Group #26132676 — subgroups

Subgroups — #26132676

Order 12032.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20244987	C₂	1	1	yes	maximal, characteristic
2	#20243513	C₂	1	2	yes	maximal
3	#20241490	C₂	1	1	yes	maximal, characteristic
4	#20241485	C₂	1	2	yes	maximal
5	C₄₇×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	#15689161	C₂²	1	2	yes	—
8	#15689078	C₂²	1	1	yes	characteristic
9	#15688898	C₄	1	2	yes	complemented
10	C₁₆×Dic₄₇	C₄	1	2	yes	complemented
11	#15688895	—	2	2	no	—
12	#15688880	—	2	2	no	—
13	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	2	yes	—
15	C₂×#10625329	D₄	1	1	yes	characteristic
16	#13093983	—	2	2	no	—
17	#13093983	D₄	1	1	yes	characteristic
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	#13093958	—	2	2	no	—
22	#13093956	—	2	2	no	—
23	#13093934	Q₈	1	2	yes	—
24	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	M₃₂×C₄₇	—	2	2	no	—
28	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
29	#108×C₄₇	—	2	2	no	—
30	#108×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₄₇	—	4	2	no	—
33	M₁₆×C₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	#41	1	2	yes	—
38	C₂×C₈×C₄₇	#41	1	2	yes	—
39	C₂²⋊Dic₄₇	—	2	1	no	—
40	C₂²⋊Dic₄₇	—	4	2	no	—
41	C₄⋊Dic₄₇	—	2	1	no	—
42	C₄⋊Dic₄₇	—	4	2	no	—
43	C₄×Dic₄₇	—	4	2	no	—
44	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
45	#10625329	—	2	2	no	—
46	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₁₆×C₄₇	—	2	2	no	—
48	C₄₇⋊C₁₆	—	2	1	no	—
49	C₄₇⋊C₁₆	—	2	1	no	—
50	C₂³×C₄₇	#135	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#124	1	2	yes	—
53	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#135	1	2	yes	—
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	—	8	2	no	—
57	C₈×C₄₇	—	8	2	no	—
58	C₈×C₄₇	#117	1	1	yes	characteristic
59	C₈×C₄₇	#117	1	1	yes	characteristic
60	C₈×C₄₇	#125	1	2	yes	—
61	C₄₇⋊C₈	—	2	2	no	—
62	#12893	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	4	1	no	—
64	C₂²×C₄₇	#427	1	1	yes	characteristic
65	C₂²×C₄₇	#488	1	2	yes	—
66	C₄×C₄₇	#382	1	1	yes	characteristic
67	C₄×C₄₇	#427	1	1	yes	characteristic
68	C₄×C₄₇	#488	1	2	yes	—
69	Dic₄₇	—	4	2	no	—
70	Dic₄₇	—	16	2	no	—
71	#2108	—	47	1	no	—
72	#1833	—	47	2	no	—
73	#1385	D₄₇	1	1	yes	characteristic
74	#1384	—	47	1	no	—
75	#1382	—	47	2	no	—
76	C₂×C₄₇	—	2	1	no	—
77	C₂×C₄₇	—	2	2	no	—
78	C₂×C₄₇	#1835	1	1	yes	characteristic
79	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
80	C₂²×C₁₆	—	47	1	no	—
81	C₂×#108	C₂×D₄₇	1	2	yes	—
82	#427	—	47	2	no	—
83	#405	—	47	1	no	—
84	#346	—	47	2	no	—
85	C₄×C₁₆	—	47	2	no	—
86	#344	—	94	2	no	—
87	H(3,ℤ/4)	—	94	2	no	—
88	C₄₇	#12893	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	47	1	no	—
91	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	#117	—	47	1	no	—
93	#117	—	94	2	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	47	1	no	—
96	C₂×C₁₆	—	47	1	no	—
97	C₂×C₁₆	—	94	2	no	—
98	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
99	#108	—	2	2	no	—
100	#108	—	2	2	no	—
101	#107	—	94	2	no	—
102	#106	—	94	2	no	—
103	#97	—	47	2	no	—
104	C₄×C₈	—	47	2	no	—
105	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	94	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	47	2	no	—
111	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
112	C₂×C₈	#10625350	1	2	yes	—
113	C₂×C₈	#10625352	1	2	yes	—
114	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
115	C₄⋊C₄	—	94	1	no	—
116	C₄⋊C₄	—	188	2	no	—
117	C₂²⋊C₄	—	94	1	no	—
118	C₂²⋊C₄	—	188	2	no	—
119	C₄²	—	47	2	no	—
120	C₄²	—	188	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	94	1	no	—
123	C₁₆	—	94	1	no	—
124	C₂³	#13094047	1	1	yes	characteristic
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	376	2	no	—
127	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
128	C₂×C₄	#13094010	1	2	yes	—
129	C₂×C₄	#13094043	1	1	yes	characteristic
130	C₂×C₄	#13094045	1	2	yes	—
131	C₈	—	8	2	no	—
132	C₈	—	94	2	no	—
133	C₈	#13093980	1	1	yes	characteristic
134	C₈	#13093981	1	1	yes	characteristic
135	C₈	#13094014	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#15689160	1	1	yes	characteristic
138	C₂²	#15689479	1	2	yes	—
139	C₄	—	188	2	no	—
140	C₄	—	752	2	no	—
141	C₄	#15688987	1	1	yes	characteristic
142	C₄	#15689158	1	1	yes	characteristic
143	C₄	#15689476	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#20243517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244987

