Group #28303616 — subgroups

Subgroups — #28303616

Order 13568.

338 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 128 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	#20290656	C₂	1	1	yes	maximal, characteristic
2	#20290654	C₂	1	1	yes	maximal, characteristic
3	#20290274	C₂	1	2	yes	maximal
4	#20290273	C₂	1	2	yes	maximal
5	#322×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133288	—	2	2	no	—
7	C₂×#13133287	—	2	2	no	—
8	#16811627	C₂²	1	1	yes	characteristic
9	#16811627	C₂²	1	1	yes	characteristic
10	#16811627	C₂²	1	2	yes	—
11	#16811487	—	2	4	no	—
12	#16811486	—	2	4	no	—
13	C₄×C₈×D₅₃	C₂²	1	2	yes	—
14	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	#16810345	C₄	1	2	yes	—
16	#16810344	C₄	1	1	yes	characteristic
17	C₅₃×#322	C₄	1	1	yes	characteristic
18	C₂×#11719936	—	2	1	no	—
19	C₂×#11719936	—	2	1	no	—
20	C₂×#11719936	—	2	2	no	—
21	#13133288	—	4	4	no	—
22	#13133287	—	4	4	no	—
23	C₄⋊#88324	—	2	2	no	—
24	C₄⋊#88324	—	2	2	no	—
25	C₄⋊#88324	—	2	2	no	—
26	C₄⋊#88324	—	2	2	no	—
27	C₂×C₈×D₅₃	D₄	1	2	yes	—
28	C₂×C₈×D₅₃	Q₈	1	4	yes	—
29	C₄²×D₅₃	C₂³	1	1	yes	characteristic
30	C₄×#88322	C₂×C₄	1	2	yes	—
31	C₄×#88322	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	D₄	1	4	yes	—
33	C₈×Dic₅₃	Q₈	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
38	C₂×#88324	—	4	1	no	—
39	C₂×#88324	—	4	2	no	—
40	C₂×#88324	—	4	1	no	—
41	#11719936	—	4	2	no	—
42	#11719936	—	4	2	no	—
43	#11719936	—	4	4	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
48	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×#88322	C₄⋊C₄	1	4	yes	—
50	C₂×#88322	C₄⋊C₄	1	2	yes	—
51	C₈×D₅₃	—	4	4	no	—
52	C₈×D₅₃	D₈	1	4	yes	—
53	C₈×D₅₃	Q₁₆	1	4	yes	—
54	C₂²×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×Dic₅₃	#127	1	1	yes	characteristic
58	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
59	C₄×D₅₃	—	2	4	no	—
60	C₄×D₅₃	C₂×D₈	1	2	yes	—
61	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
62	#88324	—	8	1	no	—
63	#88324	—	8	1	no	—
64	#88324	—	8	2	no	—
65	C₈×C₅₃	—	2	4	no	—
66	C₈×C₅₃	#107	1	4	yes	—
67	#88322	—	2	4	no	—
68	#88322	#107	1	4	yes	—
69	#8826	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	#384	1	1	yes	characteristic
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	#500	1	2	yes	—
73	C₄×C₅₃	C₂×#107	1	2	yes	—
74	C₄×C₅₃	#430	1	4	yes	—
75	Dic₅₃	—	2	2	no	—
76	Dic₅₃	#493	1	1	yes	characteristic
77	Dic₅₃	#494	1	1	yes	characteristic
78	#1848	—	53	1	no	—
79	#1848	—	53	1	no	—
80	#1567	—	53	2	no	—
81	#1567	—	53	2	no	—
82	C₂×#322	—	53	1	no	—
83	C₂×C₅₃	#1845	1	1	yes	characteristic
84	C₂×C₅₃	#1849	1	2	yes	—
85	D₅₃	—	2	2	no	—
86	C₂×#108	—	106	2	no	—
87	C₂×#108	—	106	2	no	—
88	#423	—	53	1	no	—
89	#423	—	53	2	no	—
90	#423	—	53	1	no	—
91	C₂×C₄×C₈	—	53	2	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	#335	—	106	4	no	—
94	#335	—	106	4	no	—
95	#322	—	53	1	no	—
96	#322	—	53	2	no	—
97	#322	#6157	1	1	yes	characteristic
98	C₅₃	#8826	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	106	1	no	—
100	C₂×M₁₆	—	106	1	no	—
101	C₂×M₁₆	—	106	2	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₂²×C₈	—	53	4	no	—
104	C₂×C₄²	—	53	1	no	—
105	#108	—	212	4	no	—
106	#108	—	212	4	no	—
107	C₄⋊C₈	—	106	2	no	—
108	C₄⋊C₈	—	106	2	no	—
109	C₄⋊C₈	—	106	2	no	—
110	C₄⋊C₈	—	106	2	no	—
111	C₄×C₈	—	53	4	no	—
112	C₄×C₈	—	53	2	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	2	no	—
115	C₄×C₈	C₂×#6157	1	2	yes	—
116	C₄×C₈	C₂×#6157	1	1	yes	characteristic
117	C₂²×C₄	—	53	1	no	—
118	C₂²×C₄	—	53	2	no	—
119	M₁₆	—	212	2	no	—
120	M₁₆	—	212	2	no	—
121	M₁₆	—	212	4	no	—
122	C₂×C₈	—	53	2	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	212	4	no	—
127	C₂×C₈	—	212	1	no	—
128	C₂×C₈	—	212	2	no	—
129	C₂×C₈	—	212	1	no	—
130	C₂×C₈	C₄⋊#6157	1	4	yes	—
131	C₂×C₈	C₄⋊#6157	1	2	yes	—
132	C₄²	—	53	2	no	—
133	C₄²	—	53	1	no	—
134	C₄²	C₂²×#6157	1	1	yes	characteristic
135	C₂³	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	53	2	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	106	4	no	—
141	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	53	4	no	—
145	C₈	—	106	4	no	—
146	C₈	—	424	1	no	—
147	C₈	—	424	1	no	—
148	C₈	—	424	2	no	—
149	C₈	#13133286	1	4	yes	—
150	C₂²	—	53	2	no	—
151	C₂²	—	106	2	no	—
152	C₂²	#16811569	1	1	yes	characteristic
153	C₄	—	53	1	no	—
154	C₄	—	53	1	no	—
155	C₄	—	106	2	no	—
156	C₄	C₂×#13133286	1	2	yes	—
157	C₄	#16811638	1	4	yes	—
158	C₂	—	106	2	no	—
159	C₂	#20290651	1	1	yes	characteristic
160	C₂	#20290657	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290656

