Group #26128146 — subgroups

Subgroups — #26128146

Order 12032.

236 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20245220	C₂	1	2	yes	maximal, complemented
2	#20241452	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	#20241306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	—	2	2	no	—
7	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
8	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
9	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×D₈×C₄₇	C₄	1	1	yes	characteristic
12	C₈×D₄×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#335	C₄	1	1	yes	characteristic
14	C₄₇×D₄⋊C₈	C₄	1	2	yes	—
15	C₈²×C₄₇	C₂²	1	1	yes	characteristic
16	#13093964	—	4	4	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	2	1	no	—
19	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
20	C₂×D₈×C₄₇	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	—	2	2	no	—
22	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
23	#107×C₄₇	C₈	1	1	yes	characteristic
24	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
25	#102×C₄₇	C₈	1	2	yes	—
26	C₂²⋊C₈×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	D₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	2	1	no	—
35	C₂×C₈×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
42	C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₄₇⋊C₁₆	—	2	2	no	—
45	C₄₇⋊C₁₆	—	4	1	no	—
46	C₄₇⋊C₁₆	—	4	2	no	—
47	C₂³×C₄₇	—	2	2	no	—
48	D₄×C₄₇	—	2	2	no	—
49	D₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	4	2	no	—
52	C₂×C₄×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
56	C₈×C₄₇	—	2	1	no	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	4	1	no	—
59	C₈×C₄₇	—	4	2	no	—
60	C₈×C₄₇	M₃₂	1	2	yes	complemented
61	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
62	#12064	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	4	2	no	—
64	C₂²×C₄₇	—	4	2	no	—
65	C₂²×C₄₇	—	4	2	no	—
66	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	4	2	no	—
69	C₄×C₄₇	#442	1	1	yes	characteristic
70	C₄×C₄₇	#444	1	1	yes	characteristic
71	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
72	C₄×C₄₇	#504	1	1	yes	characteristic
73	#2169	—	47	2	no	—
74	C₈×D₈	D₄₇	1	1	yes	characteristic
75	#1367	—	47	1	no	—
76	D₄⋊C₁₆	—	47	2	no	—
77	#1312	—	47	1	no	—
78	C₂×C₄₇	—	8	2	no	—
79	C₂×C₄₇	#1580	1	1	yes	characteristic
80	C₂×C₄₇	#2169	1	1	yes	characteristic
81	C₂×C₄₇	#2171	1	1	yes	characteristic
82	C₂×M₃₂	—	94	2	no	—
83	C₄×D₈	Dic₄₇	1	1	yes	characteristic
84	C₈×D₄	C₂×D₄₇	1	2	yes	—
85	C₄⋊C₁₆	—	47	2	no	—
86	C₂²⋊C₁₆	—	94	2	no	—
87	C₄×C₁₆	—	47	1	no	—
88	C₄×C₁₆	—	47	1	no	—
89	#335	Dic₄₇	1	1	yes	characteristic
90	D₄⋊C₈	Dic₄₇	1	2	yes	—
91	C₈²	C₂×D₄₇	1	1	yes	characteristic
92	C₄₇	#12064	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₈	C₄₇⋊C₈	1	1	yes	characteristic
94	C₂²×C₈	—	2	2	no	—
95	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
96	M₃₂	—	188	4	no	—
97	C₂×C₁₆	—	47	2	no	—
98	C₂×C₁₆	—	94	2	no	—
99	C₂×C₁₆	—	94	1	no	—
100	#107	C₄₇⋊C₈	1	1	yes	characteristic
101	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
102	#102	C₄₇⋊C₈	1	2	yes	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
107	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	D₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
115	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
116	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
117	C₂×C₈	#10625352	1	1	yes	characteristic
118	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
121	C₁₆	—	94	2	no	—
122	C₁₆	—	188	1	no	—
123	C₁₆	—	188	2	no	—
124	C₂³	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
132	C₂×C₄	#13094031	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	4	2	no	—
137	C₈	#13093964	1	2	yes	complemented
138	C₈	#13094048	1	2	yes	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	#15689235	1	1	yes	characteristic
146	C₄	#15689245	1	1	yes	characteristic
147	C₄	C₂×#13093964	1	1	yes	characteristic
148	C₄	#15689540	1	1	yes	characteristic
149	C₂	—	8	2	no	—
150	C₂	#20242212	1	1	yes	characteristic
151	C₂	#20245220	1	1	yes	characteristic
152	C₂	#20245230	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245220

C₂

yes

maximal, complemented

#20241452

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₇⋊C₁₆)

