Group #26109984 — subgroups

Subgroups — #26109984

Order 12032.

228 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20242151	C₂	1	1	yes	maximal, characteristic
2	#20241671	C₂	1	1	yes	maximal, characteristic
3	#20241251	C₂	1	2	yes	maximal
4	#20241249	C₂	1	2	yes	maximal
5	C₄₇×#1568	C₂	1	1	yes	maximal, characteristic
6	#15689180	C₂²	1	2	yes	—
7	#15689176	C₂²	1	1	yes	characteristic
8	#15689176	C₂²	1	1	yes	characteristic
9	#15688876	—	2	1	no	—
10	#15688874	—	2	1	no	—
11	#15688844	C₄	1	2	yes	—
12	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
13	C₄₇×#432	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#335	—	2	1	no	—
16	C₄₇×#334	—	2	1	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	#13093935	—	2	2	no	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₂²⋊C₈×C₄₇	—	2	2	no	—
31	#97×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	#41	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	1	yes	characteristic
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂³×C₄₇	#125	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
64	C₈×C₄₇	—	2	1	no	—
65	C₈×C₄₇	—	2	1	no	—
66	C₈×C₄₇	—	8	2	no	—
67	C₈×C₄₇	#108	1	2	yes	—
68	C₈×C₄₇	#108	1	2	yes	—
69	C₄₇⋊C₈	—	4	2	no	—
70	C₄₇⋊C₈	—	4	2	no	—
71	C₄₇⋊C₈	—	8	2	no	—
72	#8796	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	—	2	1	no	—
75	C₂²×C₄₇	—	2	1	no	—
76	C₂²×C₄₇	#382	1	1	yes	characteristic
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	—	2	1	no	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
83	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
84	#1568	—	47	1	no	—
85	#1568	D₄₇	1	1	yes	characteristic
86	#1450	—	47	1	no	—
87	#1287	—	47	2	no	—
88	#1287	—	47	2	no	—
89	C₂×C₄₇	—	4	1	no	—
90	C₂×C₄₇	#1833	1	1	yes	characteristic
91	C₂×C₄₇	#1833	1	1	yes	characteristic
92	C₂×C₄₇	#1837	1	1	yes	characteristic
93	#432	—	47	2	no	—
94	#432	C₂×D₄₇	1	2	yes	—
95	#431	—	47	1	no	—
96	#431	—	47	1	no	—
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	#335	—	2	1	no	—
99	#335	—	94	1	no	—
100	#334	—	2	1	no	—
101	#334	—	94	1	no	—
102	#322	—	47	2	no	—
103	C₈²	—	47	2	no	—
104	C₄₇	#8796	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
106	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
107	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	94	1	no	—
110	C₄⋊C₈	—	94	1	no	—
111	C₄⋊C₈	—	94	2	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	94	1	no	—
114	C₂²⋊C₈	—	94	1	no	—
115	C₂²⋊C₈	—	94	2	no	—
116	#97	—	2	2	no	—
117	#97	—	94	2	no	—
118	C₄×C₈	—	47	2	no	—
119	C₄×C₈	—	47	2	no	—
120	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
121	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
125	C₂²×C₄	#10625352	1	1	yes	characteristic
126	C₂²×C₄	#10625354	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	94	2	no	—
131	C₂×C₈	—	94	2	no	—
132	C₂×C₈	—	188	2	no	—
133	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
135	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
136	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₄²	#10625352	1	1	yes	characteristic
139	C₄²	#10625354	1	1	yes	characteristic
140	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
141	C₂³	#13094015	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#13093982	1	1	yes	characteristic
148	C₂×C₄	#13093982	1	1	yes	characteristic
149	C₂×C₄	#13094011	1	1	yes	characteristic
150	C₂×C₄	#13094013	1	1	yes	characteristic
151	C₂×C₄	#13094015	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	8	2	no	—
155	C₈	—	376	2	no	—
156	C₈	—	188	2	no	—
157	C₈	—	188	2	no	—
158	C₈	#13093959	1	2	yes	—
159	C₈	#13093959	1	2	yes	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	#15688988	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	C₂×#13093959	1	1	yes	characteristic
170	C₄	C₂×#13093959	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#20243511	1	1	yes	characteristic
173	C₂	#20243513	1	1	yes	characteristic
174	C₂	#20243525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal, characteristic

#20241671

C₂

yes

maximal, characteristic

#20241251

C₂

yes

maximal

#20241249

C₂

yes

maximal

C₄₇×#1568

C₂

yes

maximal, characteristic

#15689180

C₂²

yes

—

#15689176

C₂²

yes

characteristic

#15689176

C₂²

yes

—

—

C₄

yes

—

C₈×C₄₇⋊C₈

C₄

yes

—

C₄₇×#432

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

—

C₄₇×#334

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

C₂²×C₄×C₄₇

yes

C₂²×C₄×C₄₇

yes

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#108

yes

—

C₈×C₄₇

#108

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8796

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#382

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#335

—

#335

—

100

#334

—

101

#334

—

102

#322

—

103

C₈²

—

104

C₄₇

#8796

yes

characteristic, complemented, Sylow

105

C₂²×C₈

Dic₉₄

yes

characteristic

106

C₂²×C₈

Dic₉₄

yes

characteristic

107

C₂×C₄²

C₂²×D₄₇

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

114

C₂²⋊C₈

—

115

C₂²⋊C₈

—

116

#97

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₄×D₄₇

yes

characteristic

121

C₄×C₈

C₄×D₄₇

yes

characteristic

122

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

123

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

124

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

125

C₂²×C₄

#10625352

yes

characteristic

126

C₂²×C₄

#10625354

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

188

—

133

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

134

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

135

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

136

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

137

C₄²

C₂×C₄×D₄₇

yes

characteristic

138

C₄²

#10625352

yes

characteristic

139

C₄²

#10625354

yes

characteristic

140

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

141

C₂³

#13094015

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

147

C₂×C₄

#13093982

yes

characteristic

148

C₂×C₄

#13093982

yes

characteristic

149

C₂×C₄

#13094011

yes

characteristic

150

C₂×C₄

#13094013

yes

characteristic

151

C₂×C₄

#13094015

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

376

—

156

C₈

—

188

—

157

C₈

—

188

—

158

C₈

#13093959

yes

—

159

C₈

#13093959

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#15688988

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

C₂×#13093959

yes

characteristic

170

C₄

C₂×#13093959

yes

characteristic

171

C₂

—

172

C₂

#20243511

yes

characteristic

173

C₂

#20243513

yes

characteristic

174

C₂

#20243525

yes

characteristic

	subgroup	G/H
center	C₂²	#15688988
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1568	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#125
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1568	D₄₇
hypocenter	C₄₇	#8796
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#15688988

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1568

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
104. C₄₇	#8796	semidirect

H ≅ G/N

type

104. C₄₇

#8796

semidirect

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

subgroup	index
#20242151	2
#20241671	2
#20241251	2
#20241249	2
C₄₇×#1568	2
#8796	47

subgroup

index

C₄₇×#1568

Subgroups — #26109984

Conjugacy classes of subgroups

Complements

Maximal subgroups