Group #27277763 — subgroups

Subgroups — #27277763

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	#20274867	C₂	1	2	yes	maximal
2	#20272550	C₂	1	4	yes	maximal
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	2	no	—
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#336	C₄	1	4	yes	—
10	C₂²×#88322	—	2	4	no	—
11	C₄⋊#88322	—	2	8	no	—
12	C₄⋊#88322	—	4	4	no	—
13	C₄⋊#88322	—	4	4	no	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	4	no	—
21	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	—	2	8	no	—
28	C₄²×C₅₃	—	2	4	no	—
29	C₄²×C₅₃	—	4	2	no	—
30	C₄²×C₅₃	D₈	1	2	yes	—
31	C₄²×C₅₃	SD₁₆	1	4	yes	—
32	C₄²×C₅₃	Q₁₆	1	2	yes	—
33	C₄²×C₅₃	#41	1	4	yes	—
34	C₂×#88322	—	4	4	no	—
35	C₂×#88322	—	4	4	no	—
36	C₂×#88322	—	4	4	no	—
37	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
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	#106	1	4	yes	—
44	C₂×C₄×C₅₃	#107	1	4	yes	—
45	C₂×C₄×C₅₃	#117	1	2	yes	—
46	C₂×C₄×C₅₃	#124	1	2	yes	—
47	C₂×C₄×C₅₃	#125	1	2	yes	—
48	C₂×C₄×C₅₃	#131	1	4	yes	—
49	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₈×C₅₃	—	4	4	no	—
53	#88322	—	8	4	no	—
54	#8787	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#382	1	1	yes	characteristic
56	C₂²×C₅₃	#424	1	2	yes	—
57	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
59	C₂²×C₅₃	#430	1	2	yes	—
60	C₄×C₅₃	—	2	4	no	—
61	C₄×C₅₃	—	2	4	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	#433	1	4	yes	—
64	C₄×C₅₃	#486	1	2	yes	—
65	C₄×C₅₃	#487	1	2	yes	—
66	#1842	—	53	2	no	—
67	#1744	D₅₃	1	1	yes	characteristic
68	#1277	—	53	4	no	—
69	C₂×C₅₃	#1570	1	2	yes	—
70	C₂×C₅₃	#1571	1	2	yes	—
71	C₂×C₅₃	#1832	1	1	yes	characteristic
72	C₂×C₅₃	#1834	1	1	yes	characteristic
73	C₂×C₅₃	#1838	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	53	4	no	—
75	C₂×C₄⋊C₈	—	106	2	no	—
76	C₂×C₄⋊C₈	—	106	2	no	—
77	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
78	C₄³	C₂×D₅₃	1	1	yes	characteristic
79	#336	Dic₅₃	1	4	yes	—
80	C₅₃	#8787	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	106	4	no	—
82	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
83	C₂×C₄²	Dic₁₀₆	1	2	yes	—
84	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
85	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
87	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
88	C₄⋊C₈	—	106	8	no	—
89	C₄⋊C₈	—	212	4	no	—
90	C₄⋊C₈	—	212	4	no	—
91	C₄×C₈	—	2	4	no	—
92	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
96	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	8	no	—
99	C₂×C₈	—	212	4	no	—
100	C₂×C₈	—	212	4	no	—
101	C₂×C₈	—	212	4	no	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	#11719913	1	4	yes	—
105	C₄²	D₄₂₄	1	2	yes	—
106	C₄²	Dic₂₁₂	1	2	yes	—
107	C₄²	#11719945	1	4	yes	—
108	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#13133149	1	4	yes	—
115	C₂×C₄	#13133151	1	4	yes	—
116	C₂×C₄	#13133176	1	2	yes	—
117	C₂×C₄	#13133205	1	2	yes	—
118	C₂×C₄	#13133209	1	2	yes	—
119	C₂×C₄	#13133224	1	4	yes	—
120	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
121	C₂×C₄	C₂×#11719913	1	2	yes	—
122	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
123	C₈	—	4	4	no	—
124	C₈	—	424	4	no	—
125	C₂²	#16810489	1	1	yes	characteristic
126	C₂²	#16810650	1	2	yes	—
127	C₂²	C₂×#13133149	1	1	yes	characteristic
128	C₂²	C₂×#13133151	1	1	yes	characteristic
129	C₂²	#16810672	1	2	yes	—
130	C₄	—	2	4	no	—
131	C₄	—	2	4	no	—
132	C₄	—	2	2	no	—
133	C₄	#16810684	1	4	yes	—
134	C₄	#16810970	1	2	yes	—
135	C₄	#16810974	1	2	yes	—
136	C₂	#20273486	1	2	yes	—
137	C₂	#20273490	1	2	yes	—
138	C₂	#20274835	1	1	yes	characteristic
139	C₂	#20274843	1	1	yes	characteristic
140	C₂	#20274856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274867

C₂

yes

maximal

#20272550

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

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

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

#41

yes

—

C₂×#88322

—

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×C₅₃

#106

yes

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

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

#433

yes

—

C₄×C₅₃

#486

yes

—

C₄×C₅₃

#487

yes

—

#1842

—

#1744

D₅₃

yes

characteristic

#1277

—

C₂×C₅₃

#1570

yes

—

C₂×C₅₃

#1571

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

Dic₅₃

yes

—

C₅₃

#8787

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

Dic₁₀₆

yes

—

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

C₄⋊C₈

—

212

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₅₃

yes

—

C₂²×C₄

C₄⋊Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

100

C₂×C₈

—

212

—

101

C₂×C₈

—

212

—

102

C₄²

—

103

C₄²

—

104

C₄²

#11719913

yes

—

105

C₄²

D₄₂₄

yes

—

106

C₄²

Dic₂₁₂

yes

—

107

C₄²

#11719945

yes

—

108

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#13133149

yes

—

115

C₂×C₄

#13133151

yes

—

116

C₂×C₄

#13133176

yes

—

117

C₂×C₄

#13133205

yes

—

118

C₂×C₄

#13133209

yes

—

119

C₂×C₄

#13133224

yes

—

120

C₂×C₄

C₂×D₄₂₄

yes

characteristic

121

C₂×C₄

C₂×#11719913

yes

—

122

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

123

C₈

—

124

C₈

—

424

—

125

C₂²

#16810489

yes

characteristic

126

C₂²

#16810650

yes

—

127

C₂²

C₂×#13133149

yes

characteristic

128

C₂²

C₂×#13133151

yes

characteristic

129

C₂²

#16810672

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#16810684

yes

—

134

C₄

#16810970

yes

—

135

C₄

#16810974

yes

—

136

C₂

#20273486

yes

—

137

C₂

#20273490

yes

—

138

C₂

#20274835

yes

characteristic

139

C₂

#20274843

yes

characteristic

140

C₂

#20274856

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	#8787
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
80. C₅₃	#8787	semidirect

H ≅ G/N

type

80. C₅₃

#8787

semidirect

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

subgroup	index
#20274867	2
#20272550	2
C₅₃×#1744	2
#8787	53

subgroup

index

#20274867

#20272550

C₅₃×#1744

#8787

Subgroups — #27277763

Conjugacy classes of subgroups

Complements

Maximal subgroups