Group #26137185 — subgroups

Subgroups — #26137185

Order 12032.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20245665	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241590	C₂	1	1	yes	maximal, characteristic
4	#20241587	C₂	1	1	yes	maximal, characteristic
5	Dic₄₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#20241569	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1416	C₂	1	1	yes	maximal, characteristic
8	#15689524	—	2	1	no	—
9	Q₁₆×Dic₄₇	—	2	1	no	—
10	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	#15688950	—	2	1	no	—
12	#15688949	C₂²	1	1	yes	characteristic
13	#15688949	C₂²	1	1	yes	characteristic
14	#15688927	—	2	1	no	—
15	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#366	C₂²	1	1	yes	characteristic
19	#13094015	—	4	1	no	—
20	Q₈×Dic₄₇	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
22	#13093958	—	2	1	no	—
23	#13093958	—	2	1	no	—
24	#13093955	—	4	1	no	—
25	#13093948	—	4	1	no	—
26	C₈×Dic₄₇	D₄	1	1	yes	characteristic
27	C₄₇⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₄₇	—	2	1	no	—
29	Q₃₂×C₄₇	—	2	1	no	—
30	Q₃₂×C₄₇	Q₈	1	2	yes	—
31	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
32	#107×C₄₇	—	2	1	no	—
33	C₃₂×C₄₇	—	2	1	no	—
34	C₂×Q₈×C₄₇	—	4	1	no	—
35	Q₁₆×C₄₇	—	2	2	no	—
36	Q₁₆×C₄₇	—	4	1	no	—
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₄⋊Dic₄₇	—	4	1	no	—
40	C₄⋊Dic₄₇	—	4	1	no	—
41	C₄⋊Dic₄₇	—	8	1	no	—
42	C₄⋊Dic₄₇	—	8	1	no	—
43	Dic₄₇⋊C₄	—	8	1	no	—
44	C₄×Dic₄₇	—	8	1	no	—
45	C₄×Dic₄₇	D₈	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
47	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₄₇	#41	1	1	yes	characteristic
49	C₄₇⋊C₁₆	—	2	1	no	—
50	Q₈×C₄₇	—	4	2	no	—
51	Q₈×C₄₇	—	8	1	no	—
52	C₂×C₄×C₄₇	—	8	1	no	—
53	C₂×C₄×C₄₇	—	8	1	no	—
54	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₄₇	—	8	1	no	—
56	C₂×Dic₄₇	—	8	1	no	—
57	C₂×Dic₄₇	D₁₆	1	1	yes	characteristic
58	C₂×Dic₄₇	D₁₆	1	1	yes	characteristic
59	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₄₇	#137	1	1	yes	characteristic
61	C₄₇⋊C₈	—	2	1	no	—
62	#13708	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₄₇	—	8	2	no	—
65	C₄×C₄₇	—	16	1	no	—
66	C₄×C₄₇	#474	1	1	yes	characteristic
67	C₄×C₄₇	#510	1	1	yes	characteristic
68	Dic₄₇	—	2	1	no	—
69	Dic₄₇	—	2	1	no	—
70	Dic₄₇	—	16	1	no	—
71	Dic₄₇	—	16	1	no	—
72	#2253	—	47	1	no	—
73	C₄×Q₃₂	—	47	1	no	—
74	#1424	—	47	1	no	—
75	#1423	—	47	1	no	—
76	C₄⋊C₃₂	—	47	1	no	—
77	#1416	—	47	1	no	—
78	#1416	D₄₇	1	1	yes	characteristic
79	C₂×C₄₇	#2224	1	1	yes	characteristic
80	C₂×C₄₇	#2262	1	1	yes	characteristic
81	C₂×C₄₇	#2264	1	1	yes	characteristic
82	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
83	#499	—	94	1	no	—
84	C₄×Q₁₆	—	94	1	no	—
85	C₂×C₃₂	—	47	1	no	—
86	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
87	#367	—	94	1	no	—
88	#366	—	47	1	no	—
89	#366	—	47	1	no	—
90	#366	C₂×D₄₇	1	1	yes	characteristic
91	#358	—	94	1	no	—
92	C₄×C₁₆	—	47	1	no	—
93	C₄₇	#13708	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	1	no	—
95	#125	—	188	1	no	—
96	C₄×Q₈	—	188	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₉₄	1	2	yes	—
99	C₂×C₁₆	—	47	1	no	—
100	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	94	1	no	—
103	#107	—	94	1	no	—
104	#106	—	188	1	no	—
105	#103	—	188	1	no	—
106	C₄×C₈	—	47	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	94	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	2	2	no	—
111	Q₁₆	—	4	1	no	—
112	C₂×C₈	—	47	1	no	—
113	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	376	1	no	—
116	C₄⋊C₄	—	376	1	no	—
117	C₄⋊C₄	—	376	1	no	—
118	C₄⋊C₄	—	188	1	no	—
119	C₄⋊C₄	—	188	1	no	—
120	C₄²	—	47	1	no	—
121	C₄²	—	376	1	no	—
122	C₁₆	—	94	1	no	—
123	C₁₆	C₂×Dic₉₄	1	1	yes	characteristic
124	C₁₆	#10625352	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	47	1	no	—
130	C₂×C₄	—	47	1	no	—
131	C₂×C₄	—	376	1	no	—
132	C₂×C₄	—	376	1	no	—
133	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
134	C₈	—	94	1	no	—
135	C₈	#13094001	1	1	yes	characteristic
136	C₈	#13094053	1	1	yes	characteristic
137	C₂²	D₁₆×D₄₇	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	94	1	no	—
141	C₄	—	94	1	no	—
142	C₄	—	752	1	no	—
143	C₄	—	752	1	no	—
144	C₄	#15689401	1	1	yes	characteristic
145	C₄	#15689562	1	1	yes	characteristic
146	C₂	#20245523	1	1	yes	characteristic
147	C₂	#20245696	1	1	yes	characteristic
148	C₂	#20245707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245665

