Group #26116044 — subgroups

Subgroups — #26116044

Order 12032.

558 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 199 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×#15689503	C₂	1	1	yes	maximal, characteristic
2	#20244348	C₂	1	4	yes	maximal
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₈₈	—	2	2	no	—
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689503	—	2	2	no	—
8	#15689503	C₄	1	4	yes	complemented
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	#15688885	—	2	8	no	—
11	C₈×C₄₇⋊C₈	—	2	2	no	—
12	C₂×#107×C₄₇	—	2	2	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₂²	1	2	yes	—
15	C₂²×Dic₉₄	—	2	2	no	—
16	C₂×Dic₁₈₈	—	2	8	no	—
17	C₂×Dic₁₈₈	—	4	4	no	—
18	C₂×Dic₁₈₈	—	4	4	no	—
19	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
20	#13094024	—	2	4	no	—
21	#13094024	—	2	2	no	—
22	#13094024	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₄₇	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	#107×C₄₇	—	4	4	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	D₄	1	2	yes	—
33	C₂²×C₄×C₄₇	—	4	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
36	C₂²×Dic₄₇	—	4	2	no	—
37	C₂×Dic₉₄	—	2	4	no	—
38	C₂×Dic₉₄	—	4	4	no	—
39	C₂×Dic₉₄	—	4	4	no	—
40	C₂×Dic₉₄	—	4	2	no	—
41	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₄₇	D₈	1	4	yes	—
43	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
44	C₄⋊C₄×C₄₇	—	4	4	no	—
45	C₄⋊C₄×C₄₇	—	4	8	no	—
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₄⋊Dic₄₇	—	2	8	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	2	4	no	—
53	C₂×C₄₇⋊C₈	Q₁₆	1	8	yes	complemented
54	Dic₁₈₈	—	4	16	no	—
55	C₂³×C₄₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	8	2	no	—
57	C₂×C₄×C₄₇	—	8	4	no	—
58	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	#121	1	2	yes	—
61	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
65	C₂×C₄×C₄₇	#135	1	4	yes	—
66	C₂×Dic₄₇	—	4	4	no	—
67	C₂×Dic₄₇	—	8	2	no	—
68	C₂×Dic₄₇	—	8	2	no	—
69	Dic₉₄	—	4	8	no	—
70	Dic₉₄	—	8	4	no	—
71	C₈×C₄₇	#102	1	8	yes	—
72	C₄₇⋊C₈	—	2	8	no	—
73	#9825	—	47	1	no	maximal, Sylow
74	C₂²×C₄₇	#390	1	1	yes	characteristic
75	C₂²×C₄₇	#492	1	1	yes	characteristic
76	C₂²×C₄₇	#493	1	1	yes	characteristic
77	C₂²×C₄₇	#494	1	2	yes	—
78	C₂²×C₄₇	#495	1	2	yes	—
79	C₄×C₄₇	—	16	2	no	—
80	C₄×C₄₇	C₂×#102	1	2	yes	—
81	C₄×C₄₇	C₄×Q₁₆	1	4	yes	—
82	C₄×C₄₇	#467	1	4	yes	—
83	C₄×C₄₇	#486	1	2	yes	—
84	Dic₄₇	—	8	4	no	—
85	C₂×#494	—	47	1	no	—
86	#1990	—	47	4	no	—
87	#1845	D₄₇	1	1	yes	characteristic
88	C₂×C₈²	—	47	1	no	—
89	C₂×C₄₇	#1707	1	2	yes	—
90	C₂×C₄₇	#1709	1	1	yes	characteristic
91	C₂×C₄₇	#1713	1	1	yes	characteristic
92	C₂×C₄₇	#1957	1	1	yes	characteristic
93	C₂×C₄₇	#1969	1	2	yes	—
94	C₂²×Q₁₆	—	94	2	no	—
95	C₂×C₄⋊Q₈	—	47	2	no	—
96	#494	—	47	4	no	—
97	#494	—	94	2	no	—
98	C₂×#107	—	2	2	no	—
99	C₂×C₄×C₈	—	47	2	no	—
100	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	#384	C₂×D₄₇	1	2	yes	—
102	#340	—	94	8	no	—
103	C₈²	—	94	2	no	—
104	C₄₇	#9825	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	94	2	no	—
106	C₂×Q₁₆	—	94	8	no	—
107	C₂×Q₁₆	—	188	4	no	—
108	C₂×Q₁₆	—	188	4	no	—
109	C₂²×C₈	—	47	4	no	—
110	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
111	C₄⋊Q₈	—	47	4	no	—
112	C₄⋊Q₈	—	94	2	no	—
113	C₄⋊Q₈	—	94	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	94	4	no	—
117	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	94	2	no	—
120	C₄×C₈	—	94	2	no	—
121	C₄×C₈	C₄×D₄₇	1	2	yes	—
122	C₄×C₈	D₁₈₈	1	2	yes	—
123	C₂×Q₈	—	94	4	no	—
124	C₂×Q₈	—	188	4	no	—
125	C₂×Q₈	—	188	4	no	—
126	C₂×Q₈	—	188	2	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	188	2	no	—
129	C₂²×C₄	D₄×D₄₇	1	2	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
131	Q₁₆	—	188	16	no	—
132	C₂×C₈	—	47	8	no	—
133	C₂×C₈	—	94	4	no	—
134	C₂×C₈	—	94	4	no	—
135	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
136	C₂×C₈	#10625334	1	4	yes	—
137	C₂×C₈	#10625336	1	4	yes	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	94	8	no	—
141	C₄⋊C₄	—	188	4	no	—
142	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
143	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
144	C₄²	#10625354	1	2	yes	—
145	C₂³	#13094020	1	1	yes	characteristic
146	Q₈	—	376	4	no	—
147	Q₈	—	188	8	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	376	2	no	—
151	C₂×C₄	—	376	2	no	—
152	C₂×C₄	—	188	4	no	—
153	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#13093986	1	2	yes	—
155	C₂×C₄	#13093999	1	2	yes	—
156	C₂×C₄	#13094013	1	1	yes	characteristic
157	C₂×C₄	C₂×#10625334	1	2	yes	—
158	C₂×C₄	C₂×#10625336	1	2	yes	—
159	C₂×C₄	Q₁₆×D₄₇	1	4	yes	—
160	C₂×C₄	#13094043	1	4	yes	—
161	C₈	—	94	8	no	—
162	C₈	#13093944	1	8	yes	—
163	C₂²	#15689024	1	1	yes	characteristic
164	C₂²	#15689497	1	1	yes	characteristic
165	C₂²	#15689499	1	1	yes	characteristic
166	C₂²	#15689501	1	2	yes	—
167	C₂²	#15689506	1	2	yes	—
168	C₄	—	16	2	no	—
169	C₄	—	376	4	no	—
170	C₄	C₂×#13093944	1	2	yes	—
171	C₄	#15689217	1	4	yes	—
172	C₄	#15689365	1	4	yes	—
173	C₄	#15689471	1	2	yes	—
174	C₂	#20242986	1	2	yes	—
175	C₂	#20242993	1	1	yes	characteristic
176	C₂	#20243004	1	1	yes	characteristic
177	C₂	#20244193	1	1	yes	characteristic
178	C₂	#20244247	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689503

