Group #20291051 — subgroups

Subgroups — #20291051

Order 6784.

160 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 76 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×#13133292	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#16811544	C₂	1	2	yes	maximal, complemented
4	C₄⋊#11719910	C₂	1	2	yes	maximal, complemented
5	#16810580	C₂	1	1	yes	maximal, characteristic
6	#13133292	—	2	2	no	—
7	D₅₃⋊C₁₆	—	2	2	no	—
8	C₂×#11719910	C₂²	1	2	yes	—
9	C₂×#11719910	C₂²	1	2	yes	—
10	C₂×#11719912	C₄	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#13133174	C₄	1	1	yes	characteristic
13	#13133127	C₄	1	1	yes	characteristic
14	C₄×#88322	C₂²	1	1	yes	characteristic
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	#97×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	#11719921	C₈	1	2	yes	—
22	#11719919	C₈	1	2	yes	—
23	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	C₂³	1	1	yes	characteristic
26	#11719912	—	2	2	no	—
27	C₈×D₅₃	D₄	1	2	yes	—
28	C₈×D₅₃	Q₈	1	2	yes	—
29	#11719910	—	2	2	no	—
30	#11719910	—	2	2	no	—
31	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
35	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
36	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
37	C₄×D₅₃	M₁₆	1	2	yes	—
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	#88322	—	2	1	no	—
41	#88322	C₂×D₄	1	1	yes	characteristic
42	#88322	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	C₂×D₅₃	—	2	2	no	—
45	C₂×D₅₃	C₄⋊C₈	1	2	yes	—
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
49	Dic₅₃	—	2	2	no	—
50	Dic₅₃	C₄⋊C₈	1	2	yes	—
51	#2151	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₅₃	#504	1	2	yes	—
54	D₅₃	—	2	2	no	—
55	C₂×M₃₂	—	53	1	no	—
56	C₂²×C₁₆	—	53	1	no	—
57	#405	—	53	1	no	—
58	C₄⋊C₁₆	—	53	2	no	—
59	C₄⋊C₁₆	—	53	2	no	—
60	C₅₃	#2151	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	1	no	—
62	C₂²×C₈	—	53	1	no	—
63	#117	—	53	1	no	—
64	M₃₂	—	106	2	no	—
65	C₂×C₁₆	—	53	2	no	—
66	C₂×C₁₆	—	53	2	no	—
67	C₂×C₁₆	—	106	2	no	—
68	#97	—	53	1	no	—
69	#97	#6157	1	1	yes	characteristic
70	C₄×C₈	—	53	1	no	—
71	C₄×C₈	—	53	1	no	—
72	C₂²×C₄	—	53	1	no	—
73	M₁₆	—	106	2	no	—
74	C₂×C₈	—	53	1	no	—
75	C₂×C₈	—	53	2	no	—
76	C₂×C₈	—	53	2	no	—
77	C₂×C₈	—	53	1	no	—
78	C₂×C₈	C₂×#6157	1	1	yes	characteristic
79	C₂×C₈	C₂×#6157	1	1	yes	characteristic
80	C₄⋊C₄	—	53	2	no	—
81	C₂²⋊C₄	—	53	2	no	—
82	C₄²	—	53	1	no	—
83	C₄²	C₂×#6157	1	1	yes	characteristic
84	C₁₆	—	106	2	no	—
85	C₁₆	—	106	2	no	—
86	C₂³	—	53	1	no	—
87	C₂×C₄	—	53	1	no	—
88	C₂×C₄	—	53	2	no	—
89	C₂×C₄	—	53	2	no	—
90	C₂×C₄	—	53	2	no	—
91	C₂×C₄	D₅₃⋊C₈	1	2	yes	—
92	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	53	1	no	—
95	C₈	—	53	1	no	—
96	C₈	—	106	1	no	—
97	C₈	C₄⋊#6157	1	2	yes	—
98	C₂²	—	53	2	no	—
99	C₂²	—	106	2	no	—
100	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	53	2	no	—
103	C₄	—	106	2	no	—
104	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
105	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
106	C₂	—	106	2	no	—
107	C₂	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
108	C₂	#16811682	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133292

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

#16811544

C₂

yes

maximal, complemented

C₄⋊#11719910

C₂

yes

maximal, complemented

#16810580

C₂

yes

maximal, characteristic

#13133292

—

D₅₃⋊C₁₆

—

C₂×#11719910

C₂²

yes

—

C₂×#11719910

C₂²

yes

—

C₂×#11719912

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133174

C₄

yes

characteristic

#13133127

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

C₈

yes

—

#11719919

C₈

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

#11719912

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

#11719910

—

#11719910

—

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

—

#88322

C₂×D₄

yes

characteristic

#88322

C₂×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊C₈

yes

—

#2151

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

#504

yes

—

D₅₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₅₃

#2151

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

106

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₅₃⋊C₈

yes

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

106

—

C₈

C₄⋊#6157

yes

—

C₂²

—

C₂²

—

106

—

100

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

106

—

104

C₄

C₂×C₄⋊#6157

yes

characteristic

105

C₄

D₅₃⋊M₁₆

yes

characteristic

106

C₂

—

106

—

107

C₂

D₅₃⋊(C₄⋊C₈)

yes

characteristic

108

C₂

#16811682

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	#97×C₅₃	C₄
hypercenter	#97	#6157
hypocenter	C₅₃	#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

#97×C₅₃

C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
3. #16811544	C₂	semidirect
4. C₄⋊#11719910	C₂	semidirect
60. C₅₃	#2151	semidirect

H ≅ G/N

type

3. #16811544

C₂

semidirect

4. C₄⋊#11719910

C₂

semidirect

60. C₅₃

#2151

semidirect

Subgroups — #20291051

Conjugacy classes of subgroups

Complements