Group #26103378 — subgroups

Subgroups — #26103378

Order 12032.

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

normal subgroups only

This group has 190 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	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20242151	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1568	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
5	#15689180	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	#15688876	—	2	1	no	—
9	#15688874	C₄	1	2	yes	—
10	C₄₇×#432	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#335	—	2	1	no	—
13	C₄₇×#334	C₄	1	2	yes	—
14	C₂×C₄₇⋊C₁₆	—	2	2	no	—
15	C₂×C₄₇⋊C₁₆	—	4	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
18	#13093935	—	4	1	no	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	Q₈	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	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	—	4	1	no	—
35	C₂×C₈×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
43	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
44	C₂×C₄₇⋊C₈	—	4	2	no	—
45	C₄₇⋊C₁₆	—	4	2	no	—
46	C₂³×C₄₇	#107	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	2	1	no	—
59	C₈×C₄₇	—	2	1	no	—
60	C₈×C₄₇	—	8	2	no	—
61	C₈×C₄₇	#100	1	1	yes	characteristic
62	C₈×C₄₇	#101	1	1	yes	characteristic
63	C₄₇⋊C₈	—	8	2	no	—
64	#7242	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	#340	1	1	yes	characteristic
68	C₂²×C₄₇	#368	1	2	yes	—
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	—	2	1	no	—
74	C₄×C₄₇	#338	1	1	yes	characteristic
75	C₄×C₄₇	#343	1	1	yes	characteristic
76	C₂×C₄⋊C₁₆	—	47	1	no	—
77	#1568	—	47	1	no	—
78	#1568	D₄₇	1	1	yes	characteristic
79	C₂×C₄₇	—	2	2	no	—
80	C₂×C₄₇	#1292	1	1	yes	characteristic
81	C₂×C₄₇	#1381	1	1	yes	characteristic
82	C₂×C₄₇	#1384	1	1	yes	characteristic
83	C₂²×C₁₆	—	47	2	no	—
84	#432	—	2	1	no	—
85	#432	—	94	1	no	—
86	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	94	1	no	—
88	C₄⋊C₁₆	—	94	1	no	—
89	#335	—	2	1	no	—
90	#335	—	94	1	no	—
91	#334	—	47	2	no	—
92	#334	Dic₄₇	1	2	yes	—
93	C₄₇	#7242	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
95	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂×C₁₆	—	94	2	no	—
98	C₂×C₁₆	—	188	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	94	2	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	C₂²⋊C₈	—	188	1	no	—
103	#97	—	4	1	no	—
104	#97	—	188	1	no	—
105	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
106	C₄×C₈	D₁₈₈	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	Dic₄₇⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	#10625334	1	1	yes	characteristic
111	C₂²×C₄	#10625337	1	1	yes	characteristic
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	188	2	no	—
117	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
118	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
120	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
121	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
122	C₄²	#10625335	1	1	yes	characteristic
123	C₄²	#10625336	1	1	yes	characteristic
124	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
125	C₁₆	—	188	2	no	—
126	C₂³	#13093957	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
132	C₂×C₄	#13093944	1	1	yes	characteristic
133	C₂×C₄	#13093945	1	1	yes	characteristic
134	C₂×C₄	#13093947	1	1	yes	characteristic
135	C₂×C₄	#13093948	1	1	yes	characteristic
136	C₂×C₄	#13093955	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	8	2	no	—
141	C₈	—	376	2	no	—
142	C₈	#13093940	1	1	yes	characteristic
143	C₈	#13093942	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#15688885	1	1	yes	characteristic
147	C₂²	#15688952	1	2	yes	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	C₄₇⋊#338	1	1	yes	characteristic
154	C₄	#15688891	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#20241262	1	1	yes	characteristic
157	C₂	#20241481	1	1	yes	characteristic
158	C₂	#20241489	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₁₆)

C₂

yes

maximal, characteristic

#20242151

C₂

yes

maximal, characteristic

C₄₇×#1568

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₄

yes

—

#15689180

—

C₄⋊(C₄₇⋊C₁₆)

—

C₄⋊(C₄₇⋊C₁₆)

—

#15688876

—

#15688874

C₄

yes

—

C₄₇×#432

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

—

C₄₇×#334

C₄

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

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

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

C₂×C₈×C₄₇

—

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

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#107

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

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#7242

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₂²×C₄₇

#368

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₂×C₄⋊C₁₆

—

#1568

—

#1568

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₁₆

—

#432

—

#432

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

Dic₄₇

yes

—

C₄₇

#7242

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

188

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

188

—

103

#97

—

104

#97

—

188

—

105

C₄×C₈

C₄×D₄₇

yes

characteristic

106

C₄×C₈

D₁₈₈

yes

characteristic

107

C₄×C₈

C₂×Dic₄₇

yes

characteristic

108

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

109

C₂²×C₄

Dic₄₇⋊C₄

yes

characteristic

110

C₂²×C₄

#10625334

yes

characteristic

111

C₂²×C₄

#10625337

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

188

—

117

C₂×C₈

C₄×Dic₄₇

yes

characteristic

118

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

119

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

120

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

characteristic

121

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

122

C₄²

#10625335

yes

characteristic

123

C₄²

#10625336

yes

characteristic

124

C₄²

C₂²⋊Dic₄₇

yes

characteristic

125

C₁₆

—

188

—

126

C₂³

#13093957

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄₇⋊#95

yes

characteristic

132

C₂×C₄

#13093944

yes

characteristic

133

C₂×C₄

#13093945

yes

characteristic

134

C₂×C₄

#13093947

yes

characteristic

135

C₂×C₄

#13093948

yes

characteristic

136

C₂×C₄

#13093955

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

376

—

142

C₈

#13093940

yes

characteristic

143

C₈

#13093942

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

#15688885

yes

characteristic

147

C₂²

#15688952

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₄₇⋊#338

yes

characteristic

154

C₄

#15688891

yes

characteristic

155

C₂

—

156

C₂

#20241262

yes

characteristic

157

C₂

#20241481

yes

characteristic

158

C₂

#20241489

yes

characteristic

	subgroup	G/H
center	C₂²	#15688885
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1568	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#107
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1568	D₄₇
hypocenter	C₄₇	#7242
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂²

#15688885

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 ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

N	H ≅ G/N	type
93. C₄₇	#7242	semidirect

H ≅ G/N

type

93. C₄₇

#7242

semidirect

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20242151	2
C₄₇×#1568	2
#7242	47

subgroup

index

C₂×C₄⋊(C₄₇⋊C₁₆)

#20242151

C₄₇×#1568

#7242

Subgroups — #26103378

Conjugacy classes of subgroups

Complements

Maximal subgroups