Group #26131069 — subgroups

Subgroups — #26131069

Order 12032.

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

normal subgroups only

This group has 272 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	#20245670	C₂	1	2	yes	maximal, complemented
2	#20245612	C₂	1	2	yes	maximal
3	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242980	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689528	—	2	1	no	—
7	#15689528	C₂²	1	1	yes	characteristic
8	#15689528	C₂²	1	2	yes	—
9	#15689503	—	2	1	no	—
10	#15689500	C₂²	1	1	yes	characteristic
11	#15689496	—	2	1	no	—
12	#15689470	—	2	1	no	—
13	#15688951	—	2	4	no	—
14	#15688925	—	2	4	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	4	2	no	—
18	C₂×#10625325	—	4	2	no	—
19	C₂×D₃₇₆	—	2	2	no	—
20	#13094024	—	2	1	no	—
21	#13094024	—	2	2	no	—
22	#13094021	—	2	1	no	—
23	#13093958	—	2	2	no	—
24	#13093958	—	2	2	no	—
25	#13093958	—	2	2	no	—
26	#13093958	—	4	2	no	—
27	#13093946	—	4	2	no	—
28	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×D₁₈₈	—	4	1	no	—
33	C₂×D₁₈₈	—	4	1	no	—
34	C₂×D₁₈₈	—	4	1	no	—
35	C₂×Dic₉₄	—	4	1	no	—
36	C₂×Dic₉₄	—	4	2	no	—
37	C₂×C₈×C₄₇	D₈	1	2	yes	—
38	C₂×C₈×C₄₇	D₈	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
40	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₄₇	—	4	2	no	—
42	C₄⋊Dic₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	4	1	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	Dic₁₈₈	—	8	4	no	—
46	D₃₇₆	—	4	4	no	—
47	#10625325	—	8	4	no	—
48	C₁₆×C₄₇	—	2	4	no	—
49	C₂²×D₄₇	—	8	1	no	—
50	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₄₇	—	8	1	no	—
54	C₂×Dic₄₇	—	8	2	no	—
55	D₁₈₈	—	8	2	no	—
56	D₁₈₈	—	8	2	no	—
57	D₁₈₈	—	8	2	no	—
58	Dic₉₄	—	8	2	no	—
59	Dic₉₄	—	8	4	no	—
60	C₈×C₄₇	—	2	2	no	—
61	C₈×C₄₇	SD₃₂	1	4	yes	—
62	C₈×C₄₇	#137	1	4	yes	—
63	#12532	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	#493	1	1	yes	characteristic
65	C₂×D₄₇	—	16	1	no	—
66	C₂×D₄₇	—	16	1	no	—
67	C₂×D₄₇	—	16	1	no	—
68	C₄×C₄₇	#497	1	2	yes	—
69	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
70	C₄×C₄₇	#509	1	2	yes	—
71	Dic₄₇	—	16	1	no	—
72	Dic₄₇	—	16	2	no	—
73	#2254	—	47	2	no	—
74	#2240	—	47	2	no	—
75	#1713	—	47	1	no	—
76	#1706	—	47	1	no	—
77	#1312	D₄₇	1	1	yes	characteristic, complemented
78	C₂×C₄₇	#1722	1	1	yes	characteristic
79	C₂×C₄₇	#2248	1	1	yes	characteristic
80	C₂×C₄₇	#2250	1	1	yes	characteristic
81	D₄₇	—	32	1	no	—
82	#500	—	47	1	no	—
83	#500	—	47	2	no	—
84	#500	—	94	1	no	—
85	#494	—	94	1	no	—
86	#493	—	47	1	no	—
87	#492	—	94	1	no	—
88	#486	—	94	1	no	—
89	#367	—	94	4	no	—
90	#357	—	94	4	no	—
91	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
92	C₈²	C₂×D₄₇	1	1	yes	characteristic
93	C₄₇	#12532	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	188	2	no	—
95	C₂×SD₁₆	—	188	2	no	—
96	C₂×D₈	—	94	2	no	—
97	C₄⋊Q₈	—	94	1	no	—
98	C₄⋊Q₈	—	94	2	no	—
99	#127	—	94	1	no	—
100	C₂×C₁₆	D₁₈₈	1	4	yes	—
101	#107	—	94	2	no	—
102	#107	—	94	2	no	—
103	#107	—	94	2	no	—
104	#107	—	188	2	no	—
105	#102	—	188	2	no	—
106	C₄×C₈	D₁₈₈	1	1	yes	characteristic
107	C₄×C₈	D₁₈₈	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₂×Q₈	—	188	1	no	—
110	C₂×Q₈	—	188	2	no	—
111	C₂×D₄	—	188	1	no	—
112	C₂×D₄	—	188	1	no	—
113	C₂×D₄	—	188	1	no	—
114	Q₁₆	—	376	4	no	—
115	SD₁₆	—	376	4	no	—
116	D₈	—	188	4	no	—
117	C₂×C₈	D₃₇₆	1	2	yes	—
118	C₂×C₈	D₃₇₆	1	2	yes	—
119	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
120	C₄⋊C₄	—	188	2	no	—
121	C₄⋊C₄	—	188	1	no	—
122	C₄⋊C₄	—	188	1	no	—
123	C₄⋊C₄	—	188	2	no	—
124	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	376	1	no	—
127	Q₈	—	376	2	no	—
128	Q₈	—	376	4	no	—
129	D₄	—	376	2	no	—
130	D₄	—	376	2	no	—
131	D₄	—	376	2	no	—
132	C₂×C₄	—	376	1	no	—
133	C₂×C₄	—	376	2	no	—
134	C₂×C₄	#13094021	1	1	yes	characteristic
135	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
136	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#13093969	1	4	yes	—
139	C₈	#13094057	1	4	yes	—
140	C₂²	—	752	1	no	—
141	C₂²	—	752	1	no	—
142	C₂²	—	752	1	no	—
143	C₂²	#15689500	1	1	yes	characteristic
144	C₄	—	752	1	no	—
145	C₄	—	752	2	no	—
146	C₄	#15689517	1	2	yes	—
147	C₄	C₂×#13093969	1	2	yes	—
148	C₄	#15689561	1	2	yes	—
149	C₂	—	1504	1	no	—
150	C₂	#20243059	1	1	yes	characteristic
151	C₂	#20245644	1	1	yes	characteristic
152	C₂	#20245654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245670

