Group #26132666 — subgroups

Subgroups — #26132666

Order 12032.

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

normal subgroups only

This group has 282 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	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20243558	C₂	1	2	yes	maximal
3	#20241482	C₂	1	1	yes	maximal, characteristic
4	#20241481	C₂	1	2	yes	maximal
5	C₄₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	C₂×#13093959	—	2	2	no	—
9	#15689146	—	2	2	no	—
10	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₄₇	C₄	1	2	yes	—
12	C₁₆×Dic₄₇	C₄	1	2	yes	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	2	yes	—
15	C₂×#10625329	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	#13093960	—	2	2	no	—
21	#13093960	—	2	2	no	—
22	#13093959	—	4	4	no	—
23	Dic₄₇⋊C₈	—	4	4	no	—
24	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₄₇	Q₈	1	1	yes	characteristic
27	C₈×Dic₄₇	Q₈	1	1	yes	characteristic
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	—	2	2	no	—
31	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	#108×C₄₇	—	2	2	no	—
33	#108×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₄₇	Q₁₆	1	2	yes	—
36	M₁₆×C₄₇	—	4	2	no	—
37	M₁₆×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₄₇	#41	1	2	yes	—
42	C₂×C₈×C₄₇	#41	1	2	yes	—
43	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₄₇	D₈	1	2	yes	—
46	#10625329	—	4	2	no	—
47	#10625329	—	4	2	no	—
48	C₂×C₄₇⋊C₈	—	4	2	no	—
49	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
52	C₁₆×C₄₇	—	2	2	no	—
53	C₄₇⋊C₁₆	—	2	2	no	—
54	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#124	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	#107	1	2	yes	—
61	C₂×Dic₄₇	#107	1	2	yes	—
62	C₈×C₄₇	—	8	2	no	—
63	C₈×C₄₇	#117	1	1	yes	characteristic
64	C₈×C₄₇	#117	1	1	yes	characteristic
65	C₈×C₄₇	#125	1	2	yes	—
66	C₄₇⋊C₈	—	8	2	no	—
67	C₄₇⋊C₈	#106	1	2	yes	—
68	C₄₇⋊C₈	#106	1	2	yes	—
69	#12891	—	47	1	no	maximal, Sylow
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
72	C₂²×C₄₇	#487	1	2	yes	—
73	C₄×C₄₇	#382	1	1	yes	characteristic
74	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
75	C₄×C₄₇	#486	1	2	yes	—
76	Dic₄₇	—	2	2	no	—
77	Dic₄₇	#368	1	4	yes	—
78	C₂×C₄×C₁₆	—	47	1	no	—
79	#1848	—	47	2	no	—
80	#1381	—	47	1	no	—
81	#1381	—	47	2	no	—
82	#1381	D₄₇	1	1	yes	characteristic
83	C₂×C₄₇	—	2	2	no	—
84	C₂×C₄₇	#1834	1	1	yes	characteristic
85	C₂×C₄₇	C₂×#368	1	2	yes	—
86	C₂²×C₁₆	—	47	1	no	—
87	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
88	C₂×#108	—	47	2	no	—
89	C₂×#108	—	94	2	no	—
90	C₂×#108	C₂×D₄₇	1	2	yes	—
91	#423	—	94	2	no	—
92	C₂×C₄×C₈	—	47	1	no	—
93	C₄×C₁₆	—	47	2	no	—
94	C₄×C₁₆	—	47	2	no	—
95	C₄₇	#12891	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	94	2	no	—
98	C₂²×C₈	—	47	1	no	—
99	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
100	C₂×C₄²	—	47	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	47	2	no	—
103	C₂×C₁₆	—	94	2	no	—
104	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#108	—	94	2	no	—
108	#108	—	94	2	no	—
109	#108	—	188	4	no	—
110	C₄⋊C₈	—	188	4	no	—
111	C₄×C₈	—	47	1	no	—
112	C₄×C₈	—	47	1	no	—
113	C₄×C₈	—	47	1	no	—
114	C₄×C₈	—	47	1	no	—
115	C₂²×C₄	—	47	2	no	—
116	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	188	2	no	—
120	M₁₆	—	188	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	47	1	no	—
123	C₂×C₈	—	47	1	no	—
124	C₂×C₈	—	47	4	no	—
125	C₂×C₈	—	188	2	no	—
126	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
127	C₂×C₈	#10625350	1	2	yes	—
128	C₂×C₈	#10625352	1	2	yes	—
129	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
130	C₄²	—	47	1	no	—
131	C₄²	—	47	1	no	—
132	C₄²	—	47	2	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	—	94	2	no	—
135	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	47	2	no	—
138	C₂×C₄	—	94	4	no	—
139	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
140	C₂×C₄	#13094010	1	2	yes	—
141	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
142	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	47	2	no	—
145	C₈	—	47	2	no	—
146	C₈	—	376	2	no	—
147	C₈	#13093980	1	1	yes	characteristic
148	C₈	#13093981	1	1	yes	characteristic
149	C₈	#13094014	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	#107×D₄₇	1	1	yes	characteristic
152	C₂²	#15689472	1	2	yes	—
153	C₄	—	47	4	no	—
154	C₄	—	94	2	no	—
155	C₄	#15688987	1	1	yes	characteristic
156	C₄	#106×D₄₇	1	1	yes	characteristic
157	C₄	#15689468	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#20243514	1	1	yes	characteristic
160	C₂	#368×D₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₄₇

