Group #20276470 — subgroups

Subgroups — #20276470

Order 6784.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810441	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	#13133157	—	2	2	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	C₈×Dic₅₃	C₄	1	2	yes	—
10	C₈×Dic₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	M₃₂×C₅₃	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₅₃	C₈	1	2	yes	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₄	1	2	yes	—
18	C₂×C₈×C₅₃	Q₈	1	2	yes	—
19	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×Dic₅₃	C₈	1	1	yes	characteristic
21	C₄×Dic₅₃	C₈	1	1	yes	characteristic
22	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	C₂×C₄	1	2	yes	—
25	C₁₆×C₅₃	—	2	2	no	—
26	#11719908	—	2	2	no	—
27	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	—	2	4	no	—
33	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
34	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
35	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	#88322	—	2	2	no	—
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
43	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
45	Dic₅₃	—	2	2	no	—
46	Dic₅₃	M₃₂	1	4	yes	complemented
47	#2150	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	2	no	—
49	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₅₃	C₂×M₃₂	1	2	yes	—
51	C₂×M₃₂	—	53	1	no	—
52	C₂×M₃₂	D₅₃	1	1	yes	characteristic
53	C₂×C₄×C₈	—	53	1	no	—
54	C₄⋊C₁₆	—	53	4	no	—
55	C₅₃	#2150	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	53	1	no	—
57	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
58	C₂×C₄²	—	53	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	106	2	no	—
61	C₂×C₁₆	—	53	2	no	—
62	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
63	C₄×C₈	—	53	2	no	—
64	C₄×C₈	—	53	2	no	—
65	C₂²×C₄	—	53	2	no	—
66	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
67	C₂×C₈	—	53	2	no	—
68	C₂×C₈	—	106	2	no	—
69	C₂×C₈	Dic₁₀₆	1	2	yes	—
70	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
72	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
73	C₄²	—	53	2	no	—
74	C₄²	—	53	1	no	—
75	C₄²	—	53	1	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	106	2	no	—
78	C₂³	C₈×D₅₃	1	1	yes	characteristic
79	C₂×C₄	—	53	2	no	—
80	C₂×C₄	—	53	2	no	—
81	C₂×C₄	—	106	4	no	—
82	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
83	C₂×C₄	#11719912	1	2	yes	—
84	C₂×C₄	#11719919	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
86	C₈	—	106	2	no	—
87	C₈	#11719919	1	2	yes	—
88	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
89	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#13133146	1	2	yes	—
92	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
93	C₄	—	53	4	no	—
94	C₄	—	106	2	no	—
95	C₄	#13133146	1	2	yes	—
96	C₄	C₂×#11719919	1	1	yes	characteristic
97	C₄	C₂×#11719912	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×#13133146	1	1	yes	characteristic
100	C₂	M₃₂×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810441

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

#13133157

—

C₂×#11719908

C₂²

yes

—

C₈×Dic₅₃

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

M₃₂

yes

complemented

#2150

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

—

C₂×M₃₂

—

C₂×M₃₂

D₅₃

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₅₃

#2150

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

#11719912

yes

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

106

—

C₈

#11719919

yes

—

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₂²

—

C₂²

#13133146

yes

—

C₂²

C₂×C₈×D₅₃

yes

characteristic

C₄

—

C₄

—

106

—

C₄

#13133146

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

C₂×#13133146

yes

characteristic

100

C₂

M₃₂×D₅₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×M₃₂×C₅₃	C₂
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2150
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×M₃₂×C₅₃

C₂

hypercenter

C₂×M₃₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16810441	C₂	semidirect
46. Dic₅₃	M₃₂	semidirect
55. C₅₃	#2150	semidirect

H ≅ G/N

type

3. #16810441

C₂

semidirect

46. Dic₅₃

M₃₂

semidirect

55. C₅₃

#2150

semidirect

Subgroups — #20276470

Conjugacy classes of subgroups

Complements