Group #20272830 — subgroups

Subgroups — #20272830

Order 6784.

128 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 69 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#16810661	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	1	no	—
5	#13133176	—	2	1	no	—
6	#13133153	—	2	1	no	—
7	#13133153	C₄	1	2	yes	complemented
8	#13133151	—	2	1	no	—
9	#13133149	C₄	1	2	yes	—
10	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	M₃₂×C₅₃	—	2	1	no	—
12	M₃₂×C₅₃	C₄	1	2	yes	complemented
13	C₂×C₁₆×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²⋊Dic₅₃	—	4	1	no	—
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₄×Dic₅₃	—	4	1	no	—
24	#11719917	—	2	2	no	—
25	#11719917	—	4	1	no	—
26	C₂×#88322	—	4	1	no	—
27	C₁₆×C₅₃	—	2	2	no	—
28	C₂³×C₅₃	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	2	no	—
36	C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	#88322	—	4	2	no	—
41	C₂²×C₅₃	—	4	1	no	—
42	C₂²×C₅₃	#102	1	1	yes	characteristic
43	C₂²×C₅₃	#103	1	1	yes	characteristic
44	C₂²×C₅₃	#106	1	1	yes	characteristic
45	C₄×C₅₃	#95	1	1	yes	characteristic
46	C₄×C₅₃	#102	1	1	yes	characteristic
47	C₄×C₅₃	#103	1	1	yes	characteristic
48	C₄×C₅₃	#107	1	1	yes	characteristic
49	Dic₅₃	—	8	2	no	—
50	#1389	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	1	no	—
52	C₂×C₅₃	—	2	1	no	—
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	#340	1	1	yes	characteristic
55	C₂×M₃₂	D₅₃	1	1	yes	characteristic
56	C₂×#108	—	53	1	no	—
57	#427	—	53	1	no	—
58	C₅₃	#1389	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	106	1	no	—
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	#117	—	106	1	no	—
62	M₃₂	—	2	1	no	—
63	M₃₂	Dic₅₃	1	2	yes	complemented
64	C₂×C₁₆	—	2	1	no	—
65	#108	—	53	2	no	—
66	#108	—	106	1	no	—
67	#107	—	106	1	no	—
68	#106	—	53	2	no	—
69	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
70	M₁₆	—	106	2	no	—
71	M₁₆	—	212	1	no	—
72	C₂×C₈	—	212	1	no	—
73	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
74	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
75	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₄⋊C₄	—	106	2	no	—
80	C₂²⋊C₄	—	212	1	no	—
81	C₄²	—	212	1	no	—
82	C₁₆	—	2	2	no	—
83	C₂³	#11719913	1	1	yes	characteristic
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	#11719913	1	1	yes	characteristic
86	C₂×C₄	D₄₂₄	1	1	yes	characteristic
87	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
89	C₂×C₄	#11719921	1	1	yes	characteristic
90	C₂×C₄	#11719921	1	1	yes	characteristic
91	C₈	—	212	2	no	—
92	C₈	C₄×Dic₅₃	1	1	yes	characteristic
93	C₈	#11719919	1	1	yes	characteristic
94	C₈	#11719919	1	1	yes	characteristic
95	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
96	C₂²	—	4	1	no	—
97	C₂²	#13133139	1	1	yes	characteristic
98	C₂²	#13133142	1	1	yes	characteristic
99	C₂²	#13133149	1	1	yes	characteristic
100	C₄	—	424	2	no	—
101	C₄	#13133124	1	1	yes	characteristic
102	C₄	#13133139	1	1	yes	characteristic
103	C₄	#13133142	1	1	yes	characteristic
104	C₄	#13133151	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	—	2	1	no	—
108	C₂	#16810386	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810661

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#13133151

—

#13133149

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

C₄

yes

complemented

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

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

Q₈

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1389

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#340

yes

characteristic

C₂×M₃₂

D₅₃

yes

characteristic

C₂×#108

—

#427

—

C₅₃

#1389

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

106

—

M₃₂

—

M₃₂

Dic₅₃

yes

complemented

C₂×C₁₆

—

#108

—

#108

—

106

—

#107

—

106

—

#106

—

C₂²×C₄

D₂₁₂

yes

characteristic

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

212

—

C₁₆

—

C₂³

#11719913

yes

characteristic

C₂×C₄

—

212

—

C₂×C₄

#11719913

yes

characteristic

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

Dic₂₁₂

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

212

—

C₈

C₄×Dic₅₃

yes

C₈

yes

C₈

yes

C₈

C₂²⋊Dic₅₃

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

424

—

101

C₄

#13133124

yes

characteristic

102

C₄

#13133139

yes

characteristic

103

C₄

#13133142

yes

characteristic

104

C₄

#13133151

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

#16810386

yes

characteristic

	subgroup	G/H
center	C₄	#13133139
commutator	C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	#340
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#1389
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#13133139

commutator

C₈×C₅₃

C₄²

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂×C₅₃

#340

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

#1389

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
7. #13133153	C₄	semidirect
12. M₃₂×C₅₃	C₄	semidirect
58. C₅₃	#1389	semidirect
63. M₃₂	Dic₅₃	semidirect

H ≅ G/N

type

7. #13133153

C₄

semidirect

12. M₃₂×C₅₃

C₄

semidirect

58. C₅₃

#1389

semidirect

63. M₃₂

Dic₅₃

semidirect

Subgroups — #20272830

Conjugacy classes of subgroups

Complements