Group #26109981 — subgroups

Subgroups — #26109981

Order 12032.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20243552	C₂	1	1	yes	maximal, characteristic
2	#20243526	C₂	1	1	yes	maximal, characteristic
3	#20241247	C₂	1	2	yes	maximal
4	#20241247	C₂	1	2	yes	maximal
5	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	—	2	1	no	—
7	C₂×#13093956	—	2	1	no	—
8	#15689147	C₂²	1	1	yes	characteristic
9	#15689147	C₂²	1	1	yes	characteristic
10	#15688989	C₂²	1	2	yes	—
11	#15688888	—	2	1	no	—
12	#15688888	—	2	1	no	—
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#322	C₄	1	4	yes	complemented
16	C₂×#10625329	—	2	1	no	—
17	C₂×#10625329	—	2	1	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	#13093958	—	4	2	no	—
20	#13093956	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄₇⋊#95	—	2	4	no	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₂²⋊C₈×C₄₇	—	2	2	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₇	Q₈	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	#41	1	2	yes	—
34	C₂²×Dic₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	—	2	1	no	—
39	C₂×C₈×C₄₇	—	2	4	no	—
40	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₄₇	#41	1	2	yes	—
45	C₄⋊Dic₄₇	—	4	2	no	—
46	C₄⋊Dic₄₇	—	4	2	no	—
47	#10625329	—	4	1	no	—
48	#10625329	—	4	1	no	—
49	#10625329	—	4	1	no	—
50	#10625329	—	4	1	no	—
51	C₂×C₄₇⋊C₈	—	4	1	no	—
52	C₂×C₄₇⋊C₈	—	4	1	no	—
53	C₂³×C₄₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	4	2	no	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#117	1	2	yes	—
57	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#125	1	2	yes	—
59	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
63	C₂×Dic₄₇	—	8	4	no	—
64	C₂×Dic₄₇	—	8	2	no	—
65	C₈×C₄₇	—	2	2	no	—
66	C₈×C₄₇	—	2	2	no	—
67	C₈×C₄₇	—	4	4	no	—
68	C₄₇⋊C₈	—	8	1	no	—
69	C₄₇⋊C₈	—	8	1	no	—
70	#8795	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	#382	1	1	yes	characteristic
73	C₂²×C₄₇	#490	1	1	yes	characteristic
74	C₂²×C₄₇	#491	1	1	yes	characteristic
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	#428	1	1	yes	characteristic
78	C₄×C₄₇	#428	1	1	yes	characteristic
79	C₄×C₄₇	#489	1	1	yes	characteristic
80	C₄×C₄₇	#489	1	1	yes	characteristic
81	Dic₄₇	—	16	2	no	—
82	#1846	—	47	1	no	—
83	#1837	—	47	1	no	—
84	#1452	D₄₇	1	1	yes	characteristic
85	#1286	—	47	2	no	—
86	#1286	—	47	2	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	#1833	1	2	yes	—
90	C₂×C₄₇	#1836	1	1	yes	characteristic
91	#431	C₂×D₄₇	1	2	yes	—
92	C₂×#107	—	94	1	no	—
93	C₂×#106	—	94	1	no	—
94	#423	—	47	1	no	—
95	#423	—	47	1	no	—
96	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
97	#382	—	47	2	no	—
98	#341	—	94	1	no	—
99	#341	—	94	1	no	—
100	#322	Dic₄₇	1	4	yes	complemented
101	C₄₇	#8795	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	94	1	no	—
103	C₂×M₁₆	—	94	1	no	—
104	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
105	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	94	2	no	—
107	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
108	#107	—	188	2	no	—
109	#106	—	188	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	94	2	no	—
112	C₄⋊C₈	—	94	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₉₄	1	2	yes	—
115	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
116	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
117	#95	—	94	4	no	—
118	C₂²×C₄	—	188	2	no	—
119	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
120	C₂²×C₄	#10625350	1	2	yes	—
121	M₁₆	—	188	1	no	—
122	M₁₆	—	188	1	no	—
123	M₁₆	—	188	1	no	—
124	M₁₆	—	188	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	188	1	no	—
131	C₂×C₈	—	188	1	no	—
132	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
134	C₄⋊C₄	—	188	2	no	—
135	C₄⋊C₄	—	188	2	no	—
136	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
137	C₄²	#10625350	1	2	yes	—
138	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
139	C₂³	#13094012	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	376	4	no	—
142	C₂×C₄	—	376	2	no	—
143	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
144	C₂×C₄	#13093983	1	2	yes	—
145	C₂×C₄	#13094012	1	1	yes	characteristic
146	C₂×C₄	#13094016	1	2	yes	—
147	C₂×C₄	#13094052	1	1	yes	characteristic
148	C₂×C₄	#13094052	1	1	yes	characteristic
149	C₂×C₄	#13094057	1	1	yes	characteristic
150	C₂×C₄	#13094057	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	4	no	—
154	C₈	—	376	1	no	—
155	C₈	—	376	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#15688989	1	1	yes	characteristic
158	C₂²	#15689488	1	1	yes	characteristic
159	C₂²	#15689492	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	752	2	no	—
163	C₄	#15689165	1	1	yes	characteristic
164	C₄	#15689165	1	1	yes	characteristic
165	C₄	#15689483	1	1	yes	characteristic
166	C₄	#15689483	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#20243512	1	2	yes	—
170	C₂	#20243522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243552

