Group #28303531 — subgroups

Subgroups — #28303531

Order 13568.

382 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 152 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	#20290630	C₂	1	2	yes	maximal
2	#20289545	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88324	C₂²	1	4	yes	—
5	#16811489	—	2	4	no	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂²×#88324	—	2	4	no	—
12	C₄⋊#88324	—	2	8	no	—
13	C₂²×#88322	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
18	C₄×#88322	—	2	2	no	—
19	C₈×Dic₅₃	—	2	4	no	—
20	C₈×Dic₅₃	—	2	4	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
27	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
29	C₂×#88324	—	4	4	no	—
30	C₂×#88324	—	4	8	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
34	C₄²×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	M₁₆	1	2	yes	—
36	C₄×Dic₅₃	—	2	2	no	—
37	C₄×Dic₅₃	—	2	4	no	—
38	C₄×Dic₅₃	—	2	4	no	—
39	C₄×Dic₅₃	#41	1	4	yes	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	M₁₆	1	4	yes	—
43	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
47	C₂×C₄×C₅₃	#117	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	2	no	—
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	#108	1	4	yes	—
55	C₂×Dic₅₃	#124	1	2	yes	—
56	C₂×Dic₅₃	#125	1	2	yes	—
57	C₂×Dic₅₃	#135	1	4	yes	—
58	#88324	—	8	4	no	—
59	C₈×C₅₃	—	2	4	no	—
60	#88322	—	2	4	no	—
61	#8785	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	#382	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂²×C₅₃	#423	1	1	yes	characteristic
65	C₂²×C₅₃	#427	1	2	yes	—
66	C₂²×C₅₃	C₂×#108	1	2	yes	—
67	C₄×C₅₃	—	2	4	no	—
68	C₄×C₅₃	#431	1	2	yes	—
69	C₄×C₅₃	#432	1	2	yes	—
70	Dic₅₃	—	2	4	no	—
71	Dic₅₃	—	2	2	no	—
72	Dic₅₃	#488	1	4	yes	—
73	#1831	—	53	2	no	—
74	C₄²×C₈	—	53	1	no	—
75	#1277	—	53	4	no	—
76	C₂×C₅₃	#1564	1	2	yes	—
77	C₂×C₅₃	#1568	1	2	yes	—
78	C₂×C₅₃	#1831	1	1	yes	characteristic
79	C₂×C₅₃	#1835	1	1	yes	characteristic
80	C₂×C₅₃	#1837	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	53	4	no	—
82	C₂×C₄×C₈	—	53	2	no	—
83	C₂×C₄×C₈	—	53	2	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	C₂×C₄×C₈	#6157	1	1	yes	characteristic
86	C₄³	—	53	1	no	—
87	#336	—	106	4	no	—
88	C₅₃	#8785	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	53	2	no	—
90	C₂²×C₈	—	106	4	no	—
91	C₂²×C₈	C₂×#6157	1	2	yes	—
92	C₂×C₄²	—	53	1	no	—
93	C₂×C₄²	—	53	2	no	—
94	C₂×C₄²	—	53	2	no	—
95	C₂×C₄²	—	53	1	no	—
96	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
97	C₄⋊C₈	—	106	8	no	—
98	C₄×C₈	—	106	4	no	—
99	C₄×C₈	—	106	4	no	—
100	C₄×C₈	—	106	2	no	—
101	C₄×C₈	#88324	1	4	yes	complemented
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	2	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	C₄⋊#6157	1	2	yes	—
106	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	53	4	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	212	4	no	—
113	C₂×C₈	—	212	8	no	—
114	C₂×C₈	C₂×#88324	1	4	yes	—
115	C₄²	—	53	4	no	—
116	C₄²	—	106	4	no	—
117	C₄²	—	106	4	no	—
118	C₄²	—	106	2	no	—
119	C₄²	#11719936	1	2	yes	—
120	C₄²	C₂×#88324	1	2	yes	—
121	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	53	4	no	—
126	C₂×C₄	—	53	2	no	—
127	C₂×C₄	—	53	4	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	C₄⋊#88324	1	4	yes	—
132	C₂×C₄	#13133296	1	2	yes	—
133	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
134	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
135	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₈	—	106	4	no	—
138	C₈	—	424	4	no	—
139	C₂²	#16811567	1	1	yes	characteristic
140	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
141	C₂²	#16811628	1	1	yes	characteristic
142	C₂²	#16811633	1	2	yes	—
143	C₂²	D₅₃⋊#108	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	53	4	no	—
146	C₄	—	106	2	no	—
147	C₄	—	106	4	no	—
148	C₄	#16811639	1	2	yes	—
149	C₄	#16811645	1	2	yes	—
150	C₂	#20290262	1	2	yes	—
151	C₂	#20290278	1	2	yes	—
152	C₂	#20290630	1	1	yes	characteristic
153	C₂	#20290637	1	1	yes	characteristic
154	C₂	#20290640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290630

