Group #26103528 — subgroups

Subgroups — #26103528

Order 12032.

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

normal subgroups only

This group has 208 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	#20245187	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1568	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093960	—	2	1	no	—
5	C₄×#10625329	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄×C₄₇⋊C₁₆	—	2	1	no	—
9	#15688888	C₄	1	2	yes	—
10	C₄₇×#432	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#335	C₄	1	2	yes	—
13	C₄₇×#334	—	2	1	no	—
14	C₂×#10625329	—	2	2	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	#13093960	—	2	4	no	—
17	#13093935	—	4	1	no	—
18	C₄×C₄₇⋊C₈	—	4	1	no	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	—	2	2	no	—
23	C₂²⋊C₈×C₄₇	—	4	1	no	—
24	#97×C₄₇	—	4	1	no	—
25	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	4	1	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₄₇	D₈	1	1	yes	characteristic
42	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
43	#10625329	—	2	4	no	—
44	#10625329	—	4	2	no	—
45	C₂×C₄₇⋊C₈	—	4	2	no	—
46	C₄₇⋊C₁₆	—	4	2	no	—
47	C₂³×C₄₇	#103	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
58	C₈×C₄₇	—	2	1	no	—
59	C₈×C₄₇	—	2	1	no	—
60	C₈×C₄₇	—	2	1	no	—
61	C₈×C₄₇	—	8	2	no	—
62	C₈×C₄₇	#104	1	2	yes	—
63	C₄₇⋊C₈	—	4	4	no	—
64	#7297	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	#340	1	1	yes	characteristic
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	#339	1	1	yes	characteristic
74	C₄×C₄₇	#344	1	1	yes	characteristic
75	C₄×C₄₇	#359	1	2	yes	—
76	#2162	—	47	1	no	—
77	C₄×#108	—	47	1	no	—
78	#1568	D₄₇	1	1	yes	characteristic
79	C₂×C₄₇	—	4	1	no	—
80	C₂×C₄₇	#1303	1	1	yes	characteristic
81	C₂×C₄₇	#1382	1	1	yes	characteristic
82	C₂×C₄₇	#1388	1	1	yes	characteristic
83	#432	—	2	1	no	—
84	C₂×#108	—	94	1	no	—
85	C₄×M₁₆	—	94	1	no	—
86	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	47	2	no	—
88	C₂²⋊C₁₆	—	94	1	no	—
89	C₄×C₁₆	—	94	1	no	—
90	#341	—	47	2	no	—
91	#335	Dic₄₇	1	2	yes	—
92	#334	—	2	1	no	—
93	C₄₇	#7297	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₁₆	—	94	2	no	—
95	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
96	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
98	C₂×C₁₆	—	94	2	no	—
99	#108	—	94	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	#97	—	4	1	no	—
103	#97	—	188	1	no	—
104	C₄×C₈	—	188	1	no	—
105	C₄×C₈	Dic₉₄	1	1	yes	characteristic
106	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
109	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	#10625335	1	1	yes	characteristic
111	C₂²×C₄	#10625337	1	1	yes	characteristic
112	M₁₆	—	94	4	no	—
113	M₁₆	—	188	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	188	2	no	—
118	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
119	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
121	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
122	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
123	C₄²	#10625334	1	1	yes	characteristic
124	C₄²	#10625336	1	1	yes	characteristic
125	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
126	C₁₆	—	188	2	no	—
127	C₂³	#13093947	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
133	C₂×C₄	#13093944	1	1	yes	characteristic
134	C₂×C₄	#13093945	1	1	yes	characteristic
135	C₂×C₄	#13093948	1	1	yes	characteristic
136	C₂×C₄	#13093955	1	1	yes	characteristic
137	C₂×C₄	#13093957	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	188	4	no	—
143	C₈	#13093950	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#15688885	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#15688884	1	1	yes	characteristic
153	C₄	#15688894	1	1	yes	characteristic
154	C₄	#15688930	1	2	yes	—
155	C₂	—	4	1	no	—
156	C₂	#20241289	1	1	yes	characteristic
157	C₂	#20241483	1	1	yes	characteristic
158	C₂	#20241499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245187

C₂

yes

maximal, characteristic

C₄×#13093960

C₂

yes

maximal, characteristic

C₄₇×#1568

C₂

yes

maximal, characteristic

C₂×#13093960

—

C₄×#10625329

—

C₄⋊(C₄₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₇⋊C₁₆)

—

C₄×C₄₇⋊C₁₆

—

#15688888

C₄

yes

—

C₄₇×#432

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

C₄

yes

—

C₄₇×#334

—

C₂×#10625329

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

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

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#103

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

—

C₈×C₄₇

#104

yes

—

C₄₇⋊C₈

—

#7297

—

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

—

C₄×#108

—

#1568

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

—

#335

Dic₄₇

yes

—

#334

—

C₄₇

#7297

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

D₁₈₈

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

—

#108

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#97

—

188

—

104

C₄×C₈

—

188

—

105

C₄×C₈

Dic₉₄

yes

characteristic

106

C₄×C₈

C₄×D₄₇

yes

characteristic

107

C₄×C₈

C₂×Dic₄₇

yes

characteristic

108

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

109

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

110

C₂²×C₄

#10625335

yes

characteristic

111

C₂²×C₄

#10625337

yes

characteristic

112

M₁₆

—

113

M₁₆

—

188

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

188

—

118

C₂×C₈

C₄×Dic₄₇

yes

characteristic

119

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

120

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

121

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

characteristic

122

C₄²

Dic₄₇⋊C₄

yes

characteristic

123

C₄²

#10625334

yes

characteristic

124

C₄²

#10625336

yes

characteristic

125

C₄²

C₂²⋊Dic₄₇

yes

characteristic

126

C₁₆

—

188

—

127

C₂³

#13093947

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₄₇⋊#95

yes

characteristic

133

C₂×C₄

#13093944

yes

characteristic

134

C₂×C₄

#13093945

yes

characteristic

135

C₂×C₄

#13093948

yes

characteristic

136

C₂×C₄

#13093955

yes

characteristic

137

C₂×C₄

#13093957

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

188

—

143

C₈

#13093950

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#15688885

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#15688884

yes

characteristic

153

C₄

#15688894

yes

characteristic

154

C₄

#15688930

yes

—

155

C₂

—

156

C₂

#20241289

yes

characteristic

157

C₂

#20241483

yes

characteristic

158

C₂

#20241499

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093945
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1568	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#103
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1568	D₄₇
hypocenter	C₄₇	#7297
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂×C₄

#13093945

commutator

C₂×C₈×C₄₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#1568

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 ◁ #1568

N	H ≅ G/N	type
93. C₄₇	#7297	semidirect

H ≅ G/N

type

93. C₄₇

#7297

semidirect

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

subgroup	index
#20245187	2
C₄×#13093960	2
C₄₇×#1568	2
#7297	47

subgroup

index

#20245187

C₄×#13093960

C₄₇×#1568

#7297

Subgroups — #26103528

Conjugacy classes of subgroups

Complements

Maximal subgroups