C₂

yes

maximal, characteristic

#20290654

C₂

yes

maximal, characteristic

#20290274

C₂

yes

maximal

#20290273

C₂

yes

maximal

#322×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133288

—

C₂×#13133287

—

#16811627

C₂²

yes

characteristic

#16811627

C₂²

yes

characteristic

#16811627

C₂²

yes

—

#16811487

—

#16811486

—

C₄×C₈×D₅₃

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810345

C₄

yes

—

#16810344

C₄

yes

characteristic

C₅₃×#322

C₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₈×D₅₃

D₄

yes

—

C₂×C₈×D₅₃

Q₈

yes

—

C₄²×D₅₃

C₂³

yes

characteristic

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

—

—

—

—

—

—

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

D₈

yes

—

C₈×D₅₃

Q₁₆

yes

—

C₂²×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₅₃

#127

yes

characteristic

C₂×Dic₅₃

C₄⋊Q₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₈

yes

—

C₄×D₅₃

C₂×Q₁₆

yes

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

#88322

—

#88322

#107

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

#500

yes

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#430

yes

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

—

—

—

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1849

yes

—

D₅₃

—

C₂×#108

—

106

—

C₂×#108

—

106

—

#423

—

#423

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#335

—

106

—

#335

—

106

—

—

—

yes

C₅₃

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

100

C₂×M₁₆

—

106

—

101

C₂×M₁₆

—

106

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

#108

—

212

—

106

#108

—

212

—

107

C₄⋊C₈

—

106

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

—

106

—

110

C₄⋊C₈

—

106

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₂×#6157

yes

—

116

C₄×C₈

C₂×#6157

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

212

—

120

M₁₆

—

212

—

121

M₁₆

—

212

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

212

—

127

C₂×C₈

—

212

—

128

C₂×C₈

—

212

—

129

C₂×C₈

—

212

—

130

C₂×C₈

C₄⋊#6157

yes

—

131

C₂×C₈

C₄⋊#6157

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²×#6157

yes

characteristic

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

106

—

141

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#6157

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

106

—

146

C₈

—

424

—

147

C₈

—

424

—

148

C₈

—

424

—

149

C₈

#13133286

yes

—

150

C₂²

—

151

C₂²

—

106

—

152

C₂²

#16811569

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

106

—

156

C₄

C₂×#13133286

yes

—

157

C₄

#16811638

yes

—

158

C₂

—

106

—

159

C₂

#20290651

yes

characteristic

160

C₂

#20290657

yes

—

	subgroup	G/H
center	C₂²	#16811569
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#322	C₄
socle	C₂²×C₅₃	#384
prime radical	C₂²×D₅₃	C₄⋊Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#322	#6157
hypocenter	C₅₃	#8826
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16811569

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#322

C₄

socle

C₂²×C₅₃

#384

prime radical

C₂²×D₅₃

C₄⋊Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
98. C₅₃	#8826	semidirect

H ≅ G/N

type

98. C₅₃

#8826

semidirect

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

subgroup	index
#20290656	2
#20290654	2
#20290274	2
#20290273	2
#322×D₅₃	2
#8826	53

subgroup

index

#322×D₅₃

Subgroups — #28303616

Conjugacy classes of subgroups

Complements

Maximal subgroups