Group #20290673 — subgroups

Subgroups — #20290673

Order 6784.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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	#16811703	C₂	1	1	yes	maximal, characteristic
2	#16811638	C₂	1	2	yes	maximal, complemented
3	#16811598	C₂	1	1	yes	maximal, characteristic
4	#16811498	C₂	1	2	yes	maximal, complemented
5	#16810606	C₂	1	1	yes	maximal, characteristic
6	#13133305	—	2	2	no	—
7	#13133305	—	2	2	no	—
8	C₂×#11719936	C₂²	1	1	yes	characteristic
9	D₅₃⋊M₁₆	—	2	2	no	—
10	D₅₃⋊M₁₆	C₂²	1	2	yes	—
11	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
12	#13133288	—	2	2	no	—
13	#13133287	—	2	2	no	—
14	#13133274	—	2	2	no	—
15	C₂×#11719949	C₂²	1	1	yes	characteristic
16	M₁₆×D₅₃	C₂²	1	2	yes	—
17	#13133136	C₄	1	1	yes	characteristic
18	#13133133	C₄	1	2	yes	—
19	#101×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	#11719949	—	2	2	no	—
22	#11719949	—	2	2	no	—
23	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₁₂	Q₈	1	2	yes	—
25	C₂×C₄×D₅₃	D₄	1	2	yes	—
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	#11719940	—	2	2	no	—
28	C₂×#88324	—	2	1	no	—
29	C₂×#88324	—	2	1	no	—
30	#11719936	—	2	2	no	—
31	#11719936	—	2	2	no	—
32	#11719936	—	2	2	no	—
33	#11719936	D₄	1	2	yes	—
34	#11719936	D₄	1	2	yes	—
35	D₅₃⋊C₈	—	2	4	no	—
36	D₅₃⋊C₈	—	2	2	no	—
37	D₅₃⋊C₈	—	2	2	no	—
38	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	#11719917	C₂×C₄	1	2	yes	—
40	#11719912	—	2	2	no	—
41	C₈×D₅₃	—	2	2	no	—
42	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
43	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
48	D₂₁₂	—	2	4	no	—
49	D₂₁₂	—	2	2	no	—
50	C₄×D₅₃	—	2	2	no	—
51	C₄×D₅₃	—	2	2	no	—
52	C₄×D₅₃	C₂×D₄	1	2	yes	—
53	C₄×D₅₃	#41	1	2	yes	—
54	#88324	—	2	2	no	—
55	#88324	—	2	2	no	—
56	#88324	—	4	1	no	—
57	#88324	—	4	1	no	—
58	C₈×C₅₃	—	2	2	no	—
59	#88322	—	2	2	no	—
60	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×D₅₃	—	2	2	no	—
62	C₂×D₅₃	—	4	1	no	—
63	C₂×D₅₃	—	4	2	no	—
64	C₂×D₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	C₄×D₄	1	2	yes	—
67	Dic₅₃	#120	1	1	yes	characteristic
68	Dic₅₃	#123	1	1	yes	characteristic
69	#1858	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	#385	1	1	yes	characteristic
72	D₅₃	—	2	2	no	—
73	D₅₃	—	4	2	no	—
74	#568	—	53	1	no	—
75	#430	—	53	2	no	—
76	#413	—	53	1	no	—
77	#408	—	53	1	no	—
78	#341	—	53	2	no	—
79	C₅₃	#1858	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	53	1	no	—
81	#131	—	106	2	no	—
82	#131	—	106	2	no	—
83	C₂×M₁₆	—	53	2	no	—
84	C₂×M₁₆	—	53	2	no	—
85	C₂×M₁₆	—	53	1	no	—
86	C₂×M₁₆	—	106	2	no	—
87	C₂²×C₈	—	53	1	no	—
88	#108	—	106	2	no	—
89	#108	—	106	2	no	—
90	#101	—	53	1	no	—
91	#101	#6157	1	1	yes	characteristic
92	#100	—	53	2	no	—
93	C₂²⋊C₈	—	106	2	no	—
94	#41	—	106	2	no	—
95	#41	—	106	2	no	—
96	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
97	C₂×D₄	—	53	2	no	—
98	C₂×D₄	—	53	1	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	2	no	—
101	M₁₆	—	53	2	no	—
102	M₁₆	—	53	2	no	—
103	M₁₆	—	53	2	no	—
104	M₁₆	—	106	2	no	—
105	M₁₆	—	106	2	no	—
106	M₁₆	—	106	2	no	—
107	M₁₆	—	106	2	no	—
108	M₁₆	—	106	2	no	—
109	M₁₆	C₂×#6157	1	2	yes	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	106	2	no	—
112	C₂×C₈	—	106	4	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂³	—	53	1	no	—
117	C₂³	—	53	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	106	4	no	—
120	D₄	—	106	2	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	2	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	C₄⋊#6157	1	2	yes	—
127	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	106	2	no	—
130	C₈	—	106	2	no	—
131	C₈	—	106	2	no	—
132	C₈	—	212	1	no	—
133	C₈	—	212	1	no	—
134	C₂²	—	53	2	no	—
135	C₂²	—	106	2	no	—
136	C₂²	—	212	1	no	—
137	C₂²	—	212	2	no	—
138	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	53	1	no	—
141	C₄	—	53	1	no	—
142	C₄	D₄×#6157	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	106	2	no	—
145	C₂	—	212	2	no	—
146	C₂	#16811570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811703

