Group #20273490 — subgroups

Subgroups — #20273490

Order 6784.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	4	yes	maximal, complemented
4	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	C₂²×#88322	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₄⋊#88322	—	2	1	no	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄²×C₅₃	Q₈	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	#11719917	—	4	2	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	4	2	no	—
27	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	D₈	1	2	yes	—
34	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₅₃	—	2	4	no	—
39	#88322	—	4	2	no	—
40	#88322	—	4	2	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	#107	1	2	yes	—
44	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	#107	1	2	yes	—
49	C₄×C₅₃	#131	1	2	yes	—
50	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
52	#1571	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	2	no	—
54	C₂×C₅₃	#424	1	1	yes	characteristic
55	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
56	C₂×C₅₃	#430	1	1	yes	characteristic
57	#431	D₅₃	1	1	yes	characteristic
58	#423	—	53	1	no	—
59	C₂×C₄⋊C₈	—	53	1	no	—
60	#335	—	53	4	no	—
61	C₅₃	#1571	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	106	1	no	—
63	C₂²×C₈	—	106	1	no	—
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	C₄⋊C₈	—	53	2	no	—
66	C₄⋊C₈	—	53	2	no	—
67	C₄⋊C₈	—	106	1	no	—
68	C₄⋊C₈	—	106	1	no	—
69	C₄⋊C₈	Dic₅₃	1	2	yes	—
70	C₂²⋊C₈	Dic₅₃	1	2	yes	—
71	C₄×C₈	C₂×D₅₃	1	2	yes	—
72	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
73	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
75	M₁₆	—	212	2	no	—
76	C₂×C₈	—	106	2	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	212	2	no	—
79	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
80	C₄²	Dic₁₀₆	1	1	yes	characteristic
81	C₄²	D₂₁₂	1	1	yes	characteristic
82	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
83	C₄²	C₂²×D₅₃	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	D₄₂₄	1	2	yes	—
88	C₂×C₄	Dic₂₁₂	1	2	yes	—
89	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
92	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
93	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	212	2	no	—
97	C₈	—	212	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#13133151	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#13133151	1	2	yes	—
106	C₄	#13133224	1	2	yes	—
107	C₄	C₂×D₄₂₄	1	1	yes	characteristic
108	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#16810650	1	1	yes	characteristic
111	C₂	C₂×#13133151	1	1	yes	characteristic
112	C₂	#16810672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

—

C₂²⋊C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#107

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

#1571

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#335

—

C₅₃

#1571

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

Dic₅₃

yes

—

C₂²⋊C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄₂₄

yes

—

C₂×C₄

Dic₂₁₂

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

—

212

—

C₂²

—

C₂²

—

100

C₂²

#13133151

yes

—

101

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#13133151

yes

—

106

C₄

#13133224

yes

—

107

C₄

C₂×D₄₂₄

yes

characteristic

108

C₄

C₂×Dic₂₁₂

yes

characteristic

109

C₂

—

110

C₂

#16810650

yes

characteristic

111

C₂

C₂×#13133151

yes

characteristic

112

C₂

#16810672

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1571
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16810376	C₂	semidirect
61. C₅₃	#1571	semidirect

H ≅ G/N

type

3. #16810376

C₂

semidirect

61. C₅₃

#1571

semidirect

Subgroups — #20273490

Conjugacy classes of subgroups

Complements