Group #27214647 — subgroups

Subgroups — C₅₃×#7135 #27214647

Order 13568.

202 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 74 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×#1991	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1376	C₂	1	2	yes	maximal
3	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
4	C₂×M₃₂×C₅₃	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#382	C₄	1	1	yes	characteristic
7	C₅₃×#340	—	2	2	no	—
8	C₂²×C₈×C₅₃	D₄	1	2	yes	—
9	#125×C₅₃	—	2	1	no	—
10	#125×C₅₃	—	4	1	no	—
11	#125×C₅₃	C₈	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	M₃₂×C₅₃	—	2	4	no	—
17	M₃₂×C₅₃	—	4	2	no	—
18	C₂×C₁₆×C₅₃	—	4	2	no	—
19	C₄×C₈×C₅₃	—	4	1	no	—
20	#95×C₅₃	—	4	1	no	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	—	4	2	no	—
31	C₄⋊C₄×C₅₃	—	4	2	no	—
32	C₄⋊C₄×C₅₃	—	4	1	no	—
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
37	C₁₆×C₅₃	—	4	4	no	—
38	C₂³×C₅₃	#100	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	8	2	no	—
44	C₂×C₄×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	8	1	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#103	1	2	yes	—
49	C₂×C₄×C₅₃	#104	1	2	yes	—
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	—	2	2	no	—
52	#7135	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	#323	1	1	yes	characteristic
55	C₂²×C₅₃	#329	1	1	yes	characteristic
56	C₂²×C₅₃	#333	1	1	yes	characteristic
57	C₄×C₅₃	—	4	1	no	—
58	C₄×C₅₃	—	4	1	no	—
59	C₄×C₅₃	—	8	2	no	—
60	C₄×C₅₃	—	16	1	no	—
61	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
62	C₄×C₅₃	#328	1	2	yes	—
63	#1991	C₂×C₅₃	1	1	yes	characteristic
64	#1376	C₂×C₅₃	1	2	yes	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	#1272	1	1	yes	characteristic
68	C₂×C₅₃	#1366	1	2	yes	—
69	C₂×#125	C₄×C₅₃	1	1	yes	characteristic
70	C₂×M₃₂	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
72	#382	C₄×C₅₃	1	1	yes	characteristic
73	#340	—	2	2	no	—
74	C₅₃	#7135	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₈	D₄×C₅₃	1	2	yes	—
76	#125	—	2	1	no	—
77	#125	—	4	1	no	—
78	#125	C₈×C₅₃	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
83	M₃₂	—	2	4	no	—
84	M₃₂	—	4	2	no	—
85	C₂×C₁₆	—	4	2	no	—
86	C₄×C₈	—	4	1	no	—
87	#95	—	4	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
95	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
103	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
104	C₁₆	—	4	4	no	—
105	C₂³	#100×C₅₃	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
114	C₂×C₄	#99×C₅₃	1	1	yes	characteristic
115	C₂×C₄	#103×C₅₃	1	2	yes	—
116	C₂×C₄	#104×C₅₃	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₅₃×#323	1	1	yes	characteristic
121	C₂²	C₅₃×#329	1	1	yes	characteristic
122	C₂²	C₅₃×#333	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	16	1	no	—
127	C₄	C₅₃×Q₈⋊C₈	1	2	yes	—
128	C₄	C₅₃×#328	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	C₅₃×#1272	1	1	yes	characteristic
132	C₂	C₅₃×#1366	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1991

C₂

yes

maximal, characteristic

C₅₃×#1376

C₂

yes

maximal

C₂×#125×C₅₃

C₄

yes

characteristic

C₂×M₃₂×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#382

C₄

yes

characteristic

C₅₃×#340

—

C₂²×C₈×C₅₃

D₄

yes

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#7135

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₈⋊C₈

yes

—

C₄×C₅₃

#328

yes

—

#1991

C₂×C₅₃

yes

characteristic

#1376

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1272

yes

characteristic

C₂×C₅₃

#1366

yes

—

C₂×#125

C₄×C₅₃

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#382

C₄×C₅₃

yes

characteristic

#340

—

C₅₃

#7135

yes

characteristic, direct product factor, Sylow

C₂²×C₈

D₄×C₅₃

yes

—

#125

—

#125

—

#125

C₈×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

Q₁₆×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×C₈×C₅₃

yes

characteristic

103

C₄²

M₁₆×C₅₃

yes

characteristic

104

C₁₆

—

105

C₂³

#100×C₅₃

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊C₈×C₅₃

yes

characteristic

114

C₂×C₄

#99×C₅₃

yes

characteristic

115

C₂×C₄

#103×C₅₃

yes

—

116

C₂×C₄

#104×C₅₃

yes

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

C₅₃×#323

yes

characteristic

121

C₂²

C₅₃×#329

yes

characteristic

122

C₂²

C₅₃×#333

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₅₃×Q₈⋊C₈

yes

—

128

C₄

C₅₃×#328

yes

—

129

C₂

—

130

C₂

—

131

C₂

C₅₃×#1272

yes

characteristic

132

C₂

C₅₃×#1366

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#333
commutator	C₄²	C₂×C₈×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#333

commutator

C₄²

C₂×C₈×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

prime radical

C₂³×C₅₃

#100

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
52. #7135	C₅₃	direct (N × H)
74. C₅₃	#7135	direct (N × H)

H ≅ G/N

type

52. #7135

C₅₃

direct (N × H)

74. C₅₃

#7135

direct (N × H)

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

subgroup	index
C₅₃×#1991	2
C₅₃×#1376	2
#7135	53

subgroup

index

C₅₃×#1991

C₅₃×#1376

#7135

Subgroups — C₅₃×#7135 #27214647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7135 #27214647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7135 #27214647