Group #20273836 — subgroups

Subgroups — #20273836

Order 6784.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810995	C₂	1	1	yes	maximal, characteristic
2	#16810918	C₂	1	2	yes	maximal, complemented
3	#16810367	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#88322	C₂	1	2	yes	maximal, complemented
5	C₅₃×#559	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719923	—	2	2	no	—
7	#13133217	C₂²	1	1	yes	characteristic
8	#13133150	—	2	2	no	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133141	—	2	2	no	—
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	#127×C₅₃	C₂²	1	1	yes	characteristic
13	#124×C₅₃	—	2	2	no	—
14	#123×C₅₃	—	2	2	no	—
15	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₄×D₄×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	D₄	1	2	yes	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂×Dic₁₀₆	—	2	1	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
24	C₂²⋊C₄×C₅₃	—	2	4	no	—
25	C₂²⋊C₄×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#11719923	—	4	4	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂³×C₅₃	—	2	2	no	—
33	Q₈×C₅₃	—	2	2	no	—
34	Q₈×C₅₃	—	2	2	no	—
35	D₄×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	8	2	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×Dic₅₃	—	4	1	no	—
44	Dic₁₀₆	—	4	2	no	—
45	#88322	—	2	2	no	—
46	#88322	—	4	2	no	—
47	C₂²×C₅₃	—	4	4	no	—
48	C₂²×C₅₃	—	4	2	no	—
49	C₂²×C₅₃	#120	1	1	yes	characteristic
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	—	4	2	no	—
53	C₄×C₅₃	#135	1	2	yes	—
54	C₄×C₅₃	#137	1	2	yes	—
55	Dic₅₃	—	8	1	no	—
56	#1630	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	8	2	no	—
58	C₂×C₅₃	#452	1	2	yes	—
59	C₂×C₅₃	#453	1	1	yes	characteristic
60	#559	D₅₃	1	1	yes	characteristic
61	#492	—	53	1	no	—
62	#477	—	53	2	no	—
63	#331	—	53	1	no	—
64	Q₈⋊C₈	—	53	2	no	—
65	C₅₃	#1630	1	1	yes	characteristic, complemented, Sylow
66	C₂×SD₁₆	—	106	2	no	—
67	C₄⋊Q₈	—	53	1	no	—
68	#127	C₂×D₅₃	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#123	—	2	2	no	—
71	C₄×Q₈	C₂×D₅₃	1	2	yes	—
72	C₄×D₄	—	2	2	no	—
73	#107	—	106	2	no	—
74	C₄⋊C₈	—	53	2	no	—
75	#103	—	106	2	no	—
76	C₄×C₈	—	53	1	no	—
77	C₂×Q₈	—	106	1	no	—
78	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	212	4	no	—
83	C₂×C₈	—	106	1	no	—
84	C₂×C₈	—	106	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	106	2	no	—
87	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂²×D₅₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	212	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
105	C₈	—	106	2	no	—
106	C₈	—	212	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#13133186	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	424	1	no	—
114	C₄	#13133237	1	2	yes	—
115	C₄	#13133247	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#16810785	1	2	yes	—
118	C₂	#16810791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810995

C₂

yes

maximal, characteristic

#16810918

C₂

yes

maximal, complemented

#16810367

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal, complemented

C₅₃×#559

C₂

yes

maximal, characteristic

C₂×#11719923

—

#13133217

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

—

#13133141

—

C₄×#88322

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#123×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

#1630

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#452

yes

—

C₂×C₅₃

#453

yes

characteristic

#559

D₅₃

yes

—

—

—

Q₈⋊C₈

—

C₅₃

#1630

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

#127

C₂×D₅₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₅₃

yes

—

C₄×D₄

—

#107

—

106

—

C₄⋊C₈

—

#103

—

106

—

C₄×C₈

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

—

SD₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

212

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

212

—

103

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

105

C₈

—

106

—

106

C₈

—

212

—

107

C₂²

—

108

C₂²

—

109

C₂²

#13133186

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

424

—

114

C₄

#13133237

yes

—

115

C₄

#13133247

yes

—

116

C₂

—

117

C₂

#16810785

yes

—

118

C₂

#16810791

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#559	C₂
socle	C₂²×C₅₃	#120
hypercenter	#559	D₅₃
hypocenter	C₅₃	#1630
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#559

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #16810918	C₂	semidirect
4. Q₈⋊#88322	C₂	semidirect
65. C₅₃	#1630	semidirect

H ≅ G/N

type

2. #16810918

C₂

semidirect

4. Q₈⋊#88322

C₂

semidirect

65. C₅₃

#1630

semidirect

Subgroups — #20273836

Conjugacy classes of subgroups

Complements