Group #27277891 — subgroups

Subgroups — #27277891

Order 13568.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#20274887	C₂	1	1	yes	maximal, characteristic
2	#20273470	C₂	1	4	yes	maximal
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	—	2	2	no	—
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810373	—	2	8	no	—
9	C₈×#88322	C₄	1	1	yes	characteristic
10	C₈×#88322	C₄	1	2	yes	—
11	C₈×#88322	C₄	1	1	yes	characteristic
12	C₂×#108×C₅₃	—	2	2	no	—
13	C₅₃×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	2	no	—
16	C₂²×#88322	Q₈	1	4	yes	—
17	#13133153	—	4	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄×#88322	C₂×C₄	1	2	yes	—
20	C₄×#88322	C₂×C₄	1	2	yes	—
21	C₄×#88322	D₄	1	4	yes	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#108×C₅₃	—	4	4	no	—
26	C₄⋊C₈×C₅₃	—	2	4	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
32	M₁₆×C₅₃	—	4	4	no	—
33	C₂×C₈×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
37	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
38	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×D₄	1	2	yes	—
40	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	#11719917	—	4	4	no	—
42	C₂×#88322	—	2	8	no	—
43	C₂×#88322	—	4	2	no	—
44	C₂×#88322	C₄⋊C₄	1	4	yes	—
45	C₂×#88322	C₄⋊C₄	1	4	yes	—
46	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
54	C₈×C₅₃	—	8	2	no	—
55	C₈×C₅₃	#106	1	4	yes	—
56	C₈×C₅₃	#106	1	4	yes	—
57	#88322	—	2	4	no	—
58	#88322	—	8	2	no	—
59	#88322	#108	1	8	yes	complemented
60	#8822	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	#384	1	1	yes	characteristic
63	C₂²×C₅₃	#498	1	2	yes	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	C₂×#106	1	2	yes	—
66	C₄×C₅₃	C₂×#108	1	4	yes	—
67	C₄×C₅₃	#492	1	1	yes	characteristic
68	C₄×C₅₃	#492	1	1	yes	characteristic
69	#1848	—	53	1	no	—
70	#1848	D₅₃	1	1	yes	characteristic
71	#1566	—	53	4	no	—
72	C₂×C₈²	—	53	1	no	—
73	C₂×C₅₃	—	2	2	no	—
74	C₂×C₅₃	#1844	1	1	yes	characteristic
75	C₂×C₅₃	#1848	1	2	yes	—
76	C₂×#108	—	2	2	no	—
77	C₂×#108	—	106	2	no	—
78	#423	—	53	2	no	—
79	#423	C₂×D₅₃	1	2	yes	—
80	C₂×C₄×C₈	—	53	2	no	—
81	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
82	#334	—	106	8	no	—
83	C₈²	—	53	1	no	—
84	C₈²	—	53	2	no	—
85	C₈²	—	53	1	no	—
86	C₅₃	#8822	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	106	2	no	—
89	C₂²×C₈	—	53	4	no	—
90	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
91	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
92	#108	—	4	4	no	—
93	#108	—	212	4	no	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	106	4	no	—
96	C₄×C₈	—	53	2	no	—
97	C₄×C₈	—	53	2	no	—
98	C₄×C₈	—	53	4	no	—
99	C₄×C₈	Dic₁₀₆	1	2	yes	—
100	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
101	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
102	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
103	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
104	M₁₆	—	4	4	no	—
105	M₁₆	—	212	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	—	106	8	no	—
110	C₂×C₈	—	212	2	no	—
111	C₂×C₈	#11719919	1	2	yes	—
112	C₂×C₈	#11719919	1	2	yes	—
113	C₂×C₈	#11719923	1	4	yes	—
114	C₂×C₈	#11719924	1	4	yes	—
115	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₄²	D₄×D₅₃	1	2	yes	—
118	C₂³	#13133215	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
121	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
122	C₂×C₄	#13133213	1	1	yes	characteristic
123	C₂×C₄	#13133216	1	2	yes	—
124	C₂×C₄	C₂×#11719924	1	2	yes	—
125	C₂×C₄	C₂×#11719923	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	53	8	no	—
128	C₈	—	106	4	no	—
129	C₈	—	424	2	no	—
130	C₈	#13133148	1	4	yes	—
131	C₈	#13133148	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#16810492	1	1	yes	characteristic
134	C₂²	#16811019	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	C₂×#13133148	1	2	yes	—
137	C₄	#108×D₅₃	1	4	yes	—
138	C₄	#16810995	1	1	yes	characteristic
139	C₄	#16810997	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#20274871	1	1	yes	characteristic
142	C₂	#20274885	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal, characteristic

#20273470

C₂

yes

maximal

C₂×C₈×#88322

C₂

yes

maximal, characteristic

C₅₃×#1848

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810373

—

C₈×#88322

C₄

yes

characteristic

C₈×#88322

C₄

yes

—

C₈×#88322

C₄

yes

characteristic

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

C₂²×#88322

Q₈

yes

—

#13133153

—

C₄⋊#88322

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#106

yes

—

—

—

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#498

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#106

yes

—

C₄×C₅₃

C₂×#108

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

D₅₃

yes

characteristic

#1566

—

C₂×C₈²

—

C₂×C₅₃

—

C₂×C₅₃

#1844

yes

characteristic

C₂×C₅₃

#1848

yes

—

C₂×#108

—

C₂×#108

—

106

—

#423

—

#423

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#334

—

106

—

C₈²

—

C₈²

—

C₈²

—

C₅₃

#8822

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#108

—

#108

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

—

100

C₄×C₈

C₄×D₅₃

yes

characteristic

101

C₄×C₈

C₄×D₅₃

yes

characteristic

102

C₂²×C₄

Q₈×D₅₃

yes

—

103

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

104

M₁₆

—

105

M₁₆

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

212

—

111

C₂×C₈

#11719919

yes

—

112

C₂×C₈

#11719919

yes

—

113

C₂×C₈

#11719923

yes

—

114

C₂×C₈

#11719924

yes

—

115

C₄²

C₂×Dic₁₀₆

yes

characteristic

116

C₄²

C₂×C₄×D₅₃

yes

characteristic

117

C₄²

D₄×D₅₃

yes

—

118

C₂³

#13133215

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

121

C₂×C₄

C₂×#11719919

yes

characteristic

122

C₂×C₄

#13133213

yes

characteristic

123

C₂×C₄

#13133216

yes

—

124

C₂×C₄

C₂×#11719924

yes

—

125

C₂×C₄

C₂×#11719923

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

106

—

129

C₈

—

424

—

130

C₈

#13133148

yes

—

131

C₈

#13133148

yes

—

132

C₂²

—

133

C₂²

#16810492

yes

characteristic

134

C₂²

#16811019

yes

—

135

C₄

—

136

C₄

C₂×#13133148

yes

—

137

C₄

#108×D₅₃

yes

—

138

C₄

#16810995

yes

characteristic

139

C₄

#16810997

yes

characteristic

140

C₂

—

141

C₂

#20274871

yes

characteristic

142

C₂

#20274885

yes

—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#8822
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1848

C₂

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
59. #88322	#108	semidirect
86. C₅₃	#8822	semidirect

H ≅ G/N

type

59. #88322

#108

semidirect

86. C₅₃

#8822

semidirect

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

subgroup	index
#20274887	2
#20273470	2
C₂×C₈×#88322	2
C₅₃×#1848	2
#8822	53

subgroup

index

#20274887

#20273470

C₂×C₈×#88322

C₅₃×#1848

#8822

Subgroups — #27277891

Conjugacy classes of subgroups

Complements

Maximal subgroups