Group #27217291 — subgroups

Subgroups — C₅₃×#9779 #27217291

Order 13568.

454 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1546	C₂	1	4	yes	maximal
4	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₅₃	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#494	—	2	2	no	—
8	C₅₃×#494	C₄	1	4	yes	—
9	C₅₃×#431	C₂²	1	2	yes	—
10	C₅₃×#423	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#341	—	2	2	no	—
13	C₅₃×#332	—	2	8	no	—
14	C₅₃×#322	—	2	2	no	—
15	C₂²×Q₈×C₅₃	—	2	2	no	—
16	C₂×Q₁₆×C₅₃	—	2	8	no	—
17	C₂×Q₁₆×C₅₃	—	4	8	no	—
18	C₂×M₁₆×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	D₄	1	2	yes	—
20	C₄⋊Q₈×C₅₃	—	2	2	no	—
21	C₄⋊Q₈×C₅₃	—	2	4	no	—
22	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	—	2	4	no	—
26	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
27	C₂²⋊C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	D₄	1	2	yes	—
31	C₂×Q₈×C₅₃	—	2	4	no	—
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Q₈×C₅₃	—	4	8	no	—
34	C₂²×C₄×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	2	yes	—
37	Q₁₆×C₅₃	—	4	16	no	—
38	M₁₆×C₅₃	—	4	4	no	—
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₄⋊C₄×C₅₃	—	2	8	no	—
44	C₄⋊C₄×C₅₃	—	4	4	no	—
45	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×D₄	1	2	yes	—
48	C₂³×C₅₃	#124	1	1	yes	characteristic
49	Q₈×C₅₃	—	4	8	no	—
50	Q₈×C₅₃	—	8	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	—	8	4	no	—
54	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	#121	1	2	yes	—
57	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
59	C₈×C₅₃	—	2	4	no	—
60	C₈×C₅₃	—	4	4	no	—
61	C₈×C₅₃	—	8	2	no	—
62	#9779	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#390	1	1	yes	characteristic
65	C₂²×C₅₃	#490	1	2	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	8	4	no	—
68	C₄×C₅₃	#418	1	2	yes	—
69	C₄×C₅₃	#473	1	4	yes	—
70	C₄×C₅₃	#496	1	2	yes	—
71	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
72	#1837	C₂×C₅₃	1	1	yes	characteristic
73	#1546	C₂×C₅₃	1	4	yes	—
74	#1452	C₂×C₅₃	1	1	yes	characteristic
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	#1961	1	1	yes	characteristic
77	C₂×C₅₃	#1979	1	2	yes	—
78	C₂²×Q₁₆	—	2	2	no	—
79	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
80	#494	—	2	2	no	—
81	#494	C₄×C₅₃	1	4	yes	—
82	#431	C₂²×C₅₃	1	2	yes	—
83	#423	C₂²×C₅₃	1	2	yes	—
84	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
85	#341	—	2	2	no	—
86	#332	—	2	8	no	—
87	#322	—	2	2	no	—
88	C₅₃	#9779	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×Q₈	—	2	2	no	—
90	C₂×Q₁₆	—	2	8	no	—
91	C₂×Q₁₆	—	4	8	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	D₄×C₅₃	1	2	yes	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
101	C₂²⋊C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
104	C₄×C₈	D₄×C₅₃	1	2	yes	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	8	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
110	C₂²×C₄	#41×C₅₃	1	2	yes	—
111	Q₁₆	—	4	16	no	—
112	M₁₆	—	4	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
120	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
121	C₄²	C₂×D₄×C₅₃	1	2	yes	—
122	C₂³	#124×C₅₃	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
130	C₂×C₄	#121×C₅₃	1	2	yes	—
131	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
132	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	4	4	no	—
135	C₈	—	8	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₅₃×#390	1	1	yes	characteristic
138	C₂²	C₅₃×#490	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	4	no	—
141	C₄	C₅₃×#418	1	2	yes	—
142	C₄	C₅₃×#473	1	4	yes	—
143	C₄	C₅₃×#496	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₅₃×#1961	1	1	yes	characteristic
146	C₂	C₅₃×#1979	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#494

C₂

yes

maximal, characteristic

C₅₃×#1837

C₂

yes

maximal, characteristic

C₅₃×#1546

C₂

yes

maximal

C₅₃×#1452

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#494

—

C₅₃×#494

C₄

yes

—

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#341

—

C₅₃×#332

—

C₅₃×#322

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

Q₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#9779

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#490

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#418

yes

—

C₄×C₅₃

#473

yes

—

C₄×C₅₃

#496

yes

—

C₂×#494

C₂×C₅₃

yes

characteristic

#1837

C₂×C₅₃

yes

characteristic

#1546

C₂×C₅₃

yes

—

#1452

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1961

yes

characteristic

C₂×C₅₃

#1979

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

—

#494

—

#494

C₄×C₅₃

yes

—

#431

C₂²×C₅₃

yes

—

#423

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

—

—

C₅₃

#9779

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

D₄×C₅₃

yes

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×C₄×C₅₃

yes

—

104

C₄×C₈

D₄×C₅₃

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

110

C₂²×C₄

#41×C₅₃

yes

—

111

Q₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂²×C₄×C₅₃

yes

characteristic

120

C₄²

C₂×D₄×C₅₃

yes

characteristic

121

C₄²

C₂×D₄×C₅₃

yes

—

122

C₂³

#124×C₅₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

129

C₂×C₄

C₄×D₄×C₅₃

yes

—

130

C₂×C₄

#121×C₅₃

yes

—

131

C₂×C₄

#127×C₅₃

yes

characteristic

132

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂²

—

137

C₂²

C₅₃×#390

yes

characteristic

138

C₂²

C₅₃×#490

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₅₃×#418

yes

—

142

C₄

C₅₃×#473

yes

—

143

C₄

C₅₃×#496

yes

—

144

C₂

—

145

C₂

C₅₃×#1961

yes

characteristic

146

C₂

C₅₃×#1979

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#390
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#390

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
62. #9779	C₅₃	direct (N × H)
88. C₅₃	#9779	direct (N × H)

H ≅ G/N

type

62. #9779

C₅₃

direct (N × H)

88. C₅₃

#9779

direct (N × H)

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

subgroup	index
C₂×C₅₃×#494	2
C₅₃×#1837	2
C₅₃×#1546	2
C₅₃×#1452	2
#9779	53

subgroup

index

C₂×C₅₃×#494

C₅₃×#1837

C₅₃×#1546

C₅₃×#1452

#9779

Subgroups — C₅₃×#9779 #27217291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9779 #27217291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9779 #27217291