Group #26103350 — subgroups

Subgroups — #26103350

Order 12032.

244 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20243438	C₂	1	2	yes	maximal
2	C₄₇×#2113	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093960	—	2	2	no	—
4	#15689107	—	2	2	no	—
5	C₄₇⋊#338	C₄	1	4	yes	—
6	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₄₇	C₄	1	1	yes	characteristic
9	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
10	C₂×#10625329	—	2	4	no	—
11	#13093960	—	4	2	no	—
12	#13093960	—	4	2	no	—
13	#13093943	—	4	2	no	—
14	#13093941	—	4	2	no	—
15	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
16	#131×C₄₇	—	4	1	no	—
17	#131×C₄₇	—	4	1	no	—
18	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₄₇	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	D₄	1	2	yes	—
22	M₃₂×C₄₇	—	4	1	no	—
23	C₂×C₁₆×C₄₇	—	2	1	no	—
24	C₂×C₁₆×C₄₇	—	2	1	no	—
25	C₂×C₁₆×C₄₇	—	4	1	no	—
26	#41×C₄₇	—	4	1	no	—
27	#41×C₄₇	—	4	1	no	—
28	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
29	C₂×D₄×C₄₇	SD₁₆	1	2	yes	—
30	C₂×D₄×C₄₇	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
33	M₁₆×C₄₇	—	4	2	no	—
34	M₁₆×C₄₇	—	4	1	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
42	#10625329	—	4	4	no	—
43	#10625329	—	4	4	no	—
44	C₂×C₄₇⋊C₈	—	4	4	no	—
45	C₁₆×C₄₇	—	4	1	no	—
46	C₁₆×C₄₇	—	4	1	no	—
47	C₂³×C₄₇	#103	1	2	yes	—
48	C₂³×C₄₇	#106	1	1	yes	characteristic
49	Q₈×C₄₇	—	4	1	no	—
50	D₄×C₄₇	—	4	2	no	—
51	D₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#102	1	2	yes	—
58	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
59	C₈×C₄₇	—	2	1	no	—
60	C₈×C₄₇	—	2	2	no	—
61	C₈×C₄₇	#99	1	1	yes	characteristic
62	C₈×C₄₇	#99	1	1	yes	characteristic
63	C₄₇⋊C₈	—	8	4	no	—
64	#7229	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	2	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	4	2	no	—
68	C₂²×C₄₇	—	4	1	no	—
69	C₂²×C₄₇	#340	1	1	yes	characteristic
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₄₇	#342	1	1	yes	characteristic
74	#2113	D₄₇	1	1	yes	characteristic
75	#1813	—	47	2	no	—
76	C₂×C₄₇	—	2	1	no	—
77	C₂×C₄₇	—	4	2	no	—
78	C₂×C₄₇	—	4	1	no	—
79	C₂×C₄₇	#1289	1	1	yes	characteristic
80	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
81	C₂×M₃₂	Dic₄₇	1	1	yes	characteristic
82	C₂²×C₁₆	Dic₄₇	1	1	yes	characteristic
83	C₂×#108	—	94	2	no	—
84	#413	—	94	2	no	—
85	C₂²⋊C₁₆	—	2	2	no	—
86	#338	—	47	4	no	—
87	C₄₇	#7229	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	D₁₈₈	1	1	yes	characteristic
89	#131	—	4	1	no	—
90	#131	—	4	1	no	—
91	C₂×M₁₆	—	94	4	no	—
92	C₂×M₁₆	Dic₉₄	1	1	yes	characteristic
93	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
94	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	188	2	no	—
101	#108	—	188	2	no	—
102	#101	—	188	2	no	—
103	#100	—	188	2	no	—
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
107	C₂×D₄	#10625325	1	2	yes	—
108	C₂×D₄	Dic₁₈₈	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
110	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	188	4	no	—
114	M₁₆	—	188	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	188	4	no	—
120	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
121	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#13093949	1	2	yes	—
126	C₂³	#13093956	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
135	C₂×C₄	#13093946	1	2	yes	—
136	C₂×C₄	#13093958	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	376	4	no	—
140	C₈	#13093939	1	1	yes	characteristic
141	C₈	#13093939	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#15688886	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	#15688880	1	1	yes	characteristic
150	C₄	#15688890	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	4	1	no	—
154	C₂	#20241256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243438

C₂

yes

maximal

C₄₇×#2113

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689107

—

C₄₇⋊#338

C₄

yes

—

C₂×#131×C₄₇

C₂²

yes

characteristic

C₂×M₃₂×C₄₇

C₄

yes

characteristic

C₂²×C₁₆×C₄₇

C₄

yes

characteristic

C₄₇×C₂²⋊C₁₆

—

—

—

—

—

—

C₂×#41×C₄₇

D₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×M₁₆×C₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

D₈

yes

characteristic

C₂×D₄×C₄₇

SD₁₆

yes

—

C₂×D₄×C₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₂³×C₄₇

#103

yes

—

C₂³×C₄₇

#106

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#107

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#7229

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1813

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1289

yes

characteristic

C₂×#131

C₂×D₄₇

yes

characteristic

C₂×M₃₂

Dic₄₇

yes

characteristic

C₂²×C₁₆

Dic₄₇

yes

characteristic

C₂×#108

—

#413

—

C₂²⋊C₁₆

—

#338

—

C₄₇

#7229

yes

characteristic, complemented, Sylow

C₂×#41

D₁₈₈

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₉₄

yes

characteristic

C₂×M₁₆

C₄×D₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂²⋊D₄₇

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

188

—

101

#108

—

188

—

102

#101

—

188

—

103

#100

—

188

—

104

#41

—

105

#41

—

106

C₂×Q₈

D₃₇₆

yes

characteristic

107

C₂×D₄

#10625325

yes

—

108

C₂×D₄

Dic₁₈₈

yes

characteristic

109

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

110

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

188

—

114

M₁₆

—

188

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

188

—

120

C₂×C₈

C₄×Dic₄₇

yes

characteristic

121

C₂×C₈

Dic₄₇⋊C₄

yes

—

122

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₂³

#13093949

yes

—

126

C₂³

#13093956

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₄₇⋊#95

yes

characteristic

135

C₂×C₄

#13093946

yes

—

136

C₂×C₄

#13093958

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

376

—

140

C₈

#13093939

yes

characteristic

141

C₈

#13093939

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#15688886

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#15688880

yes

characteristic

150

C₄

#15688890

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#20241256

yes

characteristic

	subgroup	G/H
center	C₄	#15688880
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×#131	C₂×D₄₇
Fitting	C₄₇×#2113	C₂
socle	C₂×C₄₇	#1289
prime radical	C₂×#41×C₄₇	D₄
cosocle	C₂×#131×C₄₇	C₂²
hypercenter	#2113	D₄₇
hypocenter	C₄₇	#7229
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#15688880

commutator

C₂×C₈×C₄₇

C₄²

Frattini

C₂×#131

C₂×D₄₇

Fitting

C₄₇×#2113

C₂

socle

C₂×C₄₇

#1289

prime radical

C₂×#41×C₄₇

D₄

cosocle

C₂×#131×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
87. C₄₇	#7229	semidirect

H ≅ G/N

type

87. C₄₇

#7229

semidirect

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

subgroup	index
#20243438	2
C₄₇×#2113	2
#7229	47

subgroup

index

#20243438

C₄₇×#2113

#7229

Subgroups — #26103350

Conjugacy classes of subgroups

Complements

Maximal subgroups