Group #27270689 — subgroups

Subgroups — #27270689

Order 13568.

304 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20274568	C₂	1	2	yes	maximal
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	—	2	2	no	—
4	C₂×#13133128	C₄	1	4	yes	—
5	#16810378	—	2	2	no	—
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	—	2	8	no	—
11	#13133128	—	4	4	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	—	4	2	no	—
19	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	—	4	2	no	—
25	C₂×C₈×C₅₃	—	4	4	no	—
26	C₄²×C₅₃	—	2	1	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	2	no	—
30	C₄²×C₅₃	—	2	4	no	—
31	C₄²×C₅₃	—	2	2	no	—
32	C₄²×C₅₃	—	4	1	no	—
33	C₂×#88322	—	2	8	no	—
34	C₂×#88322	—	4	4	no	—
35	C₂×#88322	—	4	4	no	—
36	C₂³×C₅₃	#95	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	1	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	1	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	#100	1	2	yes	—
47	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#101	1	2	yes	—
49	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#104	1	4	yes	—
51	C₂×C₄×C₅₃	#108	1	2	yes	—
52	C₈×C₅₃	—	8	2	no	—
53	#88322	—	4	8	no	—
54	#7042	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
56	C₂²×C₅₃	#339	1	2	yes	—
57	C₂²×C₅₃	#341	1	1	yes	characteristic
58	C₂²×C₅₃	#343	1	2	yes	—
59	C₂²×C₅₃	#343	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	4	2	no	—
64	C₄×C₅₃	—	4	1	no	—
65	C₄×C₅₃	#329	1	2	yes	—
66	C₄×C₅₃	#330	1	2	yes	—
67	#1831	D₅₃	1	1	yes	characteristic
68	#1767	—	53	2	no	—
69	C₂×C₅₃	#1285	1	2	yes	—
70	C₂×C₅₃	#1290	1	1	yes	characteristic
71	C₂×C₅₃	#1295	1	1	yes	characteristic
72	C₂×C₅₃	#1299	1	2	yes	—
73	C₂×C₅₃	#1309	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	106	2	no	—
75	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
76	C₂×#97	—	53	4	no	—
77	C₄³	C₂×D₅₃	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	#336	—	106	2	no	—
80	C₅₃	#7042	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	106	4	no	—
83	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
84	C₂×C₄²	C₄×D₅₃	1	2	yes	—
85	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	106	8	no	—
90	#97	—	212	4	no	—
91	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
92	C₂²×C₄	#11719919	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₂²×C₄	#11719921	1	2	yes	—
95	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	106	8	no	—
99	C₂×C₈	—	212	4	no	—
100	C₂×C₈	—	212	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	1	no	—
108	C₂³	#13133124	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	#13133133	1	2	yes	—
119	C₂×C₄	#13133134	1	1	yes	characteristic
120	C₂×C₄	#13133135	1	2	yes	—
121	C₂×C₄	#13133136	1	1	yes	characteristic
122	C₂×C₄	#13133144	1	4	yes	—
123	C₂×C₄	#13133153	1	2	yes	—
124	C₈	—	8	2	no	—
125	C₈	—	212	8	no	—
126	C₂²	#16810380	1	1	yes	characteristic
127	C₂²	#16810382	1	2	yes	—
128	C₂²	#16810388	1	1	yes	characteristic
129	C₂²	#16810391	1	2	yes	—
130	C₂²	#16810392	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	1	no	—
136	C₄	#16810363	1	2	yes	—
137	C₄	#16810365	1	2	yes	—
138	C₂	#20272572	1	2	yes	—
139	C₂	#20272585	1	1	yes	characteristic
140	C₂	#20272596	1	1	yes	characteristic
141	C₂	#20272606	1	2	yes	—
142	C₂	#20272628	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274568

C₂

yes

maximal

C₅₃×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×#13133128

C₄

yes

—

#16810378

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×#88322

—

C₄⋊#88322

—

#13133128

—

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

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

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

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#100

yes

—

C₂×C₄×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

#101

yes

—

C₂×C₄×C₅₃

#101

yes

characteristic

C₂×C₄×C₅₃

#104

yes

—

C₂×C₄×C₅₃

#108

yes

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#341

yes

characteristic

C₂²×C₅₃

#343

yes

—

C₂²×C₅₃

#343

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#329

yes

—

C₄×C₅₃

#330

yes

—

#1831

D₅₃

yes

characteristic

#1767

—

C₂×C₅₃

#1285

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1309

yes

characteristic

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

Dic₅₃

yes

—

C₂×#97

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

#336

—

106

—

C₅₃

#7042

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

—

C₄⋊C₈

—

106

—

#97

—

212

—

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

C₂²×C₄

#11719919

yes

—

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂²×C₄

#11719921

yes

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

100

C₂×C₈

—

212

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

#13133124

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#13133133

yes

—

119

C₂×C₄

#13133134

yes

characteristic

120

C₂×C₄

#13133135

yes

—

121

C₂×C₄

#13133136

yes

characteristic

122

C₂×C₄

#13133144

yes

—

123

C₂×C₄

#13133153

yes

—

124

C₈

—

125

C₈

—

212

—

126

C₂²

#16810380

yes

characteristic

127

C₂²

#16810382

yes

—

128

C₂²

#16810388

yes

characteristic

129

C₂²

#16810391

yes

—

130

C₂²

#16810392

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#16810363

yes

—

137

C₄

#16810365

yes

—

138

C₂

#20272572

yes

—

139

C₂

#20272585

yes

characteristic

140

C₂

#20272596

yes

characteristic

141

C₂

#20272606

yes

—

142

C₂

#20272628

yes

characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#7042
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

#13133124

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₄³

C₂×D₅₃

Fitting

C₅₃×#1831

C₂

socle, prime radical

C₂³×C₅₃

#95

cosocle

C₄³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
80. C₅₃	#7042	semidirect

H ≅ G/N

type

80. C₅₃

#7042

semidirect

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

subgroup	index
#20274568	2
C₅₃×#1831	2
#7042	53

subgroup

index

#20274568

C₅₃×#1831

#7042

Subgroups — #27270689

Conjugacy classes of subgroups

Complements

Maximal subgroups