Group #26103505 — subgroups

Subgroups — #26103505

Order 12032.

184 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20242151	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2162	C₂	1	1	yes	maximal, characteristic
4	#15689180	—	2	1	no	—
5	#15689176	—	2	1	no	—
6	#15688876	—	2	1	no	—
7	#15688876	C₄	1	2	yes	—
8	#15688874	—	2	1	no	—
9	#15688874	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×C₄⋊C₁₆	—	2	1	no	—
12	C₄₇×C₂²⋊C₁₆	C₄	1	2	yes	—
13	C₄×C₁₆×C₄₇	—	2	1	no	—
14	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
16	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
18	#13093935	—	4	1	no	—
19	C₄×C₄₇⋊C₈	—	4	1	no	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	—	2	2	no	—
24	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	4	1	no	—
33	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₄₇	D₈	1	1	yes	characteristic
40	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	—	4	2	no	—
42	C₂×C₄₇⋊C₈	—	4	2	no	—
43	C₁₆×C₄₇	—	4	2	no	—
44	C₂³×C₄₇	#106	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
55	C₈×C₄₇	—	2	1	no	—
56	C₈×C₄₇	—	2	1	no	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	#108	1	2	yes	—
59	C₄₇⋊C₈	—	8	2	no	—
60	C₄₇⋊C₈	—	8	2	no	—
61	#7289	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	#340	1	1	yes	characteristic
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	#341	1	1	yes	characteristic
71	C₄×C₄₇	#344	1	1	yes	characteristic
72	C₄×C₄₇	#359	1	2	yes	—
73	#2162	D₄₇	1	1	yes	characteristic
74	#1568	—	47	1	no	—
75	#1564	—	47	1	no	—
76	C₂×C₄₇	—	4	1	no	—
77	C₂×C₄₇	#1301	1	1	yes	characteristic
78	C₂×C₄₇	#1382	1	1	yes	characteristic
79	C₂×C₄₇	#1389	1	1	yes	characteristic
80	#432	—	94	1	no	—
81	#431	—	94	1	no	—
82	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
83	C₄⋊C₁₆	—	2	1	no	—
84	C₂²⋊C₁₆	Dic₄₇	1	2	yes	—
85	C₄×C₁₆	—	2	1	no	—
86	#335	—	47	2	no	—
87	#335	—	94	1	no	—
88	#334	—	47	2	no	—
89	#334	—	94	1	no	—
90	C₄₇	#7289	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
93	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
94	C₂×C₁₆	—	2	2	no	—
95	C₄⋊C₈	—	94	2	no	—
96	C₄⋊C₈	—	94	2	no	—
97	C₂²⋊C₈	—	188	1	no	—
98	C₂²⋊C₈	—	188	1	no	—
99	#97	—	188	1	no	—
100	C₄×C₈	—	188	1	no	—
101	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
102	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂²×C₄	#10625325	1	1	yes	characteristic
106	C₂²×C₄	#10625325	1	1	yes	characteristic
107	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	188	2	no	—
111	C₂×C₈	—	188	2	no	—
112	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
113	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
114	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
116	C₄²	D₃₇₆	1	1	yes	characteristic
117	C₄²	Dic₁₈₈	1	1	yes	characteristic
118	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
119	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
120	C₁₆	—	4	2	no	—
121	C₂³	#13093956	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
127	C₂×C₄	#13093946	1	1	yes	characteristic
128	C₂×C₄	#13093946	1	1	yes	characteristic
129	C₂×C₄	#13093949	1	1	yes	characteristic
130	C₂×C₄	#13093949	1	1	yes	characteristic
131	C₂×C₄	#13093958	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	376	2	no	—
136	C₈	—	376	2	no	—
137	C₈	#13093959	1	2	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#15688886	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	#15688887	1	1	yes	characteristic
147	C₄	#15688893	1	1	yes	characteristic
148	C₄	#15688931	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#20241284	1	1	yes	characteristic
151	C₂	#20241484	1	1	yes	characteristic
152	C₂	#20241503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal, characteristic

#20242136

C₂

yes

maximal, characteristic

C₄₇×#2162

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#15688874

—

#15688874

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₄₇×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

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

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

Q₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#106

yes

characteristic

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

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#108

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#7289

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#2162

D₄₇

yes

characteristic

#1568

—

#1564

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₄₇

yes

—

C₄×C₁₆

—

—

—

—

—

C₄₇

#7289

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

188

—

C₂²⋊C₈

—

188

—

#97

—

188

—

100

C₄×C₈

—

188

—

101

C₄×C₈

C₄×D₄₇

yes

characteristic

102

C₄×C₈

C₄×D₄₇

yes

characteristic

103

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

104

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

105

C₂²×C₄

#10625325

yes

characteristic

106

C₂²×C₄

#10625325

yes

characteristic

107

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

188

—

111

C₂×C₈

—

188

—

112

C₂×C₈

C₄×Dic₄₇

yes

characteristic

113

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

114

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

115

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

116

C₄²

D₃₇₆

yes

characteristic

117

C₄²

Dic₁₈₈

yes

characteristic

118

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

119

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

120

C₁₆

—

121

C₂³

#13093956

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄₇⋊#95

yes

characteristic

127

C₂×C₄

#13093946

yes

characteristic

128

C₂×C₄

#13093946

yes

characteristic

129

C₂×C₄

#13093949

yes

characteristic

130

C₂×C₄

#13093949

yes

characteristic

131

C₂×C₄

#13093958

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

376

—

136

C₈

—

376

—

137

C₈

#13093959

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#15688886

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#15688887

yes

characteristic

147

C₄

#15688893

yes

characteristic

148

C₄

#15688931

yes

—

149

C₂

—

150

C₂

#20241284

yes

characteristic

151

C₂

#20241484

yes

characteristic

152

C₂

#20241503

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093946
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#2162	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#106
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#2162	D₄₇
hypocenter	C₄₇	#7289
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₂×C₄

#13093946

commutator

C₂×C₈×C₄₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#2162

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
90. C₄₇	#7289	semidirect

H ≅ G/N

type

90. C₄₇

#7289

semidirect

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

subgroup	index
#20242151	2
#20242136	2
C₄₇×#2162	2
#7289	47

subgroup

index

#20242151

#20242136

C₄₇×#2162

#7289

Subgroups — #26103505

Conjugacy classes of subgroups

Complements

Maximal subgroups