Group #27220397 — subgroups

Subgroups — C₅₃×#12885 #27220397

Order 13568.

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2153	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1379	C₂	1	2	yes	maximal
5	C₅₃×#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#405	C₂²	1	2	yes	—
10	C₅₃×#405	C₂²	1	1	yes	characteristic
11	C₅₃×#364	—	2	2	no	—
12	C₅₃×C₄⋊C₁₆	—	2	2	no	—
13	C₅₃×#338	C₄	1	1	yes	characteristic
14	C₅₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
15	C₅₃×H(3,ℤ/4)	C₄	1	2	yes	—
16	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	#117×C₅₃	C₂×C₄	1	2	yes	—
20	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₅₃	—	2	2	no	—
22	M₃₂×C₅₃	—	2	2	no	—
23	M₃₂×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	—	2	1	no	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	—	2	2	no	—
28	#97×C₅₃	—	2	2	no	—
29	#97×C₅₃	Q₈	1	2	yes	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₅₃	—	2	2	no	—
34	M₁₆×C₅₃	—	2	2	no	—
35	M₁₆×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₂×C₈×C₅₃	#41	1	2	yes	—
42	C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂²⋊C₄×C₅₃	—	2	1	no	—
45	C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₁₆×C₅₃	—	4	1	no	—
49	C₁₆×C₅₃	—	4	1	no	—
50	C₁₆×C₅₃	—	4	2	no	—
51	C₂³×C₅₃	#131	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#117	1	2	yes	—
56	C₂×C₄×C₅₃	#131	1	2	yes	—
57	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
58	C₈×C₅₃	—	2	2	no	—
59	C₈×C₅₃	—	4	2	no	—
60	C₈×C₅₃	#124	1	1	yes	characteristic
61	C₈×C₅₃	#124	1	1	yes	characteristic
62	C₈×C₅₃	#125	1	2	yes	—
63	#12885	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₅₃	—	4	1	no	—
65	C₂²×C₅₃	#424	1	1	yes	characteristic
66	C₂²×C₅₃	#433	1	2	yes	—
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	—	8	2	no	—
69	C₄×C₅₃	#382	1	1	yes	characteristic
70	C₄×C₅₃	#424	1	1	yes	characteristic
71	C₄×C₅₃	#433	1	2	yes	—
72	#2153	C₂×C₅₃	1	1	yes	characteristic
73	#2151	C₂×C₅₃	1	1	yes	characteristic
74	#1747	C₂×C₅₃	1	1	yes	characteristic
75	#1379	C₂×C₅₃	1	2	yes	—
76	#1378	C₂×C₅₃	1	2	yes	—
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	2	no	—
79	C₂×C₅₃	#1832	1	1	yes	characteristic
80	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
81	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
82	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
83	#405	C₂²×C₅₃	1	2	yes	—
84	#405	C₂²×C₅₃	1	1	yes	characteristic
85	#364	—	2	2	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	#338	C₄×C₅₃	1	1	yes	characteristic
88	H(3,ℤ/4)	C₄×C₅₃	1	1	yes	characteristic
89	H(3,ℤ/4)	C₄×C₅₃	1	2	yes	—
90	C₅₃	#12885	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
92	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
93	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
94	#117	C₂×C₄×C₅₃	1	2	yes	—
95	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	2	2	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	#97	—	2	2	no	—
104	#97	Q₈×C₅₃	1	2	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	D₄×C₅₃	1	2	yes	—
107	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
113	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
115	C₂×C₈	#41×C₅₃	1	2	yes	—
116	C₂×C₈	#41×C₅₃	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	4	2	no	—
126	C₂³	#131×C₅₃	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₄	#117×C₅₃	1	2	yes	—
131	C₂×C₄	#131×C₅₃	1	2	yes	—
132	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	#124×C₅₃	1	1	yes	characteristic
136	C₈	#124×C₅₃	1	1	yes	characteristic
137	C₈	#125×C₅₃	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	C₅₃×#424	1	1	yes	characteristic
140	C₂²	C₅₃×#433	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	C₅₃×#382	1	1	yes	characteristic
144	C₄	C₅₃×#424	1	1	yes	characteristic
145	C₄	C₅₃×#433	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	C₅₃×#1832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2153

