Group #26137073 — subgroups

Subgroups — #26137073

Order 12032.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂×#15688963	C₂	1	2	yes	maximal
2	#20245641	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#20241568	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	1	no	—
7	C₂×Dic₃₇₆	C₂²	1	2	yes	—
8	#15689503	—	2	1	no	—
9	#15688963	—	2	4	no	—
10	C₂×C₄₇⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆×C₄₇	—	2	1	no	—
13	C₄₇×#366	C₂²	1	1	yes	characteristic
14	C₄₇×#358	—	2	1	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₈₈	—	2	2	no	—
17	C₂×Dic₁₈₈	—	4	1	no	—
18	#13094024	—	4	1	no	—
19	Dic₃₇₆	—	2	2	no	—
20	Dic₃₇₆	—	2	2	no	—
21	Dic₃₇₆	—	4	2	no	—
22	C₄₇⋊C₃₂	—	2	2	no	—
23	C₂×Q₁₆×C₄₇	—	2	1	no	—
24	C₄×Q₈×C₄₇	—	4	1	no	—
25	Q₃₂×C₄₇	—	2	1	no	—
26	Q₃₂×C₄₇	D₄	1	2	yes	—
27	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
29	#107×C₄₇	—	2	1	no	—
30	#103×C₄₇	—	4	1	no	—
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂×Q₈×C₄₇	—	4	1	no	—
33	C₂×Dic₉₄	—	4	2	no	—
34	Q₁₆×C₄₇	—	2	2	no	—
35	Q₁₆×C₄₇	—	4	1	no	—
36	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₄⋊C₄×C₄₇	—	8	1	no	—
40	C₄²×C₄₇	—	8	1	no	—
41	C₄²×C₄₇	D₈	1	1	yes	characteristic
42	C₄⋊Dic₄₇	—	8	1	no	—
43	C₄⋊Dic₄₇	—	8	1	no	—
44	Dic₁₈₈	—	4	2	no	—
45	Dic₁₈₈	—	4	2	no	—
46	Dic₁₈₈	—	8	2	no	—
47	C₁₆×C₄₇	—	2	1	no	—
48	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₄₇	#41	1	1	yes	characteristic
50	Q₈×C₄₇	—	4	2	no	—
51	Q₈×C₄₇	—	8	1	no	—
52	C₂×C₄×C₄₇	—	8	1	no	—
53	C₂×C₄×C₄₇	—	8	1	no	—
54	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₈	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	#121	1	1	yes	characteristic
62	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	#13688	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	8	2	no	—
67	C₄×C₄₇	—	16	1	no	—
68	C₄×C₄₇	Q₆₄	1	2	yes	complemented
69	C₄×C₄₇	#459	1	1	yes	characteristic
70	C₄×C₄₇	#509	1	1	yes	characteristic
71	Dic₄₇	—	16	2	no	—
72	C₂×Q₆₄	—	47	2	no	—
73	#2247	—	47	1	no	—
74	C₄×Q₃₂	D₄₇	1	1	yes	characteristic
75	C₄⋊C₃₂	—	47	1	no	—
76	#1416	—	47	2	no	—
77	C₂×C₄₇	#2206	1	1	yes	characteristic
78	C₂×C₄₇	C₂×Q₆₄	1	1	yes	characteristic
79	C₂×C₄₇	#2264	1	1	yes	characteristic
80	C₂×Q₃₂	—	47	2	no	—
81	C₂×Q₃₂	—	94	1	no	—
82	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
83	#494	—	94	1	no	—
84	C₄×Q₁₆	—	2	1	no	—
85	Q₆₄	—	94	4	no	—
86	C₂×C₃₂	—	47	2	no	—
87	#366	C₂×D₄₇	1	1	yes	characteristic
88	#358	—	2	1	no	—
89	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
90	C₄₇	#13688	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	94	2	no	—
93	C₂×Q₁₆	—	188	1	no	—
94	C₄⋊Q₈	—	188	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	Q₃₂	—	2	1	no	—
97	Q₃₂	—	94	2	no	—
98	Q₃₂	—	94	2	no	—
99	Q₃₂	—	188	2	no	—
100	Q₃₂	D₁₈₈	1	2	yes	—
101	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#103	—	4	1	no	—
105	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₃₂	—	94	2	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	188	2	no	—
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	4	1	no	—
111	Q₁₆	—	376	2	no	—
112	Q₁₆	—	188	2	no	—
113	Q₁₆	—	188	2	no	—
114	C₂×C₈	#10625334	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	376	1	no	—
119	C₄⋊C₄	—	376	1	no	—
120	C₄²	—	8	1	no	—
121	C₄²	#10625334	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
124	C₁₆	#10625350	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	Q₈	—	376	2	no	—
128	Q₈	—	376	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	376	2	no	—
132	C₂×C₄	#13093965	1	1	yes	characteristic
133	C₂×C₄	#13093965	1	1	yes	characteristic
134	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#13093998	1	1	yes	characteristic
137	C₈	#13094049	1	1	yes	characteristic
138	C₂²	C₂×#13093965	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	752	2	no	—
143	C₄	#15688963	1	2	yes	complemented
144	C₄	#15689318	1	1	yes	characteristic
145	C₄	#15689558	1	1	yes	characteristic
146	C₂	#20245423	1	1	yes	characteristic
147	C₂	C₂×#15688963	1	1	yes	characteristic
148	C₂	#20245705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688963

