Group #26102812 — subgroups

Subgroups — #26102812

Order 12032.

294 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20243505	C₂	1	2	yes	maximal
2	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	#15688878	—	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)	—	4	2	no	—
11	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₄₇	—	2	2	no	—
13	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	D₄	1	2	yes	—
16	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
18	C₄×C₈×C₄₇	—	2	2	no	—
19	C₄×C₈×C₄₇	C₈	1	4	yes	complemented
20	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	—	2	4	no	—
26	C₂×C₈×C₄₇	—	4	4	no	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₄²×C₄₇	—	4	2	no	—
29	C₄²×C₄₇	—	4	2	no	—
30	C₄²×C₄₇	C₂×C₈	1	2	yes	—
31	C₄²×C₄₇	M₁₆	1	2	yes	—
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	—	4	4	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	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	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	2	no	—
46	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
49	C₂×C₄×C₄₇	#102	1	2	yes	—
50	C₂×C₄×C₄₇	#103	1	2	yes	—
51	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#108	1	2	yes	—
55	C₈×C₄₇	—	4	4	no	—
56	C₄₇⋊C₈	—	8	4	no	—
57	#7018	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	#336	1	1	yes	characteristic
59	C₂²×C₄₇	H(3,ℤ/4)	1	2	yes	—
60	C₂²×C₄₇	#340	1	1	yes	characteristic
61	C₂²×C₄₇	#341	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	—	2	2	no	—
68	C₄×C₄₇	—	4	2	no	—
69	C₄×C₄₇	#334	1	2	yes	—
70	C₄×C₄₇	#335	1	2	yes	—
71	#1831	—	47	2	no	—
72	#1744	D₄₇	1	1	yes	characteristic
73	C₂×C₄₇	#1274	1	1	yes	characteristic
74	C₂×C₄₇	#1277	1	1	yes	characteristic
75	C₂×C₄₇	#1281	1	1	yes	characteristic
76	C₂×C₄₇	#1286	1	2	yes	—
77	C₂×C₄₇	#1287	1	2	yes	—
78	C₂×C₄⋊C₈	—	94	2	no	—
79	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
80	C₄³	C₂×D₄₇	1	1	yes	characteristic
81	#336	—	2	2	no	—
82	#336	—	47	4	no	—
83	#336	—	94	2	no	—
84	C₄₇	#7018	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	—	188	2	no	—
93	C₄⋊C₈	—	188	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₄₇⋊C₈	1	4	yes	complemented
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	—	2	4	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	188	4	no	—
104	C₂×C₈	—	188	4	no	—
105	C₂×C₈	—	188	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	—	4	2	no	—
109	C₄²	#10625325	1	2	yes	—
110	C₄²	D₃₇₆	1	1	yes	characteristic
111	C₄²	Dic₁₈₈	1	1	yes	characteristic
112	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
113	C₄²	#10625329	1	2	yes	—
114	C₂³	C₄₇⋊#95	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
123	C₂×C₄	#13093935	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
125	C₂×C₄	#13093946	1	2	yes	—
126	C₂×C₄	#13093949	1	2	yes	—
127	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
128	C₂×C₄	#13093956	1	1	yes	characteristic
129	C₂×C₄	#13093958	1	1	yes	characteristic
130	C₂×C₄	#13093960	1	2	yes	—
131	C₈	—	4	4	no	—
132	C₈	—	376	4	no	—
133	C₂²	#15688878	1	1	yes	characteristic
134	C₂²	#15688879	1	2	yes	—
135	C₂²	#15688886	1	1	yes	characteristic
136	C₂²	#15688888	1	1	yes	characteristic
137	C₂²	#15688894	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	4	2	no	—
144	C₄	#15688874	1	2	yes	—
145	C₄	#15688876	1	2	yes	—
146	C₂	#20241217	1	1	yes	characteristic
147	C₂	#20241223	1	1	yes	characteristic
148	C₂	#20241233	1	1	yes	characteristic
149	C₂	#20241247	1	2	yes	—
150	C₂	#20241250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243505

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

#15688878

—

#15688878

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

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

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

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

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

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

#334

yes

—

C₄×C₄₇

#335

yes

—

#1831

—

#1744

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1287

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₄³

C₂×D₄₇

yes

—

—

—

C₄₇

#7018

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

—

188

—

C₄⋊C₈

—

188

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

complemented

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

—

188

—

104

C₂×C₈

—

188

—

105

C₂×C₈

—

188

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

#10625325

yes

—

110

C₄²

D₃₇₆

yes

characteristic

111

C₄²

Dic₁₈₈

yes

characteristic

112

C₄²

C₂×C₄₇⋊C₈

yes

—

113

C₄²

#10625329

yes

—

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

C₄×C₄₇⋊C₈

yes

characteristic

123

C₂×C₄

#13093935

yes

characteristic

124

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

125

C₂×C₄

#13093946

yes

—

126

C₂×C₄

#13093949

yes

—

127

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

128

C₂×C₄

#13093956

yes

characteristic

129

C₂×C₄

#13093958

yes

characteristic

130

C₂×C₄

#13093960

yes

—

131

C₈

—

132

C₈

—

376

—

133

C₂²

#15688878

yes

characteristic

134

C₂²

#15688879

yes

—

135

C₂²

#15688886

yes

characteristic

136

C₂²

#15688888

yes

characteristic

137

C₂²

#15688894

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#15688874

yes

—

145

C₄

#15688876

yes

—

146

C₂

#20241217

yes

characteristic

147

C₂

#20241223

yes

characteristic

148

C₂

#20241233

yes

characteristic

149

C₂

#20241247

yes

—

150

C₂

#20241250

yes

—

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#7018
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂×C₄×C₄₇

C₄×C₈

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

#95

cosocle

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
19. C₄×C₈×C₄₇	C₈	semidirect
84. C₄₇	#7018	semidirect
95. C₄×C₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

19. C₄×C₈×C₄₇

C₈

semidirect

84. C₄₇

#7018

semidirect

95. C₄×C₈

C₄₇⋊C₈

semidirect

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

subgroup	index
#20243505	2
C₄₇×#1744	2
#7018	47

subgroup

index

#20243505

C₄₇×#1744

#7018

Subgroups — #26102812

Conjugacy classes of subgroups

Complements

Maximal subgroups