C₂

yes

maximal, characteristic

#20243558

C₂

yes

maximal

#20241482

C₂

yes

maximal, characteristic

#20241481

C₂

yes

maximal

C₄₇×#1381

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

C₂×#13093959

—

#15689146

—

C₂×C₈×Dic₄₇

C₂²

yes

characteristic

C₁₆×Dic₄₇

C₄

yes

—

C₁₆×Dic₄₇

C₄

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093960

—

#13093960

—

#13093959

—

Dic₄₇⋊C₈

—

C₈×Dic₄₇

C₂×C₄

yes

characteristic

C₈×Dic₄₇

C₂×C₄

yes

characteristic

C₈×Dic₄₇

Q₈

yes

characteristic

C₈×Dic₄₇

Q₈

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#108×C₄₇

—

#108×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

Q₁₆

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄×Dic₄₇

C₄⋊C₄

yes

characteristic

C₄×Dic₄₇

C₄⋊C₄

yes

characteristic

C₄×Dic₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

SD₁₆

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

C₂×Dic₄₇

#107

yes

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#106

yes

—

C₄₇⋊C₈

#106

yes

—

#12891

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#487

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

#368

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1834

yes

characteristic

C₂×C₄₇

C₂×#368

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

#423

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄₇

#12891

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

C₄×D₄₇

yes

—

105

#108

—

106

#108

—

107

#108

—

108

#108

—

109

#108

—

188

—

110

C₄⋊C₈

—

188

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₂²×C₄

—

116

C₂²×C₄

D₄×D₄₇

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

188

—

120

M₁₆

—

188

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

188

—

126

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

127

C₂×C₈

#10625350

yes

—

128

C₂×C₈

#10625352

yes

—

129

C₂×C₈

Q₈×D₄₇

yes

characteristic

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₁₆

—

134

C₁₆

—

135

C₂³

Q₁₆×D₄₇

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

140

C₂×C₄

#13094010

yes

—

141

C₂×C₄

D₈×D₄₇

yes

characteristic

142

C₂×C₄

SD₁₆×D₄₇

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

376

—

147

C₈

#13093980

yes

characteristic

148

C₈

#13093981

yes

characteristic

149

C₈

#13094014

yes

—

150

C₂²

—

151

C₂²

#107×D₄₇

yes

characteristic

152

C₂²

#15689472

yes

—

153

C₄

—

154

C₄

—

155

C₄

#15688987

yes

characteristic

156

C₄

#106×D₄₇

yes

characteristic

157

C₄

#15689468

yes

—

158

C₂

—

159

C₂

#20243514

yes

characteristic

160

C₂

#368×D₄₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1381	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1381	D₄₇
hypocenter	C₄₇	#12891
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1381

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×C₄₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
95. C₄₇	#12891	semidirect

H ≅ G/N

type

95. C₄₇

#12891

semidirect

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

subgroup	index
C₂×C₁₆×Dic₄₇	2
#20243558	2
#20241482	2
#20241481	2
C₄₇×#1381	2
#12891	47

subgroup

index

C₂×C₁₆×Dic₄₇

C₄₇×#1381

Subgroups — #26132666

Conjugacy classes of subgroups

Complements

Maximal subgroups