Group #26116246 — subgroups

Subgroups — #26116246

Order 12032.

406 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20243560	C₂	1	1	yes	maximal, characteristic
3	#20241844	C₂	1	4	yes	maximal
4	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
5	#15689680	C₂²	1	2	yes	—
6	#15689528	C₄	1	2	yes	—
7	#15689525	—	2	2	no	—
8	#15689521	C₄	1	2	yes	—
9	C₂×#13093960	—	2	2	no	—
10	#15689161	—	2	2	no	—
11	#15689147	C₂²	1	2	yes	—
12	#15688855	—	2	8	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#322	—	2	2	no	—
16	C₂×#10625329	—	2	2	no	—
17	#13094024	C₂×C₄	1	4	yes	—
18	#13094016	—	2	2	no	—
19	#13094003	—	2	4	no	—
20	C₄×Dic₉₄	—	2	4	no	—
21	#13093983	—	2	2	no	—
22	#13093960	—	4	4	no	—
23	#13093958	—	2	4	no	—
24	#13093956	—	2	4	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	D₄	1	4	yes	—
33	C₄×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	#41	1	2	yes	—
36	C₂×Dic₉₄	—	2	4	no	—
37	C₂×C₈×C₄₇	—	2	4	no	—
38	C₂×C₈×C₄₇	—	4	4	no	—
39	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
42	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
43	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂×D₄	1	2	yes	—
46	C₂²⋊Dic₄₇	—	4	2	no	—
47	C₄⋊Dic₄₇	—	2	4	no	—
48	C₄⋊Dic₄₇	—	2	4	no	—
49	Dic₄₇⋊C₄	—	4	4	no	—
50	C₄×Dic₄₇	—	4	2	no	—
51	#10625329	—	4	4	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂³×C₄₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	#121	1	2	yes	—
58	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
61	C₂×Dic₄₇	—	4	4	no	—
62	Dic₉₄	—	4	8	no	—
63	C₈×C₄₇	—	4	4	no	—
64	C₈×C₄₇	#103	1	4	yes	—
65	C₈×C₄₇	#103	1	4	yes	—
66	C₄₇⋊C₈	—	8	2	no	—
67	#9861	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	#390	1	1	yes	characteristic
70	C₂²×C₄₇	#487	1	2	yes	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	C₂×#103	1	2	yes	—
73	C₄×C₄₇	#444	1	4	yes	—
74	C₄×C₄₇	#492	1	1	yes	characteristic
75	C₄×C₄₇	#494	1	1	yes	characteristic
76	Dic₄₇	—	8	4	no	—
77	#3160	—	47	1	no	—
78	#1848	—	47	1	no	—
79	#1493	—	47	4	no	—
80	C₂×#322	D₄₇	1	1	yes	characteristic
81	C₂×C₄₇	—	2	2	no	—
82	C₂×C₄₇	#1956	1	1	yes	characteristic
83	C₂×C₄₇	#1975	1	2	yes	—
84	#533	—	47	2	no	—
85	#500	—	47	2	no	—
86	#499	—	94	2	no	—
87	#498	—	47	2	no	—
88	C₂×#108	—	94	2	no	—
89	#427	—	94	2	no	—
90	#423	—	47	2	no	—
91	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
92	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
93	Q₈⋊C₈	—	94	8	no	—
94	#322	—	2	2	no	—
95	C₄₇	#9861	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	94	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	D₁₈₈	1	2	yes	—
99	C₄⋊Q₈	—	47	4	no	—
100	#125	—	94	2	no	—
101	C₂²⋊Q₈	—	94	4	no	—
102	C₄×Q₈	—	94	4	no	—
103	#117	—	94	2	no	—
104	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
105	#108	—	188	4	no	—
106	#107	—	94	4	no	—
107	#106	—	94	4	no	—
108	C₄⋊C₈	—	94	4	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
111	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
112	C₄×C₈	D₁₈₈	1	4	yes	—
113	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
114	C₂×Q₈	—	94	4	no	—
115	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
116	C₂²×C₄	#10625350	1	2	yes	—
117	M₁₆	—	188	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	188	2	no	—
121	C₂×C₈	#10625325	1	4	yes	—
122	C₂×C₈	Dic₁₈₈	1	4	yes	—
123	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
124	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
125	C₄⋊C₄	—	94	4	no	—
126	C₄⋊C₄	—	94	4	no	—
127	C₄⋊C₄	—	188	4	no	—
128	C₂²⋊C₄	—	188	2	no	—
129	C₄²	—	188	2	no	—
130	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
131	C₄²	C₂×D₁₈₈	1	2	yes	—
132	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
133	C₂³	#13094012	1	1	yes	characteristic
134	Q₈	—	188	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	188	4	no	—
137	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
139	C₂×C₄	#13093998	1	2	yes	—
140	C₂×C₄	#13094021	1	1	yes	characteristic
141	C₂×C₄	C₂×#10625325	1	2	yes	—
142	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	376	2	no	—
145	C₈	#13093949	1	4	yes	—
146	C₈	#13093949	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#15689025	1	1	yes	characteristic
149	C₂²	#15689474	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	376	4	no	—
152	C₄	C₂×#13093949	1	2	yes	—
153	C₄	#15689246	1	4	yes	—
154	C₄	#15689496	1	1	yes	characteristic
155	C₄	#15689503	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20244190	1	1	yes	characteristic
158	C₂	#20244281	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253902

