Group #26110003 — subgroups

Subgroups — #26110003

Order 12032.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20243137	C₂	1	1	yes	maximal, characteristic
2	#20241232	C₂	1	2	yes	maximal
3	#20241221	C₂	1	1	yes	maximal, characteristic
4	#20241218	C₂	1	2	yes	maximal
5	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₂²	1	1	yes	characteristic
8	C₂×#13093935	C₂²	1	1	yes	characteristic
9	#15688878	C₄	1	2	yes	—
10	#15688878	C₄	1	2	yes	—
11	C₄₇×#431	C₂²	1	2	yes	—
12	C₄³×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
18	#13093935	—	2	2	no	—
19	#13093935	—	2	1	no	—
20	#13093935	—	2	1	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₂×C₄²×C₄₇	—	2	2	no	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₄₇	—	2	2	no	—
26	C₂²⋊C₈×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	#41	1	2	yes	—
31	C₂×C₈×C₄₇	—	4	2	no	—
32	C₄²×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	—	4	4	no	—
35	C₄²×C₄₇	#41	1	2	yes	—
36	C₄²×C₄₇	#41	1	1	yes	characteristic
37	C₄²×C₄₇	#41	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	4	2	no	—
46	C₂³×C₄₇	#117	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#126	1	2	yes	—
54	C₂×C₄×C₄₇	#126	1	2	yes	—
55	C₂×C₄×C₄₇	#131	1	2	yes	—
56	C₂×C₄×C₄₇	#131	1	2	yes	—
57	C₈×C₄₇	—	8	2	no	—
58	C₄₇⋊C₈	—	4	1	no	—
59	C₄₇⋊C₈	—	4	2	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₄₇⋊C₈	—	8	2	no	—
62	#8802	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	#383	1	1	yes	characteristic
65	C₂²×C₄₇	#433	1	2	yes	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	4	4	no	—
69	C₄×C₄₇	#433	1	1	yes	characteristic
70	C₄×C₄₇	#433	1	2	yes	—
71	C₄×C₄₇	#433	1	1	yes	characteristic
72	#1745	—	47	1	no	—
73	#1281	—	47	2	no	—
74	#1276	—	47	1	no	—
75	#1276	D₄₇	1	1	yes	characteristic
76	#1275	—	47	2	no	—
77	C₂×C₄₇	—	2	2	no	—
78	C₂×C₄₇	#1840	1	1	yes	characteristic
79	C₂×C₄₇	#1841	1	2	yes	—
80	#431	—	47	2	no	—
81	#431	C₂×D₄₇	1	2	yes	—
82	C₂×#97	—	47	1	no	—
83	C₂×#97	—	47	1	no	—
84	C₄³	C₂×D₄₇	1	1	yes	characteristic
85	#336	—	47	2	no	—
86	#336	—	47	2	no	—
87	C₄₇	#8802	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	47	1	no	—
89	C₂²×C₈	—	47	1	no	—
90	C₂²×C₈	—	47	2	no	—
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₄×D₄₇	1	2	yes	—
93	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	94	2	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	94	2	no	—
98	#97	—	94	2	no	—
99	#97	—	94	1	no	—
100	#97	—	94	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	94	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
105	C₂²×C₄	#10625352	1	2	yes	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	94	1	no	—
108	C₂×C₈	—	94	1	no	—
109	C₂×C₈	—	94	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₂×C₈	—	94	2	no	—
112	C₂×C₈	—	94	2	no	—
113	C₂×C₈	—	94	2	no	—
114	C₂×C₈	—	188	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#10625350	1	2	yes	—
119	C₄²	#10625352	1	1	yes	characteristic
120	C₄²	#10625354	1	1	yes	characteristic
121	C₂³	#13093980	1	1	yes	characteristic
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#13093981	1	1	yes	characteristic
127	C₂×C₄	#13093982	1	1	yes	characteristic
128	C₂×C₄	#13094017	1	2	yes	—
129	C₂×C₄	#13094019	1	2	yes	—
130	C₂×C₄	#13094030	1	2	yes	—
131	C₂×C₄	#13094032	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	376	2	no	—
134	C₈	—	188	1	no	—
135	C₈	—	188	2	no	—
136	C₈	—	188	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#15688990	1	1	yes	characteristic
139	C₂²	#15689181	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	#15689182	1	1	yes	characteristic
144	C₄	#15689183	1	2	yes	—
145	C₄	#15689185	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#20243532	1	1	yes	characteristic
148	C₂	#20243534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243137

C₂

yes

maximal, characteristic

#20241232

C₂

yes

maximal

#20241221

C₂

yes

maximal, characteristic

#20241218

C₂

yes

maximal

C₄₇×#1276

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×#13093935

C₂²

yes

characteristic

C₂×#13093935

C₂²

yes

characteristic

#15688878

C₄

yes

—

#15688878

C₄

yes

—

C₄₇×#431

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#117

yes

characteristic

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

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8802

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#383

yes

characteristic

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#433

yes

characteristic

C₄×C₄₇

#433

yes

—

C₄×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

#1275

—

C₂×C₄₇

—

C₂×C₄₇

#1840

yes

characteristic

C₂×C₄₇

#1841

yes

—

#431

—

#431

C₂×D₄₇

yes

—

C₂×#97

—

C₂×#97

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

—

C₄₇

#8802

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

105

C₂²×C₄

#10625352

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

188

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

#10625350

yes

—

119

C₄²

#10625352

yes

characteristic

120

C₄²

#10625354

yes

characteristic

121

C₂³

#13093980

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#13093981

yes

characteristic

127

C₂×C₄

#13093982

yes

characteristic

128

C₂×C₄

#13094017

yes

—

129

C₂×C₄

#13094019

yes

—

130

C₂×C₄

#13094030

yes

—

131

C₂×C₄

#13094032

yes

—

132

C₈

—

133

C₈

—

376

—

134

C₈

—

188

—

135

C₈

—

188

—

136

C₈

—

188

—

137

C₂²

—

138

C₂²

#15688990

yes

characteristic

139

C₂²

#15689181

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#15689182

yes

characteristic

144

C₄

#15689183

yes

—

145

C₄

#15689185

yes

characteristic

146

C₂

—

147

C₂

#20243532

yes

characteristic

148

C₂

#20243534

yes

—

	subgroup	G/H
center	C₂×C₄	#13093982
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1276	C₂
socle	C₂²×C₄₇	#383
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1276	D₄₇
hypocenter	C₄₇	#8802
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂×C₄

#13093982

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1276

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
87. C₄₇	#8802	semidirect

H ≅ G/N

type

87. C₄₇

#8802

semidirect

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

subgroup	index
#20243137	2
#20241232	2
#20241221	2
#20241218	2
C₄₇×#1276	2
#8802	47

subgroup

index

C₄₇×#1276

Subgroups — #26110003

Conjugacy classes of subgroups

Complements

Maximal subgroups