Group #20272728 — subgroups

Subgroups — #20272728

Order 6784.

116 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 39 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	#16810864	C₂	1	1	yes	maximal, characteristic
2	#16810386	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
5	#13133196	—	2	1	no	—
6	C₂×C₄⋊Dic₅₃	—	2	1	no	—
7	#13133151	C₄	1	1	yes	characteristic, complemented
8	#13133142	—	2	1	no	—
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
12	C₂²×Dic₅₃	—	4	1	no	—
13	C₂×Dic₁₀₆	—	2	1	no	—
14	C₂×C₈×C₅₃	—	4	1	no	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²⋊Dic₅₃	—	4	1	no	—
18	C₄⋊Dic₅₃	—	2	2	no	—
19	C₄⋊Dic₅₃	—	2	1	no	—
20	C₄⋊Dic₅₃	—	4	1	no	—
21	#11719919	—	4	1	no	—
22	Dic₂₁₂	—	4	1	no	—
23	C₁₆×C₅₃	—	4	1	no	—
24	C₂³×C₅₃	D₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
29	C₂×Dic₅₃	—	4	2	no	—
30	C₂×Dic₅₃	—	4	1	no	—
31	C₂×Dic₅₃	—	4	1	no	—
32	C₂×Dic₅₃	—	8	1	no	—
33	C₂×Dic₅₃	—	8	1	no	—
34	Dic₁₀₆	—	4	1	no	—
35	Dic₁₀₆	—	4	1	no	—
36	C₈×C₅₃	—	2	1	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	#102	1	1	yes	characteristic
41	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
42	C₂²×C₅₃	Q₃₂	1	1	yes	characteristic
43	C₄×C₅₃	—	2	1	no	—
44	C₄×C₅₃	#99	1	1	yes	characteristic
45	C₄×C₅₃	#104	1	1	yes	characteristic
46	Dic₅₃	—	8	2	no	—
47	Dic₅₃	—	8	1	no	—
48	Dic₅₃	—	8	1	no	—
49	#1348	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	2	1	no	—
52	C₂×C₅₃	#327	1	1	yes	characteristic
53	C₂×C₅₃	#358	1	1	yes	characteristic
54	C₂×C₅₃	#360	1	1	yes	characteristic
55	#467	—	53	1	no	—
56	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
57	#340	—	53	1	no	—
58	C₅₃	#1348	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	53	1	no	—
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂²⋊Q₈	—	106	1	no	—
62	C₂×C₄⋊C₄	—	106	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	#107	—	53	1	no	—
65	#103	—	106	1	no	—
66	C₂×Q₈	—	106	1	no	—
67	C₂²×C₄	—	212	1	no	—
68	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
69	Q₁₆	—	212	1	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
73	C₄⋊C₄	—	106	2	no	—
74	C₄⋊C₄	—	106	1	no	—
75	C₄⋊C₄	—	212	1	no	—
76	C₄⋊C₄	—	212	1	no	—
77	C₂²⋊C₄	—	212	1	no	—
78	C₁₆	—	4	1	no	—
79	C₂³	D₄₂₄	1	1	yes	characteristic
80	Q₈	—	212	1	no	—
81	Q₈	—	212	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	424	1	no	—
85	C₂×C₄	—	424	1	no	—
86	C₂×C₄	—	212	2	no	—
87	C₂×C₄	—	212	1	no	—
88	C₂×C₄	—	212	1	no	—
89	C₂×C₄	#11719913	1	1	yes	characteristic
90	C₂×C₄	#11719921	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	#13133139	1	1	yes	characteristic
96	C₂²	#13133162	1	1	yes	characteristic
97	C₂²	Dic₄₂₄	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	424	2	no	—
100	C₄	—	424	1	no	—
101	C₄	—	424	1	no	—
102	C₄	#13133131	1	1	yes	characteristic
103	C₄	#13133143	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	2	1	no	—
106	C₂	#16810358	1	1	yes	characteristic
107	C₂	#16810428	1	1	yes	characteristic
108	C₂	#16810434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810864

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₄

yes

characteristic, complemented

#13133196

—

C₂×C₄⋊Dic₅₃

—

#13133151

C₄

yes

characteristic, complemented

#13133142

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₂²×C₅₃

SD₃₂

yes

characteristic

C₂²×C₅₃

Q₃₂

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1348

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂²⋊C₁₆

D₅₃

yes

characteristic

#340

—

C₅₃

#1348

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₁₆

—

#107

—

#103

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₁₆

—

C₂³

D₄₂₄

yes

characteristic

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

Dic₄₂₄

yes

characteristic

C₄

—

C₄

—

424

—

100

C₄

—

424

—

101

C₄

—

424

—

102

C₄

#13133131

yes

characteristic

103

C₄

#13133143

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#16810358

yes

characteristic

107

C₂

#16810428

yes

characteristic

108

C₂

#16810434

yes

characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#1348
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#13133139

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₅₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₃

#102

hypercenter

C₂²⋊C₁₆

D₅₃

hypocenter

C₅₃

#1348

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
4. C₂×Dic₂₁₂	C₄	semidirect
7. #13133151	C₄	semidirect
58. C₅₃	#1348	semidirect

H ≅ G/N

type

4. C₂×Dic₂₁₂

C₄

semidirect

7. #13133151

C₄

semidirect

58. C₅₃

#1348

semidirect

Subgroups — #20272728

Conjugacy classes of subgroups

Complements