Group #26131028 — subgroups

Subgroups — #26131028

Order 12032.

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

normal subgroups only

This group has 340 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	#20245637	C₂	1	2	yes	maximal, complemented
2	#20245613	C₂	1	2	yes	maximal, complemented
3	#20243003	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13093965	—	2	4	no	—
7	#15689500	—	2	1	no	—
8	#15689500	—	2	1	no	—
9	#15689500	C₂²	1	2	yes	—
10	#15688923	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄₇×#500	C₂²	1	1	yes	characteristic
13	C₄₇×#493	C₂²	1	1	yes	characteristic
14	C₄₇×#492	—	2	1	no	—
15	C₄₇×#486	—	2	1	no	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×D₃₇₆	—	2	2	no	—
18	C₂×D₃₇₆	—	2	2	no	—
19	C₂×D₃₇₆	—	4	2	no	—
20	C₂×D₃₇₆	—	4	2	no	—
21	#13094021	—	2	2	no	—
22	#13093965	—	4	8	no	—
23	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
24	C₂×SD₁₆×C₄₇	—	4	2	no	—
25	C₂×D₈×C₄₇	—	2	2	no	—
26	C₄⋊Q₈×C₄₇	—	2	1	no	—
27	#127×C₄₇	—	2	1	no	—
28	#107×C₄₇	—	2	2	no	—
29	#102×C₄₇	—	4	2	no	—
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₄₇	—	4	1	no	—
34	C₂×D₄×C₄₇	—	4	1	no	—
35	C₂×D₄×C₄₇	—	4	1	no	—
36	C₂×D₄×C₄₇	—	4	1	no	—
37	C₂×D₁₈₈	—	4	2	no	—
38	C₂×D₁₈₈	—	4	2	no	—
39	C₂×D₁₈₈	—	4	2	no	—
40	SD₁₆×C₄₇	—	8	4	no	—
41	D₈×C₄₇	—	4	4	no	—
42	C₂×C₈×C₄₇	D₈	1	2	yes	—
43	C₂×C₈×C₄₇	D₈	1	2	yes	—
44	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
45	C₄⋊C₄×C₄₇	—	4	1	no	—
46	C₄⋊C₄×C₄₇	—	4	1	no	—
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	D₃₇₆	—	4	4	no	—
49	D₃₇₆	—	4	4	no	—
50	D₃₇₆	—	8	4	no	—
51	D₃₇₆	—	8	4	no	—
52	C₄₇⋊C₁₆	—	2	4	no	—
53	C₂³×C₄₇	—	8	1	no	—
54	C₂²×D₄₇	—	8	2	no	—
55	Q₈×C₄₇	—	8	2	no	—
56	D₄×C₄₇	—	8	2	no	—
57	D₄×C₄₇	—	8	2	no	—
58	D₄×C₄₇	—	8	2	no	—
59	C₂×C₄×C₄₇	—	8	1	no	—
60	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
63	D₁₈₈	—	8	2	no	—
64	D₁₈₈	—	8	2	no	—
65	D₁₈₈	—	8	4	no	—
66	D₁₈₈	—	8	4	no	—
67	C₈×C₄₇	—	2	2	no	—
68	C₈×C₄₇	D₁₆	1	4	yes	complemented
69	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
70	#12525	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	—	16	1	no	—
72	C₂²×C₄₇	—	16	1	no	—
73	C₂²×C₄₇	—	16	1	no	—
74	C₂²×C₄₇	#493	1	1	yes	characteristic
75	C₂×D₄₇	—	16	2	no	—
76	C₂×D₄₇	—	16	2	no	—
77	C₂×D₄₇	—	16	2	no	—
78	C₄×C₄₇	—	16	1	no	—
79	C₄×C₄₇	#496	1	2	yes	—
80	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
81	C₄×C₄₇	#509	1	2	yes	—
