Group #20273495 — subgroups

Subgroups — #20273495

Order 6784.

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

normal subgroups only

This group has 150 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	#16810680	C₂	1	1	yes	maximal, characteristic
2	#16810676	C₂	1	1	yes	maximal, characteristic
3	#16810376	C₂	1	2	yes	maximal, complemented
4	#16810374	C₂	1	2	yes	maximal, complemented
5	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#88322	C₂²	1	2	yes	—
7	C₄⋊#88322	C₂²	1	2	yes	—
8	C₂²⋊#88322	—	2	1	no	—
9	C₂²⋊#88322	—	2	1	no	—
10	#13133128	—	2	1	no	—
11	C₄×#88322	—	2	1	no	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	Q₈	1	1	yes	characteristic
25	C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	2	no	—
39	C₈×C₅₃	—	2	2	no	—
40	#88322	—	4	2	no	—
41	#88322	—	4	2	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	#131	1	2	yes	—
51	C₄×C₅₃	#135	1	2	yes	—
52	#1572	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	4	1	no	—
54	C₂×C₅₃	#424	1	1	yes	characteristic
55	C₂×C₅₃	#427	1	1	yes	characteristic
56	C₂×C₅₃	#430	1	1	yes	characteristic
57	#432	—	53	1	no	—
58	#431	—	53	1	no	—
59	#431	D₅₃	1	1	yes	characteristic
60	#335	—	53	2	no	—
61	#334	—	53	2	no	—
62	C₅₃	#1572	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
64	C₄⋊C₈	—	53	2	no	—
65	C₄⋊C₈	—	53	2	no	—
66	C₄⋊C₈	Dic₅₃	1	2	yes	—
67	C₂²⋊C₈	—	106	1	no	—
68	C₂²⋊C₈	—	106	1	no	—
69	C₂²⋊C₈	Dic₅₃	1	2	yes	—
70	#97	—	106	1	no	—
71	C₄×C₈	—	106	1	no	—
72	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
74	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
75	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
80	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
81	C₄²	Dic₁₀₆	1	1	yes	characteristic
82	C₄²	D₂₁₂	1	1	yes	characteristic
83	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
84	C₄²	C₂²×D₅₃	1	1	yes	characteristic
85	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
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	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	212	2	no	—
98	C₈	—	212	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#13133224	1	2	yes	—
108	C₄	#13133239	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#16810650	1	1	yes	characteristic
111	C₂	#16810661	1	1	yes	characteristic
112	C₂	#16810672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

#16810676

C₂

yes

maximal, characteristic

#16810376

C₂

yes

maximal, complemented

#16810374

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

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

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

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

—

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

characteristic

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#135

yes

—

#1572

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#335

—

#334

—

C₅₃

#1572

yes

characteristic, complemented, Sylow

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₅₃

yes

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

Dic₅₃

yes

—

#97

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

—

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

—

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

—

C₈

—

212

—

C₈

—

212

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#13133224

yes

—

108

C₄

#13133239

yes

—

109

C₂

—

110

C₂

#16810650

yes

characteristic

111

C₂

#16810661

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	#1572
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
4. #16810374	C₂	semidirect
62. C₅₃	#1572	semidirect

H ≅ G/N

type

3. #16810376

C₂

semidirect

4. #16810374

C₂

semidirect

62. C₅₃

#1572

semidirect

Subgroups — #20273495

Conjugacy classes of subgroups

Complements