Group #26132701 — subgroups

Subgroups — #26132701

Order 12032.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20245153	C₂	1	2	yes	maximal
2	#20245148	C₂	1	2	yes	maximal
3	#20245147	C₂	1	1	yes	maximal, characteristic
4	#20243143	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#15689078	C₂²	1	2	yes	—
9	#15688943	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
11	Dic₄₇⋊C₁₆	—	2	4	no	—
12	C₄₇⋊#338	C₄	1	1	yes	characteristic
13	#15688880	C₄	1	1	yes	characteristic
14	#15688879	C₄	1	2	yes	—
15	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#405	C₂²	1	1	yes	characteristic
17	C₄₇×#364	—	2	2	no	—
18	C₂×#10625329	C₂×C₄	1	2	yes	—
19	#13093983	C₂×C₄	1	2	yes	—
20	#13093964	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	1	no	—
23	C₂×C₄₇⋊C₁₆	—	2	1	no	—
24	#13093934	Q₈	1	4	yes	—
25	C₈×Dic₄₇	D₄	1	4	yes	—
26	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₄₇	—	2	2	no	—
30	M₃₂×C₄₇	—	2	2	no	—
31	C₂×C₁₆×C₄₇	—	2	2	no	—
32	#97×C₄₇	Q₈	1	2	yes	—
33	C₄×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
41	C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂²⋊C₄×C₄₇	—	2	1	no	—
43	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂²⋊Dic₄₇	—	2	2	no	—
45	C₄⋊Dic₄₇	—	2	2	no	—
46	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
47	#10625329	—	2	2	no	—
48	#10625329	—	2	2	no	—
49	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
50	C₁₆×C₄₇	—	4	2	no	—
51	C₄₇⋊C₁₆	—	4	1	no	—
52	C₄₇⋊C₁₆	—	4	1	no	—
53	C₂³×C₄₇	#131	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#131	1	2	yes	—
58	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
59	C₂×Dic₄₇	—	2	4	no	—
60	C₈×C₄₇	—	2	2	no	—
61	C₈×C₄₇	#127	1	1	yes	characteristic
62	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
64	C₄₇⋊C₈	—	2	4	no	—
65	#12902	—	47	1	no	maximal, Sylow
66	C₂²×C₄₇	—	4	1	no	—
67	C₂²×C₄₇	#424	1	2	yes	—
68	C₂²×C₄₇	#424	1	1	yes	characteristic
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	#384	1	1	yes	characteristic
71	C₄×C₄₇	#424	1	2	yes	—
72	C₄×C₄₇	#424	1	1	yes	characteristic
73	Dic₄₇	—	4	4	no	—
74	#2153	—	47	2	no	—
75	#2153	D₄₇	1	1	yes	characteristic
76	#2151	—	47	2	no	—
77	#2151	—	47	1	no	—
78	#1747	—	47	1	no	—
79	C₂×C₄₇	—	2	2	no	—
80	C₂×C₄₇	—	2	1	no	—
81	C₂×C₄₇	#1843	1	1	yes	characteristic
82	C₂×M₃₂	—	47	1	no	—
83	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
84	C₂²×C₁₆	—	47	1	no	—
85	#405	—	47	2	no	—
86	#405	C₂×D₄₇	1	1	yes	characteristic
87	#364	—	2	2	no	—
88	#364	—	94	4	no	—
89	C₄⋊C₁₆	—	94	4	no	—
90	C₄⋊C₁₆	—	94	2	no	—
91	#338	—	47	1	no	—
92	H(3,ℤ/4)	—	47	1	no	—
93	H(3,ℤ/4)	—	47	2	no	—
94	C₄₇	#12902	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	47	2	no	—
96	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	#117	—	47	2	no	—
99	#117	C₄×D₄₇	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	—	94	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	94	2	no	—
105	C₂×C₁₆	—	94	1	no	—
106	C₂×C₁₆	—	94	1	no	—
107	#97	—	47	4	no	—
108	#97	Dic₉₄	1	2	yes	—
109	C₄×C₈	—	47	4	no	—
110	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	94	2	no	—
114	M₁₆	—	94	2	no	—
115	C₂×C₈	—	47	4	no	—
116	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
117	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
118	C₂×C₈	D₄×D₄₇	1	2	yes	—
119	C₂×C₈	Q₈×D₄₇	1	2	yes	—
120	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	94	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	94	2	no	—
125	C₄²	—	47	4	no	—
126	C₄²	Dic₄₇⋊C₄	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	188	1	no	—
129	C₁₆	—	188	1	no	—
130	C₂³	#13094032	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	94	4	no	—
133	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
134	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#13094030	1	2	yes	—
136	C₂×C₄	#13094032	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	94	4	no	—
139	C₈	#13094020	1	1	yes	characteristic
140	C₈	#13094022	1	1	yes	characteristic
141	C₈	#13094023	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#15689148	1	2	yes	—
144	C₂²	#15689151	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	188	4	no	—
147	C₄	#15688992	1	1	yes	characteristic
148	C₄	#15689148	1	2	yes	—
149	C₄	#15689149	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#20243541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245153