C₂

yes

maximal, characteristic

Q₃₂×Dic₄₇

C₂

yes

maximal, characteristic

#20241590

C₂

yes

maximal, characteristic

#20241587

C₂

yes

maximal, characteristic

Dic₄₇⋊C₃₂

C₂

yes

maximal, characteristic

#20241569

C₂

yes

maximal, characteristic

C₄₇×#1416

C₂

yes

maximal, characteristic

#15689524

—

Q₁₆×Dic₄₇

—

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

#15688950

—

#15688949

C₂²

yes

characteristic

#15688949

C₂²

yes

characteristic

#15688927

—

C₁₆×Dic₄₇

C₂²

yes

characteristic

C₂×Q₃₂×C₄₇

C₂²

yes

characteristic

C₂×C₃₂×C₄₇

C₂²

yes

characteristic

C₄₇×#366

C₂²

yes

characteristic

#13094015

—

Q₈×Dic₄₇

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

—

—

—

C₈×Dic₄₇

D₄

yes

characteristic

C₄₇⋊C₃₂

—

C₂×Q₁₆×C₄₇

—

Q₃₂×C₄₇

—

Q₃₂×C₄₇

Q₈

yes

—

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₃₂×C₄₇

—

C₂×Q₈×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₈

yes

characteristic

C₂×C₄₇⋊C₈

D₈

yes

characteristic

C₁₆×C₄₇

C₂×Q₈

yes

characteristic

C₁₆×C₄₇

#41

yes

characteristic

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

D₁₆

yes

characteristic

C₂×Dic₄₇

D₁₆

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

characteristic

C₈×C₄₇

#137

yes

characteristic

C₄₇⋊C₈

—

#13708

—

maximal, Sylow

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2253

—

C₄×Q₃₂

—

#1424

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

C₂×D₄₇

yes

characteristic

#499

—

C₄×Q₁₆

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₇

yes

—

—

—

C₂×D₄₇

yes

characteristic

#358

—

C₄×C₁₆

—

C₄₇

#13708

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

#125

—

188

—

C₄×Q₈

—

188

—

Q₃₂

—

Q₃₂

Dic₉₄

yes

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

101

#107

—

102

#107

—

103

#107

—

104

#106

—

188

—

105

#103

—

188

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₄₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

376

—

116

C₄⋊C₄

—

376

—

117

C₄⋊C₄

—

376

—

118

C₄⋊C₄

—

188

—

119

C₄⋊C₄

—

188

—

120

C₄²

—

121

C₄²

—

376

—

122

C₁₆

—

123

C₁₆

C₂×Dic₉₄

yes

characteristic

124

C₁₆

#10625352

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

376

—

132

C₂×C₄

—

376

—

133

C₂×C₄

D₈×D₄₇

yes

characteristic

134

C₈

—

135

C₈

#13094001

yes

characteristic

136

C₈

#13094053

yes

characteristic

137

C₂²

D₁₆×D₄₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

752

—

143

C₄

—

752

—

144

C₄

#15689401

yes

characteristic

145

C₄

#15689562

yes

characteristic

146

C₂

#20245523

yes

characteristic

147

C₂

#20245696

yes

characteristic

148

C₂

#20245707

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₇
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1416	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	#1416	D₄₇
hypocenter	C₄₇	#13708
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₄₇

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄₇×#1416

C₂

socle, prime radical

C₂²×C₄₇

C₂×D₁₆

D₄₇

C₄₇

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
93. C₄₇	#13708	semidirect

H ≅ G/N

type

93. C₄₇

#13708

semidirect

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

subgroup	index
#20245665	2
Q₃₂×Dic₄₇	2
#20241590	2
#20241587	2
Dic₄₇⋊C₃₂	2
#20241569	2
C₄₇×#1416	2
#13708	47

subgroup

index

Q₃₂×Dic₄₇

Dic₄₇⋊C₃₂

C₄₇×#1416

Subgroups — #26137185

Conjugacy classes of subgroups

Complements

Maximal subgroups