C₂

yes

maximal

#20245641

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₃₂)

C₂

yes

maximal, characteristic

#20241568

C₂

yes

maximal

C₄×Q₃₂×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

C₂×Dic₃₇₆

C₂²

yes

—

#15689503

—

#15688963

—

C₂×C₄₇⋊C₃₂

C₂²

yes

—

C₂×Q₃₂×C₄₇

C₂²

yes

characteristic

C₄×Q₁₆×C₄₇

—

C₄₇×#366

C₂²

yes

characteristic

C₄₇×#358

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

Dic₃₇₆

—

Dic₃₇₆

—

Dic₃₇₆

—

C₄₇⋊C₃₂

—

C₂×Q₁₆×C₄₇

—

C₄×Q₈×C₄₇

—

Q₃₂×C₄₇

—

Q₃₂×C₄₇

D₄

yes

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×D₄

yes

characteristic

C₁₆×C₄₇

#41

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

Q₆₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

D₄₇

yes

characteristic

C₄⋊C₃₂

—

#1416

—

C₂×C₄₇

#2206

yes

characteristic

C₂×C₄₇

C₂×Q₆₄

yes

characteristic

C₂×C₄₇

#2264

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₄₇

yes

characteristic

#494

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

—

#366

C₂×D₄₇

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13688

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

188

—

C₄⋊Q₈

—

188

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

188

—

100

Q₃₂

D₁₈₈

yes

—

101

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

102

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

103

#107

—

104

#103

—

105

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

106

C₃₂

—

107

C₂×Q₈

—

108

C₂×Q₈

—

188

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

376

—

112

Q₁₆

—

188

—

113

Q₁₆

—

188

—

114

C₂×C₈

#10625334

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

376

—

119

C₄⋊C₄

—

376

—

120

C₄²

—

121

C₄²

#10625334

yes

characteristic

122

C₁₆

—

123

C₁₆

C₂×D₁₈₈

yes

characteristic

124

C₁₆

#10625350

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

376

—

128

Q₈

—

376

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

376

—

132

C₂×C₄

#13093965

yes

characteristic

133

C₂×C₄

#13093965

yes

characteristic

134

C₂×C₄

C₂×#10625334

yes

characteristic

135

C₈

—

136

C₈

#13093998

yes

characteristic

137

C₈

#13094049

yes

characteristic

138

C₂²

C₂×#13093965

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

752

—

143

C₄

#15688963

yes

complemented

144

C₄

#15689318

yes

characteristic

145

C₄

#15689558

yes

characteristic

146

C₂

#20245423

yes

characteristic

147

C₂

C₂×#15688963

yes

characteristic

148

C₂

#20245705

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13093965
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄×Q₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₄₇
hypocenter	C₄₇	#13688
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#13093965

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄×Q₃₂×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₄₇

hypocenter

C₄₇

#13688

LCS

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
68. C₄×C₄₇	Q₆₄	semidirect
90. C₄₇	#13688	semidirect
143. C₄	#15688963	semidirect

H ≅ G/N

type

68. C₄×C₄₇

Q₆₄

semidirect

90. C₄₇

#13688

semidirect

143. C₄

#15688963

semidirect

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

subgroup	index
C₂×#15688963	2
#20245641	2
C₄⋊(C₄₇⋊C₃₂)	2
#20241568	2
C₄×Q₃₂×C₄₇	2
#13688	47

subgroup

index

C₄⋊(C₄₇⋊C₃₂)

C₄×Q₃₂×C₄₇

Subgroups — #26137073

Conjugacy classes of subgroups

Complements

Maximal subgroups