Group #27219954 — subgroups

Subgroups — C₅₃×#12442 #27219954

Order 13568.

244 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₅₃×#2246	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1668	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1372	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₅₃	—	2	1	no	—
7	C₂×D₁₆×C₅₃	—	2	1	no	—
8	C₅₃×#493	C₂²	1	2	yes	—
9	C₅₃×#460	—	2	2	no	—
10	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×C₄⋊C₁₆	—	2	1	no	—
12	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₅₃×#346	—	2	1	no	—
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#335	C₂²	1	2	yes	—
16	C₅₃×#331	—	2	2	no	—
17	C₂×SD₁₆×C₅₃	—	4	2	no	—
18	C₂×D₈×C₅₃	—	2	2	no	—
19	C₂×D₈×C₅₃	—	2	2	no	—
20	#127×C₅₃	—	2	2	no	—
21	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
22	D₁₆×C₅₃	—	4	2	no	—
23	D₁₆×C₅₃	—	4	2	no	—
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
30	#102×C₅₃	—	4	2	no	—
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
34	C₂×D₄×C₅₃	—	4	2	no	—
35	C₂×D₄×C₅₃	—	4	2	no	—
36	C₂×D₄×C₅₃	—	4	2	no	—
37	SD₁₆×C₅₃	—	8	4	no	—
38	D₈×C₅₃	—	4	2	no	—
39	D₈×C₅₃	—	4	2	no	—
40	D₈×C₅₃	—	8	2	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₅₃	—	2	1	no	—
50	C₁₆×C₅₃	—	4	1	no	—
51	C₁₆×C₅₃	SD₁₆	1	2	yes	complemented
52	C₂³×C₅₃	—	8	2	no	—
53	Q₈×C₅₃	—	2	2	no	—
54	D₄×C₅₃	—	8	2	no	—
55	D₄×C₅₃	—	8	2	no	—
56	D₄×C₅₃	—	8	2	no	—
57	D₄×C₅₃	—	8	2	no	—
58	D₄×C₅₃	—	16	2	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	4	1	no	—
66	C₈×C₅₃	—	8	2	no	—
67	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	#12442	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₅₃	—	16	2	no	—
71	C₂²×C₅₃	—	16	2	no	—
72	C₂²×C₅₃	—	16	2	no	—
73	C₂²×C₅₃	#466	1	1	yes	characteristic
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	1	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	#460	1	1	yes	characteristic
78	C₄×C₅₃	#472	1	1	yes	characteristic
79	C₄×C₅₃	#509	1	2	yes	—
80	#2246	C₂×C₅₃	1	1	yes	characteristic
81	#2183	C₂×C₅₃	1	1	yes	characteristic, complemented
82	#1668	C₂×C₅₃	1	2	yes	complemented
83	#1372	C₂×C₅₃	1	1	yes	characteristic, complemented
84	#1361	C₂×C₅₃	1	2	yes	complemented
85	C₂×C₅₃	—	32	2	no	—
86	C₂×C₅₃	#1668	1	1	yes	characteristic
87	C₂×C₅₃	#2218	1	1	yes	characteristic
88	C₂×C₅₃	#2221	1	1	yes	characteristic
89	C₂×D₁₆	—	2	1	no	—
90	C₂×D₁₆	—	2	1	no	—
91	#493	C₂²×C₅₃	1	2	yes	—
92	#460	—	2	2	no	—
93	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
96	#346	—	2	1	no	—
97	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
98	#335	C₂²×C₅₃	1	2	yes	—
99	#331	—	2	2	no	—
100	C₅₃	#12442	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×SD₁₆	—	4	2	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	—	2	2	no	—
104	#127	—	2	2	no	—
105	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
106	D₁₆	—	4	2	no	—
107	D₁₆	—	4	2	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
110	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
114	#102	—	4	2	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
117	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
118	C₂×D₄	—	4	2	no	—
119	C₂×D₄	—	4	2	no	—
120	C₂×D₄	—	4	2	no	—
121	SD₁₆	—	8	4	no	—
122	D₈	—	4	2	no	—
123	D₈	—	4	2	no	—
124	D₈	—	8	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
128	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
131	C₄²	—	2	1	no	—
132	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
133	C₁₆	—	2	1	no	—
134	C₁₆	—	4	1	no	—
135	C₁₆	SD₁₆×C₅₃	1	2	yes	complemented
136	C₂³	—	8	2	no	—
137	Q₈	—	2	2	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	2	no	—
140	D₄	—	8	2	no	—
141	D₄	—	8	2	no	—
142	D₄	—	16	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
147	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	8	2	no	—
151	C₈	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
152	C₈	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
153	C₂²	—	16	2	no	—
154	C₂²	—	16	2	no	—
155	C₂²	—	16	2	no	—
156	C₂²	C₅₃×#466	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	4	1	no	—
160	C₄	C₅₃×#460	1	1	yes	characteristic
161	C₄	C₅₃×#472	1	1	yes	characteristic
162	C₄	C₅₃×#509	1	2	yes	—
163	C₂	—	32	2	no	—
164	C₂	C₅₃×#1668	1	1	yes	characteristic
165	C₂	C₅₃×#2218	1	1	yes	characteristic
166	C₂	C₅₃×#2221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2246

