Group #26131052 — subgroups

Subgroups — #26131052

Order 12032.

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

normal subgroups only

This group has 318 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	#20245644	C₂	1	2	yes	maximal, complemented
2	#20245612	C₂	1	2	yes	maximal, complemented
3	#20243003	C₂	1	1	yes	maximal, characteristic
4	#20242985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13093969	—	2	4	no	—
7	#15689528	C₂²	1	1	yes	characteristic
8	#15689521	—	2	1	no	—
9	#15689503	C₂²	1	1	yes	characteristic
10	#15689500	—	2	1	no	—
11	#15689500	—	2	1	no	—
12	#15689500	C₂²	1	2	yes	—
13	#15689474	—	2	1	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	—	2	2	no	—
18	C₂×D₃₇₆	—	2	2	no	—
19	C₂×D₃₇₆	—	2	2	no	—
20	C₂×D₃₇₆	—	4	2	no	—
21	C₂×D₃₇₆	—	4	2	no	—
22	#13094024	—	2	1	no	—
23	#13094024	—	2	1	no	—
24	#13094021	—	2	2	no	—
25	#13093969	—	4	8	no	—
26	#13093958	—	2	2	no	—
27	#13093956	—	4	2	no	—
28	#13093949	—	4	2	no	—
29	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×D₁₈₈	—	4	2	no	—
34	C₂×D₁₈₈	—	4	2	no	—
35	C₂×D₁₈₈	—	4	2	no	—
36	C₂×Dic₉₄	—	4	1	no	—
37	C₂×Dic₉₄	—	4	1	no	—
38	C₂×C₈×C₄₇	D₈	1	2	yes	—
39	C₂×C₈×C₄₇	D₈	1	2	yes	—
40	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
41	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	4	1	no	—
44	C₄⋊Dic₄₇	—	4	1	no	—
45	C₄⋊Dic₄₇	—	4	1	no	—
46	Dic₁₈₈	—	4	4	no	—
47	D₃₇₆	—	4	4	no	—
48	D₃₇₆	—	4	4	no	—
49	D₃₇₆	—	8	4	no	—
50	D₃₇₆	—	8	4	no	—
51	C₁₆×C₄₇	—	2	4	no	—
52	C₂²×D₄₇	—	8	2	no	—
53	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₄₇	—	8	1	no	—
57	C₂×Dic₄₇	—	8	1	no	—
58	D₁₈₈	—	8	2	no	—
59	D₁₈₈	—	8	2	no	—
60	D₁₈₈	—	8	4	no	—
61	D₁₈₈	—	8	4	no	—
62	Dic₉₄	—	8	2	no	—
63	Dic₉₄	—	8	2	no	—
64	C₈×C₄₇	—	2	2	no	—
65	C₈×C₄₇	SD₃₂	1	4	yes	complemented
66	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
67	#12529	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	#493	1	1	yes	characteristic
69	C₂×D₄₇	—	16	2	no	—
70	C₂×D₄₇	—	16	2	no	—
71	C₂×D₄₇	—	16	2	no	—
72	C₄×C₄₇	#496	1	2	yes	—
73	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
74	C₄×C₄₇	#509	1	2	yes	—
75	Dic₄₇	—	16	1	no	—
76	Dic₄₇	—	16	1	no	—
77	#2248	—	47	2	no	—
78	#2240	—	47	2	no	—
79	#1712	—	47	1	no	—
80	#1707	—	47	1	no	—
81	#1312	D₄₇	1	1	yes	characteristic, complemented
82	C₂×C₄₇	#1721	1	1	yes	characteristic
83	C₂×C₄₇	#2248	1	1	yes	characteristic
84	C₂×C₄₇	#2250	1	1	yes	characteristic
85	D₄₇	—	32	2	no	—
86	C₂×SD₃₂	—	94	4	no	—
87	#500	—	47	1	no	—
88	#498	—	94	1	no	—
89	#494	—	47	1	no	—
90	#493	—	47	2	no	—
91	#493	—	94	1	no	—
92	#493	—	94	1	no	—
93	#487	—	94	1	no	—
94	#357	—	94	4	no	—
95	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
96	C₈²	C₂×D₄₇	1	1	yes	characteristic
97	C₄₇	#12529	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	94	2	no	—
99	C₂×D₈	—	94	2	no	—
100	C₂×D₈	—	94	2	no	—
101	C₂×D₈	—	188	2	no	—
102	C₂×D₈	—	188	2	no	—
103	C₄⋊Q₈	—	94	1	no	—
104	C₄⋊Q₈	—	94	1	no	—
105	#127	—	94	2	no	—
106	SD₃₂	—	188	8	no	—
107	C₂×C₁₆	D₁₈₈	1	4	yes	—
108	#107	—	94	2	no	—
109	#106	—	188	2	no	—
110	#103	—	188	2	no	—
111	C₄×C₈	D₁₈₈	1	1	yes	characteristic
112	C₄×C₈	D₁₈₈	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
114	C₂×Q₈	—	188	1	no	—
115	C₂×Q₈	—	188	1	no	—
116	C₂×D₄	—	188	2	no	—
117	C₂×D₄	—	188	2	no	—
118	C₂×D₄	—	188	2	no	—
119	Q₁₆	—	188	4	no	—
120	D₈	—	376	4	no	—
121	D₈	—	376	4	no	—
122	D₈	—	188	4	no	—
123	D₈	—	188	4	no	—
124	C₂×C₈	D₃₇₆	1	2	yes	—
125	C₂×C₈	D₃₇₆	1	2	yes	—
126	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
127	C₄⋊C₄	—	188	1	no	—
128	C₄⋊C₄	—	188	1	no	—
129	C₄⋊C₄	—	188	1	no	—
130	C₄⋊C₄	—	188	1	no	—
131	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₂³	—	376	2	no	—
134	Q₈	—	376	2	no	—
135	Q₈	—	376	2	no	—
136	D₄	—	376	2	no	—
137	D₄	—	376	2	no	—
138	D₄	—	376	4	no	—
139	D₄	—	376	4	no	—
140	C₂×C₄	—	376	1	no	—
141	C₂×C₄	—	376	1	no	—
142	C₂×C₄	#13094021	1	1	yes	characteristic
143	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
144	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#13093969	1	4	yes	complemented
147	C₈	#13094052	1	4	yes	—
148	C₂²	—	752	2	no	—
149	C₂²	—	752	2	no	—
150	C₂²	—	752	2	no	—
151	C₂²	#15689500	1	1	yes	characteristic
152	C₄	—	752	1	no	—
153	C₄	—	752	1	no	—
154	C₄	#15689512	1	2	yes	—
155	C₄	C₂×#13093969	1	2	yes	—
156	C₄	#15689561	1	2	yes	—
157	C₂	—	1504	2	no	—
158	C₂	#20243054	1	1	yes	characteristic
159	C₂	#20245644	1	1	yes	characteristic
160	C₂	#20245654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245644