82	#2246	—	47	2	no	—
83	#2240	—	47	2	no	—
84	#1712	—	47	1	no	—
85	#1706	D₄₇	1	1	yes	characteristic, complemented
86	#1312	—	47	1	no	—
87	C₂×C₄₇	—	32	1	no	—
88	C₂×C₄₇	#1721	1	1	yes	characteristic
89	C₂×C₄₇	#2246	1	1	yes	characteristic
90	C₂×C₄₇	#2250	1	1	yes	characteristic
91	D₄₇	—	32	2	no	—
92	C₂×D₁₆	—	94	4	no	—
93	#500	C₂×D₄₇	1	1	yes	characteristic
94	#493	—	47	2	no	—
95	#493	—	94	1	no	—
96	#493	—	94	1	no	—
97	#493	C₂×D₄₇	1	1	yes	characteristic
98	#492	—	2	1	no	—
99	#486	—	2	1	no	—
100	#357	—	94	4	no	—
101	C₄×C₁₆	—	47	2	no	—
102	C₈²	C₂×D₄₇	1	1	yes	characteristic
103	C₄₇	#12525	1	1	yes	characteristic, complemented, Sylow
104	C₂×SD₁₆	—	4	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	94	2	no	—
107	C₂×D₈	—	94	2	no	—
108	C₂×D₈	—	188	2	no	—
109	C₂×D₈	—	188	2	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	#127	—	2	1	no	—
112	#127	—	94	2	no	—
113	D₁₆	—	188	8	no	—
114	C₂×C₁₆	—	47	4	no	—
115	#107	—	2	2	no	—
116	#102	—	4	2	no	—
117	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
120	C₂×Q₈	—	4	1	no	—
121	C₂×D₄	—	4	1	no	—
122	C₂×D₄	—	4	1	no	—
123	C₂×D₄	—	4	1	no	—
124	C₂×D₄	—	188	2	no	—
125	C₂×D₄	—	188	2	no	—
126	C₂×D₄	—	188	2	no	—
127	SD₁₆	—	8	4	no	—
128	D₈	—	4	4	no	—
129	D₈	—	376	4	no	—
130	D₈	—	376	4	no	—
131	D₈	—	188	4	no	—
132	D₈	—	188	4	no	—
133	C₂×C₈	#10625334	1	2	yes	—
134	C₂×C₈	#10625334	1	2	yes	—
135	C₂×C₈	D₄×D₄₇	1	2	yes	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
139	C₁₆	—	94	4	no	—
140	C₂³	—	8	1	no	—
141	C₂³	—	376	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	8	2	no	—
146	D₄	—	376	2	no	—
147	D₄	—	376	2	no	—
148	D₄	—	376	4	no	—
149	D₄	—	376	4	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	#13094020	1	1	yes	characteristic
152	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
153	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	#13093965	1	4	yes	complemented
156	C₈	#13094051	1	4	yes	—
157	C₂²	—	16	1	no	—
158	C₂²	—	16	1	no	—
159	C₂²	—	16	1	no	—
160	C₂²	—	752	2	no	—
161	C₂²	—	752	2	no	—
162	C₂²	—	752	2	no	—
163	C₂²	#15689499	1	1	yes	characteristic
164	C₄	—	16	1	no	—
165	C₄	#15689509	1	2	yes	—
166	C₄	C₂×#13093965	1	2	yes	—
167	C₄	#15689558	1	2	yes	—
168	C₂	—	32	1	no	—
169	C₂	—	1504	2	no	—
170	C₂	#20243052	1	1	yes	characteristic
171	C₂	#20245637	1	1	yes	characteristic
172	C₂	#20245655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245637

