Group #26109701 — subgroups

Subgroups — #26109701

Order 12032.

630 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 195 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	#20243368	C₂	1	2	yes	maximal
2	#20241271	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
5	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
6	#15688878	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
8	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#384	—	2	2	no	—
11	C₄₇×#384	C₄	1	2	yes	—
12	C₄₇×#384	C₂²	1	2	yes	—
13	C₂²×C₄₇⋊C₈	—	2	4	no	—
14	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
15	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
16	C₂³×C₄×C₄₇	—	2	2	no	—
17	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₄₇	—	4	8	no	—
23	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
24	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₄₇	—	2	4	no	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	—	2	4	no	—
36	C₂²×C₄×C₄₇	—	4	8	no	—
37	C₂²×C₄×C₄₇	—	4	4	no	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
41	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₄₇	—	4	4	no	—
45	C₄⋊C₄×C₄₇	—	4	8	no	—
46	C₄⋊C₄×C₄₇	—	4	4	no	—
47	C₄⋊C₄×C₄₇	—	4	8	no	—
48	C₄²×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	—	2	4	no	—
50	C₄²×C₄₇	#41	1	4	yes	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂×C₄₇⋊C₈	—	4	4	no	—
53	C₂×C₄₇⋊C₈	—	4	4	no	—
54	C₂³×C₄₇	—	2	1	no	—
55	C₂³×C₄₇	—	2	1	no	—
56	C₂³×C₄₇	—	2	2	no	—
57	C₂³×C₄₇	#100	1	2	yes	—
58	C₂³×C₄₇	#103	1	4	yes	—
59	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	—	2	8	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	—	4	8	no	—
71	C₂×C₄×C₄₇	—	8	8	no	—
72	C₂×C₄×C₄₇	#101	1	2	yes	—
73	C₂×C₄×C₄₇	#103	1	4	yes	—
74	C₂×C₄×C₄₇	#117	1	2	yes	—
75	C₂×C₄×C₄₇	#123	1	4	yes	—
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
79	C₄₇⋊C₈	—	8	4	no	—
80	#8725	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	4	no	—
82	C₂²×C₄₇	—	2	2	no	—
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	2	4	no	—
85	C₂²×C₄₇	#333	1	4	yes	—
86	C₂²×C₄₇	#381	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
89	C₂²×C₄₇	C₂×#103	1	2	yes	—
90	C₂²×C₄₇	#418	1	2	yes	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	2	4	no	—
94	C₄×C₄₇	—	8	4	no	—
95	C₄×C₄₇	#481	1	4	yes	—
96	C₄×C₄₇	#484	1	4	yes	—
97	C₂×#384	D₄₇	1	1	yes	characteristic
98	#1797	—	47	2	no	—
99	#1296	—	47	4	no	—
100	C₂×C₄₇	—	2	4	no	—
101	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
102	C₂×C₄₇	#1549	1	2	yes	—
103	C₂×C₄₇	#1555	1	1	yes	characteristic
104	C₂×C₄₇	#1821	1	1	yes	characteristic
105	C₂×C₄₇	#1824	1	2	yes	—
106	C₂²×C₄⋊C₄	—	2	2	no	—
107	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
108	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	47	4	no	—
110	C₂×C₂²⋊C₈	—	94	2	no	—
111	#384	—	2	2	no	—
112	#384	Dic₄₇	1	2	yes	—
113	#384	C₂×D₄₇	1	2	yes	—
114	#336	—	94	2	no	—
115	C₄₇	#8725	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	—	2	2	no	—
117	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
118	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
119	C₂²×C₈	—	94	4	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	4	4	no	—
124	C₂×C₄⋊C₄	—	4	8	no	—
125	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
126	C₂×C₄²	—	2	4	no	—
127	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	C₄⋊C₈	—	94	8	no	—
131	C₂²⋊C₈	—	188	4	no	—
132	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	4	8	no	—
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	#10625336	1	4	yes	—
142	C₂²×C₄	#10625337	1	4	yes	—
143	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
147	C₂×C₈	—	188	4	no	—
148	C₂×C₈	—	188	4	no	—
149	C₂×C₈	—	188	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	#10625350	1	4	yes	—
157	C₂³	—	2	1	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	2	no	—
160	C₂³	#13093941	1	2	yes	—
161	C₂³	#13093948	1	4	yes	—
162	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	8	8	no	—
175	C₂×C₄	#13093943	1	2	yes	—
176	C₂×C₄	#13093948	1	4	yes	—
177	C₂×C₄	#13093983	1	2	yes	—
178	C₂×C₄	#13094007	1	4	yes	—
179	C₂×C₄	C₂×#10625336	1	2	yes	—
180	C₂×C₄	C₂×#10625337	1	2	yes	—
181	C₂×C₄	#13094050	1	4	yes	—
182	C₈	—	376	4	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#15688872	1	4	yes	—
188	C₂²	#15688986	1	1	yes	characteristic
189	C₂²	C₂×#13093941	1	1	yes	characteristic
190	C₂²	C₂×#13093943	1	1	yes	characteristic
191	C₂²	C₂×#13093948	1	2	yes	—
192	C₂²	#15689128	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	8	4	no	—
197	C₄	#15689441	1	4	yes	—
198	C₄	#15689458	1	4	yes	—
199	C₂	—	2	4	no	—
200	C₂	C₂×#15688872	1	1	yes	characteristic
201	C₂	#20242073	1	2	yes	—
202	C₂	#20242102	1	1	yes	characteristic
203	C₂	#20243468	1	1	yes	characteristic
204	C₂	#20243478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243368