C₂

yes

maximal, complemented

#20245612

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic, complemented

#20242980

C₂

yes

maximal, characteristic

C₄₇×#1312

C₂

yes

maximal, characteristic, complemented

#15689528

—

#15689528

C₂²

yes

characteristic

#15689528

C₂²

yes

—

#15689503

—

#15689500

C₂²

yes

—

—

—

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×#10625325

—

C₂×D₃₇₆

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

D₃₇₆

—

#10625325

—

C₁₆×C₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

SD₃₂

yes

—

C₈×C₄₇

#137

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#497

yes

—

C₄×C₄₇

C₂×SD₃₂

yes

—

C₄×C₄₇

#509

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12532

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₂×SD₁₆

—

188

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

100

C₂×C₁₆

D₁₈₈

yes

—

101

#107

—

102

#107

—

103

#107

—

104

#107

—

188

—

105

#102

—

188

—

106

C₄×C₈

D₁₈₈

yes

characteristic

107

C₄×C₈

D₁₈₈

yes

characteristic

108

C₄×C₈

C₂²×D₄₇

yes

characteristic

109

C₂×Q₈

—

188

—

110

C₂×Q₈

—

188

—

111

C₂×D₄

—

188

—

112

C₂×D₄

—

188

—

113

C₂×D₄

—

188

—

114

Q₁₆

—

376

—

115

SD₁₆

—

376

—

116

D₈

—

188

—

117

C₂×C₈

D₃₇₆

yes

—

118

C₂×C₈

D₃₇₆

yes

—

119

C₂×C₈

C₂×D₁₈₈

yes

—

120

C₄⋊C₄

—

188

—

121

C₄⋊C₄

—

188

—

122

C₄⋊C₄

—

188

—

123

C₄⋊C₄

—

188

—

124

C₄²

C₂×D₁₈₈

yes

characteristic

125

C₁₆

—

126

C₂³

—

376

—

127

Q₈

—

376

—

128

Q₈

—

376

—

129

D₄

—

376

—

130

D₄

—

376

—

131

D₄

—

376

—

132

C₂×C₄

—

376

—

133

C₂×C₄

—

376

—

134

C₂×C₄

#13094021

yes

characteristic

135

C₂×C₄

C₂×D₃₇₆

yes

characteristic

136

C₂×C₄

C₂×D₃₇₆

yes

characteristic

137

C₈

—

138

C₈

#13093969

yes

—

139

C₈

#13094057

yes

—

140

C₂²

—

752

—

141

C₂²

—

752

—

142

C₂²

—

752

—

143

C₂²

#15689500

yes

characteristic

144

C₄

—

752

—

145

C₄

—

752

—

146

C₄

#15689517

yes

—

147

C₄

C₂×#13093969

yes

—

148

C₄

#15689561

yes

—

149

C₂

—

1504

—

150

C₂

#20243059

yes

characteristic

151

C₂

#20245644

yes

characteristic

152

C₂

#20245654

yes

characteristic

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1312	C₂
socle	C₂²×C₄₇	#493
prime radical	#15689500	C₂²
hypercenter	#1312	D₄₇
hypocenter	C₄₇	#12532
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#15689500

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1312

C₂

C₂²×C₄₇

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #20245670	C₂	semidirect
3. #20243006	C₂	semidirect
5. C₄₇×#1312	C₂	semidirect
77. #1312	D₄₇	semidirect
93. C₄₇	#12532	semidirect

H ≅ G/N

type

1. #20245670

C₂

semidirect

3. #20243006

C₂

semidirect

5. C₄₇×#1312

C₂

semidirect

77. #1312

D₄₇

semidirect

93. C₄₇

#12532

semidirect

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

subgroup	index
#20245670	2
#20245612	2
#20243006	2
#20242980	2
C₄₇×#1312	2
#12532	47

subgroup

index

C₄₇×#1312

Subgroups — #26131069

Conjugacy classes of subgroups

Complements

Maximal subgroups