Group #20290657 — subgroups

Subgroups — #20290657

Order 6784.

212 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 86 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×#13133288	C₂	1	2	yes	maximal
2	C₂×#13133287	C₂	1	2	yes	maximal
3	#16811628	C₂	1	1	yes	maximal, characteristic
4	#16811627	C₂	1	1	yes	maximal, characteristic
5	#97×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	C₂²	1	2	yes	—
7	C₂×#11719936	C₂²	1	1	yes	characteristic
8	C₂×#11719936	C₂²	1	1	yes	characteristic
9	#13133288	—	2	4	no	—
10	#13133287	—	2	4	no	—
11	#13133284	—	2	2	no	—
12	C₄⋊#88324	—	2	2	no	—
13	C₂×C₈×D₅₃	C₂²	1	2	yes	—
14	C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#13133128	C₄	1	1	yes	characteristic
16	#13133127	C₄	1	2	yes	—
17	#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	Q₈	1	2	yes	—
19	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88324	—	2	2	no	—
21	C₂×#88324	—	2	1	no	—
22	C₂×#88324	—	2	1	no	—
23	#11719936	—	2	2	no	—
24	#11719936	—	2	2	no	—
25	#11719936	—	2	2	no	—
26	#11719936	—	2	2	no	—
27	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	D₄	1	2	yes	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₈×D₅₃	D₄	1	4	yes	—
33	C₈×D₅₃	Q₈	1	4	yes	—
34	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄×D₅₃	—	4	2	no	—
40	C₄×D₅₃	C₂×D₄	1	2	yes	—
41	C₄×D₅₃	C₂×Q₈	1	2	yes	—
42	#88324	—	4	1	no	—
43	#88324	—	4	2	no	—
44	#88324	—	4	1	no	—
45	C₈×C₅₃	C₄⋊C₄	1	4	yes	—
46	#88322	C₄⋊C₄	1	4	yes	—
47	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₅₃	—	2	2	no	—
49	C₂×D₅₃	C₄⋊Q₈	1	2	yes	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
52	Dic₅₃	—	2	2	no	—
53	Dic₅₃	#127	1	1	yes	characteristic
54	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
55	#1849	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	#384	1	1	yes	characteristic
57	C₂×C₅₃	#430	1	2	yes	—
58	D₅₃	—	2	2	no	—
59	C₂×#108	—	53	2	no	—
60	C₂×#108	—	53	2	no	—
61	#423	—	53	1	no	—
62	#423	—	53	1	no	—
63	C₂×#97	—	53	1	no	—
64	C₅₃	#1849	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	53	2	no	—
66	C₂×M₁₆	—	53	1	no	—
67	C₂×M₁₆	—	53	1	no	—
68	C₂²×C₈	—	53	2	no	—
69	C₂×C₄²	—	53	1	no	—
70	#108	—	106	4	no	—
71	#108	—	106	4	no	—
72	C₄⋊C₈	—	106	2	no	—
73	C₄⋊C₈	—	106	2	no	—
74	#97	—	53	2	no	—
75	#97	—	53	1	no	—
76	#97	#6157	1	1	yes	characteristic
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	—	53	2	no	—
79	M₁₆	—	106	2	no	—
80	M₁₆	—	106	2	no	—
81	M₁₆	—	106	2	no	—
82	M₁₆	—	106	2	no	—
83	C₂×C₈	—	53	2	no	—
84	C₂×C₈	—	53	4	no	—
85	C₂×C₈	—	53	4	no	—
86	C₂×C₈	—	106	2	no	—
87	C₂×C₈	—	106	1	no	—
88	C₂×C₈	—	106	1	no	—
89	C₂×C₈	C₂×#6157	1	2	yes	—
90	C₄²	—	53	2	no	—
91	C₄²	—	53	1	no	—
92	C₄²	C₂×#6157	1	1	yes	characteristic
93	C₂³	—	53	1	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	2	no	—
96	C₂×C₄	—	53	2	no	—
97	C₂×C₄	—	53	2	no	—
98	C₂×C₄	—	212	2	no	—
99	C₂×C₄	C₄⋊#6157	1	2	yes	—
100	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
101	C₈	—	53	4	no	—
102	C₈	—	212	1	no	—
103	C₈	—	212	2	no	—
104	C₈	—	212	1	no	—
105	C₈	C₄⋊#6157	1	4	yes	—
106	C₂²	—	53	2	no	—
107	C₂²	—	106	2	no	—
108	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	53	1	no	—
111	C₄	—	53	1	no	—
112	C₄	—	106	2	no	—
113	C₄	C₂×C₄⋊#6157	1	2	yes	—
114	C₂	—	106	2	no	—
115	C₂	#16811569	1	1	yes	characteristic
116	C₂	#16811638	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133288

C₂

yes

maximal

C₂×#13133287

C₂

yes

maximal

#16811628

C₂

yes

maximal, characteristic

#16811627

C₂

yes

maximal, characteristic

#97×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×#11719936

C₂²

yes

characteristic

C₂×#11719936

C₂²

yes

—

—

—

—

C₂×C₈×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133128

C₄

yes

characteristic

#13133127

C₄

yes

—

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

—

—

—

—

—

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

#88324

—

#88324

—

#88324

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

—

Dic₅₃

#127

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

—

—

—

—

C₂×#97

—

C₅₃

#1849

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

106

—

#108

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊#6157

yes

—

100

C₂×C₄

C₂²×#6157

yes

characteristic

101

C₈

—

102

C₈

—

212

—

103

C₈

—

212

—

104

C₈

—

212

—

105

C₈

C₄⋊#6157

yes

—

106

C₂²

—

107

C₂²

—

106

—

108

C₂²

C₂×C₄⋊#6157

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

106

—

113

C₄

C₂×C₄⋊#6157

yes

—

114

C₂

—

106

—

115

C₂

#16811569

yes

characteristic

116

C₂

#16811638

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#97	#6157
hypocenter	C₅₃	#1849
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#97×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
64. C₅₃	#1849	semidirect

H ≅ G/N

type

64. C₅₃

#1849

semidirect

Subgroups — #20290657

Conjugacy classes of subgroups

Complements