C₂

yes

maximal, complemented

#20245612

C₂

yes

maximal, complemented

#20243003

C₂

yes

maximal, characteristic

#20242985

C₂

yes

maximal, characteristic, complemented

C₄₇×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#13093969

—

#15689528

C₂²

yes

characteristic

#15689521

—

#15689503

C₂²

yes

—

—

C₂²

yes

—

#15689474

—

#15688925

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

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

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

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

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

SD₃₂

yes

complemented

C₈×C₄₇

AGL(1,ℤ/8)

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#496

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

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12529

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

100

C₂×D₈

—

101

C₂×D₈

—

188

—

102

C₂×D₈

—

188

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

#127

—

106

SD₃₂

—

188

—

107

C₂×C₁₆

D₁₈₈

yes

—

108

#107

—

109

#106

—

188

—

110

#103

—

188

—

111

C₄×C₈

D₁₈₈

yes

characteristic

112

C₄×C₈

D₁₈₈

yes

characteristic

113

C₄×C₈

C₂²×D₄₇

yes

characteristic

114

C₂×Q₈

—

188

—

115

C₂×Q₈

—

188

—

116

C₂×D₄

—

188

—

117

C₂×D₄

—

188

—

118

C₂×D₄

—

188

—

119

Q₁₆

—

188

—

120

D₈

—

376

—

121

D₈

—

376

—

122

D₈

—

188

—

123

D₈

—

188

—

124

C₂×C₈

D₃₇₆

yes

—

125

C₂×C₈

D₃₇₆

yes

—

126

C₂×C₈

C₂×D₁₈₈

yes

—

127

C₄⋊C₄

—

188

—

128

C₄⋊C₄

—

188

—

129

C₄⋊C₄

—

188

—

130

C₄⋊C₄

—

188

—

131

C₄²

C₂×D₁₈₈

yes

characteristic

132

C₁₆

—

133

C₂³

—

376

—

134

Q₈

—

376

—

135

Q₈

—

376

—

136

D₄

—

376

—

137

D₄

—

376

—

138

D₄

—

376

—

139

D₄

—

376

—

140

C₂×C₄

—

376

—

141

C₂×C₄

—

376

—

142

C₂×C₄

#13094021

yes

characteristic

143

C₂×C₄

C₂×D₃₇₆

yes

characteristic

144

C₂×C₄

C₂×D₃₇₆

yes

characteristic

145

C₈

—

146

C₈

#13093969

yes

complemented

147

C₈

#13094052

yes

—

148

C₂²

—

752

—

149

C₂²

—

752

—

150

C₂²

—

752

—

151

C₂²

#15689500

yes

characteristic

152

C₄

—

752

—

153

C₄

—

752

—

154

C₄

#15689512

yes

—

155

C₄

C₂×#13093969

yes

—

156

C₄

#15689561

yes

—

157

C₂

—

1504

—

158

C₂

#20243054

yes

characteristic

159

C₂

#20245644

yes

characteristic

160

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	#20243003	C₂
hypercenter	#1312	D₄₇
hypocenter	C₄₇	#12529
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

C₂

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #20245644	C₂	semidirect
2. #20245612	C₂	semidirect
4. #20242985	C₂	semidirect
5. C₄₇×#1312	C₂	semidirect
65. C₈×C₄₇	SD₃₂	semidirect
81. #1312	D₄₇	semidirect
97. C₄₇	#12529	semidirect
146. C₈	#13093969	semidirect

H ≅ G/N

type

1. #20245644

C₂

semidirect

2. #20245612

C₂

semidirect

4. #20242985

C₂

semidirect

5. C₄₇×#1312

C₂

semidirect

65. C₈×C₄₇

SD₃₂

semidirect

81. #1312

D₄₇

semidirect

97. C₄₇

#12529

semidirect

146. C₈

#13093969

semidirect

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

subgroup	index
#20245644	2
#20245612	2
#20243003	2
#20242985	2
C₄₇×#1312	2
#12529	47

subgroup

index

C₄₇×#1312

Subgroups — #26131052

Conjugacy classes of subgroups

Complements

Maximal subgroups