Group #26115781 — subgroups

Subgroups — #26115781

Order 12032.

558 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 119 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×#15689500	C₂	1	1	yes	maximal, characteristic
2	#20242057	C₂	1	4	yes	maximal, complemented
3	#20241680	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₇₆	—	2	2	no	—
6	C₂×#13094021	C₂²	1	2	yes	—
7	#15689500	—	2	2	no	—
8	#15689500	C₄	1	4	yes	—
9	#15689176	C₂²	1	2	yes	—
10	#15688866	—	2	8	no	—
11	#15688845	—	2	2	no	—
12	C₄₇×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#341	—	2	2	no	—
15	C₂²×D₁₈₈	—	2	2	no	—
16	C₂²×D₁₈₈	—	2	4	no	—
17	C₂×D₃₇₆	—	2	8	no	—
18	C₂×D₃₇₆	—	4	8	no	—
19	#13094021	—	2	2	no	—
20	#13094021	—	2	4	no	—
21	#13094021	C₂×C₄	1	4	yes	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	2	4	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₄₇	D₄	1	2	yes	—
31	C₂³×D₄₇	—	4	2	no	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	#41	1	2	yes	—
34	C₂×D₁₈₈	—	2	4	no	—
35	C₂×D₁₈₈	—	2	8	no	—
36	C₂×D₁₈₈	—	4	8	no	—
37	C₂×D₁₈₈	—	4	4	no	—
38	C₂×D₁₈₈	—	4	2	no	—
39	C₂×D₁₈₈	—	8	8	no	—
40	M₁₆×C₄₇	—	4	4	no	—
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
44	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₄₇⋊C₈	—	2	4	no	—
48	D₃₇₆	—	4	16	no	—
49	C₂³×C₄₇	#124	1	1	yes	characteristic
50	C₂²×D₄₇	—	4	4	no	—
51	C₂²×D₄₇	—	8	8	no	—
52	C₂²×D₄₇	—	8	4	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₄₇	#121	1	2	yes	—
57	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
59	D₁₈₈	—	4	8	no	—
60	D₁₈₈	—	8	8	no	—
61	D₁₈₈	—	8	8	no	—
62	D₁₈₈	—	8	4	no	—
63	C₈×C₄₇	—	2	4	no	—
64	C₈×C₄₇	—	8	2	no	—
65	C₄₇⋊C₈	—	4	4	no	—
66	#9780	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	#390	1	1	yes	characteristic
69	C₂²×C₄₇	#490	1	2	yes	—
70	C₂×D₄₇	—	8	4	no	—
71	C₂×D₄₇	—	8	8	no	—
72	C₂×D₄₇	—	16	8	no	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	#418	1	2	yes	—
75	C₄×C₄₇	#472	1	4	yes	—
76	C₄×C₄₇	#496	1	2	yes	—
77	C₂×#493	—	47	1	no	—
78	#1837	D₄₇	1	1	yes	characteristic, complemented
79	#1545	—	47	4	no	—
80	#1452	—	47	1	no	—
81	C₂×C₄₇	—	2	2	no	—
82	C₂×C₄₇	#1961	1	1	yes	characteristic
83	C₂×C₄₇	#1978	1	2	yes	—
84	D₄₇	—	16	4	no	—
85	C₂²×D₈	—	94	2	no	—
86	C₂×#127	—	47	2	no	—
87	#493	—	47	4	no	—
88	#493	—	94	2	no	—
89	#431	—	47	2	no	—
90	#423	C₂×D₄₇	1	2	yes	—
91	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
92	#341	—	2	2	no	—
93	#331	—	94	8	no	—
94	#322	—	94	2	no	—
95	C₄₇	#9780	1	1	yes	characteristic, complemented, Sylow
96	C₂²×D₄	—	94	4	no	—
97	C₂²×D₄	—	94	2	no	—
98	C₂×D₈	—	94	8	no	—
99	C₂×D₈	—	188	8	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
102	#127	—	47	4	no	—
103	#127	—	94	2	no	—
104	#127	—	94	4	no	—
105	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	47	4	no	—
108	C₂²⋊C₈	—	94	2	no	—
109	C₄×C₈	—	94	2	no	—
110	C₄×C₈	C₄×D₄₇	1	2	yes	—
111	C₄×C₈	D₁₈₈	1	2	yes	—
112	C₂⁴	—	188	2	no	—
113	C₂×D₄	—	94	4	no	—
114	C₂×D₄	—	94	8	no	—
115	C₂×D₄	—	376	8	no	—
116	C₂×D₄	—	188	4	no	—
117	C₂×D₄	—	188	2	no	—
118	C₂×D₄	—	188	8	no	—
119	C₂²×C₄	#10625354	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	D₈	—	188	16	no	—
122	M₁₆	—	4	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	94	4	no	—
126	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
127	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
128	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
129	C₄²	D₄×D₄₇	1	2	yes	—
130	C₂³	—	376	8	no	—
131	C₂³	—	376	4	no	—
132	C₂³	—	188	4	no	—
133	C₂³	#13094013	1	1	yes	characteristic
134	D₄	—	376	8	no	—
135	D₄	—	376	4	no	—
136	D₄	—	376	8	no	—
137	D₄	—	188	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	#13093986	1	2	yes	—
141	C₂×C₄	#13093999	1	2	yes	—
142	C₂×C₄	#13094020	1	1	yes	characteristic
143	C₂×C₄	#13094049	1	4	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	8	2	no	—
146	C₈	—	188	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	376	8	no	—
149	C₂²	—	376	4	no	—
150	C₂²	—	752	8	no	—
151	C₂²	#15689024	1	1	yes	characteristic
152	C₂²	#15689489	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#15689127	1	2	yes	—
155	C₄	#15689394	1	4	yes	—
156	C₄	#15689513	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	752	4	no	—
159	C₂	#20244211	1	1	yes	characteristic
160	C₂	#20244296	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689500

