Group #26102889 — subgroups

Subgroups — #26102889

Order 12032.

304 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20243241	C₂	1	2	yes	maximal
2	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂×#13093935	—	2	2	no	—
5	#15688878	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
7	C₄³×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
11	#13093935	—	4	2	no	—
12	#13093935	—	4	2	no	—
13	C₂²×C₈×C₄₇	—	2	2	no	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₄₇	D₄	1	2	yes	—
17	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	—	4	2	no	—
20	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	—	4	2	no	—
26	C₂×C₈×C₄₇	—	4	4	no	—
27	C₄²×C₄₇	—	2	1	no	—
28	C₄²×C₄₇	—	2	1	no	—
29	C₄²×C₄₇	—	2	4	no	—
30	C₄²×C₄₇	—	4	2	no	—
31	C₄²×C₄₇	—	4	1	no	—
32	C₄²×C₄₇	D₈	1	1	yes	characteristic
33	C₄²×C₄₇	SD₁₆	1	2	yes	—
34	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
35	C₂×C₄₇⋊C₈	—	2	8	no	—
36	C₂×C₄₇⋊C₈	—	4	4	no	—
37	C₂×C₄₇⋊C₈	—	4	4	no	—
38	C₂³×C₄₇	#95	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#102	1	2	yes	—
51	C₂×C₄×C₄₇	#103	1	2	yes	—
52	C₂×C₄×C₄₇	#104	1	4	yes	—
53	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
55	C₈×C₄₇	—	8	2	no	—
56	C₄₇⋊C₈	—	4	8	no	—
57	#7049	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	#338	1	1	yes	characteristic
59	C₂²×C₄₇	#339	1	2	yes	—
60	C₂²×C₄₇	#340	1	1	yes	characteristic
61	C₂²×C₄₇	#343	1	1	yes	characteristic
62	C₂²×C₄₇	#344	1	2	yes	—
63	C₄×C₄₇	—	2	1	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	4	2	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	#329	1	2	yes	—
70	C₄×C₄₇	#330	1	2	yes	—
71	#1831	D₄₇	1	1	yes	characteristic
72	#1767	—	47	2	no	—
73	C₂×C₄₇	#1291	1	2	yes	—
74	C₂×C₄₇	#1292	1	1	yes	characteristic
75	C₂×C₄₇	#1295	1	1	yes	characteristic
76	C₂×C₄₇	#1303	1	2	yes	—
77	C₂×C₄₇	#1307	1	1	yes	characteristic
78	C₂×C₄⋊C₈	—	94	2	no	—
79	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
80	C₂×#97	—	94	2	no	—
81	C₄³	C₂×D₄₇	1	1	yes	characteristic
82	#336	—	2	2	no	—
83	#336	—	47	4	no	—
84	C₄₇	#7049	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	94	4	no	—
87	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₄₇	1	2	yes	—
89	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
90	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	94	8	no	—
94	#97	—	188	2	no	—
95	#97	—	188	2	no	—
96	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
97	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
98	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
99	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
100	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	94	8	no	—
104	C₂×C₈	—	188	4	no	—
105	C₂×C₈	—	188	4	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	4	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	#10625325	1	2	yes	—
112	C₄²	D₃₇₆	1	1	yes	characteristic
113	C₄²	Dic₁₈₈	1	1	yes	characteristic
114	C₂³	C₄₇⋊#95	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	#13093941	1	1	yes	characteristic
125	C₂×C₄	#13093943	1	1	yes	characteristic
126	C₂×C₄	#13093946	1	2	yes	—
127	C₂×C₄	#13093949	1	2	yes	—
128	C₂×C₄	#13093951	1	4	yes	—
129	C₂×C₄	#13093956	1	1	yes	characteristic
130	C₂×C₄	#13093958	1	1	yes	characteristic
131	C₈	—	8	2	no	—
132	C₈	—	188	8	no	—
133	C₂²	C₄₇⋊#338	1	1	yes	characteristic
134	C₂²	#15688882	1	2	yes	—
135	C₂²	#15688886	1	1	yes	characteristic
136	C₂²	#15688892	1	1	yes	characteristic
137	C₂²	#15688895	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	#15688863	1	2	yes	—
145	C₄	#15688865	1	2	yes	—
146	C₂	#20241259	1	2	yes	—
147	C₂	#20241263	1	1	yes	characteristic
148	C₂	#20241269	1	1	yes	characteristic
149	C₂	#20241290	1	2	yes	—
150	C₂	#20241299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243241

C₂

yes

maximal

C₄₇×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×#13093935

—

#15688878

C₄

yes

—

C₂×C₄⋊C₈×C₄₇

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#329

yes

—

C₄×C₄₇

#330

yes

—

#1831

D₄₇

yes

characteristic

#1767

—

C₂×C₄₇

#1291

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1307

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₄₇

yes

—

C₂×#97

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

—

C₄₇

#7049

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₉₄

yes

characteristic

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

188

—

#97

—

188

—

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂²×C₄

Dic₄₇⋊C₄

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

100

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

188

—

105

C₂×C₈

—

188

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

#10625325

yes

—

112

C₄²

D₃₇₆

yes

characteristic

113

C₄²

Dic₁₈₈

yes

characteristic

114

C₂³

C₄₇⋊#95

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#13093941

yes

characteristic

125

C₂×C₄

#13093943

yes

characteristic

126

C₂×C₄

#13093946

yes

—

127

C₂×C₄

#13093949

yes

—

128

C₂×C₄

#13093951

yes

—

129

C₂×C₄

#13093956

yes

characteristic

130

C₂×C₄

#13093958

yes

characteristic

131

C₈

—

132

C₈

—

188

—

133

C₂²

C₄₇⋊#338

yes

characteristic

134

C₂²

#15688882

yes

—

135

C₂²

#15688886

yes

characteristic

136

C₂²

#15688892

yes

characteristic

137

C₂²

#15688895

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#15688863

yes

—

145

C₄

#15688865

yes

—

146

C₂

#20241259

yes

—

147

C₂

#20241263

yes

characteristic

148

C₂

#20241269

yes

characteristic

149

C₂

#20241290

yes

—

150

C₂

#20241299

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#7049
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1831

C₂

socle, prime radical

C₂³×C₄₇

#95

cosocle

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
84. C₄₇	#7049	semidirect

H ≅ G/N

type

84. C₄₇

#7049

semidirect

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

subgroup	index
#20243241	2
C₄₇×#1831	2
#7049	47

subgroup

index

#20243241

C₄₇×#1831

#7049

Subgroups — #26102889

Conjugacy classes of subgroups

Complements

Maximal subgroups