Group #26102870 — subgroups

Subgroups — #26102870

Order 12032.

292 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 131 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×#1842	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	—	2	2	no	—
7	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
8	C₄³×C₄₇	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
14	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	D₄	1	2	yes	—
17	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₄₇	—	4	2	no	—
19	C₄⋊C₈×C₄₇	—	4	2	no	—
20	C₄⋊C₈×C₄₇	—	4	2	no	—
21	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	4	2	no	—
27	C₂×C₈×C₄₇	—	4	4	no	—
28	C₄²×C₄₇	—	2	1	no	—
29	C₄²×C₄₇	—	2	1	no	—
30	C₄²×C₄₇	—	2	2	no	—
31	C₄²×C₄₇	—	4	2	no	—
32	C₄²×C₄₇	—	4	1	no	—
33	C₄²×C₄₇	D₈	1	2	yes	—
34	C₄²×C₄₇	SD₁₆	1	2	yes	—
35	C₄²×C₄₇	SD₁₆	1	2	yes	—
36	C₄²×C₄₇	Q₁₆	1	2	yes	—
37	C₂×C₄₇⋊C₈	—	4	4	no	—
38	C₂×C₄₇⋊C₈	—	4	4	no	—
39	C₂×C₄₇⋊C₈	—	4	4	no	—
40	C₂³×C₄₇	#95	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#102	1	2	yes	—
52	C₂×C₄×C₄₇	#102	1	2	yes	—
53	C₂×C₄×C₄₇	#103	1	2	yes	—
54	C₂×C₄×C₄₇	#103	1	2	yes	—
55	C₂×C₄×C₄₇	#106	1	2	yes	—
56	C₂×C₄×C₄₇	#107	1	2	yes	—
57	C₂×C₄×C₄₇	#108	1	2	yes	—
58	C₈×C₄₇	—	8	2	no	—
59	C₄₇⋊C₈	—	8	4	no	—
60	#7040	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
62	C₂²×C₄₇	#340	1	2	yes	—
63	C₂²×C₄₇	#341	1	1	yes	characteristic
64	C₂²×C₄₇	#343	1	1	yes	characteristic
65	C₂²×C₄₇	#344	1	2	yes	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	2	2	no	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	—	4	1	no	—
72	C₄×C₄₇	#331	1	1	yes	characteristic
73	C₄×C₄₇	#332	1	2	yes	—
74	C₄×C₄₇	#333	1	1	yes	characteristic
75	#1842	D₄₇	1	1	yes	characteristic
76	#1831	—	47	2	no	—
77	C₂×C₄₇	#1286	1	2	yes	—
78	C₂×C₄₇	#1290	1	1	yes	characteristic
79	C₂×C₄₇	#1296	1	1	yes	characteristic
80	C₂×C₄₇	#1301	1	2	yes	—
81	C₂×C₄₇	#1307	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	2	2	no	—
83	C₂×C₄⋊C₈	—	94	2	no	—
84	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
85	C₄³	C₂×D₄₇	1	1	yes	characteristic
86	#336	—	47	4	no	—
87	#336	—	94	2	no	—
88	C₄₇	#7040	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	94	4	no	—
91	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₄₇	1	2	yes	—
93	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	188	2	no	—
100	C₄⋊C₈	—	188	2	no	—
101	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
102	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
103	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
104	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
105	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	—	188	4	no	—
109	C₂×C₈	—	188	4	no	—
110	C₂×C₈	—	188	4	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	4	2	no	—
115	C₄²	—	4	1	no	—
116	C₄²	#10625334	1	2	yes	—
117	C₄²	#10625335	1	2	yes	—
118	C₄²	#10625336	1	2	yes	—
119	C₄²	#10625337	1	2	yes	—
120	C₂³	C₄₇⋊#95	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#13093941	1	1	yes	characteristic
130	C₂×C₄	#13093943	1	1	yes	characteristic
131	C₂×C₄	#13093944	1	2	yes	—
132	C₂×C₄	#13093945	1	2	yes	—
133	C₂×C₄	#13093947	1	2	yes	—
134	C₂×C₄	#13093948	1	2	yes	—
135	C₂×C₄	#13093955	1	2	yes	—
136	C₂×C₄	#13093957	1	2	yes	—
137	C₂×C₄	#13093960	1	2	yes	—
138	C₈	—	8	2	no	—
139	C₈	—	376	4	no	—
140	C₂²	#15688880	1	1	yes	characteristic
141	C₂²	#15688885	1	2	yes	—
142	C₂²	#15688888	1	1	yes	characteristic
143	C₂²	#15688892	1	1	yes	characteristic
144	C₂²	#15688895	1	2	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	1	no	—
151	C₄	#15688867	1	1	yes	characteristic
152	C₄	#15688869	1	2	yes	—
153	C₄	#15688872	1	1	yes	characteristic
154	C₂	#20241248	1	2	yes	—
155	C₂	#20241258	1	1	yes	characteristic
156	C₂	#20241271	1	1	yes	characteristic
157	C₂	#20241285	1	2	yes	—
158	C₂	#20241299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243505

C₂

yes

maximal

C₄₇×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

#15688878

—

#15688878

C₄

yes

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

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

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

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

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

#106

yes

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

#108

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

#331

yes

characteristic

C₄×C₄₇

#332

yes

—

C₄×C₄₇

#333

yes

characteristic

#1842

D₄₇

yes

characteristic

#1831

—

C₂×C₄₇

#1286

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1307

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

Dic₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

—

C₄₇

#7040

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

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

100

C₄⋊C₈

—

188

—

101

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

102

C₂²×C₄

Dic₄₇⋊C₄

yes

—

103

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

104

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

105

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

188

—

109

C₂×C₈

—

188

—

110

C₂×C₈

—

188

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

#10625334

yes

—

117

C₄²

#10625335

yes

—

118

C₄²

#10625336

yes

—

119

C₄²

#10625337

yes

—

120

C₂³

C₄₇⋊#95

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#13093941

yes

characteristic

130

C₂×C₄

#13093943

yes

characteristic

131

C₂×C₄

#13093944

yes

—

132

C₂×C₄

#13093945

yes

—

133

C₂×C₄

#13093947

yes

—

134

C₂×C₄

#13093948

yes

—

135

C₂×C₄

#13093955

yes

—

136

C₂×C₄

#13093957

yes

—

137

C₂×C₄

#13093960

yes

—

138

C₈

—

139

C₈

—

376

—

140

C₂²

#15688880

yes

characteristic

141

C₂²

#15688885

yes

—

142

C₂²

#15688888

yes

characteristic

143

C₂²

#15688892

yes

characteristic

144

C₂²

#15688895

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#15688867

yes

characteristic

152

C₄

#15688869

yes

—

153

C₄

#15688872

yes

characteristic

154

C₂

#20241248

yes

—

155

C₂

#20241258

yes

characteristic

156

C₂

#20241271

yes

characteristic

157

C₂

#20241285

yes

—

158

C₂

#20241299

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#7040
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1842

C₂

socle, prime radical

C₂³×C₄₇

#95

cosocle

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
88. C₄₇	#7040	semidirect

H ≅ G/N

type

88. C₄₇

#7040

semidirect

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

subgroup	index
#20243505	2
C₄₇×#1842	2
#7040	47

subgroup

index

#20243505

C₄₇×#1842

#7040

Subgroups — #26102870

Conjugacy classes of subgroups

Complements

Maximal subgroups