Group #20273855 — subgroups

Subgroups — #20273855

Order 6784.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16811002	C₂	1	1	yes	maximal, characteristic
2	#16810918	C₂	1	2	yes	maximal
3	#16810372	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#88322	C₂	1	2	yes	maximal
5	Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719925	—	2	2	no	—
7	#13133217	C₂²	1	1	yes	characteristic
8	#13133150	—	2	2	no	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	#13133141	—	2	2	no	—
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₅₃	—	2	2	no	—
13	C₄⋊Q₈×C₅₃	—	2	2	no	—
14	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₅₃	—	2	2	no	—
16	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	D₄	1	2	yes	—
19	C₂×Q₈×C₅₃	D₄	1	2	yes	—
20	C₂×Dic₁₀₆	—	2	1	no	—
21	C₄⋊C₄×C₅₃	—	2	4	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#11719925	—	4	4	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	2	no	—
32	Q₈×C₅₃	—	2	2	no	—
33	Q₈×C₅₃	—	4	2	no	—
34	Q₈×C₅₃	—	8	2	no	—
35	Q₈×C₅₃	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂×Dic₅₃	—	4	1	no	—
42	Dic₁₀₆	—	4	2	no	—
43	#88322	—	2	2	no	—
44	#88322	—	4	2	no	—
45	C₂²×C₅₃	#120	1	1	yes	characteristic
46	C₄×C₅₃	—	2	4	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	4	2	no	—
49	C₄×C₅₃	—	4	2	no	—
50	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
51	C₄×C₅₃	#137	1	2	yes	—
52	Dic₅₃	—	8	1	no	—
53	#1633	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	#451	1	2	yes	—
55	C₂×C₅₃	#453	1	1	yes	characteristic
56	Q₈²	D₅₃	1	1	yes	characteristic
57	#494	—	53	1	no	—
58	#477	—	53	2	no	—
59	#333	—	53	1	no	—
60	Q₈⋊C₈	—	53	2	no	—
61	C₅₃	#1633	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	106	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	53	1	no	—
66	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₅₃	1	2	yes	—
69	#107	—	106	2	no	—
70	C₄⋊C₈	—	53	2	no	—
71	#103	—	106	2	no	—
72	C₄×C₈	—	53	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	106	1	no	—
75	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
77	Q₁₆	—	212	4	no	—
78	C₂×C₈	—	106	1	no	—
79	C₂×C₈	—	106	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	106	2	no	—
84	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₅₃	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	212	2	no	—
92	Q₈	#11719925	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	212	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
99	C₈	—	106	2	no	—
100	C₈	—	212	2	no	—
101	C₂²	#13133186	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	424	1	no	—
107	C₄	C₂×#11719925	1	2	yes	—
108	C₄	#13133248	1	2	yes	—
109	C₂	#16810779	1	2	yes	—
110	C₂	#16810791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811002

C₂

yes

maximal, characteristic

#16810918

C₂

yes

maximal

#16810372

C₂

yes

maximal, characteristic

Q₈⋊#88322

C₂

yes

maximal

Q₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719925

—

#13133217

C₂²

yes

characteristic

#13133150

—

C₄⋊#88322

C₂²

yes

—

#13133141

—

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

Q₁₆

yes

complemented

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

C₂×Q₁₆

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

#1633

—

maximal, Sylow

C₂×C₅₃

#451

yes

—

C₂×C₅₃

#453

yes

characteristic

Q₈²

D₅₃

yes

—

—

—

Q₈⋊C₈

—

C₅₃

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

#107

—

106

—

C₄⋊C₈

—

#103

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

Q₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

212

—

Q₈

#11719925

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₈

—

106

—

100

C₈

—

212

—

101

C₂²

#13133186

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

424

—

107

C₄

C₂×#11719925

yes

—

108

C₄

#13133248

yes

—

109

C₂

#16810779

yes

—

110

C₂

#16810791

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	Q₈²×C₅₃	C₂
socle	C₂²×C₅₃	#120
hypercenter	Q₈²	D₅₃
hypocenter	C₅₃	#1633
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#13133186

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

Q₈²×C₅₃

C₂

socle

C₂²×C₅₃

#120

hypercenter

Q₈²

D₅₃

hypocenter

C₅₃

#1633

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₅₃	Q₁₆	semidirect
61. C₅₃	#1633	semidirect
92. Q₈	#11719925	semidirect

H ≅ G/N

type

35. Q₈×C₅₃

Q₁₆

semidirect

61. C₅₃

#1633

semidirect

92. Q₈

#11719925

semidirect

Subgroups — #20273855

Conjugacy classes of subgroups

Complements