C₂

yes

maximal, complemented

#20241306

C₂

yes

maximal, characteristic, complemented

C₈×D₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093964

—

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₁₆)

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₄×D₈×C₄₇

C₄

yes

characteristic

C₈×D₄×C₄₇

C₂²

yes

—

C₄₇×#335

C₄

yes

characteristic

C₄₇×D₄⋊C₈

C₄

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×D₈×C₄₇

C₈

yes

characteristic

C₂²×C₈×C₄₇

—

C₄×D₄×C₄₇

C₂×C₄

yes

—

#107×C₄₇

C₈

yes

characteristic

C₄⋊C₈×C₄₇

C₂×C₄

yes

—

#102×C₄₇

C₈

yes

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

C₂×C₈

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

M₃₂

yes

complemented

C₈×C₄₇

AGL(1,ℤ/8)

yes

—

#12064

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×M₃₂

yes

C₄×C₄₇

yes

—

C₈×D₈

D₄₇

yes

characteristic

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×M₃₂

—

C₄×D₈

Dic₄₇

yes

characteristic

C₈×D₄

C₂×D₄₇

yes

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₄₇

yes

characteristic

D₄⋊C₈

Dic₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12064

yes

characteristic, complemented, Sylow

C₂×D₈

C₄₇⋊C₈

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂×Dic₄₇

yes

—

M₃₂

—

188

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#107

C₄₇⋊C₈

yes

characteristic

101

C₄⋊C₈

C₂×Dic₄₇

yes

—

102

#102

C₄₇⋊C₈

yes

—

103

C₂²⋊C₈

—

104

C₄×C₈

C₂×Dic₄₇

yes

characteristic

105

C₄×C₈

C₂×Dic₄₇

yes

characteristic

106

C₄×C₈

C₂²×D₄₇

yes

characteristic

107

C₂×D₄

C₂×C₄₇⋊C₈

yes

—

108

C₂²×C₄

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

115

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

116

C₂×C₈

D₄×D₄₇

yes

characteristic

117

C₂×C₈

#10625352

yes

characteristic

118

C₄⋊C₄

C₂×C₄₇⋊C₈

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×Dic₄₇

yes

characteristic

121

C₁₆

—

122

C₁₆

—

188

—

123

C₁₆

—

188

—

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

D₄×Dic₄₇

yes

characteristic

131

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

132

C₂×C₄

#13094031

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

#13093964

yes

complemented

138

C₈

#13094048

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×C₄₇⋊C₈

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#15689235

yes

characteristic

146

C₄

#15689245

yes

characteristic

147

C₄

C₂×#13093964

yes

characteristic

148

C₄

#15689540

yes

characteristic

149

C₂

—

150

C₂

#20242212

yes

characteristic

151

C₂

#20245220

yes

characteristic

152

C₂

#20245230

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×D₈×C₄₇	C₂
socle	C₂²×C₄₇	C₈×D₄
prime radical	C₂×D₈×C₄₇	C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₈×D₈	D₄₇
hypocenter	C₄₇	#12064
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

subgroup

G/H

center

C₂×C₈

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₈×D₈×C₄₇

C₂

socle

C₂²×C₄₇

C₈×D₄

prime radical

C₂×D₈×C₄₇

C₈

cosocle

C₄×C₈×C₄₇

C₂³

hypercenter

C₈×D₈

D₄₇

hypocenter

C₄₇

#12064

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

N	H ≅ G/N	type
1. #20245220	C₂	semidirect
2. #20241452	C₂	semidirect
3. D₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
4. #20241306	C₂	semidirect
60. C₈×C₄₇	M₃₂	semidirect
92. C₄₇	#12064	semidirect
137. C₈	#13093964	semidirect

H ≅ G/N

type

1. #20245220

C₂

semidirect

2. #20241452

C₂

semidirect

3. D₄⋊(C₄₇⋊C₁₆)

C₂

semidirect

4. #20241306

C₂

semidirect

60. C₈×C₄₇

M₃₂

semidirect

92. C₄₇

#12064

semidirect

137. C₈

#13093964

semidirect

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

subgroup	index
#20245220	2
#20241452	2
D₄⋊(C₄₇⋊C₁₆)	2
#20241306	2
C₈×D₈×C₄₇	2
#12064	47

subgroup

index

D₄⋊(C₄₇⋊C₁₆)

C₈×D₈×C₄₇

Subgroups — #26128146

Conjugacy classes of subgroups

Complements

Maximal subgroups