Group #20276927 — subgroups

Subgroups — #20276927

Order 6784.

134 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16810672	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810445	C₂	1	2	yes	maximal, complemented
3	#16810436	C₂	1	2	yes	maximal, complemented
4	#16810405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#469	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	1	no	—
7	#13133157	C₂²	1	2	yes	—
8	#13133153	—	2	1	no	—
9	#13133153	C₂²	1	2	yes	—
10	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	#121×C₅₃	—	2	1	no	—
13	#106×C₅₃	C₂²	1	1	yes	characteristic
14	#102×C₅₃	—	2	1	no	—
15	C₂×D₄×C₅₃	—	2	1	no	—
16	C₂×D₄×C₅₃	—	4	1	no	—
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	D₈×C₅₃	—	2	2	no	—
19	M₁₆×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₂²⋊C₄×C₅₃	—	4	1	no	—
24	#11719917	—	2	2	no	—
25	#11719917	—	4	1	no	—
26	C₂×#88322	—	4	1	no	—
27	#11719908	—	2	2	no	—
28	C₂³×C₅₃	—	4	1	no	—
29	C₂³×C₅₃	SD₁₆	1	1	yes	characteristic
30	D₄×C₅₃	—	4	1	no	—
31	D₄×C₅₃	—	4	1	no	—
32	D₄×C₅₃	—	8	2	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	#41	1	2	yes	—
40	#88322	—	4	2	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	—	4	1	no	—
43	C₂²×C₅₃	—	8	1	no	—
44	C₂²×C₅₃	—	8	1	no	—
45	C₂²×C₅₃	—	8	1	no	—
46	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	8	1	no	—
49	C₄×C₅₃	#123	1	1	yes	characteristic
50	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
51	#2238	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	1	no	—
53	C₂×C₅₃	—	4	1	no	—
54	C₂×C₅₃	—	16	1	no	—
55	C₂×C₅₃	#481	1	1	yes	characteristic
56	#469	D₅₃	1	1	yes	characteristic
57	#430	—	53	1	no	—
58	#365	—	53	2	no	—
59	#361	—	53	2	no	—
60	#349	—	53	1	no	—
61	C₅₃	#2238	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂×M₁₆	—	106	1	no	—
64	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
65	#121	—	2	1	no	—
66	M₃₂	—	53	2	no	—
67	#108	—	53	2	no	—
68	#108	—	106	1	no	—
69	#106	C₂×D₅₃	1	1	yes	characteristic
70	#102	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
74	D₈	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	—	106	2	no	—
77	M₁₆	—	212	1	no	—
78	C₂×C₈	—	212	1	no	—
79	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₁₆	—	106	2	no	—
84	C₂³	—	4	1	no	—
85	C₂³	#11719923	1	1	yes	characteristic
86	D₄	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	8	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	#11719923	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	212	2	no	—
96	C₈	#11719947	1	2	yes	—
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	8	1	no	—
100	C₂²	—	8	1	no	—
101	C₂²	—	8	1	no	—
102	C₂²	C₂×#11719923	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	8	1	no	—
105	C₄	#13133198	1	1	yes	characteristic
106	C₄	#13133242	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	4	1	no	—
109	C₂	—	16	1	no	—
110	C₂	#16810939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810672

C₂

yes

maximal, characteristic, complemented

#16810445

C₂

yes

maximal, complemented

#16810436

C₂

yes

maximal, complemented

#16810405

C₂

yes

maximal, characteristic, complemented

C₅₃×#469

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133157

C₂²

yes

—

#13133153

—

#13133153

C₂²

yes

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#121×C₅₃

—

#106×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

D₈×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

SD₁₆

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#481

yes

characteristic

#469

D₅₃

yes

—

—

—

—

C₅₃

#2238

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

#121

—

M₃₂

—

#108

—

#108

—

106

—

#106

C₂×D₅₃

yes

characteristic

#102

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

106

—

C₂³

—

C₂³

#11719923

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719923

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

#11719947

yes

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×#11719923

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#13133198

yes

characteristic

106

C₄

#13133242

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

#16810939

yes

characteristic

	subgroup	G/H
center	C₂	#16810939
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#469	C₂
socle	C₂×C₅₃	#481
hypercenter	#469	D₅₃
hypocenter	C₅₃	#2238
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

subgroup

G/H

center

C₂

#16810939

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#469

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

N	H ≅ G/N	type
1. #16810672	C₂	semidirect
2. #16810445	C₂	semidirect
3. #16810436	C₂	semidirect
4. #16810405	C₂	semidirect
61. C₅₃	#2238	semidirect

H ≅ G/N

type

1. #16810672

C₂

semidirect

2. #16810445

C₂

semidirect

3. #16810436

C₂

semidirect

4. #16810405

C₂

semidirect

61. C₅₃

#2238

semidirect

Subgroups — #20276927

Conjugacy classes of subgroups

Complements