Group #27214753 — subgroups

Subgroups — C₅₃×#7241 #27214753

Order 13568.

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

normal subgroups only

This group has 206 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	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1567	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
4	C₅₃×#423	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×C₄⋊C₁₆	—	2	2	no	—
7	C₅₃×#335	—	2	2	no	—
8	C₅₃×#335	C₄	1	4	yes	—
9	C₂×M₁₆×C₅₃	—	4	2	no	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	—	2	2	no	—
14	C₂×C₁₆×C₅₃	—	4	2	no	—
15	C₄⋊C₈×C₅₃	—	2	4	no	—
16	C₄⋊C₈×C₅₃	—	4	2	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
22	M₁₆×C₅₃	—	8	2	no	—
23	M₁₆×C₅₃	—	8	2	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	4	1	no	—
27	C₂×C₈×C₅₃	—	4	4	no	—
28	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₄²×C₅₃	SD₁₆	1	2	yes	—
33	C₁₆×C₅₃	—	4	2	no	—
34	C₂³×C₅₃	#107	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#102	1	2	yes	—
40	C₂×C₄×C₅₃	#103	1	2	yes	—
41	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
44	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	1	no	—
47	C₈×C₅₃	—	8	4	no	—
48	C₈×C₅₃	#100	1	1	yes	characteristic
49	C₈×C₅₃	#101	1	1	yes	characteristic
50	#7241	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	#340	1	1	yes	characteristic
54	C₂²×C₅₃	#366	1	1	yes	characteristic
55	C₂²×C₅₃	#367	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	#338	1	1	yes	characteristic
59	C₄×C₅₃	#343	1	1	yes	characteristic
60	C₄×C₅₃	#357	1	2	yes	—
61	C₄×C₅₃	#358	1	2	yes	—
62	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
63	#1567	C₂×C₅₃	1	2	yes	—
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#1292	1	1	yes	characteristic
67	C₂×C₅₃	#1380	1	1	yes	characteristic
68	C₂×C₅₃	#1385	1	1	yes	characteristic
69	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
70	#423	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
72	C₄⋊C₁₆	—	2	2	no	—
73	#335	—	2	2	no	—
74	#335	C₄×C₅₃	1	4	yes	—
75	C₅₃	#7241	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×M₁₆	—	4	2	no	—
77	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
78	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
79	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	2	4	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
85	C₄×C₈	D₄×C₅₃	1	2	yes	—
86	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
87	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
88	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
89	M₁₆	—	8	2	no	—
90	M₁₆	—	8	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
98	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
99	C₄²	SD₁₆×C₅₃	1	2	yes	—
100	C₁₆	—	4	2	no	—
101	C₂³	#107×C₅₃	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
106	C₂×C₄	#102×C₅₃	1	2	yes	—
107	C₂×C₄	#103×C₅₃	1	2	yes	—
108	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
109	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
110	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
111	C₂×C₄	Q₃₂×C₅₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	8	4	no	—
115	C₈	#100×C₅₃	1	1	yes	characteristic
116	C₈	#101×C₅₃	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	C₅₃×#340	1	1	yes	characteristic
120	C₂²	C₅₃×#366	1	1	yes	characteristic
121	C₂²	C₅₃×#367	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	C₅₃×#338	1	1	yes	characteristic
125	C₄	C₅₃×#343	1	1	yes	characteristic
126	C₄	C₅₃×#357	1	2	yes	—
127	C₄	C₅₃×#358	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	C₅₃×#1292	1	1	yes	characteristic
131	C₂	C₅₃×#1380	1	1	yes	characteristic
132	C₂	C₅₃×#1385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₅₃×#1567

C₂

yes

maximal

C₂²×C₁₆×C₅₃

C₄

yes

—

C₅₃×#423

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×#335

—

C₅₃×#335

C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₁₆×C₅₃

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₃₂

yes

—

C₂×C₄×C₅₃

Q₃₂

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#358

yes

—

C₂×C₄⋊C₁₆

C₂×C₅₃

yes

characteristic

#1567

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

C₄×C₅₃

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

C₄×C₅₃

yes

—

C₅₃

#7241

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

Q₈×C₅₃

yes

characteristic

C₂×C₄²

D₄×C₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

characteristic

C₂²×C₄

D₈×C₅₃

yes

characteristic

C₂²×C₄

Q₁₆×C₅₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₄²

C₂²⋊C₄×C₅₃

yes

—

C₄²

SD₁₆×C₅₃

yes

—

100

C₁₆

—

101

C₂³

#107×C₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#95×C₅₃

yes

characteristic

106

C₂×C₄

#102×C₅₃

yes

—

107

C₂×C₄

#103×C₅₃

yes

—

108

C₂×C₄

#106×C₅₃

yes

characteristic

109

C₂×C₄

D₁₆×C₅₃

yes

characteristic

110

C₂×C₄

SD₃₂×C₅₃

yes

—

111

C₂×C₄

Q₃₂×C₅₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

#100×C₅₃

yes

characteristic

116

C₈

#101×C₅₃

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

C₅₃×#340

yes

characteristic

120

C₂²

C₅₃×#366

yes

characteristic

121

C₂²

C₅₃×#367

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₅₃×#338

yes

characteristic

125

C₄

C₅₃×#343

yes

characteristic

126

C₄

C₅₃×#357

yes

—

127

C₄

C₅₃×#358

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₅₃×#1292

yes

characteristic

131

C₂

C₅₃×#1380

yes

characteristic

132

C₂

C₅₃×#1385

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#340
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#340

commutator

C₂×C₈

C₄²×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

prime radical

C₂³×C₅₃

#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
50. #7241	C₅₃	direct (N × H)
75. C₅₃	#7241	direct (N × H)

H ≅ G/N

type

50. #7241

C₅₃

direct (N × H)

75. C₅₃

#7241

direct (N × H)

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₅₃×#1567	2
#7241	53

subgroup

index

C₂×C₅₃×C₄⋊C₁₆

C₅₃×#1567

#7241

Subgroups — C₅₃×#7241 #27214753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7241 #27214753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7241 #27214753