Group #26130540 — subgroups

Subgroups — #26130540

Order 12032.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20241460	C₂	1	1	yes	maximal, characteristic
4	#20241438	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₄₇	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	C₄×C₁₆×C₄₇	—	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	1	no	—
52	Q₈×C₄₇	Q₃₂	1	2	yes	complemented
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
58	C₂×Dic₄₇	—	8	2	no	—
59	Dic₉₄	—	8	2	no	—
60	Dic₉₄	—	8	2	no	—
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	—	4	1	no	—
63	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄₇⋊C₈	—	8	2	no	—
66	#12440	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	#466	1	1	yes	characteristic
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	4	1	no	—
71	C₄×C₄₇	#460	1	1	yes	characteristic
72	C₄×C₄₇	#472	1	1	yes	characteristic
73	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₄₇	#508	1	1	yes	characteristic
75	Dic₄₇	—	16	2	no	—
76	#2247	—	47	1	no	—
77	Q₈×C₁₆	D₄₇	1	1	yes	characteristic
78	#1670	—	47	2	no	—
79	#1371	—	47	1	no	—
80	#1362	—	47	2	no	—
81	C₂×C₄₇	#1668	1	1	yes	characteristic
82	C₂×C₄₇	#2216	1	1	yes	characteristic
83	C₂×C₄₇	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	94	1	no	—
85	C₂×Q₃₂	—	94	1	no	—
86	#494	—	47	2	no	—
87	#475	—	94	2	no	—
88	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
93	#335	—	47	2	no	—
94	#332	—	94	2	no	—
95	C₄₇	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	94	2	no	—
97	C₂×Q₁₆	—	94	2	no	—
98	C₄⋊Q₈	—	94	2	no	—
99	C₄×Q₈	D₁₈₈	1	1	yes	characteristic
100	Q₃₂	—	188	2	no	—
101	Q₃₂	—	188	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
105	#106	—	188	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	94	2	no	—
108	C₄⋊C₈	D₁₈₈	1	1	yes	characteristic
109	#103	—	188	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
112	C₂×Q₈	—	188	2	no	—
113	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
114	Q₁₆	—	376	2	no	—
115	Q₁₆	—	188	2	no	—
116	Q₁₆	—	188	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	188	2	no	—
119	C₂×C₈	#10625350	1	1	yes	characteristic
120	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	188	2	no	—
123	C₄⋊C₄	—	188	2	no	—
124	C₄⋊C₄	D₃₇₆	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#10625336	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	376	2	no	—
132	Q₈	—	376	2	no	—
133	Q₈	Dic₃₇₆	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	376	2	no	—
137	C₂×C₄	#13093998	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
139	C₂×C₄	#13094046	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	376	2	no	—
143	C₈	C₂×#10625336	1	1	yes	characteristic
144	C₈	#13094049	1	1	yes	characteristic
145	C₂²	#15689359	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	752	2	no	—
150	C₄	#15689323	1	1	yes	characteristic
151	C₄	#15689393	1	1	yes	characteristic
152	C₄	C₂×Dic₃₇₆	1	1	yes	characteristic
153	C₄	#15689555	1	1	yes	characteristic
154	C₂	#20242748	1	1	yes	characteristic
155	C₂	#20245479	1	1	yes	characteristic
156	C₂	#20245513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245641

C₂

yes

maximal, characteristic

#20242758

C₂

yes

maximal

#20241460

C₂

yes

maximal, characteristic

#20241438

C₂

yes

maximal

Q₈×C₁₆×C₄₇

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

C₄×C₁₆×C₄₇

—

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

—

Q₈×C₄₇

Q₃₂

yes

complemented

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

C₂×Q₃₂

yes

characteristic

C₄×C₄₇

#508

yes

characteristic

Dic₄₇

—

#2247

—

Q₈×C₁₆

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

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

#332

—

C₄₇

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₁₈₈

yes

characteristic

100

Q₃₂

—

188

—

101

Q₃₂

—

188

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

105

#106

—

188

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

D₁₈₈

yes

characteristic

109

#103

—

188

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₄₇

yes

characteristic

112

C₂×Q₈

—

188

—

113

C₂×Q₈

D₃₇₆

yes

characteristic

114

Q₁₆

—

376

—

115

Q₁₆

—

188

—

116

Q₁₆

—

188

—

117

C₂×C₈

—

118

C₂×C₈

—

188

—

119

C₂×C₈

#10625350

yes

characteristic

120

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

188

—

123

C₄⋊C₄

—

188

—

124

C₄⋊C₄

D₃₇₆

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₁₈₈

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#10625336

yes

—

130

Q₈

—

131

Q₈

—

376

—

132

Q₈

—

376

—

133

Q₈

Dic₃₇₆

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

376

—

137

C₂×C₄

#13093998

yes

characteristic

138

C₂×C₄

C₂×D₃₇₆

yes

characteristic

139

C₂×C₄

#13094046

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

376

—

143

C₈

C₂×#10625336

yes

characteristic

144

C₈

#13094049

yes

characteristic

145

C₂²

#15689359

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

752

—

150

C₄

#15689323

yes

characteristic

151

C₄

#15689393

yes

characteristic

152

C₄

C₂×Dic₃₇₆

yes

characteristic

153

C₄

#15689555

yes

characteristic

154

C₂

#20242748

yes

characteristic

155

C₂

#20245479

yes

characteristic

156

C₂

#20245513

yes

characteristic

	subgroup	G/H
center	C₂²	#15689359
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	Q₈×C₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	#466
hypercenter	Q₈×C₁₆	D₄₇
hypocenter	C₄₇	#12440
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#15689359

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

Q₈×C₁₆×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

#466

hypercenter

Q₈×C₁₆

D₄₇

hypocenter

C₄₇

#12440

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
52. Q₈×C₄₇	Q₃₂	semidirect
95. C₄₇	#12440	semidirect
133. Q₈	Dic₃₇₆	semidirect

H ≅ G/N

type

52. Q₈×C₄₇

Q₃₂

semidirect

95. C₄₇

#12440

semidirect

133. Q₈

Dic₃₇₆

semidirect

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

subgroup	index
#20245641	2
#20242758	2
#20241460	2
#20241438	2
Q₈×C₁₆×C₄₇	2
#12440	47

subgroup

index

Q₈×C₁₆×C₄₇

Subgroups — #26130540

Conjugacy classes of subgroups

Complements

Maximal subgroups