Group #27214673 — subgroups

Subgroups — C₅₃×#7161 #27214673

Order 13568.

202 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 126 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×#364	C₂	1	2	yes	maximal
2	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	—	2	2	no	—
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#364	—	2	2	no	—
8	C₅₃×#364	C₄	1	4	yes	—
9	C₅₃×#346	—	2	2	no	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
13	M₃₂×C₅₃	—	2	4	no	—
14	M₃₂×C₅₃	—	4	2	no	—
15	C₂×C₁₆×C₅₃	—	2	1	no	—
16	C₂×C₁₆×C₅₃	—	2	1	no	—
17	C₂×C₁₆×C₅₃	—	2	1	no	—
18	C₂×C₁₆×C₅₃	—	2	1	no	—
19	C₂×C₁₆×C₅₃	—	4	2	no	—
20	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
21	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
29	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
35	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
36	C₄²×C₅₃	—	2	1	no	—
37	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₅₃	—	2	2	no	—
39	C₁₆×C₅₃	—	2	2	no	—
40	C₁₆×C₅₃	—	4	4	no	—
41	C₂³×C₅₃	#108	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
50	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₈×C₅₃	#97	1	1	yes	characteristic
52	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₅₃	#102	1	2	yes	—
55	C₈×C₅₃	#103	1	2	yes	—
56	#7161	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	#334	1	1	yes	characteristic
59	C₂²×C₅₃	#335	1	1	yes	characteristic
60	C₂²×C₅₃	#341	1	1	yes	characteristic
61	C₄×C₅₃	—	4	2	no	—
62	C₄×C₅₃	#334	1	1	yes	characteristic
63	C₄×C₅₃	#335	1	1	yes	characteristic
64	C₄×C₅₃	#336	1	1	yes	characteristic
65	C₄×C₅₃	#340	1	1	yes	characteristic
66	C₂×#364	C₂×C₅₃	1	2	yes	—
67	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#1277	1	1	yes	characteristic
71	C₂×C₅₃	#1369	1	2	yes	—
72	C₂×M₃₂	—	2	2	no	—
73	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
74	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
76	#364	—	2	2	no	—
77	#364	C₄×C₅₃	1	4	yes	—
78	#346	—	2	2	no	—
79	C₅₃	#7161	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
81	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
82	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
83	M₃₂	—	2	4	no	—
84	M₃₂	—	4	2	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	4	2	no	—
90	C₂×C₁₆	C₈×C₅₃	1	2	yes	—
91	C₂×C₁₆	C₈×C₅₃	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
93	C₄×C₈	D₄×C₅₃	1	2	yes	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
96	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
99	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
101	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
102	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
103	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
104	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
105	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	4	4	no	—
111	C₂³	#108×C₅₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
115	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
116	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
117	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
118	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
119	C₂×C₄	#108×C₅₃	1	1	yes	characteristic
120	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
121	C₈	#97×C₅₃	1	1	yes	characteristic
122	C₈	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
123	C₈	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
124	C₈	#102×C₅₃	1	2	yes	—
125	C₈	#103×C₅₃	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	C₅₃×#334	1	1	yes	characteristic
128	C₂²	C₅₃×#335	1	1	yes	characteristic
129	C₂²	C₅₃×#341	1	1	yes	characteristic
130	C₄	—	4	2	no	—
131	C₄	C₅₃×#334	1	1	yes	characteristic
132	C₄	C₅₃×#335	1	1	yes	characteristic
133	C₄	C₅₃×#336	1	1	yes	characteristic
134	C₄	C₅₃×#340	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	C₅₃×#1277	1	1	yes	characteristic
138	C₂	C₅₃×#1369	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#364

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#364

—

C₅₃×#364

C₄

yes

—

C₅₃×#346

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

Q₈

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₈

yes

—

C₂×C₁₆×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#108

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

#97

yes

characteristic

C₈×C₅₃

C₂²⋊C₈

yes

characteristic

C₈×C₅₃

C₂²⋊C₈

yes

characteristic

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#103

yes

—

#7161

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#346

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1277

yes

characteristic

C₂×C₅₃

#1369

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₄×C₅₃

yes

characteristic

C₂²×C₁₆

C₄×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#364

—

#364

C₄×C₅₃

yes

—

#346

—

C₅₃

#7161

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

D₄×C₅₃

yes

characteristic

C₂×C₄²

Q₈×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₅₃

yes

—

C₂×C₁₆

C₈×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄²×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

characteristic

100

C₂×C₈

C₂×C₈×C₅₃

yes

characteristic

101

C₂×C₈

M₁₆×C₅₃

yes

characteristic

102

C₂×C₈

M₁₆×C₅₃

yes

characteristic

103

C₂×C₈

D₈×C₅₃

yes

characteristic

104

C₂×C₈

SD₁₆×C₅₃

yes

—

105

C₂×C₈

Q₁₆×C₅₃

yes

characteristic

106

C₄²

—

107

C₄²

C₄⋊C₄×C₅₃

yes

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

111

C₂³

#108×C₅₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#95×C₅₃

yes

characteristic

115

C₂×C₄

C₄⋊C₈×C₅₃

yes

characteristic

116

C₂×C₄

C₄⋊C₈×C₅₃

yes

characteristic

117

C₂×C₄

#106×C₅₃

yes

characteristic

118

C₂×C₄

#107×C₅₃

yes

characteristic

119

C₂×C₄

#108×C₅₃

yes

characteristic

120

C₈

C₄×C₈×C₅₃

yes

characteristic

121

C₈

#97×C₅₃

yes

characteristic

122

C₈

C₂²⋊C₈×C₅₃

yes

characteristic

123

C₈

C₂²⋊C₈×C₅₃

yes

characteristic

124

C₈

#102×C₅₃

yes

—

125

C₈

#103×C₅₃

yes

—

126

C₂²

—

127

C₂²

C₅₃×#334

yes

characteristic

128

C₂²

C₅₃×#335

yes

characteristic

129

C₂²

C₅₃×#341

yes

characteristic

130

C₄

—

131

C₄

C₅₃×#334

yes

characteristic

132

C₄

C₅₃×#335

yes

characteristic

133

C₄

C₅₃×#336

yes

characteristic

134

C₄

C₅₃×#340

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

C₅₃×#1277

yes

characteristic

138

C₂

C₅₃×#1369

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#106
commutator	C₈	C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#334
prime radical	C₂³×C₅₃	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#106

commutator

C₈

C₄×C₈×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
56. #7161	C₅₃	direct (N × H)
79. C₅₃	#7161	direct (N × H)

H ≅ G/N

type

56. #7161

C₅₃

direct (N × H)

79. C₅₃

#7161

direct (N × H)

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

subgroup	index
C₂×C₅₃×#364	2
C₂×C₅₃×#346	2
#7161	53

subgroup

index

C₂×C₅₃×#364

C₂×C₅₃×#346

#7161

Subgroups — C₅₃×#7161 #27214673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7161 #27214673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7161 #27214673