Group #28303548 — subgroups

Subgroups — #28303548

Order 13568.

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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	#20290655	C₂	1	1	yes	maximal, characteristic
2	#20290652	C₂	1	1	yes	maximal, characteristic
3	#20289569	C₂	1	4	yes	maximal
4	#20273000	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133288	—	2	1	no	—
6	C₂×#13133287	—	2	1	no	—
7	C₂×#13133286	—	2	1	no	—
8	C₂×#13133285	—	2	1	no	—
9	#16811628	C₂²	1	2	yes	—
10	#16811567	C₂²	1	2	yes	—
11	#16810674	C₂²	1	2	yes	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810344	C₄	1	1	yes	characteristic, complemented
14	C₈×#88322	C₄	1	2	yes	complemented
15	C₅₃×#322	C₄	1	1	yes	characteristic, complemented
16	C₂×#11719936	—	2	2	no	—
17	C₂×C₄⋊#6157	—	2	2	no	—
18	#13133288	—	4	2	no	—
19	#13133287	—	4	2	no	—
20	#13133286	—	4	2	no	—
21	#13133285	—	4	2	no	—
22	#13133284	—	2	4	no	—
23	(C₂×#6157)⋊C₄	—	2	4	no	—
24	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
26	C₄²×D₅₃	C₂³	1	1	yes	characteristic
27	#13133146	—	2	2	no	—
28	#13133129	—	2	2	no	—
29	C₄×#88322	C₂×C₄	1	2	yes	—
30	C₄×#88322	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#6157	—	4	2	no	—
36	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	#41	1	2	yes	—
38	C₂×#88324	—	4	2	no	—
39	C₄⋊#6157	—	4	2	no	—
40	C₄⋊#6157	—	4	2	no	—
41	#11719936	—	4	2	no	—
42	#11719936	—	4	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×Dic₅₃	#41	1	2	yes	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
52	C₈×D₅₃	D₈	1	2	yes	—
53	C₈×D₅₃	SD₁₆	1	2	yes	—
54	C₈×D₅₃	SD₁₆	1	2	yes	—
55	C₈×D₅₃	Q₁₆	1	2	yes	—
56	C₂²×D₅₃	#124	1	1	yes	characteristic
57	C₂×#6157	—	8	4	no	—
58	C₂×#6157	—	8	2	no	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	2	yes	—
61	C₂×Dic₅₃	#124	1	1	yes	characteristic
62	C₂×Dic₅₃	#125	1	2	yes	—
63	C₄×D₅₃	—	4	2	no	—
64	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
65	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	#88324	—	8	2	no	—
69	C₈×C₅₃	—	4	2	no	—
70	C₈×C₅₃	#106	1	2	yes	—
71	C₈×C₅₃	#107	1	2	yes	—
72	#88322	—	4	2	no	—
73	#88322	#106	1	2	yes	—
74	#88322	#107	1	2	yes	—
75	#8793	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	#382	1	1	yes	characteristic
77	C₂×D₅₃	—	2	2	no	—
78	C₂×D₅₃	#486	1	1	yes	characteristic
79	C₂×D₅₃	#487	1	1	yes	characteristic
80	#6157	—	16	2	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
83	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
84	Dic₅₃	—	2	2	no	—
85	Dic₅₃	#486	1	1	yes	characteristic
86	Dic₅₃	#487	1	1	yes	characteristic
87	#1848	—	53	1	no	—
88	#1846	—	53	1	no	—
89	#1450	—	53	1	no	—
90	#1286	—	53	4	no	—
91	C₂×C₅₃	#1833	1	2	yes	—
92	C₂×C₅₃	#1834	1	1	yes	characteristic
93	D₅₃	—	2	1	no	—
94	D₅₃	—	2	1	no	—
95	#431	—	53	2	no	—
96	C₂×#108	—	106	1	no	—
97	C₂×#108	—	106	1	no	—
98	C₂×#107	—	106	1	no	—
99	C₂×#106	—	106	1	no	—
100	#423	—	53	2	no	—
101	C₂×C₄×C₈	—	53	1	no	—
102	#382	—	53	2	no	—
103	#322	—	53	1	no	—
104	#322	#6157	1	1	yes	characteristic, complemented
105	C₈²	—	53	2	no	—
106	C₅₃	#8793	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	106	2	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	—	53	1	no	—
110	C₂×C₄⋊C₄	—	106	2	no	—
111	C₂×C₄²	—	53	1	no	—
112	#108	—	212	2	no	—
113	#108	—	212	2	no	—
114	#107	—	212	2	no	—
115	#106	—	212	2	no	—
116	C₄⋊C₈	—	106	2	no	—
117	C₄⋊C₈	—	106	4	no	—
118	C₂²⋊C₈	—	106	2	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	—	53	1	no	—
122	C₄×C₈	—	53	2	no	—
123	C₄×C₈	C₂×#6157	1	2	yes	—
124	C₄×C₈	C₂×#6157	1	1	yes	characteristic
125	#95	—	106	4	no	—
126	C₂²×C₄	—	53	1	no	—
127	C₂²×C₄	—	53	2	no	—
128	C₂²×C₄	—	212	2	no	—
129	M₁₆	—	212	2	no	—
130	M₁₆	—	212	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	53	1	no	—
133	C₂×C₈	—	53	1	no	—
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	—	53	2	no	—
136	C₂×C₈	—	53	2	no	—
137	C₂×C₈	—	53	2	no	—
138	C₂×C₈	—	106	2	no	—
139	C₂×C₈	—	212	2	no	—
140	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
141	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
142	C₄⋊C₄	—	212	2	no	—
143	C₄⋊C₄	—	212	2	no	—
144	C₄²	—	53	1	no	—
145	C₄²	—	53	2	no	—
146	C₄²	C₂²×#6157	1	1	yes	characteristic
147	C₂³	—	53	1	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	53	1	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	2	no	—
154	C₂×C₄	—	424	2	no	—
155	C₂×C₄	—	424	4	no	—
156	C₂×C₄	—	212	2	no	—
157	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
158	C₂×C₄	#13133296	1	2	yes	—
159	C₈	—	4	2	no	—
160	C₈	—	53	2	no	—
161	C₈	—	53	2	no	—
162	C₈	—	424	2	no	—
163	C₈	—	212	2	no	—
164	C₈	#13133285	1	2	yes	—
165	C₈	#13133286	1	2	yes	—
166	C₂²	—	53	1	no	—
167	C₂²	—	53	1	no	—
168	C₂²	—	106	2	no	—
169	C₂²	#16811567	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	53	1	no	—
172	C₄	—	53	1	no	—
173	C₄	—	848	2	no	—
174	C₄	—	106	2	no	—
175	C₄	C₂×#13133285	1	1	yes	characteristic
176	C₄	C₂×#13133286	1	1	yes	characteristic
177	C₂	—	106	1	no	—
178	C₂	—	106	1	no	—
179	C₂	#20290635	1	2	yes	—
180	C₂	#20290636	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290655