C₂

yes

maximal

#20241271

C₂

yes

maximal, complemented

C₂×C₄₇×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₂²⋊(C₄₇⋊C₈)

—

#15688878

—

C₂²×C₄⋊C₄×C₄₇

—

C₂²×C₄⋊C₄×C₄₇

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#384

—

C₄₇×#384

C₄

yes

—

C₄₇×#384

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂⁴×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#100

yes

—

C₂³×C₄₇

#103

yes

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄₇⋊C₈

—

#8725

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#333

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#418

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#481

yes

—

C₄×C₄₇

#484

yes

—

C₂×#384

D₄₇

yes

characteristic

#1797

—

#1296

—

100

C₂×C₄₇

—

101

C₂×C₄₇

C₂×#333

yes

characteristic

102

C₂×C₄₇

#1549

yes

—

103

C₂×C₄₇

#1555

yes

characteristic

104

C₂×C₄₇

#1821

yes

characteristic

105

C₂×C₄₇

#1824

yes

—

106

C₂²×C₄⋊C₄

—

107

C₂²×C₄⋊C₄

Dic₄₇

yes

—

108

C₂²×C₄²

C₂×D₄₇

yes

characteristic

109

C₂×C₄⋊C₈

—

110

C₂×C₂²⋊C₈

—

111

#384

—

112

#384

Dic₄₇

yes

—

113

#384

C₂×D₄₇

yes

—

114

#336

—

115

C₄₇

#8725

yes

characteristic, complemented, Sylow

116

C₂³×C₄

—

117

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

118

C₂³×C₄

C₂²⋊D₄₇

yes

—

119

C₂²×C₈

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

126

C₂×C₄²

—

127

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

128

C₂×C₄²

C₂²⋊D₄₇

yes

—

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

C₄⋊C₈

—

131

C₂²⋊C₈

—

188

—

132

C₂⁴

C₂²⋊Dic₄₇

yes

characteristic

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

#10625336

yes

—

142

C₂²×C₄

#10625337

yes

—

143

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

144

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

145

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

147

C₂×C₈

—

188

—

148

C₂×C₈

—

188

—

149

C₂×C₈

—

188

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

#10625350

yes

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

#13093941

yes

—

161

C₂³

#13093948

yes

—

162

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#13093943

yes

—

176

C₂×C₄

#13093948

yes

—

177

C₂×C₄

#13093983

yes

—

178

C₂×C₄

#13094007

yes

—

179

C₂×C₄

C₂×#10625336

yes

—

180

C₂×C₄

C₂×#10625337

yes

—

181

C₂×C₄

#13094050

yes

—

182

C₈

—

376

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#15688872

yes

—

188

C₂²

#15688986

yes

characteristic

189

C₂²

C₂×#13093941

yes

characteristic

190

C₂²

C₂×#13093943

yes

characteristic

191

C₂²

C₂×#13093948

yes

—

192

C₂²

#15689128

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#15689441

yes

—

198

C₄

#15689458

yes

—

199

C₂

—

200

C₂

C₂×#15688872

yes

characteristic

201

C₂

#20242073

yes

—

202

C₂

#20242102

yes

characteristic

203

C₂

#20243468

yes

characteristic

204

C₂

#20243478

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	#8725
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#384

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#384

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₄₇

C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#384

N	H ≅ G/N	type
2. #20241271	C₂	semidirect
115. C₄₇	#8725	semidirect

H ≅ G/N

type

2. #20241271

C₂

semidirect

115. C₄₇

#8725

semidirect

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

subgroup	index
#20243368	2
#20241271	2
C₂×C₄₇×#384	2
#8725	47

subgroup

index

#20243368

#20241271

C₂×C₄₇×#384

#8725

Subgroups — #26109701

Conjugacy classes of subgroups

Complements

Maximal subgroups