Group #27217879 — subgroups

Subgroups — C₅₃×#10367 #27217879

Order 13568.

364 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2864	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1518	C₂	1	2	yes	maximal
4	C₅₃×#1286	C₂	1	2	yes	maximal
5	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₅₃×#423	C₂²	1	1	yes	characteristic
9	C₅₃×#389	—	2	2	no	—
10	C₅₃×#387	—	2	4	no	—
11	C₅₃×#382	C₂²	1	2	yes	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#339	—	2	2	no	—
14	C₅₃×#328	—	2	4	no	—
15	C₂×M₁₆×C₅₃	—	2	1	no	—
16	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
17	#125×C₅₃	—	2	1	no	—
18	C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₄×Q₈×C₅₃	—	2	2	no	—
20	#117×C₅₃	—	2	1	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	D₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₂²⋊C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	#95×C₅₃	—	2	4	no	—
31	#95×C₅₃	—	2	2	no	—
32	#95×C₅₃	D₄	1	4	yes	—
33	C₂×Q₈×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	4	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	—	4	1	no	—
41	C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₄⋊C₄×C₅₃	—	4	8	no	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄⋊C₄×C₅₃	—	4	2	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₂²⋊C₄×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	4	4	no	—
51	C₄²×C₅₃	—	4	1	no	—
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂³×C₅₃	#120	1	1	yes	characteristic
56	Q₈×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	8	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#124	1	2	yes	—
67	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#135	1	2	yes	—
69	C₂×C₄×C₅₃	#137	1	2	yes	—
70	C₈×C₅₃	—	8	2	no	—
71	C₈×C₅₃	—	8	1	no	—
72	#10367	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#393	1	1	yes	characteristic
75	C₂²×C₅₃	#452	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	4	4	no	—
79	C₄×C₅₃	—	8	4	no	—
80	C₄×C₅₃	—	8	2	no	—
81	C₄×C₅₃	#479	1	2	yes	—
82	C₄×C₅₃	#488	1	1	yes	characteristic
83	C₄×C₅₃	#491	1	1	yes	characteristic
84	#2864	C₂×C₅₃	1	1	yes	characteristic
85	#1994	C₂×C₅₃	1	1	yes	characteristic
86	#1518	C₂×C₅₃	1	2	yes	—
87	#1286	C₂×C₅₃	1	2	yes	—
88	#1276	C₂×C₅₃	1	1	yes	characteristic
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#2026	1	1	yes	characteristic
91	C₂×C₅₃	#2031	1	2	yes	—
92	#533	C₂²×C₅₃	1	1	yes	characteristic
93	#431	C₂²×C₅₃	1	2	yes	—
94	#423	C₂²×C₅₃	1	1	yes	characteristic
95	#389	—	2	2	no	—
96	#387	—	2	4	no	—
97	#382	C₂²×C₅₃	1	2	yes	—
98	C₄³	C₂²×C₅₃	1	1	yes	characteristic
99	#339	—	2	2	no	—
100	#328	—	2	4	no	—
101	C₅₃	#10367	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×M₁₆	—	2	1	no	—
103	C₄⋊Q₈	Q₈×C₅₃	1	2	yes	—
104	#125	—	2	1	no	—
105	C₂²⋊Q₈	—	2	2	no	—
106	C₄×Q₈	—	2	2	no	—
107	#117	—	2	1	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₄×C₅₃	1	2	yes	—
112	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	2	4	no	—
118	#95	—	2	2	no	—
119	#95	D₄×C₅₃	1	4	yes	—
120	C₂×Q₈	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	4	1	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	8	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	4	1	no	—
139	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
140	C₄²	#41×C₅₃	1	1	yes	characteristic
141	C₄²	#41×C₅₃	1	2	yes	—
142	C₂³	#120×C₅₃	1	1	yes	characteristic
143	Q₈	—	8	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
152	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
153	C₂×C₄	#124×C₅₃	1	2	yes	—
154	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#135×C₅₃	1	2	yes	—
156	C₂×C₄	#137×C₅₃	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	8	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	C₅₃×#393	1	1	yes	characteristic
161	C₂²	C₅₃×#452	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	4	no	—
166	C₄	—	8	2	no	—
167	C₄	C₅₃×#479	1	2	yes	—
168	C₄	C₅₃×#488	1	1	yes	characteristic
169	C₄	C₅₃×#491	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	C₅₃×#2026	1	1	yes	characteristic
172	C₂	C₅₃×#2031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2864

C₂

yes

maximal, characteristic

C₅₃×#1994

C₂

yes

maximal, characteristic

C₅₃×#1518

C₂

yes

maximal

C₅₃×#1286

C₂

yes

maximal

C₅₃×#1276

C₂

yes

maximal, characteristic

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#389

—

C₅₃×#387

—

C₅₃×#382

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#339

—

C₅₃×#328

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

—

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₂³×C₅₃

#120

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#10367

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#393

yes

characteristic

C₂²×C₅₃

#452

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#479

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1994

C₂×C₅₃

yes

characteristic

#1518

C₂×C₅₃

yes

—

#1286

C₂×C₅₃

yes

—

#1276

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#2026

yes

characteristic

C₂×C₅₃

#2031

yes

—

#533

C₂²×C₅₃

yes

characteristic

#431

C₂²×C₅₃

yes

—

#423

C₂²×C₅₃

yes

—

—

C₂²×C₅₃

yes

—

C₄³

C₂²×C₅₃

yes

characteristic

#339

—

100

#328

—

101

C₅₃

#10367

yes

characteristic, direct product factor, Sylow

102

C₂×M₁₆

—

103

C₄⋊Q₈

Q₈×C₅₃

yes

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

D₄×C₅₃

yes

—

112

C₂×C₄²

C₂³×C₅₃

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

119

#95

D₄×C₅₃

yes

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×D₄×C₅₃

yes

—

124

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂×Q₈×C₅₃

yes

characteristic

140

C₄²

#41×C₅₃

yes

characteristic

141

C₄²

#41×C₅₃

yes

—

142

C₂³

#120×C₅₃

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

152

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

characteristic

153

C₂×C₄

#124×C₅₃

yes

—

154

C₂×C₄

#124×C₅₃

yes

characteristic

155

C₂×C₄

#135×C₅₃

yes

—

156

C₂×C₄

#137×C₅₃

yes

—

157

C₈

—

158

C₈

—

159

C₂²

—

160

C₂²

C₅₃×#393

yes

characteristic

161

C₂²

C₅₃×#452

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

C₅₃×#479

yes

—

168

C₄

C₅₃×#488

yes

characteristic

169

C₄

C₅₃×#491

yes

characteristic

170

C₂

—

171

C₂

C₅₃×#2026

yes

characteristic

172

C₂

C₅₃×#2031

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#393

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂³×C₅₃

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
72. #10367	C₅₃	direct (N × H)
101. C₅₃	#10367	direct (N × H)

H ≅ G/N

type

72. #10367

C₅₃

direct (N × H)

101. C₅₃

#10367

direct (N × H)

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

subgroup	index
C₅₃×#2864	2
C₅₃×#1994	2
C₅₃×#1518	2
C₅₃×#1286	2
C₅₃×#1276	2
#10367	53

subgroup

index

C₅₃×#2864

C₅₃×#1994

C₅₃×#1518

C₅₃×#1286

C₅₃×#1276

#10367

Subgroups — C₅₃×#10367 #27217879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10367 #27217879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10367 #27217879