Group #20274622 — subgroups

Subgroups — #20274622

Order 6784.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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×#13133152	C₂	1	2	yes	maximal
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
4	#16810387	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	#13133152	C₄	1	8	yes	complemented
10	#13133128	—	2	1	no	—
11	#13133128	C₄	1	4	yes	—
12	C₄×#88322	—	2	1	no	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	#97×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	M₁₆×C₅₃	—	2	2	no	—
20	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	D₄	1	1	yes	characteristic
25	#11719917	—	2	2	no	—
26	#11719917	C₂×C₄	1	4	yes	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	C₂×C₄	1	2	yes	—
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	C₂×#88322	C₂×C₄	1	4	yes	—
32	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#41	1	2	yes	—
40	C₈×C₅₃	—	2	4	no	—
41	#88322	—	2	2	no	—
42	#88322	—	2	4	no	—
43	#88322	C₄²	1	4	yes	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₅₃	#117	1	1	yes	characteristic
51	C₄×C₅₃	C₄×D₄	1	2	yes	—
52	#1778	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	2	no	—
54	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅₃	#430	1	2	yes	—
56	C₂×#108	—	53	2	no	—
57	C₄×M₁₆	—	53	1	no	—
58	C₄×M₁₆	D₅₃	1	1	yes	characteristic
59	C₂×#97	—	53	1	no	—
60	#341	—	53	2	no	—
61	C₅₃	#1778	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	53	2	no	—
63	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
64	C₂²×C₈	—	53	1	no	—
65	C₂²×C₈	—	53	1	no	—
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	#108	—	53	8	no	—
68	#97	—	2	1	no	—
69	#97	—	53	4	no	—
70	#97	—	106	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	106	1	no	—
73	C₂²×C₄	Dic₁₀₆	1	2	yes	—
74	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	53	4	no	—
77	M₁₆	—	106	2	no	—
78	M₁₆	C₄×D₅₃	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	53	2	no	—
81	C₂×C₈	—	53	4	no	—
82	C₂×C₈	—	53	2	no	—
83	C₂×C₈	—	106	2	no	—
84	C₂×C₈	—	106	2	no	—
85	C₄²	C₄×D₅₃	1	2	yes	—
86	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#11719919	1	2	yes	—
91	C₂×C₄	#11719919	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
94	C₂×C₄	#11719945	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	53	4	no	—
98	C₈	—	106	2	no	—
99	C₈	—	106	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#11719919	1	1	yes	characteristic
102	C₂²	C₄×Dic₁₀₆	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₅₃	1	1	yes	characteristic
106	C₄	#13133174	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#11719919	1	1	yes	characteristic
110	C₂	#16810671	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133152

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal

C₄×M₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133152

C₄

yes

complemented

#13133128

—

#13133128

C₄

yes

—

C₄×#88322

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

C₄²

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

#1778

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#430

yes

—

C₂×#108

—

C₄×M₁₆

—

C₄×M₁₆

D₅₃

yes

characteristic

C₂×#97

—

#341

—

C₅₃

#1778

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

C₄×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

C₄×D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

—

C₂×C₄

#11719919

yes

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

#11719945

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₈

—

106

—

100

C₂²

—

101

C₂²

C₂×#11719919

yes

characteristic

102

C₂²

C₄×Dic₁₀₆

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₅₃

yes

characteristic

106

C₄

#13133174

yes

characteristic

107

C₄

C₄×C₂²⋊D₅₃

yes

—

108

C₂

—

109

C₂

C₄×#11719919

yes

characteristic

110

C₂

#16810671

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	#1778
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₄×M₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₅₃

hypocenter

C₅₃

#1778

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #13133152	C₄	semidirect
61. C₅₃	#1778	semidirect

H ≅ G/N

type

9. #13133152

C₄

semidirect

61. C₅₃

#1778

semidirect

Subgroups — #20274622

Conjugacy classes of subgroups

Complements