄

—

188

—

133

#95

—

134

C₂⁴

—

135

C₂⁴

—

136

C₂⁴

—

137

C₂⁴

C₂²×Dic₄₇

yes

—

138

C₂⁴

C₂×C₂²⋊D₄₇

yes

—

139

C₂⁴

C₂³×D₄₇

yes

characteristic

140

#41

—

256

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂²⋊Dic₄₇

yes

complemented

153

C₂²×C₄

C₂²×Dic₄₇

yes

—

154

C₂²×C₄

C₂²×Dic₄₇

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

156

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

157

C₄⋊C₄

—

188

—

158

C₂²⋊C₄

—

188

—

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

169

C₂³

C₂³×Dic₄₇

yes

characteristic

170

C₂³

C₂²×C₂²⋊D₄₇

yes

—

171

C₂³

#13094070

yes

—

172

C₂³

#13094071

yes

—

173

Q₈

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

188

—

179

C₂×C₄

—

188

—

180

C₂×C₄

—

188

—

181

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

182

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

183

C₂×C₄

#13093993

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₂²×C₂²⋊Dic₄₇

yes

—

190

C₂²

#15689606

yes

—

191

C₂²

C₂×#13093993

yes

—

192

C₂²

#15689687

yes

—

193

C₂²

#15689694

yes

—

194

C₂²

#15689701

yes

—

195

C₄

—

196

C₄

—

376

—

197

C₄

#15688979

yes

—

198

C₂

—

199

C₂

C₂×#15688979

yes

characteristic

200

C₂

#20245980

yes

—

201

C₂

#20249112

yes

—

202

C₂

#20249117

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting, prime radical	C₂³×#41×C₄₇	C₂
socle, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂³×#41	D₄₇
hypocenter	C₄₇	#14337
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting, prime radical

C₂³×#41×C₄₇

C₂

socle, cosocle

C₂⁴×C₄₇

C₂⁴

hypercenter

C₂³×#41

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂³×#41

N	H ≅ G/N	type
1. C₂×#15688983	C₂	semidirect
2. #20241659	C₂	semidirect
3. #20241629	C₂	semidirect
10. C₂²×#41×C₄₇	C₄	semidirect
47. C₂²×C₄×C₄₇	C₂²⋊C₄	semidirect
107. C₂²×#41	Dic₄₇	semidirect
115. C₄₇	#14337	semidirect
152. C₂²×C₄	C₂²⋊Dic₄₇	semidirect

H ≅ G/N

type

1. C₂×#15688983

C₂

semidirect

2. #20241659

C₂

semidirect

3. #20241629

C₂

semidirect

10. C₂²×#41×C₄₇

C₄

semidirect

47. C₂²×C₄×C₄₇

C₂²⋊C₄

semidirect

107. C₂²×#41

Dic₄₇

semidirect

115. C₄₇

#14337

semidirect

152. C₂²×C₄

C₂²⋊Dic₄₇

semidirect

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

subgroup	index
C₂×#15688983	2
#20241659	2
#20241629	2
C₂³×#41×C₄₇	2
#14337	47

subgroup

index

C₂³×#41×C₄₇

Subgroups — #26143443

Conjugacy classes of subgroups

Complements

Maximal subgroups