Group #27221052 — subgroups

Subgroups — C₅₃×#13540 #27221052

Order 13568.

234 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 102 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×#1946	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1389	C₂	1	2	yes	maximal
4	C₅₃×#1388	C₂	1	2	yes	maximal
5	C₅₃×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
7	C₂×#108×C₅₃	—	2	1	no	—
8	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#108×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#427	C₂²	1	1	yes	characteristic
11	C₅₃×#424	—	2	1	no	—
12	C₅₃×#423	—	2	1	no	—
13	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#339	—	2	1	no	—
15	C₂×M₁₆×C₅₃	—	2	1	no	—
16	C₂×M₁₆×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	#117×C₅₃	—	2	1	no	—
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	M₃₂×C₅₃	—	2	2	no	—
22	M₃₂×C₅₃	—	2	2	no	—
23	C₂×C₁₆×C₅₃	—	2	2	no	—
24	#108×C₅₃	—	2	2	no	—
25	#108×C₅₃	—	2	4	no	—
26	#108×C₅₃	—	4	2	no	—
27	#107×C₅₃	—	2	1	no	—
28	#106×C₅₃	Q₈	1	2	yes	—
29	C₄⋊C₈×C₅₃	—	4	2	no	—
30	C₄⋊C₈×C₅₃	—	4	2	no	—
31	#97×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	M₁₆×C₅₃	—	4	1	no	—
35	M₁₆×C₅₃	—	4	2	no	—
36	M₁₆×C₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	4	1	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	D₈	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂²⋊C₄×C₅₃	—	4	1	no	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	—	4	1	no	—
51	C₄²×C₅₃	SD₁₆	1	2	yes	—
52	C₁₆×C₅₃	—	4	2	no	—
53	C₂³×C₅₃	#137	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	—	8	1	no	—
57	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	—	8	1	no	—
65	C₈×C₅₃	—	8	2	no	—
66	C₈×C₅₃	#125	1	2	yes	—
67	C₈×C₅₃	#126	1	2	yes	—
68	#13540	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₅₃	—	4	1	no	—
70	C₂²×C₅₃	#474	1	1	yes	characteristic
71	C₂²×C₅₃	#483	1	1	yes	characteristic
72	C₂²×C₅₃	#491	1	1	yes	characteristic
73	C₄×C₅₃	—	4	2	no	—
74	C₄×C₅₃	—	8	2	no	—
75	C₄×C₅₃	#400	1	1	yes	characteristic
76	C₄×C₅₃	#475	1	1	yes	characteristic
77	C₄×C₅₃	#480	1	1	yes	characteristic
78	C₄×C₅₃	#486	1	1	yes	characteristic
79	#1946	C₂×C₅₃	1	1	yes	characteristic
80	#1849	C₂×C₅₃	1	1	yes	characteristic
81	#1389	C₂×C₅₃	1	2	yes	—
82	#1388	C₂×C₅₃	1	2	yes	—
83	#1375	C₂×C₅₃	1	1	yes	characteristic
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	#2099	1	1	yes	characteristic
88	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
89	C₂×#108	—	2	1	no	—
90	C₂×#108	C₂²×C₅₃	1	2	yes	—
91	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
92	#427	C₂²×C₅₃	1	1	yes	characteristic
93	#424	—	2	1	no	—
94	#423	—	2	1	no	—
95	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
96	#339	—	2	1	no	—
97	C₅₃	#13540	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
101	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	2	2	no	—
108	#108	—	2	4	no	—
109	#108	—	4	2	no	—
110	#107	—	2	1	no	—
111	#106	Q₈×C₅₃	1	2	yes	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	D₈×C₅₃	1	2	yes	—
126	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
127	C₂×C₈	#41×C₅₃	1	2	yes	—
128	C₂×C₈	#41×C₅₃	1	2	yes	—
129	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	4	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	SD₁₆×C₅₃	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂³	#137×C₅₃	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
141	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
142	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
145	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	8	2	no	—
148	C₈	—	8	1	no	—
149	C₈	#125×C₅₃	1	2	yes	—
150	C₈	#126×C₅₃	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	C₅₃×#474	1	1	yes	characteristic
153	C₂²	C₅₃×#483	1	1	yes	characteristic
154	C₂²	C₅₃×#491	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	C₅₃×#400	1	1	yes	characteristic
158	C₄	C₅₃×#475	1	1	yes	characteristic
159	C₄	C₅₃×#480	1	1	yes	characteristic
160	C₄	C₅₃×#486	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	C₅₃×#2099	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1946

