Group #27220016 — subgroups

Subgroups — C₅₃×#12504 #27220016

Order 13568.

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2252	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2247	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2241	C₂	1	2	yes	maximal
4	C₅₃×#1707	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₅₃	—	2	2	no	—
7	C₅₃×#500	C₂²	1	2	yes	—
8	C₅₃×#498	—	2	2	no	—
9	C₅₃×#494	C₂²	1	2	yes	—
10	C₅₃×#487	—	2	2	no	—
11	C₅₃×#366	—	2	2	no	—
12	C₅₃×#358	—	2	4	no	—
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₈²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₅₃	—	2	2	no	—
17	C₂×Q₁₆×C₅₃	—	2	2	no	—
18	C₄⋊Q₈×C₅₃	—	2	2	no	—
19	C₄⋊Q₈×C₅₃	—	2	2	no	—
20	Q₃₂×C₅₃	—	4	4	no	—
21	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
22	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
23	#107×C₅₃	—	2	2	no	—
24	#107×C₅₃	—	2	2	no	—
25	#106×C₅₃	—	4	4	no	—
26	#103×C₅₃	—	4	4	no	—
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	—	4	2	no	—
31	C₂×Q₈×C₅₃	—	4	2	no	—
32	Q₁₆×C₅₃	—	4	4	no	—
33	Q₁₆×C₅₃	—	4	4	no	—
34	C₂×C₈×C₅₃	D₈	1	2	yes	—
35	C₂×C₈×C₅₃	D₈	1	2	yes	—
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	—	4	2	no	—
39	C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₄⋊C₄×C₅₃	—	4	2	no	—
41	C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₅₃	—	2	2	no	—
44	C₁₆×C₅₃	SD₁₆	1	4	yes	—
45	Q₈×C₅₃	—	8	2	no	—
46	Q₈×C₅₃	—	8	2	no	—
47	Q₈×C₅₃	—	8	4	no	—
48	C₂×C₄×C₅₃	—	8	2	no	—
49	C₂×C₄×C₅₃	—	8	2	no	—
50	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	Q₃₂	1	4	yes	complemented
55	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
56	C₈×C₅₃	#135	1	2	yes	—
57	#12504	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	—	16	2	no	—
60	C₄×C₅₃	—	16	2	no	—
61	C₄×C₅₃	#492	1	1	yes	characteristic
62	C₄×C₅₃	#495	1	1	yes	characteristic
63	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
64	C₄×C₅₃	#508	1	2	yes	—
65	#2252	C₂×C₅₃	1	1	yes	characteristic
66	#2247	C₂×C₅₃	1	1	yes	characteristic
67	#2241	C₂×C₅₃	1	2	yes	—
68	#1707	C₂×C₅₃	1	2	yes	—
69	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
70	C₂×C₅₃	#1706	1	1	yes	characteristic
71	C₂×C₅₃	#2247	1	1	yes	characteristic
72	C₂×C₅₃	#2249	1	1	yes	characteristic
73	C₂×Q₃₂	—	2	2	no	—
74	#500	C₂²×C₅₃	1	2	yes	—
75	#498	—	2	2	no	—
76	#494	C₂²×C₅₃	1	2	yes	—
77	#487	—	2	2	no	—
78	#366	—	2	2	no	—
79	#358	—	2	4	no	—
80	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
82	C₈²	C₂²×C₅₃	1	1	yes	characteristic
83	C₅₃	#12504	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	Q₃₂	—	4	4	no	—
89	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
90	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#106	—	4	4	no	—
94	#103	—	4	4	no	—
95	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
96	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	C₂×C₈	D₈×C₅₃	1	2	yes	—
103	C₂×C₈	D₈×C₅₃	1	2	yes	—
104	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	SD₁₆×C₅₃	1	4	yes	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	8	4	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Q₃₂×C₅₃	1	4	yes	complemented
123	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
124	C₈	#135×C₅₃	1	2	yes	—
125	C₂²	C₅₃×#493	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	16	2	no	—
128	C₄	C₅₃×#492	1	1	yes	characteristic
129	C₄	C₅₃×#495	1	1	yes	characteristic
130	C₄	C₂×Q₃₂×C₅₃	1	2	yes	—
131	C₄	C₅₃×#508	1	2	yes	—
132	C₂	C₅₃×#1706	1	1	yes	characteristic
133	C₂	C₅₃×#2247	1	1	yes	characteristic
134	C₂	C₅₃×#2249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2252

C₂

yes

maximal, characteristic

C₅₃×#2247

C₂

yes

maximal, characteristic

C₅₃×#2241

C₂

yes

maximal

C₅₃×#1707

C₂

yes

maximal

C₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₅₃

—

C₅₃×#500

C₂²

yes

—

C₅₃×#498

—

C₅₃×#494

C₂²

yes

—

C₅₃×#487

—

C₅₃×#366

—

C₅₃×#358

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

—

#107×C₅₃

—

#107×C₅₃

—

#106×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

SD₁₆

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

Q₃₂

yes

complemented

C₈×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#135

yes

—

#12504

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

—

C₄×C₅₃

#508

yes

—

#2252

C₂×C₅₃

yes

characteristic

#2247

C₂×C₅₃

yes

characteristic

#2241

C₂×C₅₃

yes

—

#1707

C₂×C₅₃

yes

—

C₈×C₁₆

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

#500

C₂²×C₅₃

yes

—

#498

—

#494

C₂²×C₅₃

yes

—

#487

—

#366

—

#358

—

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

C₈²

C₂²×C₅₃

yes

characteristic

C₅₃

#12504

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₄×C₅₃

yes

—

C₂×C₁₆

D₄×C₅₃

yes

—

—

—

—

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

C₂³×C₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

D₈×C₅₃

yes

—

103

C₂×C₈

D₈×C₅₃

yes

—

104

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

105

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

C₂×D₄×C₅₃

yes

characteristic

111

C₁₆

—

112

C₁₆

SD₁₆×C₅₃

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#127×C₅₃

yes

characteristic

119

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

120

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

121

C₈

—

122

C₈

Q₃₂×C₅₃

yes

complemented

123

C₈

C₂×SD₁₆×C₅₃

yes

—

124

C₈

#135×C₅₃

yes

—

125

C₂²

C₅₃×#493

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₅₃×#492

yes

characteristic

129

C₄

C₅₃×#495

yes

characteristic

130

C₄

C₂×Q₃₂×C₅₃

yes

—

131

C₄

C₅₃×#508

yes

—

132

C₂

C₅₃×#1706

yes

characteristic

133

C₂

C₅₃×#2247

yes

characteristic

134

C₂

C₅₃×#2249

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₅₃

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₅₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

N	H ≅ G/N	type
54. C₈×C₅₃	Q₃₂	semidirect
57. #12504	C₅₃	direct (N × H)
83. C₅₃	#12504	direct (N × H)
122. C₈	Q₃₂×C₅₃	semidirect

H ≅ G/N

type

54. C₈×C₅₃

Q₃₂

semidirect

57. #12504

C₅₃

direct (N × H)

83. C₅₃

#12504

direct (N × H)

122. C₈

Q₃₂×C₅₃

semidirect

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

subgroup	index
C₅₃×#2252	2
C₅₃×#2247	2
C₅₃×#2241	2
C₅₃×#1707	2
C₈×C₁₆×C₅₃	2
#12504	53

subgroup

index

C₅₃×#2252

C₅₃×#2247

C₅₃×#2241

C₅₃×#1707

C₈×C₁₆×C₅₃

#12504

Subgroups — C₅₃×#12504 #27220016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12504 #27220016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12504 #27220016