Group #20276222 — subgroups

Subgroups — #20276222

Order 6784.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16810665	C₂	1	2	yes	maximal
2	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
3	#16810649	C₂	1	1	yes	maximal, characteristic
4	#16810391	C₂	1	2	yes	maximal
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	1	yes	characteristic
7	#13133176	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	#13133151	—	2	2	no	—
10	#13133151	—	2	2	no	—
11	#13133149	—	2	2	no	—
12	#13133146	—	2	2	no	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	—	2	2	no	—
17	M₁₆×C₅₃	—	2	2	no	—
18	M₁₆×C₅₃	Q₈	1	4	yes	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	D₄	1	2	yes	—
22	C₂²⋊Dic₅₃	—	2	1	no	—
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
26	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
27	C₄×Dic₅₃	—	2	1	no	—
28	#11719917	—	2	2	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂³×C₅₃	#41	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₅₃	#41	1	2	yes	—
35	C₂×Dic₅₃	—	2	1	no	—
36	C₂×Dic₅₃	—	2	1	no	—
37	C₂×Dic₅₃	—	4	2	no	—
38	C₂×Dic₅₃	—	8	2	no	—
39	C₈×C₅₃	—	2	4	no	—
40	C₈×C₅₃	—	2	2	no	—
41	#88322	—	4	1	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	#121	1	1	yes	characteristic
45	C₂²×C₅₃	#125	1	2	yes	—
46	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₅₃	#123	1	1	yes	characteristic
48	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₅₃	—	4	1	no	—
50	Dic₅₃	—	4	1	no	—
51	Dic₅₃	—	8	2	no	—
52	#2089	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	2	no	—
54	C₂×C₅₃	#398	1	1	yes	characteristic
55	C₂×C₅₃	#472	1	1	yes	characteristic
56	C₂×C₅₃	#473	1	1	yes	characteristic
57	#428	—	53	2	no	—
58	C₂×#107	—	53	1	no	—
59	#424	—	53	1	no	—
60	#343	—	53	2	no	—
61	#341	D₅₃	1	1	yes	characteristic
62	C₅₃	#2089	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	53	1	no	—
64	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
65	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#117	—	53	1	no	—
67	C₂×C₄⋊C₄	—	53	2	no	—
68	#107	—	106	2	no	—
69	#107	—	106	2	no	—
70	#106	—	106	2	no	—
71	C₄⋊C₈	—	106	2	no	—
72	C₂²×C₄	—	106	2	no	—
73	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	106	2	no	—
76	M₁₆	Dic₁₀₆	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	106	1	no	—
80	C₂×C₈	D₂₁₂	1	2	yes	—
81	C₄⋊C₄	—	53	1	no	—
82	C₄⋊C₄	—	53	1	no	—
83	C₄⋊C₄	—	106	2	no	—
84	C₄⋊C₄	—	106	2	no	—
85	C₂²⋊C₄	—	106	1	no	—
86	C₄²	—	106	1	no	—
87	C₂³	#11719949	1	1	yes	characteristic
88	C₂×C₄	—	106	1	no	—
89	C₂×C₄	—	106	1	no	—
90	C₂×C₄	—	424	2	no	—
91	C₂×C₄	—	212	2	no	—
92	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
93	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
94	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
95	C₂×C₄	#11719947	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	212	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#13133192	1	1	yes	characteristic
102	C₂²	#13133208	1	2	yes	—
103	C₄	—	424	2	no	—
104	C₄	—	212	1	no	—
105	C₄	—	212	1	no	—
106	C₄	#13133194	1	2	yes	—
107	C₄	#13133201	1	1	yes	characteristic
108	C₄	#13133217	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#16810555	1	1	yes	characteristic
111	C₂	#16810894	1	1	yes	characteristic
112	C₂	#16810899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810665

C₂

yes

maximal

C₂×#13133151

C₂

yes

maximal, characteristic

#16810649

C₂

yes

maximal, characteristic

#16810391

C₂

yes

maximal

C₅₃×#341

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#125

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2089

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2089

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#107

—

106

—

#107

—

106

—

#106

—

106

—

C₄⋊C₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

Dic₁₀₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₂³

#11719949

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

—

C₈

—

C₈

—

C₈

—

212

—

C₂²

—

100

C₂²

—

101

C₂²

#13133192

yes

characteristic

102

C₂²

#13133208

yes

—

103

C₄

—

424

—

104

C₄

—

212

—

105

C₄

—

212

—

106

C₄

#13133194

yes

—

107

C₄

#13133201

yes

characteristic

108

C₄

#13133217

yes

characteristic

109

C₂

—

110

C₂

#16810555

yes

characteristic

111

C₂

#16810894

yes

characteristic

112

C₂

#16810899

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	#121
hypercenter	#341	D₅₃
hypocenter	C₅₃	#2089
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#13133192

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#341

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
62. C₅₃	#2089	semidirect

H ≅ G/N

type

62. C₅₃

#2089

semidirect

Subgroups — #20276222

Conjugacy classes of subgroups

Complements