Group #20275498 — subgroups

Subgroups — #20275498

Order 6784.

196 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#16810663	C₂	1	2	yes	maximal
3	#16810578	C₂	1	1	yes	maximal, characteristic
4	#16810392	C₂	1	2	yes	maximal
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×#11719917	C₂²	1	1	yes	characteristic
8	#13133176	C₂²	1	1	yes	characteristic
9	#13133153	C₄	1	4	yes	complemented
10	#13133150	—	2	2	no	—
11	#13133148	—	2	2	no	—
12	#13133127	—	2	1	no	—
13	C₈×Dic₅₃	—	2	1	no	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	#107×C₅₃	—	2	1	no	—
17	#107×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	Q₈	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂²⋊Dic₅₃	—	2	1	no	—
27	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
28	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
29	C₄×Dic₅₃	—	2	1	no	—
30	#11719917	—	2	2	no	—
31	#11719917	—	2	1	no	—
32	#11719917	C₂×C₄	1	4	yes	—
33	#11719917	Q₈	1	2	yes	—
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	—	8	2	no	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	#88322	—	2	4	no	—
50	#88322	—	2	2	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₅₃	#121	1	1	yes	characteristic
55	C₂²×C₅₃	#125	1	1	yes	characteristic
56	C₄×C₅₃	—	8	2	no	—
57	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₅₃	—	4	1	no	—
62	Dic₅₃	—	4	1	no	—
63	#1952	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#389	1	1	yes	characteristic
67	C₂×C₅₃	#472	1	1	yes	characteristic
68	C₂×C₅₃	#473	1	1	yes	characteristic
69	C₂×#108	—	53	1	no	—
70	#428	—	53	2	no	—
71	C₂×#107	D₅₃	1	1	yes	characteristic
72	#405	—	53	1	no	—
73	#343	—	53	2	no	—
74	C₅₃	#1952	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	53	1	no	—
76	C₂×M₁₆	—	53	2	no	—
77	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
78	#117	—	53	1	no	—
79	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
80	#108	—	53	4	no	—
81	#107	—	2	1	no	—
82	#107	—	2	1	no	—
83	#107	—	106	2	no	—
84	#106	—	106	2	no	—
85	#97	—	106	1	no	—
86	C₄×C₈	—	106	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
89	M₁₆	—	53	4	no	—
90	M₁₆	—	53	2	no	—
91	M₁₆	—	106	2	no	—
92	M₁₆	—	106	1	no	—
93	C₂×C₈	—	106	2	no	—
94	C₂×C₈	—	106	1	no	—
95	C₂×C₈	Dic₁₀₆	1	2	yes	—
96	C₂×C₈	C₄×D₅₃	1	2	yes	—
97	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	53	1	no	—
102	C₄⋊C₄	—	53	1	no	—
103	C₂²⋊C₄	—	106	1	no	—
104	C₄²	—	106	1	no	—
105	C₂³	#11719947	1	1	yes	characteristic
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	106	1	no	—
109	C₂×C₄	—	106	1	no	—
110	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
113	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
114	C₂×C₄	#11719949	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
116	C₈	—	106	4	no	—
117	C₈	—	106	2	no	—
118	C₈	#11719919	1	2	yes	—
119	C₈	#11719919	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#13133182	1	1	yes	characteristic
123	C₂²	#13133188	1	1	yes	characteristic
124	C₂²	#13133208	1	1	yes	characteristic
125	C₄	—	8	2	no	—
126	C₄	—	212	1	no	—
127	C₄	—	212	1	no	—
128	C₄	C₂×#11719919	1	1	yes	characteristic
129	C₄	#13133179	1	1	yes	characteristic
130	C₄	#13133193	1	1	yes	characteristic
131	C₄	#13133216	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16810519	1	1	yes	characteristic
135	C₂	#16810890	1	1	yes	characteristic
136	C₂	#16810895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810663

C₂

yes

maximal

#16810578

C₂

yes

maximal, characteristic

#16810392

C₂

yes

maximal

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

#13133153

C₄

yes

—

—

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

#107×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

—

#11719917

—

#11719917

—

#11719917

C₂×C₄

yes

—

#11719917

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1952

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#405

—

#343

—

C₅₃

#1952

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

—

—

—

—

106

—

#106

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

106

—

104

C₄²

—

106

—

105

C₂³

#11719947

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

111

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

112

C₂×C₄

D₄×D₅₃

yes

characteristic

113

C₂×C₄

Q₈×D₅₃

yes

characteristic

114

C₂×C₄

#11719949

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

116

C₈

—

106

—

117

C₈

—

106

—

118

C₈

#11719919

yes

—

119

C₈

#11719919

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#13133182

yes

characteristic

123

C₂²

#13133188

yes

characteristic

124

C₂²

#13133208

yes

characteristic

125

C₄

—

126

C₄

—

212

—

127

C₄

—

212

—

128

C₄

C₂×#11719919

yes

characteristic

129

C₄

#13133179

yes

characteristic

130

C₄

#13133193

yes

characteristic

131

C₄

#13133216

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#16810519

yes

characteristic

135

C₂

#16810890

yes

characteristic

136

C₂

#16810895

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1952
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂²

#13133188

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. #13133153	C₄	semidirect
74. C₅₃	#1952	semidirect

H ≅ G/N

type

9. #13133153

C₄

semidirect

74. C₅₃

#1952

semidirect

Subgroups — #20275498

Conjugacy classes of subgroups

Complements