Group #20290824 — subgroups

Subgroups — #20290824

Order 6784.

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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×#13133287	C₂	1	1	yes	maximal, characteristic
2	#16811630	C₂	1	2	yes	maximal
3	C₂×#13133271	C₂	1	1	yes	maximal, characteristic
4	#16811493	C₂	1	2	yes	maximal
5	#16810660	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	C₂²	1	2	yes	—
7	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#6157	C₂²	1	1	yes	characteristic
9	#13133287	—	2	2	no	—
10	#13133284	—	2	2	no	—
11	C₄⋊#88324	—	2	2	no	—
12	#13133271	—	2	2	no	—
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133175	C₂²	1	2	yes	—
15	#13133151	C₄	1	1	yes	characteristic, complemented
16	#13133148	C₄	1	2	yes	complemented
17	#107×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂²×#6157	—	2	1	no	—
19	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88324	—	2	2	no	—
21	C₂×#88324	D₄	1	2	yes	—
22	C₄×#6157	—	2	1	no	—
23	C₄×#6157	Q₈	1	2	yes	—
24	#11719936	—	2	2	no	—
25	#11719936	—	2	2	no	—
26	D₅₃⋊C₈	—	2	1	no	—
27	D₅₃⋊C₈	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
30	#11719921	—	2	2	no	—
31	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₂×#88322	C₂×C₄	1	1	yes	characteristic
34	C₈×D₅₃	D₄	1	2	yes	—
35	C₈×D₅₃	Q₈	1	2	yes	—
36	C₂²×D₅₃	#41	1	1	yes	characteristic
37	C₂×#6157	—	2	2	no	—
38	C₂×#6157	—	4	1	no	—
39	C₂×#6157	—	4	1	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄×D₅₃	#41	1	1	yes	characteristic
48	#88324	—	2	2	no	—
49	#88324	—	4	2	no	—
50	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	#88322	C₄⋊C₄	1	2	yes	—
52	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₂×D₅₃	—	2	1	no	—
54	C₂×D₅₃	—	2	1	no	—
55	C₂×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂×D₅₃	#125	1	1	yes	characteristic
57	#6157	—	4	2	no	—
58	C₄×C₅₃	—	4	2	no	—
59	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	Dic₅₃	—	4	2	no	—
62	Dic₅₃	#121	1	1	yes	characteristic
63	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	#1946	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	#389	1	1	yes	characteristic
66	C₂×C₅₃	#444	1	1	yes	characteristic
67	C₂×C₅₃	#444	1	1	yes	characteristic
68	D₅₃	—	2	1	no	—
69	D₅₃	—	2	1	no	—
70	C₂×#108	—	53	1	no	—
71	#427	—	53	1	no	—
72	#424	—	53	2	no	—
73	C₂×#97	—	53	1	no	—
74	#339	—	53	2	no	—
75	C₅₃	#1946	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	53	2	no	—
77	C₂²×C₈	—	53	1	no	—
78	C₂²×C₈	—	53	1	no	—
79	#117	—	53	2	no	—
80	C₂×C₄²	—	53	1	no	—
81	#108	—	106	2	no	—
82	#107	—	53	1	no	—
83	#107	#6157	1	1	yes	characteristic, complemented
84	#106	—	53	2	no	—
85	C₄⋊C₈	—	106	2	no	—
86	C₄⋊C₈	—	106	2	no	—
87	#97	—	106	2	no	—
88	C₂²×C₄	—	53	1	no	—
89	C₂²×C₄	—	106	1	no	—
90	M₁₆	—	106	2	no	—
91	M₁₆	—	106	2	no	—
92	C₂×C₈	—	53	1	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	53	2	no	—
96	C₂×C₈	—	106	1	no	—
97	C₂×C₈	—	106	1	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	C₂×#6157	1	1	yes	characteristic
100	C₄⋊C₄	—	53	2	no	—
101	C₄⋊C₄	C₂×#6157	1	2	yes	—
102	C₂²⋊C₄	—	106	2	no	—
103	C₄²	—	53	2	no	—
104	C₄²	—	106	2	no	—
105	C₄²	—	106	1	no	—
106	C₂³	—	53	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	53	1	no	—
109	C₂×C₄	—	53	1	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	106	2	no	—
114	C₂×C₄	—	106	2	no	—
115	C₂×C₄	—	212	1	no	—
116	C₂×C₄	—	212	1	no	—
117	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
118	C₈	—	53	2	no	—
119	C₈	—	106	2	no	—
120	C₈	—	212	2	no	—
121	C₈	C₄⋊#6157	1	2	yes	—
122	C₂²	—	53	1	no	—
123	C₂²	—	53	1	no	—
124	C₂²	—	106	1	no	—
125	C₂²	—	106	1	no	—
126	C₂²	D₄×#6157	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	53	1	no	—
129	C₄	—	53	1	no	—
130	C₄	—	212	2	no	—
131	C₄	—	212	2	no	—
132	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
133	C₄	Q₈×#6157	1	1	yes	characteristic
134	C₂	—	106	1	no	—
135	C₂	—	106	1	no	—
136	C₂	#16811574	1	1	yes	characteristic
137	C₂	#16811669	1	1	yes	characteristic
138	C₂	#16811670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133287

C₂

yes

maximal, characteristic

#16811630

C₂

yes

maximal

C₂×#13133271

C₂

yes

maximal, characteristic

#16811493

C₂

yes

maximal

#16810660

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×#6157

C₂²

yes

—

—

—

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133175

C₂²

yes

—

#13133151

C₄

yes

characteristic, complemented

#13133148

C₄

yes

complemented

#107×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×#88324

—

C₂×#88324

D₄

yes

—

C₄×#6157

—

C₄×#6157

Q₈

yes

—

#11719936

—

#11719936

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

yes

—

—

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

C₄×D₅₃

yes

—

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊Q₈

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

#121

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×#108

—

#427

—

#424

—

C₂×#97

—

#339

—

C₅₃

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

#108

—

106

—

#107

—

#107

#6157

yes

characteristic, complemented

#106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#97

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×#6157

yes

—

102

C₂²⋊C₄

—

106

—

103

C₄²

—

104

C₄²

—

106

—

105

C₄²

—

106

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₂²×#6157

yes

characteristic

118

C₈

—

119

C₈

—

106

—

120

C₈

—

212

—

121

C₈

C₄⋊#6157

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

106

—

125

C₂²

—

106

—

126

C₂²

D₄×#6157

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

212

—

131

C₄

—

212

—

132

C₄

C₂×C₄⋊#6157

yes

characteristic

133

C₄

Q₈×#6157

yes

characteristic

134

C₂

—

106

—

135

C₂

—

106

—

136

C₂

#16811574

yes

characteristic

137

C₂

#16811669

yes

characteristic

138

C₂

#16811670

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#107×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#107	#6157
hypocenter	C₅₃	#1946
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#107×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
15. #13133151	C₄	semidirect
16. #13133148	C₄	semidirect
17. #107×C₅₃	C₄	semidirect
75. C₅₃	#1946	semidirect
83. #107	#6157	semidirect

H ≅ G/N

type

15. #13133151

C₄

semidirect

16. #13133148

C₄

semidirect

17. #107×C₅₃

C₄

semidirect

75. C₅₃

#1946

semidirect

83. #107

#6157

semidirect

Subgroups — #20290824

Conjugacy classes of subgroups

Complements