Group #27277864 — subgroups

Subgroups — #27277864

Order 13568.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#20274872	C₂	1	1	yes	maximal, characteristic
2	#20274842	C₂	1	4	yes	maximal
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133149	—	2	2	no	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	#16810493	C₂²	1	2	yes	—
8	#16810385	—	2	8	no	—
9	C₈×#88322	C₄	1	4	yes	complemented
10	C₂×#106×C₅₃	—	2	2	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#384	C₂²	1	2	yes	—
13	C₂²×#88322	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₅₃	—	2	4	no	—
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	#13133149	—	4	4	no	—
17	C₄×#88322	C₂×C₄	1	4	yes	—
18	C₄×#88322	Q₈	1	4	yes	—
19	C₂²×C₈×C₅₃	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	#106×C₅₃	—	4	4	no	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₃	—	4	2	no	—
27	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
29	C₂²×Dic₅₃	—	4	2	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
32	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
33	C₄⋊C₄×C₅₃	—	4	4	no	—
34	C₄⋊C₄×C₅₃	—	4	8	no	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
38	C₄⋊Dic₅₃	—	4	4	no	—
39	C₄⋊Dic₅₃	—	4	8	no	—
40	C₂×#88322	—	2	8	no	—
41	C₂×#88322	C₄⋊C₄	1	8	yes	—
42	C₂³×C₅₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	8	2	no	—
44	C₂×C₄×C₅₃	—	8	4	no	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₅₃	#135	1	8	yes	—
52	C₂×Dic₅₃	—	8	2	no	—
53	C₂×Dic₅₃	—	8	4	no	—
54	C₈×C₅₃	#106	1	8	yes	—
55	#88322	—	2	8	no	—
56	#8815	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	#384	1	1	yes	characteristic
58	C₂²×C₅₃	#492	1	1	yes	characteristic
59	C₂²×C₅₃	#492	1	1	yes	characteristic
60	C₂²×C₅₃	#495	1	4	yes	—
61	C₄×C₅₃	—	16	2	no	—
62	C₄×C₅₃	C₂×#106	1	2	yes	—
63	C₄×C₅₃	#427	1	4	yes	—
64	C₄×C₅₃	#498	1	2	yes	—
65	C₄×C₅₃	#499	1	4	yes	—
66	Dic₅₃	—	16	2	no	—
67	#1844	—	53	1	no	—
68	#1844	D₅₃	1	1	yes	characteristic
69	#1834	—	53	4	no	—
70	C₂×C₈²	—	53	1	no	—
71	C₂×C₅₃	#1708	1	2	yes	—
72	C₂×C₅₃	#1708	1	2	yes	—
73	C₂×C₅₃	#1844	1	1	yes	characteristic
74	C₂×C₅₃	#1846	1	2	yes	—
75	C₂×#106	—	2	2	no	—
76	C₂×#106	—	106	2	no	—
77	C₂×C₄×C₈	—	53	2	no	—
78	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
79	#384	—	53	2	no	—
80	#384	C₂×D₅₃	1	2	yes	—
81	#340	—	106	8	no	—
82	C₈²	—	53	4	no	—
83	C₅₃	#8815	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	53	4	no	—
85	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	106	4	no	—
89	C₂×C₄⋊C₄	—	106	2	no	—
90	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	212	4	no	—
93	C₄×C₈	—	53	4	no	—
94	C₄×C₈	—	53	4	no	—
95	C₄×C₈	Dic₁₀₆	1	2	yes	—
96	C₄×C₈	C₄×D₅₃	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	212	2	no	—
99	C₂²×C₄	D₄×D₅₃	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₂×C₈	—	53	8	no	—
102	C₂×C₈	—	106	8	no	—
103	C₂×C₈	#11719919	1	4	yes	—
104	C₂×C₈	#11719923	1	4	yes	—
105	C₂×C₈	#11719924	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	212	4	no	—
109	C₄⋊C₄	—	212	8	no	—
110	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
111	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₄²	Q₈×D₅₃	1	2	yes	—
113	C₂³	#13133213	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	424	2	no	—
117	C₂×C₄	—	424	4	no	—
118	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
119	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
120	C₂×C₄	#13133215	1	1	yes	characteristic
121	C₂×C₄	#13133216	1	2	yes	—
122	C₂×C₄	C₂×#11719924	1	2	yes	—
123	C₂×C₄	C₂×#11719923	1	2	yes	—
124	C₂×C₄	#13133238	1	8	yes	—
125	C₈	—	106	8	no	—
126	C₈	#13133148	1	8	yes	—
127	C₂²	#16810492	1	1	yes	characteristic
128	C₂²	#16810995	1	1	yes	characteristic
129	C₂²	#16810997	1	1	yes	characteristic
130	C₂²	#16811004	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	848	2	no	—
133	C₄	C₂×#13133148	1	2	yes	—
134	C₄	#16810658	1	4	yes	—
135	C₄	#16811019	1	2	yes	—
136	C₄	#16811023	1	4	yes	—
137	C₂	#20274314	1	2	yes	—
138	C₂	#20274317	1	2	yes	—
139	C₂	#20274871	1	1	yes	characteristic
140	C₂	#20274876	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274872

