Group #20290534 — subgroups

Subgroups — #108×#6157 #20290534

Order 6784.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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	D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#6157	C₂	1	2	yes	maximal
3	C₂×C₈×#6157	C₂	1	1	yes	maximal, characteristic
4	#16811498	C₂	1	2	yes	maximal
5	#108×D₅₃	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₁₆	C₂²	1	2	yes	—
7	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#6157	C₂²	1	1	yes	characteristic
9	#13133288	C₄	1	2	yes	complemented
10	#13133287	C₄	1	2	yes	complemented
11	#13133271	—	2	2	no	—
12	C₈×#6157	—	2	2	no	—
13	C₈×#6157	C₄	1	2	yes	—
14	C₈×#6157	C₄	1	2	yes	—
15	M₁₆×D₅₃	C₂²	1	2	yes	—
16	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#13133153	C₄	1	1	yes	characteristic, complemented
18	#13133152	C₄	1	2	yes	complemented
19	#108×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6157	Q₈	1	2	yes	—
21	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
22	#11719940	—	2	2	no	—
23	C₂×#88324	—	2	1	no	—
24	C₄×#6157	C₂×C₄	1	2	yes	—
25	C₄×#6157	D₄	1	1	yes	characteristic
26	C₄×#6157	D₄	1	1	yes	characteristic
27	#11719936	C₂×C₄	1	4	yes	—
28	D₅₃⋊C₈	—	2	2	no	—
29	D₅₃⋊C₈	—	2	1	no	—
30	D₅₃⋊C₈	C₂×C₄	1	2	yes	—
31	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	#11719917	C₂×C₄	1	2	yes	—
34	C₂×#88322	C₂×C₄	1	1	yes	characteristic
35	#11719912	—	2	2	no	—
36	C₈×D₅₃	—	2	2	no	—
37	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×#6157	—	2	4	no	—
43	C₂×#6157	C₄⋊C₄	1	2	yes	—
44	C₂×#6157	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	#41	1	1	yes	characteristic
47	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₅₃	#41	1	1	yes	characteristic
51	#88324	—	2	2	no	—
52	#88324	—	2	4	no	—
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₈×C₅₃	C₄²	1	1	yes	characteristic
56	#88322	—	2	2	no	—
57	#88322	C₄²	1	1	yes	characteristic
58	#88322	C₄²	1	1	yes	characteristic
59	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
60	C₂×D₅₃	—	2	2	no	—
61	C₂×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₅₃	C₄×Q₈	1	1	yes	characteristic
63	#6157	—	2	2	no	—
64	#6157	#108	1	4	yes	direct product factor
65	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	Dic₅₃	#117	1	1	yes	characteristic
68	Dic₅₃	C₄×D₄	1	1	yes	characteristic
69	C₄×#108	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
72	D₅₃	—	2	1	no	—
73	D₅₃	C₂×#108	1	2	yes	—
74	C₂×#108	—	53	1	no	—
75	C₂×#108	—	53	1	no	—
76	C₄×M₁₆	—	53	2	no	—
77	C₂×C₄×C₈	—	53	1	no	—
78	#341	—	53	2	no	—
79	C₅₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₁₆	—	53	2	no	—
81	C₂×M₁₆	—	53	2	no	—
82	C₂²×C₈	—	53	1	no	—
83	C₂²×C₈	—	53	1	no	—
84	C₂×C₄²	—	53	1	no	—
85	#108	—	53	2	no	—
86	#108	—	53	2	no	—
87	#108	—	53	2	no	—
88	#108	—	53	1	no	—
89	#108	#6157	1	1	yes	characteristic, direct product factor
90	#97	—	106	2	no	—
91	C₄×C₈	—	53	2	no	—
92	C₄×C₈	—	53	2	no	—
93	C₄×C₈	—	106	2	no	—
94	C₂²×C₄	—	53	1	no	—
95	C₂²×C₄	—	53	2	no	—
96	M₁₆	—	53	4	no	—
97	M₁₆	—	53	2	no	—
98	M₁₆	—	106	2	no	—
99	M₁₆	—	106	2	no	—
100	M₁₆	C₂×#6157	1	2	yes	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	C₂×#6157	1	1	yes	characteristic
112	C₄²	—	53	2	no	—
113	C₄²	—	53	1	no	—
114	C₄²	—	53	1	no	—
115	C₂³	—	53	1	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	2	no	—
123	C₂×C₄	—	106	4	no	—
124	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	53	1	no	—
127	C₈	—	53	1	no	—
128	C₈	—	106	2	no	—
129	C₈	—	106	4	no	—
130	C₈	—	106	2	no	—
131	C₈	C₄×#6157	1	1	yes	characteristic
132	C₈	C₄×#6157	1	1	yes	characteristic
133	C₂²	—	53	1	no	—
134	C₂²	—	53	1	no	—
135	C₂²	—	106	2	no	—
136	C₂²	Q₈×#6157	1	1	yes	characteristic
137	C₄	—	53	4	no	—
138	C₄	—	53	1	no	—
139	C₄	—	53	1	no	—
140	C₄	—	106	2	no	—
141	C₄	C₂×C₄×#6157	1	1	yes	characteristic
142	C₄	D₄×#6157	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	53	2	no	—
145	C₂	—	106	1	no	—
146	C₂	C₄⋊C₄×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊#108

