Group #20272804 — subgroups

Subgroups — #20272804

Order 6784.

136 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
4	#13133157	—	2	1	no	—
5	#13133157	C₄	1	2	yes	complemented
6	C₂×#11719908	—	2	1	no	—
7	C₂×#11719908	—	2	1	no	—
8	C₂×#11719908	—	2	1	no	—
9	C₂×#11719908	C₄	1	2	yes	complemented
10	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	#117×C₅₃	C₄	1	1	yes	characteristic
13	#97×C₅₃	—	2	1	no	—
14	C₄×C₈×C₅₃	—	2	1	no	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	M₁₆×C₅₃	—	2	1	no	—
17	M₁₆×C₅₃	C₈	1	2	yes	—
18	C₂×C₈×C₅₃	—	2	1	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₄⋊C₄×C₅₃	C₈	1	2	yes	—
26	C₂²⋊C₄×C₅₃	—	2	1	no	—
27	C₄²×C₅₃	—	2	1	no	—
28	#11719908	—	2	2	no	—
29	#11719908	—	2	2	no	—
30	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
38	C₈×C₅₃	—	2	2	no	—
39	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₅₃	—	2	2	no	—
44	C₂²×C₅₃	#97	1	1	yes	characteristic
45	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₅₃	—	4	2	no	—
48	C₄×C₅₃	#95	1	1	yes	characteristic
49	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	#1379	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	#336	1	1	yes	characteristic
56	C₂×C₅₃	#350	1	2	yes	—
57	C₂×M₃₂	—	53	1	no	—
58	C₂²×C₁₆	—	53	1	no	—
59	#405	D₅₃	1	1	yes	characteristic
60	C₅₃	#1379	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	Dic₅₃	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
63	#117	Dic₅₃	1	1	yes	characteristic
64	M₃₂	—	53	2	no	—
65	M₃₂	—	106	1	no	—
66	C₂×C₁₆	—	53	2	no	—
67	C₂×C₁₆	—	106	1	no	—
68	C₂×C₁₆	—	106	1	no	—
69	C₂×C₁₆	—	106	1	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
73	M₁₆	—	2	1	no	—
74	M₁₆	#88322	1	2	yes	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
77	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
78	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
79	C₂×C₈	D₂₁₂	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
82	C₄⋊C₄	#88322	1	2	yes	—
83	C₂²⋊C₄	—	2	1	no	—
84	C₄²	—	2	1	no	—
85	C₁₆	—	106	2	no	—
86	C₁₆	—	106	2	no	—
87	C₂³	#11719917	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	C₂×#88322	1	1	yes	characteristic
90	C₂×C₄	C₂×#88322	1	1	yes	characteristic
91	C₂×C₄	#11719917	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
93	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	#11719919	1	1	yes	characteristic
97	C₈	#11719919	1	1	yes	characteristic
98	C₈	#11719921	1	1	yes	characteristic
99	C₈	#11719921	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	#13133128	1	1	yes	characteristic
102	C₂²	C₂²⋊#88322	1	1	yes	characteristic
103	C₂²	C₄⋊#88322	1	1	yes	characteristic
104	C₄	—	4	2	no	—
105	C₄	#13133124	1	1	yes	characteristic
106	C₄	C₄×#88322	1	1	yes	characteristic
107	C₄	C₂²⋊#88322	1	1	yes	characteristic
108	C₄	C₄⋊#88322	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#16810378	1	1	yes	characteristic
112	C₂	#16810407	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂

yes

maximal, characteristic

C₅₃×#405

C₂

yes

maximal, characteristic

#13133157

—

#13133157

C₄

yes

—

—

—

C₄

yes

complemented

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₄

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₄⋊C₄×C₅₃

C₈

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719908

—

#11719908

—

C₂³×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#97

yes

characteristic

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#95

yes

characteristic

C₄×C₅₃

C₄×C₈

yes

characteristic

C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

#1379

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#336

yes

characteristic

C₂×C₅₃

#350

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

D₅₃

yes

characteristic

C₅₃

#1379

yes

characteristic, complemented, Sylow

C₂×M₁₆

Dic₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

Dic₅₃

yes

characteristic

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

M₁₆

#88322

yes

—

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

#88322

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

#11719917

yes

characteristic

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

yes

C₈

yes

100

C₂²

—

101

C₂²

#13133128

yes

characteristic

102

C₂²

C₂²⋊#88322

yes

characteristic

103

C₂²

C₄⋊#88322

yes

characteristic

104

C₄

—

105

C₄

#13133124

yes

characteristic

106

C₄

C₄×#88322

yes

characteristic

107

C₄

C₂²⋊#88322

yes

characteristic

108

C₄

C₄⋊#88322

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#16810378

yes

characteristic

112

C₂

#16810407

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	#405	D₅₃
hypocenter	C₅₃	#1379
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₂×C₈

C₂²⋊D₅₃

commutator

C₄×C₅₃

C₄×C₈

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₅₃×#405

C₂

socle

C₂²×C₅₃

C₂²⋊C₈

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
5. #13133157	C₄	semidirect
9. C₂×#11719908	C₄	semidirect
60. C₅₃	#1379	semidirect

H ≅ G/N

type

5. #13133157

C₄

semidirect

9. C₂×#11719908

C₄

semidirect

60. C₅₃

#1379

semidirect

Subgroups — #20272804

Conjugacy classes of subgroups

Complements