Group #27270848 — subgroups

Subgroups — #27270848

Order 13568.

242 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20272797	C₂	1	2	yes	maximal
2	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×#103×C₅₃	—	2	2	no	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#384	C₄	1	1	yes	characteristic
8	#13133157	—	2	4	no	—
9	#13133157	—	4	2	no	—
10	C₂×#11719908	—	4	2	no	—
11	C₂²×Q₈×C₅₃	—	2	1	no	—
12	C₂²×C₈×C₅₃	D₄	1	2	yes	—
13	C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₄⋊Q₈×C₅₃	—	4	1	no	—
15	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#103×C₅₃	—	4	2	no	—
21	#103×C₅₃	—	4	2	no	—
22	C₄×C₈×C₅₃	—	4	1	no	—
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	4	1	no	—
25	C₂×Q₈×C₅₃	—	4	2	no	—
26	C₂×Q₈×C₅₃	—	4	2	no	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	—	4	1	no	—
29	C₂²×C₄×C₅₃	—	4	1	no	—
30	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	D₈	1	2	yes	—
34	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
35	C₄⋊C₄×C₅₃	—	4	2	no	—
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₄⋊C₄×C₅₃	—	8	2	no	—
39	C₄²×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
42	#11719908	—	4	4	no	—
43	C₂³×C₅₃	#100	1	1	yes	characteristic
44	Q₈×C₅₃	—	4	4	no	—
45	Q₈×C₅₃	—	8	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	—	8	1	no	—
52	C₂×C₄×C₅₃	—	8	1	no	—
53	C₂×C₄×C₅₃	—	8	2	no	—
54	C₂×C₄×C₅₃	—	8	1	no	—
55	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#102	1	2	yes	—
58	C₂×C₄×C₅₃	#104	1	2	yes	—
59	C₈×C₅₃	—	2	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	#7108	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	#323	1	1	yes	characteristic
65	C₂²×C₅₃	#329	1	1	yes	characteristic
66	C₂²×C₅₃	#331	1	1	yes	characteristic
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	8	2	no	—
70	C₄×C₅₃	—	16	1	no	—
71	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
72	C₄×C₅₃	#327	1	2	yes	—
73	#1956	D₅₃	1	1	yes	characteristic
74	#1376	—	53	2	no	—
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#1270	1	1	yes	characteristic
78	C₂×C₅₃	#1358	1	1	yes	characteristic
79	C₂×C₅₃	#1359	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
81	C₂×M₃₂	—	106	2	no	—
82	C₂×#103	—	2	2	no	—
83	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	#384	Dic₅₃	1	1	yes	characteristic
85	C₅₃	#7108	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	2	1	no	—
87	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
88	C₄⋊Q₈	—	2	1	no	—
89	C₄⋊Q₈	—	4	1	no	—
90	C₄⋊Q₈	#88322	1	2	yes	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	4	1	no	—
93	C₂×C₄⋊C₄	—	4	1	no	—
94	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	M₃₂	—	106	4	no	—
96	M₃₂	—	212	2	no	—
97	C₂×C₁₆	—	212	2	no	—
98	#103	—	4	2	no	—
99	#103	—	4	2	no	—
100	C₄×C₈	—	4	1	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	4	1	no	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	#11719922	1	2	yes	—
112	C₂×C₈	#11719923	1	2	yes	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂×#88322	1	1	yes	characteristic
119	C₄²	#11719917	1	1	yes	characteristic
120	C₁₆	—	212	4	no	—
121	C₂³	#13133134	1	1	yes	characteristic
122	Q₈	—	4	4	no	—
123	Q₈	—	8	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
134	C₂×C₄	#13133132	1	1	yes	characteristic
135	C₂×C₄	#13133138	1	2	yes	—
136	C₂×C₄	#13133144	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#16810347	1	1	yes	characteristic
142	C₂²	#16810363	1	1	yes	characteristic
143	C₂²	#16810367	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	8	2	no	—
147	C₄	—	16	1	no	—
148	C₄	D₄⋊#88322	1	2	yes	—
149	C₄	#16810357	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	#20272536	1	1	yes	characteristic
153	C₂	#20272754	1	1	yes	characteristic
154	C₂	#20272756	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272797

C₂

yes

maximal

C₅₃×#1956

C₂

yes

maximal, characteristic

C₂×#13133157

—

C₂×C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₄

yes

—

—

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#103×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

#100

yes

characteristic

Q₈×C₅₃

—

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

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#7108

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

D₄⋊C₈

yes

—

C₄×C₅₃

#327

yes

—

#1956

D₅₃

yes

characteristic

#1376

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

Dic₅₃

yes

characteristic

C₂×M₃₂

—

106

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

Dic₅₃

yes

characteristic

C₅₃

#7108

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#88322

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

212

—

#103

—

#103

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#11719922

yes

—

112

C₂×C₈

#11719923

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

C₂×#88322

yes

characteristic

119

C₄²

#11719917

yes

characteristic

120

C₁₆

—

212

—

121

C₂³

#13133134

yes

characteristic

122

Q₈

—

123

Q₈

—

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

—

133

C₂×C₄

C₂²⋊#88322

yes

characteristic

134

C₂×C₄

#13133132

yes

characteristic

135

C₂×C₄

#13133138

yes

—

136

C₂×C₄

#13133144

yes

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

#16810347

yes

characteristic

142

C₂²

#16810363

yes

characteristic

143

C₂²

#16810367

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

D₄⋊#88322

yes

—

149

C₄

#16810357

yes

—

150

C₂

—

151

C₂

—

152

C₂

#20272536

yes

characteristic

153

C₂

#20272754

yes

characteristic

154

C₂

#20272756

yes

characteristic

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1956	C₂
socle	C₂²×C₅₃	#331
prime radical	C₂³×C₅₃	#100
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1956	D₅₃
hypocenter	C₅₃	#7108
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1956

subgroup

G/H

center

C₂²

#16810367

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1956

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1956

N	H ≅ G/N	type
85. C₅₃	#7108	semidirect

H ≅ G/N

type

85. C₅₃

#7108

semidirect

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

subgroup	index
#20272797	2
C₅₃×#1956	2
#7108	53

subgroup

index

#20272797

C₅₃×#1956

#7108

Subgroups — #27270848

Conjugacy classes of subgroups

Complements

Maximal subgroups