Group #27273435 — subgroups

Subgroups — #27273435

Order 13568.

414 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276253	C₂	1	3	yes	maximal
2	#20272541	C₂	1	3	yes	maximal
3	C₅₃×#2811	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133150	—	2	3	no	—
5	C₂×#13133141	—	2	3	no	—
6	#16810493	C₂²	1	1	yes	characteristic
7	#16810378	C₂²	1	3	yes	—
8	C₅₃×#395	—	2	3	no	—
9	C₅₃×#391	—	2	3	no	—
10	C₅₃×#391	C₂²	1	3	yes	—
11	C₂²×#88322	—	2	3	no	—
12	C₂×C₄⋊Dic₅₃	—	2	3	no	—
13	#13133150	—	4	6	no	—
14	#13133141	—	4	6	no	—
15	C₂²×Q₈×C₅₃	—	2	1	no	—
16	C₂²×Q₈×C₅₃	D₄	1	3	yes	—
17	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
18	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
19	C₂×C₄²×C₅₃	—	2	1	no	—
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	#95×C₅₃	—	2	3	no	—
22	#95×C₅₃	—	2	6	no	—
23	C₂×Q₈×C₅₃	—	2	3	no	—
24	C₂×Q₈×C₅₃	—	4	3	no	—
25	C₂×Q₈×C₅₃	—	4	3	no	—
26	C₂×Q₈×C₅₃	—	8	3	no	—
27	C₂×Q₈×C₅₃	Q₁₆	1	6	yes	—
28	C₂²×C₄×C₅₃	—	2	3	no	—
29	C₂²×C₄×C₅₃	—	2	3	no	—
30	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
31	C₂²×Dic₅₃	—	4	1	no	—
32	C₄⋊C₄×C₅₃	—	4	3	no	—
33	C₄⋊C₄×C₅₃	—	8	3	no	—
34	C₄²×C₅₃	—	4	1	no	—
35	C₄²×C₅₃	—	8	1	no	—
36	C₄⋊Dic₅₃	—	4	3	no	—
37	C₄⋊Dic₅₃	—	4	3	no	—
38	C₂×#88322	—	2	6	no	—
39	C₂×#88322	—	4	3	no	—
40	C₂×#88322	—	4	3	no	—
41	C₂³×C₅₃	#120	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	6	no	—
43	Q₈×C₅₃	—	8	3	no	—
44	Q₈×C₅₃	—	8	4	no	—
45	C₂×C₄×C₅₃	—	2	3	no	—
46	C₂×C₄×C₅₃	—	2	3	no	—
47	C₂×C₄×C₅₃	—	2	6	no	—
48	C₂×C₄×C₅₃	—	4	3	no	—
49	C₂×C₄×C₅₃	—	4	3	no	—
50	C₂×C₄×C₅₃	—	4	3	no	—
51	C₂×C₄×C₅₃	—	4	6	no	—
52	C₂×C₄×C₅₃	C₂×Q₁₆	1	3	yes	—
53	C₂×C₄×C₅₃	#137	1	3	yes	—
54	C₂×Dic₅₃	—	8	3	no	—
55	#88322	—	4	6	no	—
56	#7837	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	#451	1	3	yes	—
58	C₂²×C₅₃	#453	1	3	yes	—
59	C₂²×C₅₃	#458	1	1	yes	characteristic
60	C₄×C₅₃	—	2	3	no	—
61	C₄×C₅₃	—	2	3	no	—
62	C₄×C₅₃	—	4	6	no	—
63	C₄×C₅₃	—	8	3	no	—
64	Dic₅₃	—	16	1	no	—
65	#2811	D₅₃	1	1	yes	characteristic
66	#2095	—	53	3	no	—
67	#1272	—	53	3	no	—
68	C₂×C₅₃	#1614	1	3	yes	—
69	C₂×C₅₃	#1633	1	3	yes	—
70	C₂×C₅₃	#1655	1	1	yes	characteristic
71	C₂×#107	—	106	3	no	—
72	C₂×#103	—	106	3	no	—
73	#395	—	2	3	no	—
74	#391	—	2	3	no	—
75	#391	C₂×D₅₃	1	3	yes	—
76	#384	—	53	1	no	—
77	#336	—	53	3	no	—
78	C₅₃	#7837	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×Q₈	C₂²⋊D₅₃	1	3	yes	—
81	C₂²×C₈	—	106	3	no	—
82	C₂×C₄⋊C₄	—	2	3	no	—
83	C₂×C₄⋊C₄	—	106	3	no	—
84	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
87	#107	—	212	6	no	—
88	#103	—	212	6	no	—
89	#95	—	2	3	no	—
90	#95	—	2	6	no	—
91	C₂×Q₈	—	2	3	no	—
92	C₂×Q₈	—	4	3	no	—
93	C₂×Q₈	—	4	3	no	—
94	C₂×Q₈	—	8	3	no	—
95	C₂×Q₈	#11719925	1	6	yes	—
96	C₂²×C₄	—	2	3	no	—
97	C₂²×C₄	—	2	3	no	—
98	C₂²×C₄	—	212	1	no	—
99	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
100	C₂×C₈	—	106	6	no	—
101	C₂×C₈	—	212	3	no	—
102	C₂×C₈	—	212	3	no	—
103	C₄⋊C₄	—	4	3	no	—
104	C₄⋊C₄	—	8	3	no	—
105	C₄⋊C₄	—	212	3	no	—
106	C₄⋊C₄	—	212	3	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	8	1	no	—
109	C₂³	#13133186	1	1	yes	characteristic
110	Q₈	—	4	6	no	—
111	Q₈	—	8	3	no	—
112	Q₈	—	8	4	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	4	3	no	—
117	C₂×C₄	—	4	3	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	6	no	—
120	C₂×C₄	—	424	3	no	—
121	C₂×C₄	C₂×#11719925	1	3	yes	—
122	C₂×C₄	#13133248	1	3	yes	—
123	C₈	—	212	6	no	—
124	C₂²	#16810779	1	3	yes	—
125	C₂²	#16810791	1	3	yes	—
126	C₂²	#16810813	1	1	yes	characteristic
127	C₄	—	2	3	no	—
128	C₄	—	2	3	no	—
129	C₄	—	4	6	no	—
130	C₄	—	8	3	no	—
131	C₄	—	848	1	no	—
132	C₂	#20273740	1	3	yes	—
133	C₂	#20273855	1	3	yes	—
134	C₂	#20274004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276253