C₂

yes

maximal, characteristic

C₅₃×#2151

C₂

yes

maximal, characteristic

C₅₃×#1747

C₂

yes

maximal, characteristic

C₅₃×#1379

C₂

yes

maximal

C₅₃×#1378

C₂

yes

maximal

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#405

C₂²

yes

—

C₅₃×#405

C₂²

yes

characteristic

C₅₃×#364

—

C₅₃×C₄⋊C₁₆

—

C₅₃×#338

C₄

yes

characteristic

C₅₃×H(3,ℤ/4)

C₄

yes

characteristic

C₅₃×H(3,ℤ/4)

C₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

C₂×C₄

yes

—

#117×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

#97×C₅₃

—

#97×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#131

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

#12885

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#424

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2153

C₂×C₅₃

yes

characteristic

#2151

C₂×C₅₃

yes

characteristic

#1747

C₂×C₅₃

yes

characteristic

#1379

C₂×C₅₃

yes

—

#1378

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1832

yes

characteristic

C₂×M₃₂

C₂²×C₅₃

yes

characteristic

C₂×M₃₂

C₂²×C₅₃

yes

—

C₂²×C₁₆

C₂²×C₅₃

yes

characteristic

#405

C₂²×C₅₃

yes

—

#405

C₂²×C₅₃

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

C₄×C₅₃

yes

characteristic

H(3,ℤ/4)

C₄×C₅₃

yes

characteristic

H(3,ℤ/4)

C₄×C₅₃

yes

—

C₅₃

#12885

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₂×C₄×C₅₃

yes

—

C₂×M₁₆

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

C₂³×C₅₃

yes

characteristic

#117

C₂×C₄×C₅₃

yes

—

#117

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#97

—

104

#97

Q₈×C₅₃

yes

—

105

C₄×C₈

—

106

C₄×C₈

D₄×C₅₃

yes

—

107

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

113

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

114

C₂×C₈

C₂×Q₈×C₅₃

yes

characteristic

115

C₂×C₈

#41×C₅₃

yes

—

116

C₂×C₈

#41×C₅₃

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₄⋊C₄×C₅₃

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

#131×C₅₃

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

130

C₂×C₄

#117×C₅₃

yes

—

131

C₂×C₄

#131×C₅₃

yes

—

132

C₂×C₄

#131×C₅₃

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

#124×C₅₃

yes

characteristic

136

C₈

#124×C₅₃

yes

characteristic

137

C₈

#125×C₅₃

yes

—

138

C₂²

—

139

C₂²

C₅₃×#424

yes

characteristic

140

C₂²

C₅₃×#433

yes

—

141

C₄

—

142

C₄

—

143

C₄

C₅₃×#382

yes

characteristic

144

C₄

C₅₃×#424

yes

characteristic

145

C₄

C₅₃×#433

yes

—

146

C₂

—

147

C₂

—

148

C₂

C₅₃×#1832

yes

characteristic

	subgroup	G/H
center	C₈×C₅₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1832
prime radical	C₂³×C₅₃	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₈×C₅₃

#124

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

C₂×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
63. #12885	C₅₃	direct (N × H)
90. C₅₃	#12885	direct (N × H)

H ≅ G/N

type

63. #12885

C₅₃

direct (N × H)

90. C₅₃

#12885

direct (N × H)

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

subgroup	index
C₅₃×#2153	2
C₅₃×#2151	2
C₅₃×#1747	2
C₅₃×#1379	2
C₅₃×#1378	2
#12885	53

subgroup

index

C₅₃×#2153

C₅₃×#2151

C₅₃×#1747

C₅₃×#1379

C₅₃×#1378

#12885

Subgroups — C₅₃×#12885 #27220397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12885 #27220397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12885 #27220397