Group #20289609 — subgroups

Subgroups — #20289609

Order 6784.

144 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 44 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	#16811645	C₂	1	1	yes	maximal, characteristic
2	#16811641	C₂	1	1	yes	maximal, characteristic
3	#97×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133284	—	2	1	no	—
5	C₄⋊#88324	C₄	1	2	yes	—
6	#13133274	—	2	1	no	—
7	#13133274	C₄	1	2	yes	—
8	#13133272	—	2	1	no	—
9	C₄×#88324	—	2	1	no	—
10	C₂×C₈×D₅₃	—	2	1	no	—
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#13133128	C₄	1	1	yes	characteristic
13	#13133127	—	2	1	no	—
14	#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
18	C₂×#88324	—	2	2	no	—
19	C₂×#88324	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₅₃	D₄	1	1	yes	characteristic
24	C₄×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂×#88322	—	2	1	no	—
26	C₈×D₅₃	—	2	4	no	—
27	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄×D₅₃	—	2	2	no	—
35	C₄×D₅₃	—	2	2	no	—
36	C₄×D₅₃	—	4	1	no	—
37	#88324	—	4	2	no	—
38	#88324	—	4	2	no	—
39	C₈×C₅₃	—	2	2	no	—
40	#88322	—	2	2	no	—
41	C₂²×C₅₃	#95	1	1	yes	characteristic
42	C₂×D₅₃	—	2	1	no	—
43	C₂×D₅₃	—	2	1	no	—
44	C₂×D₅₃	#108	1	2	yes	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	Dic₅₃	—	2	1	no	—
49	Dic₅₃	—	2	1	no	—
50	Dic₅₃	#104	1	2	yes	—
51	#1302	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#339	1	1	yes	characteristic
53	C₂×C₅₃	#341	1	1	yes	characteristic
54	C₂×C₅₃	#344	1	1	yes	characteristic
55	D₅₃	—	2	2	no	—
56	#432	—	53	1	no	—
57	#431	—	53	1	no	—
58	C₂×#97	—	53	1	no	—
59	C₅₃	#1302	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	106	1	no	—
61	C₂×C₄²	—	53	1	no	—
62	C₄⋊C₈	—	53	2	no	—
63	C₄⋊C₈	—	106	1	no	—
64	C₂²⋊C₈	—	53	2	no	—
65	C₂²⋊C₈	—	106	1	no	—
66	#97	—	53	1	no	—
67	#97	—	106	1	no	—
68	#97	—	106	1	no	—
69	#97	#6157	1	1	yes	characteristic
70	C₄×C₈	—	106	1	no	—
71	C₂²×C₄	—	53	1	no	—
72	C₂²×C₄	—	53	1	no	—
73	C₂²×C₄	—	53	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	106	4	no	—
76	C₂×C₈	—	106	1	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	106	2	no	—
79	C₄²	—	53	1	no	—
80	C₄²	—	53	1	no	—
81	C₄²	—	53	1	no	—
82	C₄²	C₂×#6157	1	1	yes	characteristic
83	C₂³	—	53	1	no	—
84	C₂×C₄	—	53	1	no	—
85	C₂×C₄	—	53	1	no	—
86	C₂×C₄	—	53	1	no	—
87	C₂×C₄	—	106	2	no	—
88	C₂×C₄	—	106	2	no	—
89	C₂×C₄	—	212	1	no	—
90	C₂×C₄	C₄×#6157	1	1	yes	characteristic
91	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	106	2	no	—
95	C₈	—	212	2	no	—
96	C₈	—	212	2	no	—
97	C₂²	—	53	2	no	—
98	C₂²	—	106	1	no	—
99	C₂²	—	106	1	no	—
100	C₂²	(C₂×#6157)⋊C₄	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	53	2	no	—
105	C₄	—	106	1	no	—
106	C₄	—	106	1	no	—
107	C₂	—	106	2	no	—
108	C₂	#16811494	1	1	yes	characteristic
109	C₂	#16811498	1	1	yes	characteristic
110	C₂	#16811506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811645

C₂

yes

maximal, characteristic

#16811641

C₂

yes

maximal, characteristic

#97×D₅₃

C₂

yes

maximal, characteristic

#13133284

—

C₄⋊#88324

C₄

yes

—

#13133274

—

#13133274

C₄

yes

—

#13133272

—

C₄×#88324

—

C₂×C₈×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133128

C₄

yes

characteristic

#13133127

—

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×#88324

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#95

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#104

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#432

—

#431

—

C₂×#97

—

C₅₃

#1302

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

#97

—

106

—

#97

—

106

—

#97

#6157

yes

characteristic

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

212

—

C₈

—

212

—

C₂²

—

C₂²

—

106

—

C₂²

—

106

—

100

C₂²

(C₂×#6157)⋊C₄

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

—

106

C₄

—

106

—

107

C₂

—

106

—

108

C₂

#16811494

yes

characteristic

109

C₂

#16811498

yes

characteristic

110

C₂

#16811506

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#6157)⋊C₄
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₄²	C₂×#6157
Fitting	#97×C₅₃	C₄
socle	C₂²×C₅₃	#95
hypercenter	#97	#6157
hypocenter	C₅₃	#1302
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

(C₂×#6157)⋊C₄

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₄²

C₂×#6157

Fitting

#97×C₅₃

C₄

C₂²×C₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
59. C₅₃	#1302	semidirect

H ≅ G/N

type

59. C₅₃

#1302

semidirect

Subgroups — #20289609

Conjugacy classes of subgroups

Complements