C₂

yes

maximal

#20289545

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

C₂²

yes

—

#16811489

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂²×#88324

—

C₄⋊#88324

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

C₂×C₄×Dic₅₃

Q₈

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

C₄×#88322

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₄⋊C₄

yes

—

C₂²×Dic₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88324

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

M₁₆

yes

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#108

yes

—

C₂×Dic₅₃

#124

yes

—

C₂×Dic₅₃

#125

yes

—

C₂×Dic₅₃

#135

yes

—

#88324

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

—

C₄×C₅₃

#432

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#488

yes

—

#1831

—

C₄²×C₈

—

#1277

—

C₂×C₅₃

#1564

yes

—

C₂×C₅₃

#1568

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄³

—

#336

—

106

—

C₅₃

#8785

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×#6157

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₄⋊C₈

—

106

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

100

C₄×C₈

—

106

—

101

C₄×C₈

#88324

yes

complemented

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₄⋊#6157

yes

—

106

C₂²×C₄

C₂²×#6157

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

212

—

113

C₂×C₈

—

212

—

114

C₂×C₈

C₂×#88324

yes

—

115

C₄²

—

116

C₄²

—

106

—

117

C₄²

—

106

—

118

C₄²

—

106

—

119

C₄²

#11719936

yes

—

120

C₄²

C₂×#88324

yes

—

121

C₂³

C₂×C₄⋊#6157

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

106

—

131

C₂×C₄

C₄⋊#88324

yes

—

132

C₂×C₄

#13133296

yes

—

133

C₂×C₄

C₂²×#88324

yes

characteristic

134

C₂×C₄

D₅₃⋊M₁₆

yes

—

135

C₂×C₄

C₂×#11719936

yes

characteristic

136

C₈

—

137

C₈

—

106

—

138

C₈

—

424

—

139

C₂²

#16811567

yes

characteristic

140

C₂²

C₂×C₄⋊#88324

yes

characteristic

141

C₂²

#16811628

yes

characteristic

142

C₂²

#16811633

yes

—

143

C₂²

D₅₃⋊#108

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

106

—

147

C₄

—

106

—

148

C₄

#16811639

yes

—

149

C₄

#16811645

yes

—

150

C₂

#20290262

yes

—

151

C₂

#20290278

yes

—

152

C₂

#20290630

yes

characteristic

153

C₂

#20290637

yes

characteristic

154

C₂

#20290640

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8785
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₅₃

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
23. C₄×C₈×C₅₃	C₈	semidirect
88. C₅₃	#8785	semidirect
101. C₄×C₈	#88324	semidirect

H ≅ G/N

type

23. C₄×C₈×C₅₃

C₈

semidirect

88. C₅₃

#8785

semidirect

101. C₄×C₈

#88324

semidirect

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

subgroup	index
#20290630	2
#20289545	2
C₄×C₈×Dic₅₃	2
#8785	53

subgroup

index

#20290630

#20289545

C₄×C₈×Dic₅₃

#8785

Subgroups — #28303531

Conjugacy classes of subgroups

Complements

Maximal subgroups