Group #26131033 — subgroups

Subgroups — #26131033

Order 12032.

296 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245643	C₂	1	2	yes	maximal
2	#20245617	C₂	1	2	yes	maximal, complemented
3	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1706	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093968	—	2	4	no	—
7	#15689528	—	2	1	no	—
8	#15689528	—	2	1	no	—
9	#15689503	C₂²	1	2	yes	—
10	#15688926	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄₇×#500	C₂²	1	1	yes	characteristic
13	C₄₇×#493	C₂²	1	1	yes	characteristic
14	C₄₇×#492	—	2	1	no	—
15	C₄₇×#486	—	2	1	no	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	2	no	—
19	#13094024	—	2	2	no	—
20	#13093968	—	4	8	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
22	#13093958	—	4	2	no	—
23	#13093958	—	4	2	no	—
24	C₂×SD₁₆×C₄₇	—	4	2	no	—
25	C₂×D₈×C₄₇	—	2	2	no	—
26	C₄⋊Q₈×C₄₇	—	2	1	no	—
27	#127×C₄₇	—	2	1	no	—
28	#107×C₄₇	—	2	2	no	—
29	#102×C₄₇	—	4	2	no	—
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×Q₈×C₄₇	—	4	1	no	—
34	C₂×D₄×C₄₇	—	4	1	no	—
35	C₂×D₄×C₄₇	—	4	1	no	—
36	C₂×D₄×C₄₇	—	4	1	no	—
37	C₂×Dic₉₄	—	4	2	no	—
38	SD₁₆×C₄₇	—	8	4	no	—
39	D₈×C₄₇	—	4	4	no	—
40	C₂×C₈×C₄₇	D₈	1	2	yes	—
41	C₂×C₈×C₄₇	D₈	1	2	yes	—
42	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₄₇	—	4	1	no	—
44	C₄⋊C₄×C₄₇	—	4	1	no	—
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₄₇	—	4	2	no	—
47	C₄⋊Dic₄₇	—	4	2	no	—
48	Dic₁₈₈	—	4	4	no	—
49	Dic₁₈₈	—	4	4	no	—
50	C₄₇⋊C₁₆	—	2	4	no	—
51	C₂³×C₄₇	—	8	1	no	—
52	Q₈×C₄₇	—	8	2	no	—
53	D₄×C₄₇	—	8	2	no	—
54	D₄×C₄₇	—	8	2	no	—
55	D₄×C₄₇	—	8	2	no	—
56	C₂×C₄×C₄₇	—	8	1	no	—
57	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₄₇	—	8	2	no	—
61	Dic₉₄	—	8	2	no	—
62	Dic₉₄	—	8	2	no	—
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	SD₃₂	1	4	yes	complemented
65	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
66	#12526	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	16	1	no	—
68	C₂²×C₄₇	—	16	1	no	—
69	C₂²×C₄₇	—	16	1	no	—
70	C₂²×C₄₇	#493	1	1	yes	characteristic
71	C₄×C₄₇	—	16	1	no	—
72	C₄×C₄₇	#496	1	2	yes	—
73	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
74	C₄×C₄₇	#510	1	2	yes	—
75	Dic₄₇	—	16	2	no	—
76	#2248	—	47	2	no	—
77	#2241	—	47	2	no	—
78	#1713	—	47	1	no	—
79	#1706	D₄₇	1	1	yes	characteristic
80	#1312	—	47	1	no	—
81	C₂×C₄₇	—	32	1	no	—
82	C₂×C₄₇	#1721	1	1	yes	characteristic
83	C₂×C₄₇	#2248	1	1	yes	characteristic
84	C₂×C₄₇	#2251	1	1	yes	characteristic
85	C₂×SD₃₂	—	94	4	no	—
86	#500	—	94	1	no	—
87	#500	—	94	1	no	—
88	#500	C₂×D₄₇	1	1	yes	characteristic
89	#494	—	47	2	no	—
90	#493	C₂×D₄₇	1	1	yes	characteristic
91	#492	—	2	1	no	—
92	#486	—	2	1	no	—
93	#358	—	94	4	no	—
94	C₄×C₁₆	—	47	2	no	—
95	C₈²	C₂×D₄₇	1	1	yes	characteristic
96	C₄₇	#12526	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₁₆	—	94	2	no	—
98	C₂×Q₁₆	—	94	2	no	—
99	C₂×SD₁₆	—	4	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	94	2	no	—
103	#127	—	2	1	no	—
104	SD₃₂	—	188	8	no	—
105	C₂×C₁₆	—	47	4	no	—
106	#107	—	2	2	no	—
107	#107	—	188	2	no	—
108	#107	—	188	2	no	—
109	#102	—	4	2	no	—
110	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	C₂×Q₈	—	4	1	no	—
114	C₂×Q₈	—	188	2	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	1	no	—
118	Q₁₆	—	188	4	no	—
119	Q₁₆	—	188	4	no	—
120	SD₁₆	—	8	4	no	—
121	D₈	—	4	4	no	—
122	C₂×C₈	#10625334	1	2	yes	—
123	C₂×C₈	#10625334	1	2	yes	—
124	C₂×C₈	D₄×D₄₇	1	2	yes	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	188	2	no	—
128	C₄⋊C₄	—	188	2	no	—
129	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
130	C₁₆	—	94	4	no	—
131	C₂³	—	8	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	376	2	no	—
134	Q₈	—	376	2	no	—
135	D₄	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	376	2	no	—
140	C₂×C₄	#13094020	1	1	yes	characteristic
141	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
142	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#13093968	1	4	yes	complemented
145	C₈	#13094051	1	4	yes	—
146	C₂²	—	16	1	no	—
147	C₂²	—	16	1	no	—
148	C₂²	—	16	1	no	—
149	C₂²	#15689499	1	1	yes	characteristic
150	C₄	—	16	1	no	—
151	C₄	—	752	2	no	—
152	C₄	#15689509	1	2	yes	—
153	C₄	C₂×#13093968	1	2	yes	—
154	C₄	#15689563	1	2	yes	—
155	C₂	—	32	1	no	—
156	C₂	#20243052	1	1	yes	characteristic
157	C₂	#20245643	1	1	yes	characteristic
158	C₂	#20245660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245643

