Group #26130557 — subgroups

Subgroups — #26130557

Order 12032.

212 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20245641	C₂	1	1	yes	maximal, characteristic
2	#20242758	C₂	1	2	yes	maximal
3	#20241462	C₂	1	1	yes	maximal, characteristic
4	#20241438	C₂	1	2	yes	maximal
5	C₄₇×#2183	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	1	no	—
7	C₂×Dic₃₇₆	—	2	1	no	—
8	#15689503	C₂²	1	2	yes	—
9	#15689408	—	2	2	no	—
10	#15688876	C₂²	1	2	yes	—
11	#15688870	—	2	2	no	—
12	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×C₄⋊C₁₆	—	2	1	no	—
14	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄₇×#346	—	2	1	no	—
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	2	no	—
19	#13094024	—	2	2	no	—
20	Dic₃₇₆	—	4	2	no	—
21	Dic₃₇₆	—	4	2	no	—
22	#13093955	—	4	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	#13093948	—	4	2	no	—
25	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	—	2	1	no	—
27	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	1	no	—
30	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	—	2	1	no	—
32	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
34	C₂×Dic₉₄	—	4	2	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
40	C₄²×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₇	—	4	2	no	—
43	C₄⋊Dic₄₇	—	4	2	no	—
44	C₂×C₄₇⋊C₈	—	4	2	no	—
45	Dic₁₈₈	—	4	2	no	—
46	Dic₁₈₈	—	4	2	no	—
47	Dic₁₈₈	—	8	2	no	—
48	C₁₆×C₄₇	—	2	1	no	—
49	C₁₆×C₄₇	—	4	1	no	—
50	C₁₆×C₄₇	SD₁₆	1	2	yes	—
51	Q₈×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
57	C₂×Dic₄₇	—	8	2	no	—
58	Dic₉₄	—	8	2	no	—
59	Dic₉₄	—	8	2	no	—
60	C₈×C₄₇	—	2	1	no	—
61	C₈×C₄₇	—	4	1	no	—
62	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄₇⋊C₈	—	8	2	no	—
65	#12443	—	47	1	no	maximal, Sylow
66	C₂²×C₄₇	#466	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	—	4	1	no	—
70	C₄×C₄₇	#460	1	1	yes	characteristic
71	C₄×C₄₇	#472	1	1	yes	characteristic
72	C₄×C₄₇	#510	1	2	yes	—
73	Dic₄₇	—	16	2	no	—
74	#2247	—	47	1	no	—
75	#2183	D₄₇	1	1	yes	characteristic
76	#1670	—	47	2	no	—
77	#1372	—	47	1	no	—
78	#1362	—	47	2	no	—
79	C₂×C₄₇	#1668	1	1	yes	characteristic
80	C₂×C₄₇	#2219	1	1	yes	characteristic
81	C₂×C₄₇	#2222	1	1	yes	characteristic
82	C₂×Q₃₂	—	94	1	no	—
83	C₂×Q₃₂	—	94	1	no	—
84	#494	—	47	2	no	—
85	#475	—	94	2	no	—
86	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	1	no	—
88	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
89	#346	—	2	1	no	—
90	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
91	#335	—	47	2	no	—
92	#332	—	94	2	no	—
93	C₄₇	#12443	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	94	2	no	—
95	C₂×Q₁₆	—	94	2	no	—
96	C₄⋊Q₈	—	94	2	no	—
97	C₄×Q₈	D₁₈₈	1	1	yes	characteristic
98	Q₃₂	—	188	2	no	—
99	Q₃₂	—	188	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
103	#106	—	188	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	94	2	no	—
106	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
107	#103	—	188	2	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
110	C₂×Q₈	—	188	2	no	—
111	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
112	Q₁₆	—	376	2	no	—
113	Q₁₆	—	188	2	no	—
114	Q₁₆	—	188	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	188	2	no	—
117	C₂×C₈	#10625350	1	1	yes	characteristic
118	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	188	2	no	—
121	C₄⋊C₄	—	188	2	no	—
122	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#10625336	1	2	yes	—
128	Q₈	—	2	2	no	—
129	Q₈	—	376	2	no	—
130	Q₈	—	376	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	376	2	no	—
134	C₂×C₄	#13093998	1	1	yes	characteristic
135	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
136	C₂×C₄	#13094046	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	376	2	no	—
140	C₈	C₂×#10625336	1	1	yes	characteristic
141	C₈	#13094049	1	1	yes	characteristic
142	C₂²	#15689359	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	752	2	no	—
147	C₄	#15689323	1	1	yes	characteristic
148	C₄	#15689393	1	1	yes	characteristic
149	C₄	#15689566	1	2	yes	—
150	C₂	#20242748	1	1	yes	characteristic
151	C₂	#20245496	1	1	yes	characteristic
152	C₂	#20245513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245641