C₂

yes

maximal, characteristic

#20244348

C₂

yes

maximal

C₂×C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#1845

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

—

C₂×#13094024

C₂²

yes

—

#15689503

—

#15689503

C₄

yes

complemented

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688885

—

C₈×C₄₇⋊C₈

—

C₂×#107×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₂²×Dic₉₄

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂²×C₄₇⋊C₈

D₄

yes

—

#13094024

—

#13094024

—

#13094024

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

Q₁₆

yes

complemented

Dic₁₈₈

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

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

#124

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

#102

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#495

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#102

yes

—

C₄×C₄₇

C₄×Q₁₆

yes

—

C₄×C₄₇

#467

yes

—

C₄×C₄₇

#486

yes

—

Dic₄₇

—

C₂×#494

—

#1990

—

#1845

D₄₇

yes

characteristic

C₂×C₈²

—

C₂×C₄₇

#1707

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×#107

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

101

#384

C₂×D₄₇

yes

—

102

#340

—

103

C₈²

—

104

C₄₇

#9825

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

188

—

108

C₂×Q₁₆

—

188

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂²⋊D₄₇

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

C₂²×D₄₇

yes

characteristic

118

#107

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₄×D₄₇

yes

—

122

C₄×C₈

D₁₈₈

yes

—

123

C₂×Q₈

—

124

C₂×Q₈

—

188

—

125

C₂×Q₈

—

188

—

126

C₂×Q₈

—

188

—

127

C₂²×C₄

—

128

C₂²×C₄

—

188

—

129

C₂²×C₄

D₄×D₄₇

yes

—

130

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

131

Q₁₆

—

188

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

136

C₂×C₈

#10625334

yes

—

137

C₂×C₈

#10625336

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

188

—

142

C₄²

C₂×C₄×D₄₇

yes

characteristic

143

C₄²

C₂×D₁₈₈

yes

characteristic

144

C₄²

#10625354

yes

—

145

C₂³

#13094020

yes

characteristic

146

Q₈

—

376

—

147

Q₈

—

188

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

376

—

151

C₂×C₄

—

376

—

152

C₂×C₄

—

188

—

153

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

154

C₂×C₄

#13093986

yes

—

155

C₂×C₄

#13093999

yes

—

156

C₂×C₄

#13094013

yes

characteristic

157

C₂×C₄

C₂×#10625334

yes

—

158

C₂×C₄

C₂×#10625336

yes

—

159

C₂×C₄

Q₁₆×D₄₇

yes

—

160

C₂×C₄

#13094043

yes

—

161

C₈

—

162

C₈

#13093944

yes

—

163

C₂²

#15689024

yes

characteristic

164

C₂²

#15689497

yes

characteristic

165

C₂²

#15689499

yes

characteristic

166

C₂²

#15689501

yes

—

167

C₂²

#15689506

yes

—

168

C₄

—

169

C₄

—

376

—

170

C₄

C₂×#13093944

yes

—

171

C₄

#15689217

yes

—

172

C₄

#15689365

yes

—

173

C₄

#15689471

yes

—

174

C₂

#20242986

yes

—

175

C₂

#20242993

yes

characteristic

176

C₂

#20243004

yes

characteristic

177

C₂

#20244193

yes

characteristic

178

C₂

#20244247

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#9825
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1845

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
8. #15689503	C₄	semidirect
53. C₂×C₄₇⋊C₈	Q₁₆	semidirect
104. C₄₇	#9825	semidirect

H ≅ G/N

type

8. #15689503

C₄

semidirect

53. C₂×C₄₇⋊C₈

Q₁₆

semidirect

104. C₄₇

#9825

semidirect

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

subgroup	index
C₂×#15689503	2
#20244348	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1845	2
#9825	47

subgroup

index

C₂×#15689503

#20244348

C₂×C₈×C₄₇⋊C₈

C₄₇×#1845

#9825

Subgroups — #26116044

Conjugacy classes of subgroups

Complements

Maximal subgroups