C₂

yes

maximal

#20272541

C₂

yes

maximal

C₅₃×#2811

C₂

yes

maximal, characteristic

C₂×#13133150

—

C₂×#13133141

—

#16810493

C₂²

yes

characteristic

#16810378

C₂²

yes

—

C₅₃×#395

—

C₅₃×#391

—

C₅₃×#391

C₂²

yes

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

#13133150

—

#13133141

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

—

—

C₂²×C₅₃

yes

—

C₂²×C₅₃

#453

yes

—

C₂²×C₅₃

#458

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2811

D₅₃

yes

characteristic

#2095

—

#1272

—

C₂×C₅₃

#1614

yes

—

C₂×C₅₃

#1633

yes

—

C₂×C₅₃

#1655

yes

characteristic

C₂×#107

—

106

—

C₂×#103

—

106

—

#395

—

#391

—

#391

C₂×D₅₃

yes

—

#384

—

#336

—

C₅₃

#7837

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂²⋊D₅₃

yes

—

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#107

—

212

—

#103

—

212

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#11719925

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

212

—

102

C₂×C₈

—

212

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

212

—

106

C₄⋊C₄

—

212

—

107

C₄²

—

108

C₄²

—

109

C₂³

#13133186

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

424

—

121

C₂×C₄

C₂×#11719925

yes

—

122

C₂×C₄

#13133248

yes

—

123

C₈

—

212

—

124

C₂²

#16810779

yes

—

125

C₂²

#16810791

yes

—

126

C₂²

#16810813

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

848

—

132

C₂

#20273740

yes

—

133

C₂

#20273855

yes

—

134

C₂

#20274004

yes

characteristic

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2811	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2811	D₅₃
hypocenter	C₅₃	#7837
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2811

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂³×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

N	H ≅ G/N	type
78. C₅₃	#7837	semidirect

H ≅ G/N

type

78. C₅₃

#7837

semidirect

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

subgroup	index
#20276253	2
#20272541	2
C₅₃×#2811	2
#7837	53

subgroup

index

#20276253

#20272541

C₅₃×#2811

#7837

Subgroups — #27273435

Conjugacy classes of subgroups

Complements

Maximal subgroups