C₂

yes

maximal, characteristic

#20243560

C₂

yes

maximal, characteristic

#20241844

C₂

yes

maximal

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

#15689528

C₄

yes

—

#15689525

—

#15689521

C₄

yes

—

C₂×#13093960

—

#15689161

—

#15689147

C₂²

yes

—

#15688855

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

C₂×#10625329

—

#13094024

C₂×C₄

yes

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

—

—

—

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

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

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#103

yes

—

C₈×C₄₇

#103

yes

—

C₄₇⋊C₈

—

#9861

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#487

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#103

yes

—

C₄×C₄₇

#444

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1956

yes

characteristic

C₂×C₄₇

#1975

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

Q₈⋊C₈

—

#322

—

C₄₇

#9861

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₁₈₈

yes

—

C₄⋊Q₈

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₄₇

yes

characteristic

105

#108

—

188

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₄₇

yes

characteristic

111

C₄×C₈

C₄×D₄₇

yes

characteristic

112

C₄×C₈

D₁₈₈

yes

—

113

C₄×C₈

C₂²⋊D₄₇

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

116

C₂²×C₄

#10625350

yes

—

117

M₁₆

—

188

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

188

—

121

C₂×C₈

#10625325

yes

—

122

C₂×C₈

Dic₁₈₈

yes

—

123

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

124

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

188

—

128

C₂²⋊C₄

—

188

—

129

C₄²

—

188

—

130

C₄²

C₂×C₄×D₄₇

yes

characteristic

131

C₄²

C₂×D₁₈₈

yes

—

132

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

133

C₂³

#13094012

yes

characteristic

134

Q₈

—

188

—

135

C₂×C₄

—

136

C₂×C₄

—

188

—

137

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

138

C₂×C₄

C₄×D₁₈₈

yes

—

139

C₂×C₄

#13093998

yes

—

140

C₂×C₄

#13094021

yes

characteristic

141

C₂×C₄

C₂×#10625325

yes

—

142

C₂×C₄

C₂×Dic₁₈₈

yes

—

143

C₈

—

144

C₈

—

376

—

145

C₈

#13093949

yes

—

146

C₈

#13093949

yes

—

147

C₂²

—

148

C₂²

#15689025

yes

characteristic

149

C₂²

#15689474

yes

—

150

C₄

—

151

C₄

—

376

—

152

C₄

C₂×#13093949

yes

—

153

C₄

#15689246

yes

—

154

C₄

#15689496

yes

characteristic

155

C₄

#15689503

yes

characteristic

156

C₂

—

157

C₂

#20244190

yes

characteristic

158

C₂

#20244281

yes

—

	subgroup	G/H
center	C₂×C₄	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#9861
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#322

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
95. C₄₇	#9861	semidirect

H ≅ G/N

type

95. C₄₇

#9861

semidirect

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

subgroup	index
#20253902	2
#20243560	2
#20241844	2
C₂×C₄₇×#322	2
#9861	47

subgroup

index

C₂×C₄₇×#322

Subgroups — #26116246

Conjugacy classes of subgroups

Complements

Maximal subgroups