C₂

yes

maximal, characteristic

#20243526

C₂

yes

maximal, characteristic

#20241247

C₂

yes

maximal

#20241247

C₂

yes

maximal

C₄₇×#1452

C₂

yes

maximal, characteristic

C₂×#13093958

—

C₂×#13093956

—

#15689147

C₂²

yes

characteristic

#15689147

C₂²

yes

characteristic

#15688989

C₂²

yes

—

#15688888

—

#15688888

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₄

yes

complemented

C₂×#10625329

—

C₂×#10625329

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093956

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

—

—

—

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8795

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1846

—

#1837

—

#1452

D₄₇

yes

characteristic

#1286

—

#1286

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1833

yes

—

C₂×C₄₇

#1836

yes

characteristic

#431

C₂×D₄₇

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

100

#322

Dic₄₇

yes

complemented

101

C₄₇

#8795

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

D₁₈₈

yes

characteristic

105

C₂²×C₈

D₁₈₈

yes

characteristic

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₄₇

yes

characteristic

108

#107

—

188

—

109

#106

—

188

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

C₄×C₈

Dic₉₄

yes

—

115

C₄×C₈

C₂×Dic₄₇

yes

—

116

C₄×C₈

C₂×Dic₄₇

yes

—

117

#95

—

118

C₂²×C₄

—

188

—

119

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

120

C₂²×C₄

#10625350

yes

—

121

M₁₆

—

188

—

122

M₁₆

—

188

—

123

M₁₆

—

188

—

124

M₁₆

—

188

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

188

—

131

C₂×C₈

—

188

—

132

C₂×C₈

C₄⋊Dic₄₇

yes

—

133

C₂×C₈

C₄⋊Dic₄₇

yes

—

134

C₄⋊C₄

—

188

—

135

C₄⋊C₄

—

188

—

136

C₄²

C₂×Dic₉₄

yes

characteristic

137

C₄²

#10625350

yes

—

138

C₄²

C₂²×Dic₄₇

yes

characteristic

139

C₂³

#13094012

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

376

—

142

C₂×C₄

—

376

—

143

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

144

C₂×C₄

#13093983

yes

—

145

C₂×C₄

#13094012

yes

characteristic

146

C₂×C₄

#13094016

yes

—

147

C₂×C₄

#13094052

yes

characteristic

148

C₂×C₄

#13094052

yes

characteristic

149

C₂×C₄

#13094057

yes

characteristic

150

C₂×C₄

#13094057

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

376

—

155

C₈

—

376

—

156

C₂²

—

157

C₂²

#15688989

yes

characteristic

158

C₂²

#15689488

yes

characteristic

159

C₂²

#15689492

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

752

—

163

C₄

#15689165

yes

characteristic

164

C₄

#15689165

yes

characteristic

165

C₄

#15689483

yes

characteristic

166

C₄

#15689483

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#20243512

yes

—

170

C₂

#20243522

yes

characteristic

	subgroup	G/H
center	C₂²	#15688989
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1452	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1452	D₄₇
hypocenter	C₄₇	#8795
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#15688989

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1452

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
15. C₄₇×#322	C₄	semidirect
100. #322	Dic₄₇	semidirect
101. C₄₇	#8795	semidirect

H ≅ G/N

type

15. C₄₇×#322

C₄

semidirect

100. #322

Dic₄₇

semidirect

101. C₄₇

#8795

semidirect

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

subgroup	index
#20243552	2
#20243526	2
#20241247	2
C₄₇×#1452	2
#8795	47

subgroup

index

C₄₇×#1452

Subgroups — #26109981

Conjugacy classes of subgroups

Complements

Maximal subgroups