Group #20273796 — subgroups

Subgroups — #20273796

Order 6784.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811339	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810815	C₂	1	2	yes	maximal, complemented
3	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810352	C₂	1	2	yes	maximal, complemented
5	C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719922	—	2	2	no	—
7	#13133206	—	2	2	no	—
8	#13133192	—	2	2	no	—
9	C₄×D₂₁₂	C₂²	1	2	yes	—
10	#13133175	—	2	2	no	—
11	C₄⋊#88322	C₂²	1	2	yes	—
12	#13133138	—	2	2	no	—
13	C₂×SD₁₆×C₅₃	—	2	2	no	—
14	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	#127×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	D₄	1	2	yes	—
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	C₂×D₂₁₂	—	2	2	no	—
21	C₂×D₂₁₂	D₄	1	2	yes	—
22	C₂×C₄×D₅₃	—	2	2	no	—
23	SD₁₆×C₅₃	—	4	4	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719922	—	4	4	no	—
28	#11719921	—	2	4	no	—
29	#11719921	—	2	2	no	—
30	C₄⋊Dic₅₃	D₄	1	2	yes	—
31	C₄×Dic₅₃	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂³×C₅₃	—	4	1	no	—
34	C₂²×D₅₃	—	2	2	no	—
35	Q₈×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	4	2	no	—
38	D₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	2	no	—
43	D₂₁₂	—	2	2	no	—
44	D₂₁₂	—	2	2	no	—
45	D₂₁₂	—	8	2	no	—
46	C₄×D₅₃	—	4	2	no	—
47	C₄×D₅₃	—	4	2	no	—
48	C₈×C₅₃	—	2	2	no	—
49	#88322	—	4	2	no	—
50	C₂²×C₅₃	—	8	2	no	—
51	C₂²×C₅₃	—	8	1	no	—
52	C₂²×C₅₃	#120	1	1	yes	characteristic
53	C₂×D₅₃	—	4	2	no	—
54	C₂×D₅₃	—	4	2	no	—
55	C₂×D₅₃	—	4	2	no	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	4	2	no	—
58	C₄×C₅₃	#135	1	2	yes	—
59	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₅₃	—	4	2	no	—
61	#1624	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	16	1	no	—
63	C₂×C₅₃	#449	1	2	yes	—
64	C₂×C₅₃	#453	1	1	yes	characteristic
65	D₅₃	—	8	2	no	—
66	#555	—	53	1	no	—
67	#492	D₅₃	1	1	yes	characteristic, complemented
68	#459	—	53	2	no	—
69	#333	—	53	1	no	—
70	D₄⋊C₈	—	53	2	no	—
71	C₅₃	#1624	1	1	yes	characteristic, complemented, Sylow
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	106	2	no	—
74	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
75	#127	C₂×D₅₃	1	1	yes	characteristic
76	#124	—	106	2	no	—
77	#121	—	106	2	no	—
78	C₄×D₄	—	53	2	no	—
79	#117	—	106	2	no	—
80	C₄⋊C₈	—	53	2	no	—
81	#102	—	106	2	no	—
82	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	53	2	no	—
87	C₂×D₄	—	106	2	no	—
88	C₂²×C₄	—	106	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	212	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	106	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	53	2	no	—
95	C₂²⋊C₄	—	106	4	no	—
96	C₂²⋊C₄	—	106	2	no	—
97	C₄²	—	106	2	no	—
98	C₄²	C₂²×D₅₃	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	106	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	106	2	no	—
106	D₄	—	106	2	no	—
107	D₄	—	424	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	106	2	no	—
110	C₂×C₄	—	212	2	no	—
111	C₂×C₄	—	212	2	no	—
112	C₂×C₄	D₄×D₅₃	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	212	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	212	2	no	—
119	C₂²	—	212	2	no	—
120	C₂²	—	212	2	no	—
121	C₂²	#13133185	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	212	2	no	—
125	C₄	#13133238	1	2	yes	—
126	C₄	#13133244	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	424	2	no	—
129	C₂	#16810767	1	2	yes	—
130	C₂	#16810790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811339

C₂

yes

maximal, characteristic, complemented

#16810815

C₂

yes

maximal, complemented

#16810372

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, complemented

C₅₃×#492

C₂

yes

maximal, characteristic, complemented

C₂×#11719922

—

#13133206

—

#13133192

—

C₄×D₂₁₂

C₂²

yes

—

#13133175

—

C₄⋊#88322

C₂²

yes

—

#13133138

—

C₂×SD₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

C₄⋊Dic₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

Dic₅₃

—

#1624

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#449

yes

—

C₂×C₅₃

#453

yes

characteristic

D₅₃

—

#555

—

#492

D₅₃

yes

characteristic, complemented

#459

—

#333

—

D₄⋊C₈

—

C₅₃

#1624

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

#124

—

106

—

#121

—

106

—

C₄×D₄

—

#117

—

106

—

C₄⋊C₈

—

#102

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

SD₁₆

—

D₈

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

100

C₂³

—

106

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

—

106

D₄

—

106

—

107

D₄

—

424

—

108

C₂×C₄

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

D₄×D₅₃

yes

—

113

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

114

C₈

—

115

C₈

—

212

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

212

—

119

C₂²

—

212

—

120

C₂²

—

212

—

121

C₂²

#13133185

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

212

—

125

C₄

#13133238

yes

—

126

C₄

#13133244

yes

—

127

C₂

—

128

C₂

—

424

—

129

C₂

#16810767

yes

—

130

C₂

#16810790

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂²×C₅₃	#120
hypercenter	#492	D₅₃
hypocenter	C₅₃	#1624
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#492

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16811339	C₂	semidirect
2. #16810815	C₂	semidirect
3. #16810372	C₂	semidirect
4. #16810352	C₂	semidirect
5. C₅₃×#492	C₂	semidirect
67. #492	D₅₃	semidirect
71. C₅₃	#1624	semidirect

H ≅ G/N

type

1. #16811339

C₂

semidirect

2. #16810815

C₂

semidirect

3. #16810372

C₂

semidirect

4. #16810352

C₂

semidirect

5. C₅₃×#492

C₂

semidirect

67. #492

D₅₃

semidirect

71. C₅₃

#1624

semidirect

Subgroups — #20273796

Conjugacy classes of subgroups

Complements