C₂

yes

maximal

#20245148

C₂

yes

maximal

#20245147

C₂

yes

maximal, characteristic

#20243143

C₂

yes

maximal, characteristic

C₄₇×#2153

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15689078

C₂²

yes

—

#15688943

—

C₄⋊(C₄₇⋊C₁₆)

—

Dic₄₇⋊C₁₆

—

C₄₇⋊#338

C₄

yes

characteristic

#15688880

C₄

yes

characteristic

#15688879

C₄

yes

—

C₂×M₃₂×C₄₇

C₂²

yes

—

C₄₇×#405

C₂²

yes

characteristic

C₄₇×#364

—

C₂×#10625329

C₂×C₄

yes

—

#13093983

C₂×C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

#13093934

Q₈

yes

—

C₈×Dic₄₇

D₄

yes

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

C₂×C₄

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

#97×C₄₇

Q₈

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#131

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

characteristic

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#127

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

—

C₄₇⋊C₈

—

#12902

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#424

yes

—

C₂²×C₄₇

#424

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#384

yes

characteristic

C₄×C₄₇

#424

yes

—

C₄×C₄₇

#424

yes

characteristic

Dic₄₇

—

#2153

—

#2153

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1843

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₄₇

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₄₇

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₄₇

#12902

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

#117

—

#117

C₄×D₄₇

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

M₃₂

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

#97

—

108

#97

Dic₉₄

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₄₇

yes

—

111

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

Dic₄₇⋊C₄

yes

—

117

C₂×C₈

C₂×Dic₉₄

yes

characteristic

118

C₂×C₈

D₄×D₄₇

yes

—

119

C₂×C₈

Q₈×D₄₇

yes

—

120

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

Dic₄₇⋊C₄

yes

—

127

C₁₆

—

128

C₁₆

—

188

—

129

C₁₆

—

188

—

130

C₂³

#13094032

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

134

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

135

C₂×C₄

#13094030

yes

—

136

C₂×C₄

#13094032

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#13094020

yes

characteristic

140

C₈

#13094022

yes

characteristic

141

C₈

#13094023

yes

—

142

C₂²

—

143

C₂²

#15689148

yes

—

144

C₂²

#15689151

yes

characteristic

145

C₄

—

146

C₄

—

188

—

147

C₄

#15688992

yes

characteristic

148

C₄

#15689148

yes

—

149

C₄

#15689149

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#20243541

yes

characteristic

	subgroup	G/H
center	C₈	#13094020
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2153	C₂
socle	C₂×C₄₇	#1843
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2153	D₄₇
hypocenter	C₄₇	#12902
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#13094020

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2153

C₂

C₂×C₄₇

C₂³×C₄₇

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
94. C₄₇	#12902	semidirect

H ≅ G/N

type

94. C₄₇

#12902

semidirect

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

subgroup	index
#20245153	2
#20245148	2
#20245147	2
#20243143	2
C₄₇×#2153	2
#12902	47

subgroup

index

C₄₇×#2153

Subgroups — #26132701

Conjugacy classes of subgroups

Complements

Maximal subgroups