Group #26121088 — subgroups

Subgroups — #26121088

Order 12032.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20243529	C₂	1	1	yes	maximal, characteristic
2	#20242060	C₂	1	2	yes	maximal
3	#20241246	C₂	1	2	yes	maximal
4	#20241233	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1960	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689147	C₂²	1	2	yes	—
8	#15688888	—	2	2	no	—
9	#15688869	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×#103×C₄₇	—	2	2	no	—
12	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#382	C₂²	1	1	yes	characteristic
14	C₂×#10625329	—	2	2	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×Q₈×C₄₇	—	2	1	no	—
21	C₂²×C₈×C₄₇	D₄	1	2	yes	—
22	C₄⋊Q₈×C₄₇	—	2	2	no	—
23	C₄⋊Q₈×C₄₇	—	2	1	no	—
24	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	#103×C₄₇	—	4	4	no	—
29	#97×C₄₇	—	2	2	no	—
30	#95×C₄₇	—	2	2	no	—
31	C₂×Q₈×C₄₇	—	2	2	no	—
32	C₂×Q₈×C₄₇	—	4	1	no	—
33	C₂×Q₈×C₄₇	—	4	2	no	—
34	C₂×Q₈×C₄₇	—	4	2	no	—
35	C₂²×C₄×C₄₇	—	4	1	no	—
36	C₂²×C₄×C₄₇	—	4	1	no	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	#41	1	2	yes	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₄⋊C₄×C₄₇	—	4	2	no	—
44	C₄⋊C₄×C₄₇	—	4	2	no	—
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₇	#41	1	1	yes	characteristic
47	C₄²×C₄₇	#41	1	2	yes	—
48	#10625329	—	4	2	no	—
49	#10625329	—	4	2	no	—
50	C₂×C₄₇⋊C₈	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂³×C₄₇	#124	1	1	yes	characteristic
53	Q₈×C₄₇	—	4	4	no	—
54	Q₈×C₄₇	—	8	2	no	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	—	4	2	no	—
57	C₂×C₄×C₄₇	—	8	1	no	—
58	C₂×C₄×C₄₇	—	8	1	no	—
59	C₂×C₄×C₄₇	—	8	2	no	—
60	C₂×C₄×C₄₇	—	8	1	no	—
61	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#123	1	2	yes	—
64	C₂×C₄×C₄₇	#126	1	2	yes	—
65	C₂×C₄×C₄₇	#135	1	2	yes	—
66	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
67	C₈×C₄₇	—	4	2	no	—
68	C₄₇⋊C₈	—	8	2	no	—
69	C₄₇⋊C₈	—	8	2	no	—
70	#10635	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	#397	1	1	yes	characteristic
73	C₂²×C₄₇	#488	1	1	yes	characteristic
74	C₂²×C₄₇	#490	1	1	yes	characteristic
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	8	2	no	—
78	C₄×C₄₇	—	16	1	no	—
79	C₄×C₄₇	#463	1	2	yes	—
80	C₄×C₄₇	#482	1	2	yes	—
81	#1960	D₄₇	1	1	yes	characteristic
82	#1838	—	47	1	no	—
83	#1546	—	47	2	no	—
84	#1286	—	47	2	no	—
85	#1281	—	47	1	no	—
86	C₂×C₄₇	—	2	1	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	#2073	1	1	yes	characteristic
89	C₂×C₄₇	#2082	1	2	yes	—
90	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
91	#431	—	47	2	no	—
92	#423	—	47	2	no	—
93	C₂×#103	—	2	2	no	—
94	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
95	#382	C₂×D₄₇	1	1	yes	characteristic
96	#341	—	94	2	no	—
97	#332	—	94	4	no	—
98	C₄₇	#10635	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	2	1	no	—
100	C₂×M₁₆	—	94	2	no	—
101	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	—	2	2	no	—
104	C₄⋊Q₈	D₁₈₈	1	2	yes	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
108	C₄⋊C₈	—	94	2	no	—
109	C₄⋊C₈	—	94	2	no	—
110	C₄⋊C₈	—	94	2	no	—
111	#103	—	4	4	no	—
112	C₂²⋊C₈	—	94	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	94	2	no	—
115	#95	—	2	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	#10625352	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
124	M₁₆	—	188	2	no	—
125	M₁₆	—	188	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	188	2	no	—
130	C₂×C₈	—	188	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
135	C₄²	#10625350	1	1	yes	characteristic
136	C₄²	#10625352	1	2	yes	—
137	C₂³	#13094011	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	#13093998	1	1	yes	characteristic
147	C₂×C₄	#13094005	1	1	yes	characteristic
148	C₂×C₄	#13094008	1	2	yes	—
149	C₂×C₄	#13094017	1	2	yes	—
150	C₂×C₄	#13094044	1	2	yes	—
151	C₂×C₄	#13094050	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	376	2	no	—
154	C₈	—	376	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#15689048	1	1	yes	characteristic
157	C₂²	#15689477	1	1	yes	characteristic
158	C₂²	#15689487	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	#15689338	1	2	yes	—
164	C₄	#15689444	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#20244801	1	1	yes	characteristic
168	C₂	#20244852	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243529

C₂

yes

maximal, characteristic

#20242060

C₂

yes

maximal

#20241246

C₂

yes

maximal

#20241233

C₂

yes

maximal, characteristic

C₄₇×#1960

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

#15689147

C₂²

yes

—

#15688888

—

#15688869

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₂×#103×C₄₇

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#382

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#103×C₄₇

—

#97×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10635

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#463

yes

—

C₄×C₄₇

#482

yes

—

#1960

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2073

yes

characteristic

C₂×C₄₇

#2082

yes

—

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

—

—

C₂×#97

C₂×D₄₇

yes

characteristic

#382

C₂×D₄₇

yes

characteristic

#341

—

#332

—

C₄₇

#10635

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₇

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

D₁₈₈

yes

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₄₇

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

#10625352

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

124

M₁₆

—

188

—

125

M₁₆

—

188

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

188

—

130

C₂×C₈

—

188

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₁₈₈

yes

characteristic

135

C₄²

#10625350

yes

characteristic

136

C₄²

#10625352

yes

—

137

C₂³

#13094011

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#13093998

yes

characteristic

147

C₂×C₄

#13094005

yes

characteristic

148

C₂×C₄

#13094008

yes

—

149

C₂×C₄

#13094017

yes

—

150

C₂×C₄

#13094044

yes

—

151

C₂×C₄

#13094050

yes

—

152

C₈

—

153

C₈

—

376

—

154

C₈

—

376

—

155

C₂²

—

156

C₂²

#15689048

yes

characteristic

157

C₂²

#15689477

yes

characteristic

158

C₂²

#15689487

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#15689338

yes

—

164

C₄

#15689444

yes

—

165

C₂

—

166

C₂

—

167

C₂

#20244801

yes

characteristic

168

C₂

#20244852

yes

—

	subgroup	G/H
center	C₂²	#15689048
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1960	C₂
socle	C₂²×C₄₇	#397
prime radical	C₂³×C₄₇	#124
hypercenter	#1960	D₄₇
hypocenter	C₄₇	#10635
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#15689048

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1960

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
98. C₄₇	#10635	semidirect

H ≅ G/N

type

98. C₄₇

#10635

semidirect

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

subgroup	index
#20243529	2
#20242060	2
#20241246	2
#20241233	2
C₄₇×#1960	2
#10635	47

subgroup

index

C₄₇×#1960

Subgroups — #26121088

Conjugacy classes of subgroups

Complements

Maximal subgroups