Group #27270626 — subgroups

Subgroups — #27270626

Order 13568.

294 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20274832	C₂	1	2	yes	maximal
2	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	—	2	2	no	—
4	#16810378	—	2	2	no	—
5	#16810378	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₄³×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#336	—	2	2	no	—
9	C₂²×#88322	—	2	4	no	—
10	C₄⋊#88322	—	4	2	no	—
11	C₄⋊#88322	—	4	2	no	—
12	C₂²×C₈×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	D₄	1	2	yes	—
17	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
18	C₄×C₈×C₅₃	—	2	2	no	—
19	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
20	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	4	no	—
26	C₂×C₈×C₅₃	—	4	4	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	—	4	2	no	—
29	C₄²×C₅₃	—	4	2	no	—
30	C₄²×C₅₃	C₂×C₈	1	2	yes	—
31	C₄²×C₅₃	M₁₆	1	2	yes	—
32	C₄²×C₅₃	D₈	1	1	yes	characteristic
33	C₄²×C₅₃	SD₁₆	1	2	yes	—
34	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
35	C₂×#88322	—	4	4	no	—
36	C₂×#88322	—	4	4	no	—
37	C₂×#88322	—	4	4	no	—
38	C₂³×C₅₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
49	C₂×C₄×C₅₃	#102	1	2	yes	—
50	C₂×C₄×C₅₃	#103	1	2	yes	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#108	1	2	yes	—
55	C₈×C₅₃	—	4	4	no	—
56	#88322	—	8	4	no	—
57	#7018	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	#336	1	1	yes	characteristic
59	C₂²×C₅₃	H(3,ℤ/4)	1	2	yes	—
60	C₂²×C₅₃	#340	1	1	yes	characteristic
61	C₂²×C₅₃	#341	1	1	yes	characteristic
62	C₂²×C₅₃	#344	1	2	yes	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	#334	1	2	yes	—
70	C₄×C₅₃	#335	1	2	yes	—
71	#1831	—	53	2	no	—
72	#1744	D₅₃	1	1	yes	characteristic
73	C₂×C₅₃	#1274	1	1	yes	characteristic
74	C₂×C₅₃	#1277	1	1	yes	characteristic
75	C₂×C₅₃	#1281	1	1	yes	characteristic
76	C₂×C₅₃	#1286	1	2	yes	—
77	C₂×C₅₃	#1287	1	2	yes	—
78	C₂×C₄⋊C₈	—	106	2	no	—
79	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
80	C₄³	C₂×D₅₃	1	1	yes	characteristic
81	#336	—	2	2	no	—
82	#336	—	53	4	no	—
83	#336	—	106	2	no	—
84	C₅₃	#7018	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	106	4	no	—
87	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₅₃	1	2	yes	—
89	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
90	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
92	C₄⋊C₈	—	212	2	no	—
93	C₄⋊C₈	—	212	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	#88322	1	4	yes	complemented
96	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
97	C₂²×C₄	#11719919	1	2	yes	—
98	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
99	C₂²×C₄	#11719921	1	2	yes	—
100	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	212	4	no	—
104	C₂×C₈	—	212	4	no	—
105	C₂×C₈	—	212	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	—	4	2	no	—
109	C₄²	#11719913	1	2	yes	—
110	C₄²	D₄₂₄	1	1	yes	characteristic
111	C₄²	Dic₂₁₂	1	1	yes	characteristic
112	C₄²	C₂×#88322	1	2	yes	—
113	C₄²	#11719917	1	2	yes	—
114	C₂³	#13133124	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₄×#88322	1	1	yes	characteristic
123	C₂×C₄	#13133128	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊#88322	1	2	yes	—
125	C₂×C₄	#13133139	1	2	yes	—
126	C₂×C₄	#13133142	1	2	yes	—
127	C₂×C₄	C₄⋊#88322	1	2	yes	—
128	C₂×C₄	#13133149	1	1	yes	characteristic
129	C₂×C₄	#13133151	1	1	yes	characteristic
130	C₂×C₄	#13133153	1	2	yes	—
131	C₈	—	4	4	no	—
132	C₈	—	424	4	no	—
133	C₂²	#16810378	1	1	yes	characteristic
134	C₂²	#16810379	1	2	yes	—
135	C₂²	#16810386	1	1	yes	characteristic
136	C₂²	#16810388	1	1	yes	characteristic
137	C₂²	#16810394	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	4	2	no	—
144	C₄	#16810374	1	2	yes	—
145	C₄	#16810376	1	2	yes	—
146	C₂	#20272544	1	1	yes	characteristic
147	C₂	#20272550	1	1	yes	characteristic
148	C₂	#20272560	1	1	yes	characteristic
149	C₂	#20272574	1	2	yes	—
150	C₂	#20272577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

#16810378

—

#16810378

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

—

—

C₂³×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#334

yes

—

C₄×C₅₃

#335

yes

—

#1831

—

#1744

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1287

yes

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

Dic₅₃

yes

—

C₄³

C₂×D₅₃

yes

—

—

—

106

—

C₅₃

#7018

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

Dic₁₀₆

yes

characteristic

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₄⋊C₈

—

212

—

C₄⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

#88322

yes

complemented

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

C₂²×C₄

#11719919

yes

—

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂²×C₄

#11719921

yes

—

100

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

212

—

104

C₂×C₈

—

212

—

105

C₂×C₈

—

212

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

#11719913

yes

—

110

C₄²

D₄₂₄

yes

characteristic

111

C₄²

Dic₂₁₂

yes

characteristic

112

C₄²

C₂×#88322

yes

—

113

C₄²

#11719917

yes

—

114

C₂³

#13133124

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×#88322

yes

characteristic

123

C₂×C₄

#13133128

yes

characteristic

124

C₂×C₄

C₂²⋊#88322

yes

—

125

C₂×C₄

#13133139

yes

—

126

C₂×C₄

#13133142

yes

—

127

C₂×C₄

C₄⋊#88322

yes

—

128

C₂×C₄

#13133149

yes

characteristic

129

C₂×C₄

#13133151

yes

characteristic

130

C₂×C₄

#13133153

yes

—

131

C₈

—

132

C₈

—

424

—

133

C₂²

#16810378

yes

characteristic

134

C₂²

#16810379

yes

—

135

C₂²

#16810386

yes

characteristic

136

C₂²

#16810388

yes

characteristic

137

C₂²

#16810394

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#16810374

yes

—

145

C₄

#16810376

yes

—

146

C₂

#20272544

yes

characteristic

147

C₂

#20272550

yes

characteristic

148

C₂

#20272560

yes

characteristic

149

C₂

#20272574

yes

—

150

C₂

#20272577

yes

—

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#7018
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂³

#13133124

commutator

C₂×C₄×C₅₃

C₄×C₈

Frattini

C₄³

C₂×D₅₃

Fitting

C₅₃×#1744

C₂

socle, prime radical

C₂³×C₅₃

#95

cosocle

C₄³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
19. C₄×C₈×C₅₃	C₈	semidirect
84. C₅₃	#7018	semidirect
95. C₄×C₈	#88322	semidirect

H ≅ G/N

type

19. C₄×C₈×C₅₃

C₈

semidirect

84. C₅₃

#7018

semidirect

95. C₄×C₈

#88322

semidirect

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

subgroup	index
#20274832	2
C₅₃×#1744	2
#7018	53

subgroup

index

#20274832

C₅₃×#1744

#7018

Subgroups — #27270626

Conjugacy classes of subgroups

Complements

Maximal subgroups