Group #27277868 — subgroups

Subgroups — #27277868

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	#20274875	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×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	—	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×#107×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	#13133151	—	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	#107×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	D₈	1	4	yes	—
32	C₂×C₈×C₅₃	Q₁₆	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	4	no	—
41	C₂×#88322	—	2	4	no	—
42	C₂×#88322	C₄⋊C₄	1	8	yes	—
43	C₂³×C₅₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	8	4	no	—
46	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
52	C₂×C₄×C₅₃	#135	1	4	yes	—
53	C₂×C₄×C₅₃	#135	1	4	yes	—
54	C₂×Dic₅₃	—	8	2	no	—
55	C₂×Dic₅₃	—	8	4	no	—
56	C₈×C₅₃	#107	1	8	yes	—
57	#88322	—	2	8	no	—
58	#8816	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	#384	1	1	yes	characteristic
60	C₂²×C₅₃	#493	1	1	yes	characteristic
61	C₂²×C₅₃	#494	1	1	yes	characteristic
62	C₂²×C₅₃	#495	1	2	yes	—
63	C₂²×C₅₃	#495	1	2	yes	—
64	C₄×C₅₃	—	16	2	no	—
65	C₄×C₅₃	C₂×#107	1	2	yes	—
66	C₄×C₅₃	#427	1	4	yes	—
67	C₄×C₅₃	#499	1	4	yes	—
68	C₄×C₅₃	#500	1	2	yes	—
69	Dic₅₃	—	16	2	no	—
70	#1845	—	53	1	no	—
71	#1845	D₅₃	1	1	yes	characteristic
72	#1834	—	53	4	no	—
73	C₂×C₈²	—	53	1	no	—
74	C₂×C₅₃	#1709	1	1	yes	characteristic
75	C₂×C₅₃	#1709	1	1	yes	characteristic
76	C₂×C₅₃	#1710	1	2	yes	—
77	C₂×C₅₃	#1845	1	1	yes	characteristic
78	C₂×C₅₃	#1846	1	2	yes	—
79	C₂×#107	—	2	2	no	—
80	C₂×#107	—	106	2	no	—
81	C₂×C₄×C₈	—	53	2	no	—
82	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	#384	—	53	2	no	—
84	#384	C₂×D₅₃	1	2	yes	—
85	#340	—	106	8	no	—
86	C₈²	—	53	4	no	—
87	C₅₃	#8816	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	53	4	no	—
89	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	106	4	no	—
93	C₂×C₄⋊C₄	—	106	2	no	—
94	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
95	#107	—	4	4	no	—
96	#107	—	212	4	no	—
97	C₄×C₈	—	53	4	no	—
98	C₄×C₈	—	53	4	no	—
99	C₄×C₈	Dic₁₀₆	1	2	yes	—
100	C₄×C₈	C₄×D₅₃	1	2	yes	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	—	212	2	no	—
103	C₂²×C₄	D₄×D₅₃	1	2	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×C₈	—	53	8	no	—
106	C₂×C₈	—	106	4	no	—
107	C₂×C₈	—	106	4	no	—
108	C₂×C₈	#11719919	1	4	yes	—
109	C₂×C₈	#11719922	1	4	yes	—
110	C₂×C₈	#11719925	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	212	4	no	—
114	C₄⋊C₄	—	212	8	no	—
115	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₄²	Q₈×D₅₃	1	2	yes	—
118	C₂³	#13133213	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	424	4	no	—
122	C₂×C₄	—	424	2	no	—
123	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
124	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
125	C₂×C₄	#13133215	1	1	yes	characteristic
126	C₂×C₄	#13133216	1	2	yes	—
127	C₂×C₄	C₂×#11719922	1	2	yes	—
128	C₂×C₄	C₂×#11719925	1	2	yes	—
129	C₂×C₄	#13133236	1	4	yes	—
130	C₂×C₄	#13133240	1	4	yes	—
131	C₈	—	106	8	no	—
132	C₈	#13133150	1	8	yes	—
133	C₂²	#16810492	1	1	yes	characteristic
134	C₂²	#16810999	1	1	yes	characteristic
135	C₂²	#16811002	1	1	yes	characteristic
136	C₂²	#16811006	1	2	yes	—
137	C₂²	#16811008	1	2	yes	—
138	C₄	—	16	2	no	—
139	C₄	—	848	2	no	—
140	C₄	C₂×#13133150	1	2	yes	—
141	C₄	#16810660	1	4	yes	—
142	C₄	#16811024	1	4	yes	—
143	C₄	#16811026	1	2	yes	—
144	C₂	#20274320	1	1	yes	characteristic
145	C₂	#20274322	1	1	yes	characteristic
146	C₂	#20274324	1	2	yes	—
147	C₂	#20274874	1	1	yes	characteristic
148	C₂	#20274878	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274875

