Group #27214780 — subgroups

Subgroups — C₅₃×#7268 #27214780

Order 13568.

238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	—	2	2	no	—
4	C₂×#106×C₅₃	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#384	C₄	1	2	yes	—
7	C₅₃×#364	—	2	2	no	—
8	C₅₃×#364	C₄	1	4	yes	complemented
9	C₂²×C₈×C₅₃	D₄	1	2	yes	—
10	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
13	M₃₂×C₅₃	—	2	4	no	—
14	M₃₂×C₅₃	—	4	2	no	—
15	C₂×C₁₆×C₅₃	—	4	2	no	—
16	#106×C₅₃	—	4	2	no	—
17	#106×C₅₃	—	4	2	no	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	4	2	no	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	D₈	1	2	yes	—
27	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	4	4	no	—
31	C₄⋊C₄×C₅₃	—	8	4	no	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₄²	1	1	yes	characteristic
34	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₁₆×C₅₃	—	4	4	no	—
36	C₂³×C₅₃	#100	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	8	4	no	—
41	C₂×C₄×C₅₃	—	8	2	no	—
42	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#102	1	2	yes	—
45	C₂×C₄×C₅₃	#103	1	2	yes	—
46	C₈×C₅₃	#102	1	2	yes	—
47	C₈×C₅₃	#103	1	2	yes	—
48	C₈×C₅₃	#106	1	2	yes	—
49	C₈×C₅₃	#107	1	2	yes	—
50	#7268	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#331	1	1	yes	characteristic
53	C₂²×C₅₃	#333	1	1	yes	characteristic
54	C₂²×C₅₃	#343	1	1	yes	characteristic
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	16	2	no	—
58	C₄×C₅₃	#332	1	2	yes	—
59	C₄×C₅₃	#340	1	2	yes	—
60	C₂×#364	C₂×C₅₃	1	2	yes	—
61	#1844	C₂×C₅₃	1	1	yes	characteristic
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	#1296	1	1	yes	characteristic
65	C₂×C₅₃	#1359	1	2	yes	—
66	C₂×M₃₂	—	2	2	no	—
67	C₂×#106	—	2	2	no	—
68	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
69	#384	C₄×C₅₃	1	2	yes	—
70	#364	—	2	2	no	—
71	#364	C₄×C₅₃	1	4	yes	complemented
72	C₅₃	#7268	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₈	D₄×C₅₃	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	4	2	no	—
76	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
77	M₃₂	—	2	4	no	—
78	M₃₂	—	4	2	no	—
79	C₂×C₁₆	—	4	2	no	—
80	#106	—	4	2	no	—
81	#106	—	4	2	no	—
82	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
83	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
84	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
89	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
90	C₂×C₈	D₈×C₅₃	1	2	yes	—
91	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
92	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
93	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	C₄²×C₅₃	1	1	yes	characteristic
98	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
99	C₁₆	—	4	4	no	—
100	C₂³	#100×C₅₃	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
107	C₂×C₄	#101×C₅₃	1	1	yes	characteristic
108	C₂×C₄	#102×C₅₃	1	2	yes	—
109	C₂×C₄	#103×C₅₃	1	2	yes	—
110	C₈	#102×C₅₃	1	2	yes	—
111	C₈	#103×C₅₃	1	2	yes	—
112	C₈	#106×C₅₃	1	2	yes	—
113	C₈	#107×C₅₃	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	C₅₃×#331	1	1	yes	characteristic
116	C₂²	C₅₃×#333	1	1	yes	characteristic
117	C₂²	C₅₃×#343	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	C₅₃×#332	1	2	yes	—
122	C₄	C₅₃×#340	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	C₅₃×#1296	1	1	yes	characteristic
126	C₂	C₅₃×#1359	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#364

C₂

yes

maximal

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

—

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₄

yes

—

C₅₃×#364

—

C₅₃×#364

C₄

yes

complemented

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#106×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

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

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#107

yes

—

#7268

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

#332

yes

—

C₄×C₅₃

#340

yes

—

C₂×#364

C₂×C₅₃

yes

—

#1844

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1296

yes

characteristic

C₂×C₅₃

#1359

yes

—

C₂×M₃₂

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#384

C₄×C₅₃

yes

—

#364

—

#364

C₄×C₅₃

yes

complemented

C₅₃

#7268

yes

characteristic, direct product factor, Sylow

C₂²×C₈

D₄×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#106

—

#106

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

Q₈×C₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

D₈×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

Q₁₆×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄²×C₅₃

yes

characteristic

C₄²

C₄⋊C₄×C₅₃

yes

characteristic

C₁₆

—

100

C₂³

#100×C₅₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#95×C₅₃

yes

characteristic

107

C₂×C₄

#101×C₅₃

yes

characteristic

108

C₂×C₄

#102×C₅₃

yes

—

109

C₂×C₄

#103×C₅₃

yes

—

110

C₈

#102×C₅₃

yes

—

111

C₈

#103×C₅₃

yes

—

112

C₈

#106×C₅₃

yes

—

113

C₈

#107×C₅₃

yes

—

114

C₂²

—

115

C₂²

C₅₃×#331

yes

characteristic

116

C₂²

C₅₃×#333

yes

characteristic

117

C₂²

C₅₃×#343

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₅₃×#332

yes

—

122

C₄

C₅₃×#340

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₅₃×#1296

yes

characteristic

126

C₂

C₅₃×#1359

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#331
commutator	C₄²	C₄²×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

#331

commutator

C₄²

C₄²×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
8. C₅₃×#364	C₄	semidirect
50. #7268	C₅₃	direct (N × H)
71. #364	C₄×C₅₃	semidirect
72. C₅₃	#7268	direct (N × H)

H ≅ G/N

type

8. C₅₃×#364

C₄

semidirect

50. #7268

C₅₃

direct (N × H)

71. #364

C₄×C₅₃

semidirect

72. C₅₃

#7268

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
C₅₃×#1844	2
#7268	53

subgroup

index

C₂×C₅₃×#364

C₅₃×#1844

#7268

Subgroups — C₅₃×#7268 #27214780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7268 #27214780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7268 #27214780