C₂

yes

maximal, characteristic

#16811638

C₂

yes

maximal, complemented

#16811598

C₂

yes

maximal, characteristic

#16811498

C₂

yes

maximal, complemented

#16810606

C₂

yes

maximal, characteristic

#13133305

—

#13133305

—

C₂×#11719936

C₂²

yes

characteristic

D₅₃⋊M₁₆

—

D₅₃⋊M₁₆

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

M₁₆×D₅₃

C₂²

yes

—

#13133136

C₄

yes

characteristic

#13133133

C₄

yes

—

#101×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

#11719949

—

#11719949

—

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×D₂₁₂

Q₈

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

—

—

—

—

—

D₄

yes

—

#11719936

D₄

yes

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

M₁₆×C₅₃

C₂×C₄

yes

—

#11719917

C₂×C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

C₂×Q₈

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

#41

yes

—

—

—

—

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#385

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

—

—

C₅₃

#1858

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

#108

—

106

—

#108

—

106

—

—

yes

—

C₂²⋊C₈

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

C₂×#6157

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

106

—

105

M₁₆

—

106

—

106

M₁₆

—

106

—

107

M₁₆

—

106

—

108

M₁₆

—

106

—

109

M₁₆

C₂×#6157

yes

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂³

—

117

C₂³

—

118

Q₈

—

119

D₄

—

106

—

120

D₄

—

106

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

C₄⋊#6157

yes

—

127

C₂×C₄

C₂²×#6157

yes

characteristic

128

C₈

—

129

C₈

—

106

—

130

C₈

—

106

—

131

C₈

—

106

—

132

C₈

—

212

—

133

C₈

—

212

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

—

212

—

137

C₂²

—

212

—

138

C₂²

C₂×C₄⋊#6157

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

D₄×#6157

yes

—

143

C₂

—

144

C₂

—

106

—

145

C₂

—

212

—

146

C₂

#16811570

yes

characteristic

	subgroup	G/H
center	C₂	#16811570
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#101×C₅₃	C₄
socle	C₂×C₅₃	#385
hypercenter	#101	#6157
hypocenter	C₅₃	#1858
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

#16811570

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#101×C₅₃

C₄

C₂×C₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
2. #16811638	C₂	semidirect
4. #16811498	C₂	semidirect
79. C₅₃	#1858	semidirect

H ≅ G/N

type

2. #16811638

C₂

semidirect

4. #16811498

C₂

semidirect

79. C₅₃

#1858

semidirect

Subgroups — #20290673

Conjugacy classes of subgroups

Complements