Group #27271191 — subgroups

Subgroups — #27271191

Order 13568.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20273474	C₂	1	2	yes	maximal
2	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#16810647	—	2	2	no	—
4	#16810376	—	2	2	no	—
5	#16810376	C₄	1	4	yes	—
6	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₄⋊C₁₆	—	2	2	no	—
9	C₂×#11719917	—	4	2	no	—
10	C₄⋊#88322	—	2	4	no	—
11	C₄⋊#88322	—	4	2	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	—	2	2	no	—
16	C₂×C₁₆×C₅₃	—	4	2	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	4	1	no	—
25	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₅₃	SD₁₆	1	2	yes	—
30	#11719917	—	8	2	no	—
31	#11719917	—	8	2	no	—
32	C₂×#88322	—	4	4	no	—
33	C₁₆×C₅₃	—	4	2	no	—
34	C₂³×C₅₃	#107	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#102	1	2	yes	—
40	C₂×C₄×C₅₃	#103	1	2	yes	—
41	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
44	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	1	no	—
47	C₈×C₅₃	#100	1	1	yes	characteristic
48	C₈×C₅₃	#101	1	1	yes	characteristic
49	#88322	—	8	4	no	—
50	#7241	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	#340	1	1	yes	characteristic
54	C₂²×C₅₃	#366	1	1	yes	characteristic
55	C₂²×C₅₃	#367	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	#338	1	1	yes	characteristic
59	C₄×C₅₃	#343	1	1	yes	characteristic
60	C₄×C₅₃	#357	1	2	yes	—
61	C₄×C₅₃	#358	1	2	yes	—
62	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
63	#1567	—	53	2	no	—
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#1292	1	1	yes	characteristic
67	C₂×C₅₃	#1380	1	1	yes	characteristic
68	C₂×C₅₃	#1385	1	1	yes	characteristic
69	C₂²×C₁₆	Dic₅₃	1	2	yes	—
70	#423	—	106	2	no	—
71	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₄⋊C₁₆	—	2	2	no	—
73	#335	—	53	4	no	—
74	#335	—	106	2	no	—
75	C₅₃	#7241	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	212	2	no	—
77	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	106	4	no	—
83	C₄⋊C₈	—	212	2	no	—
84	C₄×C₈	C₄×D₅₃	1	2	yes	—
85	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
86	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
87	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
88	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
89	M₁₆	—	424	2	no	—
90	M₁₆	—	424	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	212	4	no	—
95	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	#11719919	1	2	yes	—
97	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
98	C₄²	#11719913	1	2	yes	—
99	C₄²	#11719921	1	2	yes	—
100	C₁₆	—	4	2	no	—
101	C₂³	#13133151	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	#13133124	1	1	yes	characteristic
106	C₂×C₄	#13133139	1	2	yes	—
107	C₂×C₄	#13133142	1	2	yes	—
108	C₂×C₄	#13133149	1	1	yes	characteristic
109	C₂×C₄	D₈₄₈	1	1	yes	characteristic
110	C₂×C₄	#13133162	1	2	yes	—
111	C₂×C₄	Dic₄₂₄	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	424	4	no	—
115	C₈	#13133134	1	1	yes	characteristic
116	C₈	#13133136	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16810386	1	1	yes	characteristic
120	C₂²	#16810449	1	1	yes	characteristic
121	C₂²	#16810451	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	#16810381	1	1	yes	characteristic
125	C₄	#16810392	1	1	yes	characteristic
126	C₄	#16810425	1	2	yes	—
127	C₄	#16810428	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#20272590	1	1	yes	characteristic
131	C₂	#20272807	1	1	yes	characteristic
132	C₂	#20272819	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273474

C₂

yes

maximal

C₂×C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#16810647

—

#16810376

—

#16810376

C₄

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₃₂

yes

—

C₂×C₄×C₅₃

Q₃₂

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#358

yes

—

C₂×C₄⋊C₁₆

D₅₃

yes

characteristic

#1567

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

Dic₅₃

yes

—

#423

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

106

—

C₅₃

#7241

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

D₄₂₄

yes

characteristic

C₂²×C₄

Dic₂₁₂

yes

characteristic

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

M₁₆

—

424

—

M₁₆

—

424

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₄×Dic₅₃

yes

characteristic

C₂×C₈

#11719919

yes

—

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

C₄²

#11719913

yes

—

C₄²

#11719921

yes

—

100

C₁₆

—

101

C₂³

#13133151

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#13133124

yes

characteristic

106

C₂×C₄

#13133139

yes

—

107

C₂×C₄

#13133142

yes

—

108

C₂×C₄

#13133149

yes

characteristic

109

C₂×C₄

D₈₄₈

yes

characteristic

110

C₂×C₄

#13133162

yes

—

111

C₂×C₄

Dic₄₂₄

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

424

—

115

C₈

#13133134

yes

characteristic

116

C₈

#13133136

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

#16810386

yes

characteristic

120

C₂²

#16810449

yes

characteristic

121

C₂²

#16810451

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#16810381

yes

characteristic

125

C₄

#16810392

yes

characteristic

126

C₄

#16810425

yes

—

127

C₄

#16810428

yes

—

128

C₂

—

129

C₂

—

130

C₂

#20272590

yes

characteristic

131

C₂

#20272807

yes

characteristic

132

C₂

#20272819

yes

characteristic

	subgroup	G/H
center	C₂²	#16810386
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#7241
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#16810386

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×C₄⋊C₁₆

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₅₃

hypocenter

C₅₃

#7241

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
75. C₅₃	#7241	semidirect

H ≅ G/N

type

75. C₅₃

#7241

semidirect

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

subgroup	index
#20273474	2
C₂×C₅₃×C₄⋊C₁₆	2
#7241	53

subgroup

index

#20273474

C₂×C₅₃×C₄⋊C₁₆

#7241

Subgroups — #27271191

Conjugacy classes of subgroups

Complements

Maximal subgroups