C₂

yes

maximal, characteristic

#20274842

C₂

yes

maximal

C₂×C₈×#88322

C₂

yes

maximal, characteristic

C₅₃×#1845

C₂

yes

maximal, characteristic

C₂×#13133151

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810385

—

C₈×#88322

C₄

yes

complemented

C₂×#107×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

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

#107×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

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

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

—

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×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#107

yes

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#495

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#499

yes

—

C₄×C₅₃

#500

yes

—

Dic₅₃

—

#1845

—

#1845

D₅₃

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1846

yes

—

C₂×#107

—

C₂×#107

—

106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

#384

C₂×D₅₃

yes

—

#340

—

106

—

C₈²

—

C₅₃

#8816

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

#107

—

#107

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

—

100

C₄×C₈

C₄×D₅₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

212

—

103

C₂²×C₄

D₄×D₅₃

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

#11719919

yes

—

109

C₂×C₈

#11719922

yes

—

110

C₂×C₈

#11719925

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

212

—

114

C₄⋊C₄

—

212

—

115

C₄²

C₂×Dic₁₀₆

yes

characteristic

116

C₄²

C₂×C₄×D₅₃

yes

characteristic

117

C₄²

Q₈×D₅₃

yes

—

118

C₂³

#13133213

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

424

—

122

C₂×C₄

—

424

—

123

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

124

C₂×C₄

C₂×#11719919

yes

characteristic

125

C₂×C₄

#13133215

yes

characteristic

126

C₂×C₄

#13133216

yes

—

127

C₂×C₄

C₂×#11719922

yes

—

128

C₂×C₄

C₂×#11719925

yes

—

129

C₂×C₄

#13133236

yes

—

130

C₂×C₄

#13133240

yes

—

131

C₈

—

106

—

132

C₈

#13133150

yes

—

133

C₂²

#16810492

yes

characteristic

134

C₂²

#16810999

yes

characteristic

135

C₂²

#16811002

yes

characteristic

136

C₂²

#16811006

yes

—

137

C₂²

#16811008

yes

—

138

C₄

—

139

C₄

—

848

—

140

C₄

C₂×#13133150

yes

—

141

C₄

#16810660

yes

—

142

C₄

#16811024

yes

—

143

C₄

#16811026

yes

—

144

C₂

#20274320

yes

characteristic

145

C₂

#20274322

yes

characteristic

146

C₂

#20274324

yes

—

147

C₂

#20274874

yes

characteristic

148

C₂

#20274878

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#8816
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1845

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 ◁ #1845

N	H ≅ G/N	type
9. C₈×#88322	C₄	semidirect
87. C₅₃	#8816	semidirect

H ≅ G/N

type

9. C₈×#88322

C₄

semidirect

87. C₅₃

#8816

semidirect

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

subgroup	index
#20274875	2
#20274842	2
C₂×C₈×#88322	2
C₅₃×#1845	2
#8816	53

subgroup

index

#20274875

#20274842

C₂×C₈×#88322

C₅₃×#1845

#8816

Subgroups — #27277868

Conjugacy classes of subgroups

Complements

Maximal subgroups