C₂

yes

maximal, characteristic

C₅₃×#2183

C₂

yes

maximal, characteristic, complemented

C₅₃×#1668

C₂

yes

maximal, complemented

C₅₃×#1372

C₂

yes

maximal, characteristic, complemented

C₅₃×#1361

C₂

yes

maximal, complemented

C₂×D₁₆×C₅₃

—

C₂×D₁₆×C₅₃

—

C₅₃×#493

C₂²

yes

—

C₅₃×#460

—

C₈×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₃×#346

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₂²

yes

—

C₅₃×#331

—

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

C₄×Q₈×C₅₃

D₄

yes

characteristic

D₁₆×C₅₃

—

D₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

SD₁₆

yes

complemented

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×SD₁₆

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#466

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2246

C₂×C₅₃

yes

characteristic

#2183

C₂×C₅₃

yes

characteristic, complemented

#1668

C₂×C₅₃

yes

complemented

#1372

C₂×C₅₃

yes

characteristic, complemented

#1361

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×D₁₆

—

C₂×D₁₆

—

#493

C₂²×C₅₃

yes

—

#460

—

C₈×Q₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₅₃

yes

characteristic

#346

—

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

#335

C₂²×C₅₃

yes

—

#331

—

100

C₅₃

#12442

yes

characteristic, direct product factor, Sylow

101

C₂×SD₁₆

—

102

C₂×D₈

—

103

C₂×D₈

—

104

#127

—

105

C₄×Q₈

D₄×C₅₃

yes

characteristic

106

D₁₆

—

107

D₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

D₄×C₅₃

yes

characteristic

110

C₂×C₁₆

D₄×C₅₃

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

D₄×C₅₃

yes

characteristic

114

#102

—

115

C₄×C₈

—

116

C₄×C₈

C₂³×C₅₃

yes

characteristic

117

C₂×Q₈

D₈×C₅₃

yes

characteristic

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

SD₁₆

—

122

D₈

—

123

D₈

—

124

D₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

128

C₂×C₈

#41×C₅₃

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

D₈×C₅₃

yes

characteristic

131

C₄²

—

132

C₄²

C₂×D₄×C₅₃

yes

characteristic

133

C₁₆

—

134

C₁₆

—

135

C₁₆

SD₁₆×C₅₃

yes

complemented

136

C₂³

—

137

Q₈

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#121×C₅₃

yes

characteristic

146

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

147

C₂×C₄

#135×C₅₃

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

C₂×SD₁₆×C₅₃

yes

characteristic

152

C₈

AGL(1,ℤ/8)×C₅₃

yes

characteristic

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₅₃×#466

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₅₃×#460

yes

characteristic

161

C₄

C₅₃×#472

yes

characteristic

162

C₄

C₅₃×#509

yes

—

163

C₂

—

164

C₂

C₅₃×#1668

yes

characteristic

165

C₂

C₅₃×#2218

yes

characteristic

166

C₂

C₅₃×#2221

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₅₃×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈×Q₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₅₃

prime radical

C₅₃×#2246

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈×Q₈×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#2183	C₂	semidirect
3. C₅₃×#1668	C₂	semidirect
4. C₅₃×#1372	C₂	semidirect
5. C₅₃×#1361	C₂	semidirect
51. C₁₆×C₅₃	SD₁₆	semidirect
69. #12442	C₅₃	direct (N × H)
81. #2183	C₂×C₅₃	semidirect
82. #1668	C₂×C₅₃	semidirect
83. #1372	C₂×C₅₃	semidirect
84. #1361	C₂×C₅₃	semidirect
100. C₅₃	#12442	direct (N × H)
135. C₁₆	SD₁₆×C₅₃	semidirect

H ≅ G/N

type

2. C₅₃×#2183

C₂

semidirect

3. C₅₃×#1668

C₂

semidirect

4. C₅₃×#1372

C₂

semidirect

5. C₅₃×#1361

C₂

semidirect

51. C₁₆×C₅₃

SD₁₆

semidirect

69. #12442

C₅₃

direct (N × H)

81. #2183

C₂×C₅₃

semidirect

82. #1668

C₂×C₅₃

semidirect

83. #1372

C₂×C₅₃

semidirect

84. #1361

C₂×C₅₃

semidirect

100. C₅₃

#12442

direct (N × H)

135. C₁₆

SD₁₆×C₅₃

semidirect

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

subgroup	index
C₅₃×#2246	2
C₅₃×#2183	2
C₅₃×#1668	2
C₅₃×#1372	2
C₅₃×#1361	2
#12442	53

subgroup

index

C₅₃×#2246

C₅₃×#2183

C₅₃×#1668

C₅₃×#1372

C₅₃×#1361

#12442

Subgroups — C₅₃×#12442 #27219954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12442 #27219954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12442 #27219954