C₂

yes

maximal, characteristic

M₁₆×#6157

C₂

yes

maximal

C₂×C₈×#6157

C₂

yes

maximal, characteristic

#16811498

C₂

yes

maximal

#108×D₅₃

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×#6157

C₂²

yes

characteristic

#13133288

C₄

yes

complemented

#13133287

C₄

yes

—

—

C₄

yes

—

C₈×#6157

C₄

yes

—

M₁₆×D₅₃

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133153

C₄

yes

characteristic, complemented

#13133152

C₄

yes

complemented

#108×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

Q₈

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#6157

D₄

yes

characteristic

C₄×#6157

D₄

yes

characteristic

#11719936

C₂×C₄

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

D₅₃⋊C₈

C₂×C₄

yes

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

#11719917

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₂²×D₅₃

C₂×Q₈

yes

characteristic

C₂×#6157

—

C₂×#6157

C₄⋊C₄

yes

—

C₂×#6157

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

#41

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

C₄×D₅₃

yes

—

—

C₈×C₅₃

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₄²

yes

characteristic

#88322

—

#88322

C₄²

yes

characteristic

#88322

C₄²

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

C₄×Q₈

yes

—

yes

direct product factor

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

Dic₅₃

#117

yes

characteristic

Dic₅₃

C₄×D₄

yes

characteristic

C₄×#108

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

D₅₃

—

D₅₃

C₂×#108

yes

—

C₂×#108

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

C₅₃

C₄×#108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

—

—

—

—

yes

characteristic, direct product factor

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

100

M₁₆

C₂×#6157

yes

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

C₂×#6157

yes

characteristic

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

C₂²×#6157

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

106

—

129

C₈

—

106

—

130

C₈

—

106

—

131

C₈

C₄×#6157

yes

characteristic

132

C₈

C₄×#6157

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

Q₈×#6157

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

C₂×C₄×#6157

yes

characteristic

142

C₄

D₄×#6157

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

106

—

146

C₂

C₄⋊C₄×#6157

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6157
Fitting	#108×C₅₃	C₄
socle	C₂×C₅₃	C₄×C₄⋊C₄
hypercenter	#108	#6157
hypocenter	C₅₃	C₄×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄

C₂²×#6157

Fitting

#108×C₅₃

C₄

socle

C₂×C₅₃

C₄×C₄⋊C₄

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
9. #13133288	C₄	semidirect
10. #13133287	C₄	semidirect
17. #13133153	C₄	semidirect
18. #13133152	C₄	semidirect
19. #108×C₅₃	C₄	semidirect
64. #6157	#108	direct (N × H)
79. C₅₃	C₄×#108	semidirect
89. #108	#6157	direct (N × H)

H ≅ G/N

type

9. #13133288

C₄

semidirect

10. #13133287

C₄

semidirect

17. #13133153

C₄

semidirect

18. #13133152

C₄

semidirect

19. #108×C₅₃

C₄

semidirect

64. #6157

#108

direct (N × H)

79. C₅₃

C₄×#108

semidirect

89. #108

#6157

direct (N × H)

Subgroups — #108×#6157 #20290534

Conjugacy classes of subgroups

Complements