C₂

yes

maximal, characteristic

#20290652

C₂

yes

maximal, characteristic

#20289569

C₂

yes

maximal

#20273000

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#16811567

C₂²

yes

—

#16810674

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810344

C₄

yes

characteristic, complemented

C₈×#88322

C₄

yes

complemented

C₅₃×#322

C₄

yes

characteristic, complemented

C₂×#11719936

—

C₂×C₄⋊#6157

—

—

—

—

—

—

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

#41

yes

—

—

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

D₈

yes

—

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

Q₁₆

yes

—

C₂²×D₅₃

yes

—

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×Dic₅₃

#124

yes

characteristic

C₂×Dic₅₃

#125

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₈

yes

characteristic

C₄×D₅₃

C₂×SD₁₆

yes

characteristic

C₄×D₅₃

C₂×SD₁₆

yes

characteristic

C₄×D₅₃

C₂×Q₁₆

yes

characteristic

#88324

—

C₈×C₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#107

yes

—

#88322

—

#88322

#106

yes

—

#88322

#107

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

—

—

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

#1834

yes

characteristic

D₅₃

—

D₅₃

—

#431

—

C₂×#108

—

106

—

C₂×#108

—

106

—

C₂×#107

—

106

—

C₂×#106

—

106

—

100

#423

—

101

C₂×C₄×C₈

—

102

#382

—

103

#322

—

104

#322

#6157

yes

characteristic, complemented

105

C₈²

—

106

C₅₃

#8793

yes

characteristic, complemented, Sylow

107

C₂×M₁₆

—

106

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂×C₄⋊C₄

—

106

—

111

C₂×C₄²

—

112

#108

—

212

—

113

#108

—

212

—

114

#107

—

212

—

115

#106

—

212

—

116

C₄⋊C₈

—

106

—

117

C₄⋊C₈

—

106

—

118

C₂²⋊C₈

—

106

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

C₂×#6157

yes

—

124

C₄×C₈

C₂×#6157

yes

characteristic

125

#95

—

106

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

212

—

129

M₁₆

—

212

—

130

M₁₆

—

212

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

106

—

139

C₂×C₈

—

212

—

140

C₂×C₈

C₄⋊#6157

yes

characteristic

141

C₂×C₈

C₄⋊#6157

yes

characteristic

142

C₄⋊C₄

—

212

—

143

C₄⋊C₄

—

212

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×#6157

yes

characteristic

147

C₂³

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

424

—

155

C₂×C₄

—

424

—

156

C₂×C₄

—

212

—

157

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

158

C₂×C₄

#13133296

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

424

—

163

C₈

—

212

—

164

C₈

#13133285

yes

—

165

C₈

#13133286

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

106

—

169

C₂²

#16811567

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

848

—

174

C₄

—

106

—

175

C₄

C₂×#13133285

yes

characteristic

176

C₄

C₂×#13133286

yes

characteristic

177

C₂

—

106

—

178

C₂

—

106

—

179

C₂

#20290635

yes

—

180

C₂

#20290636

yes

characteristic

	subgroup	G/H
center	C₂²	#16811567
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#322	C₄
socle	C₂²×C₅₃	#382
prime radical	C₂²×D₅₃	#124
cosocle	C₄²×D₅₃	C₂³
hypercenter	#322	#6157
hypocenter	C₅₃	#8793
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16811567

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#322

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
13. #16810344	C₄	semidirect
14. C₈×#88322	C₄	semidirect
15. C₅₃×#322	C₄	semidirect
104. #322	#6157	semidirect
106. C₅₃	#8793	semidirect

H ≅ G/N

type

13. #16810344

C₄

semidirect

14. C₈×#88322

C₄

semidirect

15. C₅₃×#322

C₄

semidirect

104. #322

#6157

semidirect

106. C₅₃

#8793

semidirect

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

subgroup	index
#20290655	2
#20290652	2
#20289569	2
#20273000	2
#8793	53

subgroup

index

Subgroups — #28303548

Conjugacy classes of subgroups

Complements

Maximal subgroups