C₂

yes

maximal, characteristic

#20242057

C₂

yes

maximal, complemented

#20241680

C₂

yes

maximal, characteristic, complemented

C₄₇×#1837

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₇₆

—

C₂×#13094021

C₂²

yes

—

#15689500

—

#15689500

C₄

yes

—

#15689176

C₂²

yes

—

#15688866

—

#15688845

—

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

—

#13094021

—

#13094021

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

C₂³×C₄₇

#124

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#9780

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#490

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#418

yes

—

C₄×C₄₇

#472

yes

—

C₄×C₄₇

#496

yes

—

C₂×#493

—

#1837

D₄₇

yes

characteristic, complemented

#1545

—

#1452

—

C₂×C₄₇

—

C₂×C₄₇

#1961

yes

characteristic

C₂×C₄₇

#1978

yes

—

D₄₇

—

C₂²×D₈

—

—

—

—

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

C₄₇

#9780

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

188

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₇

yes

—

102

#127

—

103

#127

—

104

#127

—

105

C₂×C₄²

C₂²×D₄₇

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₄₇

yes

—

111

C₄×C₈

D₁₈₈

yes

—

112

C₂⁴

—

188

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

376

—

116

C₂×D₄

—

188

—

117

C₂×D₄

—

188

—

118

C₂×D₄

—

188

—

119

C₂²×C₄

#10625354

yes

—

120

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

121

D₈

—

188

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

127

C₄²

C₂×C₄×D₄₇

yes

characteristic

128

C₄²

C₂×D₁₈₈

yes

characteristic

129

C₄²

D₄×D₄₇

yes

—

130

C₂³

—

376

—

131

C₂³

—

376

—

132

C₂³

—

188

—

133

C₂³

#13094013

yes

characteristic

134

D₄

—

376

—

135

D₄

—

376

—

136

D₄

—

376

—

137

D₄

—

188

—

138

C₂×C₄

—

139

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

140

C₂×C₄

#13093986

yes

—

141

C₂×C₄

#13093999

yes

—

142

C₂×C₄

#13094020

yes

characteristic

143

C₂×C₄

#13094049

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

188

—

147

C₂²

—

148

C₂²

—

376

—

149

C₂²

—

376

—

150

C₂²

—

752

—

151

C₂²

#15689024

yes

characteristic

152

C₂²

#15689489

yes

—

153

C₄

—

154

C₄

#15689127

yes

—

155

C₄

#15689394

yes

—

156

C₄

#15689513

yes

—

157

C₂

—

158

C₂

—

752

—

159

C₂

#20244211

yes

characteristic

160

C₂

#20244296

yes

—

	subgroup	G/H
center	C₂²	#15689024
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1837	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂×#15689500	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1837	D₄₇
hypocenter	C₄₇	#9780
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#15689024

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1837

C₂

C₂²×C₄₇

C₂

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
2. #20242057	C₂	semidirect
3. #20241680	C₂	semidirect
4. C₄₇×#1837	C₂	semidirect
78. #1837	D₄₇	semidirect
95. C₄₇	#9780	semidirect

H ≅ G/N

type

2. #20242057

C₂

semidirect

3. #20241680

C₂

semidirect

4. C₄₇×#1837

C₂

semidirect

78. #1837

D₄₇

semidirect

95. C₄₇

#9780

semidirect

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

subgroup	index
C₂×#15689500	2
#20242057	2
#20241680	2
C₄₇×#1837	2
#9780	47

subgroup

index

C₄₇×#1837

Subgroups — #26115781

Conjugacy classes of subgroups

Complements

Maximal subgroups