C₂

yes

maximal

#20245617

C₂

yes

maximal, complemented

#20243006

C₂

yes

maximal, characteristic, complemented

#20241306

C₂

yes

maximal, characteristic, complemented

C₄₇×#1706

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#15688926

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#500

C₂²

yes

characteristic

C₄₇×#493

C₂²

yes

characteristic

C₄₇×#492

—

C₄₇×#486

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13093968

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

C₂×SD₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#127×C₄₇

—

#107×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

SD₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

SD₃₂

yes

complemented

C₈×C₄₇

AGL(1,ℤ/8)

yes

—

#12526

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#496

yes

—

C₄×C₄₇

C₂×SD₃₂

yes

—

C₄×C₄₇

#510

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1312

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×SD₃₂

—

#500

—

#500

—

#500

C₂×D₄₇

yes

characteristic

#494

—

#493

C₂×D₄₇

yes

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12526

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

100

C₂×D₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

#127

—

104

SD₃₂

—

188

—

105

C₂×C₁₆

—

106

#107

—

107

#107

—

188

—

108

#107

—

188

—

109

#102

—

110

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

111

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

112

C₄×C₈

C₂²×D₄₇

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

188

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

Q₁₆

—

188

—

119

Q₁₆

—

188

—

120

SD₁₆

—

121

D₈

—

122

C₂×C₈

#10625334

yes

—

123

C₂×C₈

#10625334

yes

—

124

C₂×C₈

D₄×D₄₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

188

—

128

C₄⋊C₄

—

188

—

129

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

130

C₁₆

—

131

C₂³

—

132

Q₈

—

133

Q₈

—

376

—

134

Q₈

—

376

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

376

—

140

C₂×C₄

#13094020

yes

characteristic

141

C₂×C₄

C₂×#10625334

yes

characteristic

142

C₂×C₄

C₂×#10625334

yes

characteristic

143

C₈

—

144

C₈

#13093968

yes

complemented

145

C₈

#13094051

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#15689499

yes

characteristic

150

C₄

—

151

C₄

—

752

—

152

C₄

#15689509

yes

—

153

C₄

C₂×#13093968

yes

—

154

C₄

#15689563

yes

—

155

C₂

—

156

C₂

#20243052

yes

characteristic

157

C₂

#20245643

yes

characteristic

158

C₂

#20245660

yes

characteristic

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1706	C₂
socle	C₂²×C₄₇	#493
prime radical	C₄₇×#493	C₂²
hypercenter	#1706	D₄₇
hypocenter	C₄₇	#12526
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1706

C₂

socle

C₂²×C₄₇

#493

prime radical

C₄₇×#493

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

N	H ≅ G/N	type
2. #20245617	C₂	semidirect
3. #20243006	C₂	semidirect
4. #20241306	C₂	semidirect
64. C₈×C₄₇	SD₃₂	semidirect
96. C₄₇	#12526	semidirect
144. C₈	#13093968	semidirect

H ≅ G/N

type

2. #20245617

C₂

semidirect

3. #20243006

C₂

semidirect

4. #20241306

C₂

semidirect

64. C₈×C₄₇

SD₃₂

semidirect

96. C₄₇

#12526

semidirect

144. C₈

#13093968

semidirect

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

subgroup	index
#20245643	2
#20245617	2
#20243006	2
#20241306	2
C₄₇×#1706	2
#12526	47

subgroup

index

C₄₇×#1706

Subgroups — #26131033

Conjugacy classes of subgroups

Complements

Maximal subgroups