Group #20273858 — subgroups

Subgroups — #20273858

Order 6784.

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

normal subgroups only

This group has 218 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
2	#16810910	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×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133206	—	2	2	no	—
7	#13133192	—	2	2	no	—
8	C₄×D₂₁₂	C₂²	1	2	yes	—
9	#13133175	—	2	2	no	—
10	#13133148	—	2	2	no	—
11	C₄⋊#88322	C₂²	1	2	yes	—
12	#13133137	—	2	2	no	—
13	C₂×Q₁₆×C₅₃	—	2	2	no	—
14	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	1	no	—
18	C₂×Q₈×C₅₃	D₄	1	2	yes	—
19	C₂×D₂₁₂	—	2	2	no	—
20	C₂×D₂₁₂	D₄	1	2	yes	—
21	C₂×C₄×D₅₃	—	2	2	no	—
22	Q₁₆×C₅₃	—	4	4	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719921	—	2	4	no	—
28	#11719921	—	2	2	no	—
29	C₄⋊Dic₅₃	D₄	1	2	yes	—
30	C₄×Dic₅₃	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂²×D₅₃	—	2	2	no	—
33	Q₈×C₅₃	—	4	2	no	—
34	Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₅₃	—	2	2	no	—
40	D₂₁₂	—	2	2	no	—
41	D₂₁₂	—	2	2	no	—
42	D₂₁₂	—	8	2	no	—
43	C₄×D₅₃	—	4	2	no	—
44	C₄×D₅₃	—	4	2	no	—
45	C₈×C₅₃	—	2	2	no	—
46	#88322	—	4	2	no	—
47	C₂²×C₅₃	#120	1	1	yes	characteristic
48	C₂×D₅₃	—	4	2	no	—
49	C₂×D₅₃	—	4	2	no	—
50	C₂×D₅₃	—	4	2	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	4	2	no	—
53	C₄×C₅₃	—	8	1	no	—
54	C₄×C₅₃	#135	1	2	yes	—
55	C₄×C₅₃	#137	1	2	yes	—
56	Dic₅₃	—	4	2	no	—
57	#1634	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#452	1	2	yes	—
59	C₂×C₅₃	#453	1	1	yes	characteristic
60	D₅₃	—	8	2	no	—
61	#555	—	53	1	no	—
62	#494	D₅₃	1	1	yes	characteristic, complemented
63	#476	—	53	2	no	—
64	#333	—	53	1	no	—
65	D₄⋊C₈	—	53	2	no	—
66	C₅₃	#1634	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
70	#124	—	106	2	no	—
71	#121	—	106	2	no	—
72	C₄×D₄	—	53	2	no	—
73	#117	—	106	2	no	—
74	#106	—	106	2	no	—
75	C₄⋊C₈	—	53	2	no	—
76	#102	—	106	2	no	—
77	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
80	C₂×D₄	—	53	2	no	—
81	C₂×D₄	—	106	2	no	—
82	C₂²×C₄	—	106	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	106	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	53	2	no	—
89	C₂²⋊C₄	—	106	4	no	—
90	C₂²⋊C₄	—	106	2	no	—
91	C₄²	—	106	2	no	—
92	C₄²	C₂²×D₅₃	1	1	yes	characteristic
93	C₂³	—	106	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	106	2	no	—
97	D₄	—	106	2	no	—
98	D₄	—	424	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	106	2	no	—
102	C₂×C₄	—	212	2	no	—
103	C₂×C₄	—	212	2	no	—
104	C₂×C₄	D₄×D₅₃	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	212	2	no	—
108	C₂²	—	212	2	no	—
109	C₂²	—	212	2	no	—
110	C₂²	—	212	2	no	—
111	C₂²	#13133185	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	212	2	no	—
116	C₄	#13133240	1	2	yes	—
117	C₄	#13133249	1	2	yes	—
118	C₂	—	424	2	no	—
119	C₂	#16810784	1	2	yes	—
120	C₂	#16810790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811339

C₂

yes

maximal, characteristic

#16810910

C₂

yes

maximal, complemented

#16810372

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, complemented

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

#13133206

—

#13133192

—

C₄×D₂₁₂

C₂²

yes

—

#13133175

—

#13133148

—

C₄⋊#88322

C₂²

yes

—

#13133137

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×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

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

#1634

—

maximal, Sylow

C₂×C₅₃

#452

yes

—

C₂×C₅₃

#453

yes

characteristic

D₅₃

—

#555

—

#494

D₅₃

yes

characteristic, complemented

#476

—

#333

—

D₄⋊C₈

—

C₅₃

#1634

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#124

—

106

—

#121

—

106

—

C₄×D₄

—

#117

—

106

—

#106

—

106

—

C₄⋊C₈

—

#102

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

Q₈

—

Q₈

—

D₄

—

106

—

D₄

—

106

—

D₄

—

424

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

212

—

103

C₂×C₄

—

212

—

104

C₂×C₄

D₄×D₅₃

yes

—

105

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

106

C₈

—

107

C₈

—

212

—

108

C₂²

—

212

—

109

C₂²

—

212

—

110

C₂²

—

212

—

111

C₂²

#13133185

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

212

—

116

C₄

#13133240

yes

—

117

C₄

#13133249

yes

—

118

C₂

—

424

—

119

C₂

#16810784

yes

—

120

C₂

#16810790

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	#120
hypercenter	#494	D₅₃
hypocenter	C₅₃	#1634
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#494

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #16810910	C₂	semidirect
3. #16810372	C₂	semidirect
4. #16810352	C₂	semidirect
5. C₅₃×#494	C₂	semidirect
62. #494	D₅₃	semidirect
66. C₅₃	#1634	semidirect

H ≅ G/N

type

2. #16810910

C₂

semidirect

3. #16810372

C₂

semidirect

4. #16810352

C₂

semidirect

5. C₅₃×#494

C₂

semidirect

62. #494

D₅₃

semidirect

66. C₅₃

#1634

semidirect

Subgroups — #20273858

Conjugacy classes of subgroups

Complements