C₂

yes

maximal, characteristic

C₅₃×#1849

C₂

yes

maximal, characteristic

C₅₃×#1389

C₂

yes

maximal

C₅₃×#1388

C₂

yes

maximal

C₅₃×#1375

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×#108×C₅₃

—

C₂×#108×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#427

C₂²

yes

characteristic

C₅₃×#424

—

C₅₃×#423

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#339

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#108×C₅₃

—

#108×C₅₃

—

#108×C₅₃

—

#107×C₅₃

—

#106×C₅₃

Q₈

yes

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₁₆×C₅₃

—

C₂³×C₅₃

#137

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#125

yes

—

C₈×C₅₃

#126

yes

—

#13540

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1849

C₂×C₅₃

yes

characteristic

#1389

C₂×C₅₃

yes

—

#1388

C₂×C₅₃

yes

—

#1375

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2099

yes

characteristic

C₂×M₃₂

C₂²×C₅₃

yes

—

C₂×#108

—

C₂×#108

C₂²×C₅₃

yes

—

C₂×#108

C₂²×C₅₃

yes

characteristic

#427

C₂²×C₅₃

yes

characteristic

#424

—

#423

—

C₂×#97

C₂²×C₅₃

yes

characteristic

#339

—

C₅₃

#13540

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

D₄×C₅₃

yes

characteristic

101

C₂²×C₈

C₂³×C₅₃

yes

characteristic

102

#117

—

103

C₂×C₄²

D₄×C₅₃

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

108

#108

—

109

#108

—

110

#107

—

111

#106

Q₈×C₅₃

yes

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

D₈×C₅₃

yes

—

126

C₂×C₈

C₂×Q₈×C₅₃

yes

characteristic

127

C₂×C₈

#41×C₅₃

yes

—

128

C₂×C₈

#41×C₅₃

yes

—

129

C₂×C₈

#41×C₅₃

yes

characteristic

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₄²

SD₁₆×C₅₃

yes

—

135

C₁₆

—

136

C₂³

#137×C₅₃

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

characteristic

141

C₂×C₄

#123×C₅₃

yes

characteristic

142

C₂×C₄

#124×C₅₃

yes

characteristic

143

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

144

C₂×C₄

C₂×SD₁₆×C₅₃

yes

characteristic

145

C₂×C₄

#137×C₅₃

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

#125×C₅₃

yes

—

150

C₈

#126×C₅₃

yes

—

151

C₂²

—

152

C₂²

C₅₃×#474

yes

characteristic

153

C₂²

C₅₃×#483

yes

characteristic

154

C₂²

C₅₃×#491

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

C₅₃×#400

yes

characteristic

158

C₄

C₅₃×#475

yes

characteristic

159

C₄

C₅₃×#480

yes

characteristic

160

C₄

C₅₃×#486

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

C₅₃×#2099

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2099
prime radical	C₂³×C₅₃	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

#486

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

C₂×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

N	H ≅ G/N	type
68. #13540	C₅₃	direct (N × H)
97. C₅₃	#13540	direct (N × H)

H ≅ G/N

type

68. #13540

C₅₃

direct (N × H)

97. C₅₃

#13540

direct (N × H)

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

subgroup	index
C₅₃×#1946	2
C₅₃×#1849	2
C₅₃×#1389	2
C₅₃×#1388	2
C₅₃×#1375	2
#13540	53

subgroup

index

C₅₃×#1946

C₅₃×#1849

C₅₃×#1389

C₅₃×#1388

C₅₃×#1375

#13540

Subgroups — C₅₃×#13540 #27221052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13540 #27221052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13540 #27221052