C₂

yes

maximal, characteristic

#20243513

C₂

yes

maximal

#20241490

C₂

yes

maximal, characteristic

#20241485

C₂

yes

maximal

C₄₇×#1385

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15689161

C₂²

yes

—

#15689078

C₂²

yes

characteristic

#15688898

C₄

yes

complemented

C₁₆×Dic₄₇

C₄

yes

complemented

#15688895

—

#15688880

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₂×#10625329

D₄

yes

characteristic

#13093983

—

#13093983

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₄₇

C₂×C₄

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#108×C₄₇

—

#108×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

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

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

#12893

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#427

yes

characteristic

C₂²×C₄₇

#488

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

#2108

—

#1833

—

#1385

D₄₇

yes

characteristic

#1384

—

#1382

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1835

yes

characteristic

C₂×M₃₂

C₂×D₄₇

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₄₇

yes

—

#427

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

H(3,ℤ/4)

—

C₄₇

#12893

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

#117

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

—

#108

—

100

#108

—

101

#107

—

102

#106

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₄₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

112

C₂×C₈

#10625350

yes

—

113

C₂×C₈

#10625352

yes

—

114

C₂×C₈

Q₈×D₄₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

188

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

188

—

119

C₄²

—

120

C₄²

—

188

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#13094047

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

376

—

127

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

128

C₂×C₄

#13094010

yes

—

129

C₂×C₄

#13094043

yes

characteristic

130

C₂×C₄

#13094045

yes

—

131

C₈

—

132

C₈

—

133

C₈

#13093980

yes

characteristic

134

C₈

#13093981

yes

characteristic

135

C₈

#13094014

yes

—

136

C₂²

—

137

C₂²

#15689160

yes

characteristic

138

C₂²

#15689479

yes

—

139

C₄

—

188

—

140

C₄

—

752

—

141

C₄

#15688987

yes

characteristic

142

C₄

#15689158

yes

characteristic

143

C₄

#15689476

yes

—

144

C₂

—

145

C₂

—

146

C₂

#20243517

yes

characteristic

	subgroup	G/H
center	C₄	#15689158
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1385	C₂
socle	C₂×C₄₇	#1835
prime radical	C₂³×C₄₇	#135
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1385	D₄₇
hypocenter	C₄₇	#12893
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#15689158

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1385

C₂

C₂×C₄₇

C₂³×C₄₇

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
9. #15688898	C₄	semidirect
10. C₁₆×Dic₄₇	C₄	semidirect
88. C₄₇	#12893	semidirect

H ≅ G/N

type

9. #15688898

C₄

semidirect

10. C₁₆×Dic₄₇

C₄

semidirect

88. C₄₇

#12893

semidirect

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

subgroup	index
#20244987	2
#20243513	2
#20241490	2
#20241485	2
C₄₇×#1385	2
#12893	47

subgroup

index

C₄₇×#1385

Subgroups — #26132676

Conjugacy classes of subgroups

Complements

Maximal subgroups