Group #20276948 — subgroups

Subgroups — #20276948

Order 6784.

138 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16811024	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811008	C₂	1	1	yes	maximal, characteristic
3	#16810427	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810425	C₂	1	1	yes	maximal, characteristic
5	#16810423	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	—	2	1	no	—
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	#13133208	—	2	1	no	—
11	#13133206	—	2	1	no	—
12	C₂×#11719908	C₂²	1	1	yes	characteristic
13	#13133151	C₂²	1	1	yes	characteristic
14	#13133148	—	2	1	no	—
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	1	no	—
20	C₂×D₂₁₂	—	2	1	no	—
21	Q₁₆×C₅₃	—	2	1	no	—
22	Q₁₆×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	#11719924	—	4	2	no	—
26	#11719921	—	4	1	no	—
27	#11719921	—	4	1	no	—
28	C₄⋊Dic₅₃	—	2	1	no	—
29	C₄⋊Dic₅₃	—	4	1	no	—
30	#11719919	—	4	1	no	—
31	C₄×Dic₅₃	D₄	1	1	yes	characteristic
32	C₂×#88322	D₄	1	1	yes	characteristic
33	D₄₂₄	—	2	1	no	—
34	D₄₂₄	—	2	1	no	—
35	C₁₆×C₅₃	—	2	1	no	—
36	#11719908	—	2	1	no	—
37	C₂²×D₅₃	—	4	1	no	—
38	Q₈×C₅₃	—	4	1	no	—
39	Q₈×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	D₈	1	1	yes	characteristic
45	C₂×Dic₅₃	D₈	1	1	yes	characteristic
46	D₂₁₂	—	4	1	no	—
47	D₂₁₂	—	4	1	no	—
48	C₈×C₅₃	#41	1	1	yes	characteristic
49	C₈×C₅₃	#41	1	1	yes	characteristic
50	#88322	SD₁₆	1	2	yes	—
51	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×D₅₃	—	8	1	no	—
53	C₂×D₅₃	—	8	1	no	—
54	C₂×D₅₃	—	8	1	no	—
55	C₄×C₅₃	—	8	1	no	—
56	C₄×C₅₃	—	8	1	no	—
57	C₄×C₅₃	#124	1	1	yes	characteristic
58	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	—	2	1	no	—
61	Dic₅₃	—	8	1	no	—
62	#2242	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#486	1	1	yes	characteristic
64	C₂×C₅₃	#508	1	1	yes	characteristic
65	C₂×C₅₃	#508	1	1	yes	characteristic
66	D₅₃	—	16	1	no	—
67	#499	—	53	1	no	—
68	#495	—	53	1	no	—
69	#358	—	53	1	no	—
70	#358	D₅₃	1	1	yes	characteristic, complemented
71	#357	—	53	1	no	—
72	#357	—	53	1	no	—
73	C₄×C₁₆	—	53	1	no	—
74	C₅₃	#2242	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
76	C₂×SD₁₆	—	106	1	no	—
77	C₂×D₈	—	53	1	no	—
78	#125	—	106	1	no	—
79	#124	—	106	1	no	—
80	C₂×C₁₆	—	53	1	no	—
81	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	#107	—	53	1	no	—
83	#107	C₂×D₅₃	1	1	yes	characteristic
84	#106	—	106	1	no	—
85	C₄×C₈	—	53	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×D₄	—	106	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	2	1	no	—
90	SD₁₆	—	212	2	no	—
91	D₈	—	106	1	no	—
92	D₈	—	106	1	no	—
93	C₂×C₈	—	53	1	no	—
94	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	106	1	no	—
97	C₄⋊C₄	—	212	1	no	—
98	C₄⋊C₄	—	212	1	no	—
99	C₂²⋊C₄	—	212	1	no	—
100	C₂²⋊C₄	—	212	1	no	—
101	C₄²	—	53	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	106	1	no	—
104	C₂³	—	212	1	no	—
105	Q₈	—	4	1	no	—
106	Q₈	—	4	1	no	—
107	D₄	—	212	1	no	—
108	D₄	—	212	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
115	C₈	—	53	2	no	—
116	C₈	#11719945	1	1	yes	characteristic
117	C₈	#11719947	1	1	yes	characteristic
118	C₂²	—	424	1	no	—
119	C₂²	—	424	1	no	—
120	C₂²	—	424	1	no	—
121	C₂²	D₈×D₅₃	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	106	1	no	—
125	C₄	—	106	1	no	—
126	C₄	—	424	1	no	—
127	C₄	#13133203	1	1	yes	characteristic
128	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
129	C₂	—	848	1	no	—
130	C₂	#16810968	1	1	yes	characteristic
131	C₂	#16811054	1	1	yes	characteristic
132	C₂	#16811056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811024

C₂

yes

maximal, characteristic, complemented

#16811008

C₂

yes

maximal, characteristic

#16810427

C₂

yes

maximal, characteristic, complemented

#16810425

C₂

yes

maximal, characteristic

#16810423

C₂

yes

maximal, characteristic

C₁₆×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#358

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

—

C₂×D₄₂₄

C₂²

yes

—

—

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133148

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719924

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

yes

C₈×C₅₃

yes

SD₁₆

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#357

—

#357

—

C₄×C₁₆

—

C₅₃

#2242

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×SD₁₆

—

106

—

C₂×D₈

—

#125

—

106

—

#124

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#107

C₂×D₅₃

yes

characteristic

#106

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×D₄

—

106

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

212

—

D₈

—

106

—

D₈

—

106

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

100

C₂²⋊C₄

—

212

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

—

106

—

104

C₂³

—

212

—

105

Q₈

—

106

Q₈

—

107

D₄

—

212

—

108

D₄

—

212

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

212

—

114

C₂×C₄

D₄×D₅₃

yes

characteristic

115

C₈

—

116

C₈

#11719945

yes

characteristic

117

C₈

#11719947

yes

characteristic

118

C₂²

—

424

—

119

C₂²

—

424

—

120

C₂²

—

424

—

121

C₂²

D₈×D₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

—

424

—

127

C₄

#13133203

yes

characteristic

128

C₄

SD₁₆×D₅₃

yes

characteristic

129

C₂

—

848

—

130

C₂

#16810968

yes

characteristic

131

C₂

#16811054

yes

characteristic

132

C₂

#16811056

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#358	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#358	D₅₃
hypocenter	C₅₃	#2242
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#358

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
1. #16811024	C₂	semidirect
3. #16810427	C₂	semidirect
6. C₁₆×Dic₅₃	C₂	semidirect
7. C₅₃×#358	C₂	semidirect
70. #358	D₅₃	semidirect
74. C₅₃	#2242	semidirect

H ≅ G/N

type

1. #16811024

C₂

semidirect

3. #16810427

C₂

semidirect

6. C₁₆×Dic₅₃

C₂

semidirect

7. C₅₃×#358

C₂

semidirect

70. #358

D₅₃

semidirect

74. C₅₃

#2242

semidirect

Subgroups — #20276948

Conjugacy classes of subgroups

Complements