C₂

yes

maximal, characteristic

#20242758

C₂

yes

maximal

#20241462

C₂

yes

maximal, characteristic

#20241438

C₂

yes

maximal

C₄₇×#2183

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

C₂×Dic₃₇₆

—

#15689503

C₂²

yes

—

#15689408

—

#15688876

C₂²

yes

—

#15688870

—

C₈×Q₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₄₇×C₄⋊C₁₆

C₂²

yes

characteristic

C₄₇×#346

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

Dic₃₇₆

—

Dic₃₇₆

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

—

#13093948

—

C₄×Q₈×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

D₈

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

SD₁₆

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×SD₁₆

yes

characteristic

C₈×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2247

—

#2183

D₄₇

yes

—

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

#475

—

C₈×Q₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₄₇

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

#332

—

C₄₇

#12443

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₁₈₈

yes

characteristic

Q₃₂

—

188

—

Q₃₂

—

188

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

103

#106

—

188

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

D₁₈₈

yes

characteristic

107

#103

—

188

—

108

C₄×C₈

—

109

C₄×C₈

C₂²×D₄₇

yes

characteristic

110

C₂×Q₈

—

188

—

111

C₂×Q₈

D₃₇₆

yes

characteristic

112

Q₁₆

—

376

—

113

Q₁₆

—

188

—

114

Q₁₆

—

188

—

115

C₂×C₈

—

116

C₂×C₈

—

188

—

117

C₂×C₈

#10625350

yes

characteristic

118

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

188

—

121

C₄⋊C₄

—

188

—

122

C₄⋊C₄

D₃₇₆

yes

characteristic

123

C₄²

—

124

C₄²

C₂×D₁₈₈

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#10625336

yes

—

128

Q₈

—

129

Q₈

—

376

—

130

Q₈

—

376

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

376

—

134

C₂×C₄

#13093998

yes

characteristic

135

C₂×C₄

C₂×D₃₇₆

yes

characteristic

136

C₂×C₄

#13094046

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

376

—

140

C₈

C₂×#10625336

yes

characteristic

141

C₈

#13094049

yes

characteristic

142

C₂²

#15689359

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

752

—

147

C₄

#15689323

yes

characteristic

148

C₄

#15689393

yes

characteristic

149

C₄

#15689566

yes

—

150

C₂

#20242748

yes

characteristic

151

C₂

#20245496

yes

characteristic

152

C₂

#20245513

yes

characteristic

	subgroup	G/H
center	C₂²	#15689359
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2183	C₂
socle, prime radical	C₂²×C₄₇	#466
hypercenter	#2183	D₄₇
hypocenter	C₄₇	#12443
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#15689359

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2183

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
93. C₄₇	#12443	semidirect

H ≅ G/N

type

93. C₄₇

#12443

semidirect

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

subgroup	index
#20245641	2
#20242758	2
#20241462	2
#20241438	2
C₄₇×#2183	2
#12443	47

subgroup

index

C₄₇×#2183

Subgroups — #26130557

Conjugacy classes of subgroups

Complements

Maximal subgroups