C₂

yes

maximal, characteristic

#20274842

C₂

yes

maximal

C₂×C₈×#88322

C₂

yes

maximal, characteristic

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×#13133149

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810385

—

C₈×#88322

C₄

yes

complemented

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133149

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

Q₈

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#106

yes

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#106

yes

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#498

yes

—

C₄×C₅₃

#499

yes

—

Dic₅₃

—

#1844

—

#1844

D₅₃

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₅₃

#1708

yes

—

C₂×C₅₃

#1708

yes

—

C₂×C₅₃

#1844

yes

characteristic

C₂×C₅₃

#1846

yes

—

C₂×#106

—

C₂×#106

—

106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

#384

C₂×D₅₃

yes

—

#340

—

106

—

C₈²

—

C₅₃

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#106

—

#106

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₄×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

D₄×D₅₃

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

106

—

103

C₂×C₈

#11719919

yes

—

104

C₂×C₈

#11719923

yes

—

105

C₂×C₈

#11719924

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

212

—

109

C₄⋊C₄

—

212

—

110

C₄²

C₂×Dic₁₀₆

yes

characteristic

111

C₄²

C₂×C₄×D₅₃

yes

characteristic

112

C₄²

Q₈×D₅₃

yes

—

113

C₂³

#13133213

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

424

—

117

C₂×C₄

—

424

—

118

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

119

C₂×C₄

C₂×#11719919

yes

characteristic

120

C₂×C₄

#13133215

yes

characteristic

121

C₂×C₄

#13133216

yes

—

122

C₂×C₄

C₂×#11719924

yes

—

123

C₂×C₄

C₂×#11719923

yes

—

124

C₂×C₄

#13133238

yes

—

125

C₈

—

106

—

126

C₈

#13133148

yes

—

127

C₂²

#16810492

yes

characteristic

128

C₂²

#16810995

yes

characteristic

129

C₂²

#16810997

yes

characteristic

130

C₂²

#16811004

yes

—

131

C₄

—

132

C₄

—

848

—

133

C₄

C₂×#13133148

yes

—

134

C₄

#16810658

yes

—

135

C₄

#16811019

yes

—

136

C₄

#16811023

yes

—

137

C₂

#20274314

yes

—

138

C₂

#20274317

yes

—

139

C₂

#20274871

yes

characteristic

140

C₂

#20274876

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#8815
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1844

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
9. C₈×#88322	C₄	semidirect
83. C₅₃	#8815	semidirect

H ≅ G/N

type

9. C₈×#88322

C₄

semidirect

83. C₅₃

#8815

semidirect

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

subgroup	index
#20274872	2
#20274842	2
C₂×C₈×#88322	2
C₅₃×#1844	2
#8815	53

subgroup

index

#20274872

#20274842

C₂×C₈×#88322

C₅₃×#1844

#8815

Subgroups — #27277864

Conjugacy classes of subgroups

Complements

Maximal subgroups