Group #27271311 — subgroups

Subgroups — #27271311

Order 13568.

250 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20273039	C₂	1	2	yes	maximal
2	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic
3	#16810676	—	2	2	no	—
4	#16810349	—	2	2	no	—
5	#16810349	C₄	1	4	yes	—
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#336	—	2	2	no	—
10	C₄⋊#88322	—	4	2	no	—
11	C₂²⋊#88322	—	2	4	no	—
12	C₄×#88322	—	4	2	no	—
13	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	—	2	1	no	—
15	C₂²×C₈×C₅₃	—	2	1	no	—
16	C₄×Q₈×C₅₃	—	4	1	no	—
17	C₄×Q₈×C₅₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	2	yes	—
22	C₄⋊C₈×C₅₃	—	4	1	no	—
23	#97×C₅₃	—	4	1	no	—
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	—	4	1	no	—
26	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	4	1	no	—
33	C₂×C₈×C₅₃	—	4	1	no	—
34	C₂×C₈×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	4	1	no	—
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₄²×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₂×#88322	—	4	4	no	—
43	C₂³×C₅₃	#95	1	1	yes	characteristic
44	Q₈×C₅₃	—	4	4	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#108	1	4	yes	—
58	C₈×C₅₃	—	4	2	no	—
59	C₈×C₅₃	—	8	1	no	—
60	#88322	—	8	4	no	—
61	#7286	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	#341	1	2	yes	—
65	C₂²×C₅₃	#342	1	1	yes	characteristic
66	C₄×C₅₃	—	2	4	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	4	1	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	—	4	1	no	—
71	C₄×C₅₃	#355	1	1	yes	characteristic
72	C₄×C₅₃	#356	1	1	yes	characteristic
73	#1985	D₅₃	1	1	yes	characteristic
74	#1461	—	53	2	no	—
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	#1298	1	1	yes	characteristic
77	C₂×C₅₃	#1391	1	1	yes	characteristic
78	C₂×C₅₃	#1394	1	1	yes	characteristic
79	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
80	#431	—	106	2	no	—
81	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
82	C₂×#97	Dic₅₃	1	1	yes	characteristic
83	#336	—	2	2	no	—
84	#324	—	53	4	no	—
85	#324	—	106	2	no	—
86	C₅₃	#7286	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	—	2	1	no	—
90	C₄×Q₈	—	4	1	no	—
91	C₄×Q₈	—	4	1	no	—
92	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
93	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
96	C₄⋊C₈	—	4	1	no	—
97	C₄⋊C₈	—	212	2	no	—
98	C₂²⋊C₈	—	106	4	no	—
99	#97	—	4	1	no	—
100	C₄×C₈	—	212	2	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
104	C₂²×C₄	#11719919	1	2	yes	—
105	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
106	C₂²×C₄	#11719921	1	2	yes	—
107	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	212	4	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#13133124	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	#13133132	1	1	yes	characteristic
133	C₂×C₄	#13133132	1	1	yes	characteristic
134	C₂×C₄	#13133152	1	4	yes	—
135	C₈	—	4	2	no	—
136	C₈	—	8	1	no	—
137	C₈	—	424	4	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#16810387	1	2	yes	—
141	C₂²	#16810390	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	4	no	—
144	C₄	—	4	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	#16810420	1	1	yes	characteristic
148	C₄	#16810422	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#20272602	1	1	yes	characteristic
151	C₂	#20272836	1	1	yes	characteristic
152	C₂	#20272842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273039

C₂

yes

maximal

C₅₃×#1985

C₂

yes

maximal, characteristic

#16810676

—

#16810349

—

#16810349

C₄

yes

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×#97×C₅₃

C₄

yes

characteristic

C₅₃×#336

—

C₄⋊#88322

—

C₂²⋊#88322

—

C₄×#88322

—

C₂²×Q₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

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

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

#95

yes

characteristic

Q₈×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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#7286

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#341

yes

—

C₂²×C₅₃

#342

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1461

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

#431

—

106

—

C₂×C₄⋊C₈

Dic₅₃

yes

characteristic

C₂×#97

Dic₅₃

yes

—

—

—

106

—

C₅₃

#7286

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×D₅₃

yes

—

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

212

—

C₂²⋊C₈

—

106

—

#97

—

100

C₄×C₈

—

212

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

104

C₂²×C₄

#11719919

yes

—

105

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

106

C₂²×C₄

#11719921

yes

—

107

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

212

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂³

#13133124

yes

characteristic

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#13133132

yes

characteristic

133

C₂×C₄

#13133132

yes

characteristic

134

C₂×C₄

#13133152

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

424

—

138

C₂²

—

139

C₂²

—

140

C₂²

#16810387

yes

—

141

C₂²

#16810390

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#16810420

yes

characteristic

148

C₄

#16810422

yes

characteristic

149

C₂

—

150

C₂

#20272602

yes

characteristic

151

C₂

#20272836

yes

characteristic

152

C₂

#20272842

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₅₃
Fitting	C₅₃×#1985	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂³×C₅₃	#95
cosocle	C₂×C₄×Q₈×C₅₃	C₂²
hypercenter	#1985	D₅₃
hypocenter	C₅₃	#7286
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

subgroup

G/H

center

C₂×C₄

#13133132

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₅₃

Fitting

C₅₃×#1985

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×Q₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

N	H ≅ G/N	type
86. C₅₃	#7286	semidirect

H ≅ G/N

type

86. C₅₃

#7286

semidirect

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

subgroup	index
#20273039	2
C₅₃×#1985	2
#7286	53

subgroup

index

#20273039

C₅₃×#1985

#7286

Subgroups — #27271311

Conjugacy classes of subgroups

Complements

Maximal subgroups