C₂

yes

maximal, complemented

#20245613

C₂

yes

maximal, complemented

#20243003

C₂

yes

maximal, characteristic, complemented

#20241306

C₂

yes

maximal, characteristic, complemented

C₄₇×#1706

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

#15688923

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#500

C₂²

yes

characteristic

C₄₇×#493

C₂²

yes

characteristic

C₄₇×#492

—

C₄₇×#486

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

—

#13093965

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×SD₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#127×C₄₇

—

#107×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

SD₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

D₁₆

yes

complemented

C₈×C₄₇

AGL(1,ℤ/8)

yes

—

#12525

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#496

yes

—

C₄×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

#509

yes

—

—

—

—

D₄₇

yes

characteristic, complemented

#1312

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×D₁₆

—

#500

C₂×D₄₇

yes

—

—

—

C₂×D₄₇

yes

characteristic

#492

—

#486

—

100

#357

—

101

C₄×C₁₆

—

102

C₈²

C₂×D₄₇

yes

characteristic

103

C₄₇

#12525

yes

characteristic, complemented, Sylow

104

C₂×SD₁₆

—

105

C₂×D₈

—

106

C₂×D₈

—

107

C₂×D₈

—

108

C₂×D₈

—

188

—

109

C₂×D₈

—

188

—

110

C₄⋊Q₈

—

111

#127

—

112

#127

—

113

D₁₆

—

188

—

114

C₂×C₁₆

—

115

#107

—

116

#102

—

117

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

118

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

119

C₄×C₈

C₂²×D₄₇

yes

characteristic

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

188

—

125

C₂×D₄

—

188

—

126

C₂×D₄

—

188

—

127

SD₁₆

—

128

D₈

—

129

D₈

—

376

—

130

D₈

—

376

—

131

D₈

—

188

—

132

D₈

—

188

—

133

C₂×C₈

#10625334

yes

—

134

C₂×C₈

#10625334

yes

—

135

C₂×C₈

D₄×D₄₇

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

139

C₁₆

—

140

C₂³

—

141

C₂³

—

376

—

142

Q₈

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

376

—

147

D₄

—

376

—

148

D₄

—

376

—

149

D₄

—

376

—

150

C₂×C₄

—

151

C₂×C₄

#13094020

yes

characteristic

152

C₂×C₄

C₂×#10625334

yes

characteristic

153

C₂×C₄

C₂×#10625334

yes

characteristic

154

C₈

—

155

C₈

#13093965

yes

complemented

156

C₈

#13094051

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

752

—

161

C₂²

—

752

—

162

C₂²

—

752

—

163

C₂²

#15689499

yes

characteristic

164

C₄

—

165

C₄

#15689509

yes

—

166

C₄

C₂×#13093965

yes

—

167

C₄

#15689558

yes

—

168

C₂

—

169

C₂

—

1504

—

170

C₂

#20243052

yes

characteristic

171

C₂

#20245637

yes

characteristic

172

C₂

#20245655

yes

characteristic

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1706	C₂
socle	C₂²×C₄₇	#493
hypercenter	#1706	D₄₇
hypocenter	C₄₇	#12525
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1706

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

N	H ≅ G/N	type
1. #20245637	C₂	semidirect
2. #20245613	C₂	semidirect
3. #20243003	C₂	semidirect
4. #20241306	C₂	semidirect
5. C₄₇×#1706	C₂	semidirect
68. C₈×C₄₇	D₁₆	semidirect
85. #1706	D₄₇	semidirect
103. C₄₇	#12525	semidirect
155. C₈	#13093965	semidirect

H ≅ G/N

type

1. #20245637

C₂

semidirect

2. #20245613

C₂

semidirect

3. #20243003

C₂

semidirect

4. #20241306

C₂

semidirect

5. C₄₇×#1706

C₂

semidirect

68. C₈×C₄₇

D₁₆

semidirect

85. #1706

D₄₇

semidirect

103. C₄₇

#12525

semidirect

155. C₈

#13093965

semidirect

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

subgroup	index
#20245637	2
#20245613	2
#20243003	2
#20241306	2
C₄₇×#1706	2
#12525	47

subgroup

index

C₄₇×#1706

Subgroups — #26131028

Conjugacy classes of subgroups

Complements

Maximal subgroups