Group #20290636 — subgroups

Subgroups — #20290636

Order 6784.

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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	#16811569	C₂	1	2	yes	maximal
2	#16811497	C₂	1	4	yes	maximal
3	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#6157	—	2	2	no	—
5	C₂×C₄⋊#6157	—	2	2	no	—
6	C₂×C₄⋊#6157	C₂²	1	4	yes	—
7	C₂×C₈×D₅₃	C₂²	1	2	yes	—
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#88322	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₅₃	C₄	1	2	yes	complemented
11	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂²×#6157	—	2	4	no	—
13	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
14	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
16	C₄⋊#6157	—	2	8	no	—
17	C₄⋊#6157	—	4	4	no	—
18	C₄⋊#6157	—	4	4	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
23	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	2	yes	—
25	C₈×D₅₃	—	2	4	no	—
26	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×#6157	—	4	4	no	—
28	C₂×#6157	—	4	4	no	—
29	C₂×#6157	—	4	4	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	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₄×D₅₃	D₈	1	2	yes	—
37	C₄×D₅₃	SD₁₆	1	4	yes	—
38	C₄×D₅₃	Q₁₆	1	2	yes	—
39	C₄×D₅₃	#41	1	4	yes	—
40	C₈×C₅₃	—	2	2	no	—
41	#88322	—	2	2	no	—
42	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×D₅₃	#124	1	2	yes	—
44	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
45	C₂×D₅₃	C₂×SD₁₆	1	2	yes	—
46	C₂×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
47	#6157	—	8	4	no	—
48	C₄×C₅₃	#106	1	2	yes	—
49	C₄×C₅₃	#107	1	2	yes	—
50	C₄×C₅₃	#117	1	2	yes	—
51	Dic₅₃	#106	1	2	yes	—
52	Dic₅₃	#107	1	2	yes	—
53	Dic₅₃	#125	1	2	yes	—
54	#1834	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#382	1	1	yes	characteristic
56	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
57	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
58	D₅₃	#486	1	2	yes	—
59	D₅₃	#487	1	2	yes	—
60	C₂×C₄×C₈	—	53	1	no	—
61	#384	—	53	2	no	—
62	#340	—	53	4	no	—
63	C₅₃	#1834	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	53	2	no	—
65	C₂×C₄⋊C₄	—	53	4	no	—
66	C₂×C₄⋊C₄	—	106	2	no	—
67	C₂×C₄⋊C₄	—	106	2	no	—
68	C₂×C₄²	—	53	1	no	—
69	C₄×C₈	—	53	2	no	—
70	C₄×C₈	—	53	1	no	—
71	C₄×C₈	#6157	1	1	yes	characteristic, complemented
72	C₂²×C₄	—	53	1	no	—
73	C₂²×C₄	—	53	1	no	—
74	C₂²×C₄	—	53	1	no	—
75	C₂²×C₄	—	106	4	no	—
76	C₂×C₈	—	53	2	no	—
77	C₂×C₈	—	106	4	no	—
78	C₂×C₈	C₂×#6157	1	2	yes	—
79	C₄⋊C₄	—	106	8	no	—
80	C₄⋊C₄	—	212	4	no	—
81	C₄⋊C₄	—	212	4	no	—
82	C₄²	—	53	1	no	—
83	C₄²	—	53	1	no	—
84	C₄²	—	53	1	no	—
85	C₄²	C₂×#6157	1	1	yes	characteristic
86	C₂³	—	53	1	no	—
87	C₂×C₄	—	53	1	no	—
88	C₂×C₄	—	53	1	no	—
89	C₂×C₄	—	53	1	no	—
90	C₂×C₄	—	53	4	no	—
91	C₂×C₄	—	53	2	no	—
92	C₂×C₄	—	53	2	no	—
93	C₂×C₄	—	53	4	no	—
94	C₂×C₄	—	212	4	no	—
95	C₂×C₄	—	212	4	no	—
96	C₂×C₄	—	212	4	no	—
97	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
98	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
99	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
100	C₈	—	2	2	no	—
101	C₈	—	106	2	no	—
102	C₂²	—	53	2	no	—
103	C₂²	—	53	2	no	—
104	C₂²	—	53	1	no	—
105	C₂²	—	53	1	no	—
106	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
107	C₄	—	53	2	no	—
108	C₄	—	53	2	no	—
109	C₄	—	53	2	no	—
110	C₄	—	424	4	no	—
111	C₄	#13133285	1	2	yes	—
112	C₄	#13133286	1	2	yes	—
113	C₄	#13133296	1	2	yes	—
114	C₂	—	53	2	no	—
115	C₂	—	53	2	no	—
116	C₂	#16811567	1	1	yes	characteristic
117	C₂	C₂×#13133285	1	1	yes	characteristic
118	C₂	C₂×#13133286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811569

C₂

yes

maximal

#16811497

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic, complemented

C₈×Dic₅₃

C₄

yes

complemented

C₄×C₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

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

Q₈

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

—

—

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

Q₁₆

yes

—

C₄×D₅₃

#41

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

#124

yes

—

C₂×D₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

—

C₂×D₅₃

C₂×Q₁₆

yes

characteristic

#6157

—

C₄×C₅₃

#106

yes

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

#117

yes

—

Dic₅₃

#106

yes

—

Dic₅₃

#107

yes

—

Dic₅₃

#125

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

D₅₃

#487

yes

—

C₂×C₄×C₈

—

#384

—

#340

—

C₅₃

#1834

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#6157

yes

characteristic

100

C₈

—

101

C₈

—

106

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×C₄⋊#6157

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

424

—

111

C₄

#13133285

yes

—

112

C₄

#13133286

yes

—

113

C₄

#13133296

yes

—

114

C₂

—

115

C₂

—

116

C₂

#16811567

yes

characteristic

117

C₂

C₂×#13133285

yes

characteristic

118

C₂

C₂×#13133286

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1834
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
9. C₄×#88322	C₄	semidirect
10. C₈×Dic₅₃	C₄	semidirect
11. C₄×C₈×C₅₃	C₄	semidirect
63. C₅₃	#1834	semidirect
71. C₄×C₈	#6157	semidirect

H ≅ G/N

type

9. C₄×#88322

C₄

semidirect

10. C₈×Dic₅₃

C₄

semidirect

11. C₄×C₈×C₅₃

C₄

semidirect

63. C₅₃

#1834

semidirect

71. C₄×C₈

#6157

semidirect

Subgroups — #20290636

Conjugacy classes of subgroups

Complements