Group #20274043 — subgroups

Subgroups — #20274043

Order 6784.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16811033	C₂	1	2	yes	maximal
2	#16810993	C₂	1	1	yes	maximal, characteristic
3	#16810365	C₂	1	2	yes	maximal
4	#16810365	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#564	C₂	1	1	yes	maximal, characteristic
6	#13133217	C₂²	1	1	yes	characteristic
7	#13133150	—	2	2	no	—
8	#13133148	—	2	2	no	—
9	#13133140	—	2	2	no	—
10	#13133128	C₂²	1	2	yes	—
11	#13133128	C₂²	1	1	yes	characteristic
12	#125×C₅₃	—	2	2	no	—
13	#125×C₅₃	—	2	2	no	—
14	#125×C₅₃	—	2	2	no	—
15	#125×C₅₃	C₂²	1	2	yes	—
16	#125×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀₆	—	2	1	no	—
18	C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂×Dic₅₃	—	4	1	no	—
38	Dic₁₀₆	—	4	2	no	—
39	#88322	—	2	4	no	—
40	#88322	—	4	1	no	—
41	C₂²×C₅₃	#120	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	4	2	no	—
45	C₄×C₅₃	—	4	2	no	—
46	C₄×C₅₃	—	4	2	no	—
47	Dic₅₃	—	8	1	no	—
48	#1663	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#454	1	1	yes	characteristic
50	C₂×C₅₃	#456	1	2	yes	—
51	#564	D₅₃	1	1	yes	characteristic
52	#501	—	53	2	no	—
53	#491	—	53	1	no	—
54	#330	—	53	2	no	—
55	#330	—	53	1	no	—
56	C₅₃	#1663	1	1	yes	characteristic, complemented, Sylow
57	C₄⋊Q₈	—	53	1	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	C₂×D₅₃	1	1	yes	characteristic
62	#125	C₂×D₅₃	1	2	yes	—
63	#107	—	106	2	no	—
64	#106	—	106	2	no	—
65	#103	—	106	2	no	—
66	#97	—	53	1	no	—
67	#97	—	53	2	no	—
68	C₂×Q₈	—	106	1	no	—
69	C₂×C₈	—	106	2	no	—
70	C₂×C₈	—	106	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	106	2	no	—
78	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₅₃	1	1	yes	characteristic
83	Q₈	—	212	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	212	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	C₈	—	106	4	no	—
91	C₈	—	212	1	no	—
92	C₂²	#13133186	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	424	1	no	—
99	C₂	#16810796	1	1	yes	characteristic
100	C₂	#16810806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811033

C₂

yes

maximal

#16810993

C₂

yes

maximal, characteristic

#16810365

C₂

yes

maximal

#16810365

C₂

yes

maximal, characteristic

C₅₃×#564

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

—

—

—

C₂²

yes

—

#13133128

C₂²

yes

characteristic

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

#564

D₅₃

yes

—

—

—

—

C₅₃

#1663

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

C₂×D₅₃

yes

characteristic

#125

C₂×D₅₃

yes

—

#107

—

106

—

#106

—

106

—

#103

—

106

—

#97

—

#97

—

C₂×Q₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

106

—

C₈

—

212

—

C₂²

#13133186

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

424

—

C₂

#16810796

yes

characteristic

100

C₂

#16810806

yes

—

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#564	C₂
socle	C₂²×C₅₃	#120
hypercenter	#564	D₅₃
hypocenter	C₅₃	#1663
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#564

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
56. C₅₃	#1663	semidirect

H ≅ G/N

type

56. C₅₃

#1663

semidirect

Subgroups — #20274043

Conjugacy classes of subgroups

Complements