Group #27219952 — subgroups

Subgroups — C₅₃×#12440 #27219952

Order 13568.

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2247	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1670	C₂	1	2	yes	maximal
4	C₅₃×#1371	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1362	C₂	1	2	yes	maximal
6	C₂×Q₃₂×C₅₃	—	2	1	no	—
7	C₂×Q₃₂×C₅₃	—	2	1	no	—
8	C₅₃×#494	C₂²	1	2	yes	—
9	C₅₃×#475	—	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	C₄×C₁₆×C₅₃	—	2	1	no	—
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#335	C₂²	1	2	yes	—
16	C₅₃×#332	—	2	2	no	—
17	C₂×Q₁₆×C₅₃	—	2	2	no	—
18	C₂×Q₁₆×C₅₃	—	2	2	no	—
19	C₄⋊Q₈×C₅₃	—	2	2	no	—
20	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
21	Q₃₂×C₅₃	—	4	2	no	—
22	Q₃₂×C₅₃	—	4	2	no	—
23	C₂×C₁₆×C₅₃	—	2	1	no	—
24	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
26	#106×C₅₃	—	4	2	no	—
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
30	#103×C₅₃	—	4	2	no	—
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	—	4	2	no	—
34	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
35	Q₁₆×C₅₃	—	4	2	no	—
36	Q₁₆×C₅₃	—	4	2	no	—
37	Q₁₆×C₅₃	—	8	2	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
46	C₄²×C₅₃	—	2	1	no	—
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₅₃	—	2	1	no	—
49	C₁₆×C₅₃	—	4	1	no	—
50	C₁₆×C₅₃	SD₁₆	1	2	yes	—
51	Q₈×C₅₃	—	2	1	no	—
52	Q₈×C₅₃	—	8	2	no	—
53	Q₈×C₅₃	—	8	2	no	—
54	Q₈×C₅₃	Q₃₂	1	2	yes	complemented
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	4	1	no	—
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	#12440	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₃	#466	1	1	yes	characteristic
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	4	1	no	—
71	C₄×C₅₃	—	16	2	no	—
72	C₄×C₅₃	#460	1	1	yes	characteristic
73	C₄×C₅₃	#472	1	1	yes	characteristic
74	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
75	C₄×C₅₃	#508	1	1	yes	characteristic
76	#2247	C₂×C₅₃	1	1	yes	characteristic
77	Q₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
78	#1670	C₂×C₅₃	1	2	yes	—
79	#1371	C₂×C₅₃	1	1	yes	characteristic
80	#1362	C₂×C₅₃	1	2	yes	—
81	C₂×C₅₃	#1668	1	1	yes	characteristic
82	C₂×C₅₃	#2216	1	1	yes	characteristic
83	C₂×C₅₃	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	1	no	—
86	#494	C₂²×C₅₃	1	2	yes	—
87	#475	—	2	2	no	—
88	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
93	#335	C₂²×C₅₃	1	2	yes	—
94	#332	—	2	2	no	—
95	C₅₃	#12440	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
100	Q₃₂	—	4	2	no	—
101	Q₃₂	—	4	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
104	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
105	#106	—	4	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
109	#103	—	4	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
120	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	SD₁₆×C₅₃	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	8	2	no	—
133	Q₈	Q₃₂×C₅₃	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
139	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	8	2	no	—
143	C₈	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
144	C₈	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
145	C₂²	C₅₃×#466	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	16	2	no	—
150	C₄	C₅₃×#460	1	1	yes	characteristic
151	C₄	C₅₃×#472	1	1	yes	characteristic
152	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
153	C₄	C₅₃×#508	1	1	yes	characteristic
154	C₂	C₅₃×#1668	1	1	yes	characteristic
155	C₂	C₅₃×#2216	1	1	yes	characteristic
156	C₂	C₅₃×#2222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2247

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#1670

C₂

yes

maximal

C₅₃×#1371

C₂

yes

maximal, characteristic

C₅₃×#1362

C₂

yes

maximal

C₂×Q₃₂×C₅₃

—

C₂×Q₃₂×C₅₃

—

C₅₃×#494

C₂²

yes

—

C₅₃×#475

—

C₈×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₂²

yes

—

C₅₃×#332

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

D₄

yes

characteristic

Q₃₂×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

#106×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

characteristic

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×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

—

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

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

Q₃₂

yes

complemented

C₂×C₄×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

#466

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

Q₈×C₁₆

C₂×C₅₃

yes

characteristic

#1670

C₂×C₅₃

yes

—

#1371

C₂×C₅₃

yes

characteristic

#1362

C₂×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂²×C₅₃

yes

—

#475

—

C₈×Q₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₅₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

#335

C₂²×C₅₃

yes

—

#332

—

C₅₃

#12440

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₄×C₅₃

yes

characteristic

100

Q₃₂

—

101

Q₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

D₄×C₅₃

yes

characteristic

104

C₂×C₁₆

D₄×C₅₃

yes

characteristic

105

#106

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

D₄×C₅₃

yes

characteristic

109

#103

—

110

C₄×C₈

—

111

C₄×C₈

C₂³×C₅₃

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

D₈×C₅₃

yes

characteristic

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

120

C₂×C₈

#41×C₅₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

D₈×C₅₃

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₄×C₅₃

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

SD₁₆×C₅₃

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

Q₃₂×C₅₃

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#121×C₅₃

yes

characteristic

138

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

139

C₂×C₄

#135×C₅₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

C₂×SD₁₆×C₅₃

yes

characteristic

144

C₈

AGL(1,ℤ/8)×C₅₃

yes

characteristic

145

C₂²

C₅₃×#466

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₅₃×#460

yes

characteristic

151

C₄

C₅₃×#472

yes

characteristic

152

C₄

C₂×Q₃₂×C₅₃

yes

characteristic

153

C₄

C₅₃×#508

yes

characteristic

154

C₂

C₅₃×#1668

yes

characteristic

155

C₂

C₅₃×#2216

yes

characteristic

156

C₂

C₅₃×#2222

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂²×C₅₃

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₅₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
54. Q₈×C₅₃	Q₃₂	semidirect
66. #12440	C₅₃	direct (N × H)
95. C₅₃	#12440	direct (N × H)
133. Q₈	Q₃₂×C₅₃	semidirect

H ≅ G/N

type

54. Q₈×C₅₃

Q₃₂

semidirect

66. #12440

C₅₃

direct (N × H)

95. C₅₃

#12440

direct (N × H)

133. Q₈

Q₃₂×C₅₃

semidirect

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

subgroup	index
C₅₃×#2247	2
Q₈×C₁₆×C₅₃	2
C₅₃×#1670	2
C₅₃×#1371	2
C₅₃×#1362	2
#12440	53

subgroup

index

C₅₃×#2247

Q₈×C₁₆×C₅₃

C₅₃×#1670

C₅₃×#1371

C₅₃×#1362

#12440

Subgroups — C₅₃×#12440 #27219952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12440 #27219952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12440 #27219952