Group #27277765 — subgroups

Subgroups — #27277765

Order 13568.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal
4	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	#16810378	—	2	2	no	—
8	#16810378	C₄	1	2	yes	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	2	no	—
13	C₂²×#88322	—	2	2	no	—
14	C₂²×#88322	C₂×C₄	1	4	yes	—
15	C₄⋊#88322	—	2	4	no	—
16	C₄×#88322	—	2	2	no	—
17	C₄×#88322	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	2	yes	—
22	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	—	2	4	no	—
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	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	—	4	2	no	—
31	C₂×C₈×C₅₃	—	4	4	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	—	4	2	no	—
36	C₄²×C₅₃	#41	1	2	yes	—
37	C₄²×C₅₃	#41	1	2	yes	—
38	C₂×#88322	—	2	4	no	—
39	C₂×#88322	—	2	4	no	—
40	C₂×#88322	—	2	4	no	—
41	C₂×#88322	—	4	2	no	—
42	C₂×#88322	—	4	4	no	—
43	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	#117	1	2	yes	—
52	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#125	1	2	yes	—
55	C₂×C₄×C₅₃	#131	1	2	yes	—
56	C₂×C₄×C₅₃	#131	1	2	yes	—
57	C₂×C₄×C₅₃	#135	1	4	yes	—
58	C₈×C₅₃	—	8	2	no	—
59	#88322	—	4	4	no	—
60	#88322	—	8	2	no	—
61	#8788	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	#382	1	1	yes	characteristic
63	C₂²×C₅₃	#424	1	1	yes	characteristic
64	C₂²×C₅₃	#424	1	1	yes	characteristic
65	C₂²×C₅₃	#427	1	2	yes	—
66	C₂²×C₅₃	#430	1	2	yes	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#433	1	4	yes	—
71	C₄×C₅₃	#488	1	2	yes	—
72	C₄×C₅₃	#488	1	2	yes	—
73	#1831	—	53	1	no	—
74	#1831	D₅₃	1	1	yes	characteristic
75	#1744	—	53	1	no	—
76	#1277	—	53	4	no	—
77	C₂×C₅₃	#1572	1	2	yes	—
78	C₂×C₅₃	#1572	1	2	yes	—
79	C₂×C₅₃	#1832	1	1	yes	characteristic
80	C₂×C₅₃	#1835	1	1	yes	characteristic
81	C₂×C₅₃	#1838	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	53	2	no	—
83	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
84	C₂×C₄×C₈	—	53	2	no	—
85	C₄³	C₂×D₅₃	1	1	yes	characteristic
86	#336	—	2	2	no	—
87	#336	—	53	2	no	—
88	#336	—	53	2	no	—
89	#336	—	106	2	no	—
90	C₅₃	#8788	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	53	4	no	—
93	C₂²×C₈	—	106	2	no	—
94	C₂×C₄²	Dic₁₀₆	1	2	yes	—
95	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
98	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	106	4	no	—
101	C₄×C₈	—	106	2	no	—
102	C₄×C₈	—	106	2	no	—
103	C₂²×C₄	#11719919	1	2	yes	—
104	C₂²×C₄	#11719919	1	2	yes	—
105	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	106	4	no	—
111	C₂×C₈	—	106	4	no	—
112	C₂×C₈	—	106	4	no	—
113	C₂×C₈	—	212	2	no	—
114	C₂×C₈	—	212	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	4	2	no	—
119	C₄²	#11719947	1	2	yes	—
120	C₄²	#11719949	1	2	yes	—
121	C₂³	C₂×#11719919	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#13133175	1	2	yes	—
130	C₂×C₄	#13133204	1	1	yes	characteristic
131	C₂×C₄	#13133206	1	1	yes	characteristic
132	C₂×C₄	#13133208	1	2	yes	—
133	C₂×C₄	#13133223	1	2	yes	—
134	C₂×C₄	#13133225	1	2	yes	—
135	C₂×C₄	#13133237	1	4	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	424	2	no	—
138	C₈	—	212	4	no	—
139	C₂²	#16810488	1	1	yes	characteristic
140	C₂²	#16810649	1	1	yes	characteristic
141	C₂²	#16810651	1	1	yes	characteristic
142	C₂²	#16810659	1	2	yes	—
143	C₂²	#16810671	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#16810683	1	4	yes	—
148	C₄	#16810977	1	2	yes	—
149	C₄	#16810980	1	2	yes	—
150	C₂	#20273493	1	2	yes	—
151	C₂	#20273496	1	2	yes	—
152	C₂	#20274834	1	1	yes	characteristic
153	C₂	#20274845	1	1	yes	characteristic
154	C₂	#20274855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal, characteristic

#20274461

C₂

yes

maximal, characteristic

#20272549

C₂

yes

maximal

C₅₃×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810378

—

#16810378

C₄

yes

—

#16810378

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

—

C₄⋊#88322

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

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

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#430

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#488

yes

—

C₄×C₅₃

#488

yes

—

#1831

—

#1831

D₅₃

yes

characteristic

#1744

—

#1277

—

C₂×C₅₃

#1572

yes

—

C₂×C₅₃

#1572

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₅₃

yes

—

—

—

—

106

—

C₅₃

#8788

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

Dic₁₀₆

yes

—

C₂×C₄²

C₄×D₅₃

yes

characteristic

C₂×C₄²

C₄×D₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

106

—

101

C₄×C₈

—

106

—

102

C₄×C₈

—

106

—

103

C₂²×C₄

#11719919

yes

—

104

C₂²×C₄

#11719919

yes

—

105

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

106

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

107

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

212

—

114

C₂×C₈

—

212

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

#11719947

yes

—

120

C₄²

#11719949

yes

—

121

C₂³

C₂×#11719919

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#13133175

yes

—

130

C₂×C₄

#13133204

yes

characteristic

131

C₂×C₄

#13133206

yes

characteristic

132

C₂×C₄

#13133208

yes

—

133

C₂×C₄

#13133223

yes

—

134

C₂×C₄

#13133225

yes

—

135

C₂×C₄

#13133237

yes

—

136

C₈

—

137

C₈

—

424

—

138

C₈

—

212

—

139

C₂²

#16810488

yes

characteristic

140

C₂²

#16810649

yes

characteristic

141

C₂²

#16810651

yes

characteristic

142

C₂²

#16810659

yes

—

143

C₂²

#16810671

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#16810683

yes

—

148

C₄

#16810977

yes

—

149

C₄

#16810980

yes

—

150

C₂

#20273493

yes

—

151

C₂

#20273496

yes

—

152

C₂

#20274834

yes

characteristic

153

C₂

#20274845

yes

characteristic

154

C₂

#20274855

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#8788
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1831

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 ◁ #1831

N	H ≅ G/N	type
90. C₅₃	#8788	semidirect

H ≅ G/N

type

90. C₅₃

#8788

semidirect

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

subgroup	index
#20274832	2
#20274461	2
#20272549	2
C₅₃×#1831	2
#8788	53

subgroup

index

C₅₃×#1831

Subgroups — #27277765

Conjugacy classes of subgroups

Complements

Maximal subgroups