Group #27277852 — subgroups

Subgroups — #27277852

Order 13568.

362 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#16810376	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	2	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	#16810378	—	2	4	no	—
7	#16810376	—	2	4	no	—
8	#16810374	—	2	4	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	C₄	1	4	yes	—
13	C₂²×#88322	—	2	4	no	—
14	C₄⋊#88322	—	2	4	no	—
15	C₄⋊#88322	—	2	4	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₃	D₄	1	2	yes	—
20	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	2	yes	—
23	C₄×C₈×C₅₃	D₄	1	2	yes	—
24	C₄×C₈×C₅₃	Q₈	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
27	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	—	2	4	no	—
31	C₂×C₈×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
34	C₄²×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	—	4	2	no	—
37	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×#88322	—	4	4	no	—
42	C₂×#88322	—	4	8	no	—
43	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#131	1	4	yes	—
55	C₂×C₄×C₅₃	#135	1	4	yes	—
56	C₈×C₅₃	—	2	4	no	—
57	C₈×C₅₃	—	2	4	no	—
58	#88322	—	8	4	no	—
59	#8811	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	#384	1	1	yes	characteristic
61	C₂²×C₅₃	#424	1	2	yes	—
62	C₂²×C₅₃	#427	1	2	yes	—
63	C₂²×C₅₃	#430	1	2	yes	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	4	no	—
67	C₄×C₅₃	#424	1	2	yes	—
68	C₄×C₅₃	#424	1	2	yes	—
69	C₄×C₅₃	#495	1	2	yes	—
70	C₄×C₅₃	#499	1	2	yes	—
71	#1831	—	53	2	no	—
72	#1744	D₅₃	1	1	yes	characteristic
73	C₂×#335	—	53	2	no	—
74	C₂×#334	—	53	2	no	—
75	C₂×C₅₃	#1572	1	4	yes	—
76	C₂×C₅₃	#1843	1	1	yes	characteristic
77	C₂×C₅₃	#1846	1	1	yes	characteristic
78	C₂×C₅₃	#1849	1	1	yes	characteristic
79	C₂×C₄⋊C₈	—	53	4	no	—
80	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
82	C₄³	C₂×D₅₃	1	1	yes	characteristic
83	#336	—	106	4	no	—
84	#336	Dic₅₃	1	4	yes	—
85	#335	—	106	4	no	—
86	#334	—	106	4	no	—
87	C₅₃	#8811	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	106	4	no	—
89	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
90	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
91	C₂×C₄²	Dic₁₀₆	1	2	yes	—
92	C₂×C₄²	D₂₁₂	1	2	yes	—
93	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
94	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	C₄⋊C₈	—	106	4	no	—
96	C₄⋊C₈	—	106	4	no	—
97	C₄×C₈	Dic₁₀₆	1	2	yes	—
98	C₄×C₈	Dic₁₀₆	1	2	yes	—
99	C₄×C₈	D₂₁₂	1	2	yes	—
100	C₄×C₈	D₂₁₂	1	2	yes	—
101	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
102	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	212	4	no	—
108	C₂×C₈	—	212	8	no	—
109	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
110	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	4	2	no	—
114	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
115	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
116	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
117	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
118	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
125	C₂×C₄	#13133214	1	1	yes	characteristic
126	C₂×C₄	#13133217	1	1	yes	characteristic
127	C₂×C₄	#13133217	1	1	yes	characteristic
128	C₂×C₄	#13133217	1	1	yes	characteristic
129	C₂×C₄	#13133224	1	4	yes	—
130	C₂×C₄	#13133239	1	4	yes	—
131	C₈	—	2	4	no	—
132	C₈	—	2	4	no	—
133	C₈	—	424	4	no	—
134	C₂²	#16810493	1	1	yes	characteristic
135	C₂²	#16810650	1	2	yes	—
136	C₂²	#16810661	1	2	yes	—
137	C₂²	#16810672	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	#16810650	1	2	yes	—
142	C₄	#16810650	1	2	yes	—
143	C₄	#16811007	1	2	yes	—
144	C₄	#16811025	1	2	yes	—
145	C₂	#20273495	1	4	yes	—
146	C₂	#20274869	1	1	yes	characteristic
147	C₂	#20274879	1	1	yes	characteristic
148	C₂	#20274890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal

C₂×#16810376

C₂

yes

maximal

C₂×#16810374

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

#16810378

—

#16810376

—

#16810374

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#427

yes

—

C₂²×C₅₃

#430

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#499

yes

—

#1831

—

#1744

D₅₃

yes

characteristic

C₂×#335

—

C₂×#334

—

C₂×C₅₃

#1572

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄³

C₂×D₅₃

yes

characteristic

#336

—

106

—

#336

Dic₅₃

yes

—

#335

—

106

—

#334

—

106

—

C₅₃

#8811

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

Dic₁₀₆

yes

—

C₂×C₄²

D₂₁₂

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

—

100

C₄×C₈

D₂₁₂

yes

—

101

C₂²×C₄

C₄⋊Dic₅₃

yes

—

102

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

103

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

104

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

212

—

108

C₂×C₈

—

212

—

109

C₂×C₈

C₄⋊Dic₅₃

yes

—

110

C₂×C₈

C₄⋊Dic₅₃

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×Dic₁₀₆

yes

characteristic

115

C₄²

C₂×Dic₁₀₆

yes

characteristic

116

C₄²

C₂×D₂₁₂

yes

characteristic

117

C₄²

C₂×D₂₁₂

yes

characteristic

118

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

124

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

125

C₂×C₄

#13133214

yes

characteristic

126

C₂×C₄

#13133217

yes

characteristic

127

C₂×C₄

#13133217

yes

characteristic

128

C₂×C₄

#13133217

yes

characteristic

129

C₂×C₄

#13133224

yes

—

130

C₂×C₄

#13133239

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

424

—

134

C₂²

#16810493

yes

characteristic

135

C₂²

#16810650

yes

—

136

C₂²

#16810661

yes

—

137

C₂²

#16810672

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#16810650

yes

—

142

C₄

#16810650

yes

—

143

C₄

#16811007

yes

—

144

C₄

#16811025

yes

—

145

C₂

#20273495

yes

—

146

C₂

#20274869

yes

characteristic

147

C₂

#20274879

yes

characteristic

148

C₂

#20274890

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8811
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1744

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
87. C₅₃	#8811	semidirect

H ≅ G/N

type

87. C₅₃

#8811

semidirect

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

subgroup	index
#20274832	2
C₂×#16810376	2
C₂×#16810374	2
C₅₃×#1744	2
#8811	53

subgroup

index

C₅₃×#1744

Subgroups — #27277852

Conjugacy classes of subgroups

Complements

Maximal subgroups