Group #20273039 — subgroups

Subgroups — #20273039

Order 6784.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16810676	C₂	1	2	yes	maximal
2	#16810349	C₂	1	4	yes	maximal
3	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	—	2	2	no	—
5	C₂²⋊#88322	C₂²	1	4	yes	—
6	C₄×#88322	—	2	2	no	—
7	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₅₃	—	2	2	no	—
9	C₄×Q₈×C₅₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₄	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	2	no	—
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×Q₈×C₅₃	C₈	1	4	yes	complemented
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₄²×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×#88322	—	2	4	no	—
29	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₅₃	—	2	8	no	—
31	C₂×C₄×C₅₃	—	2	2	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	—	4	1	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	#88322	—	4	4	no	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₅₃	—	2	4	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	4	2	no	—
54	C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
55	C₄×C₅₃	#101	1	2	yes	—
56	#1461	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	1	no	—
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₅₃	#410	1	1	yes	characteristic
61	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
62	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
63	#431	—	53	2	no	—
64	#324	—	53	4	no	—
65	C₅₃	#1461	1	1	yes	characteristic, complemented, Sylow
66	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
71	C₂×C₄²	Dic₅₃	1	2	yes	—
72	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
73	C₄⋊C₈	—	106	2	no	—
74	C₂²⋊C₈	—	53	4	no	—
75	C₄×C₈	—	106	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	#88322	1	4	yes	complemented
79	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
80	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
82	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×C₈	—	106	4	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	C₂²⋊D₅₃	1	2	yes	—
90	C₄²	C₂²⋊D₅₃	1	2	yes	—
91	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₂×#88322	1	4	yes	—
100	C₂×C₄	C₂×#88322	1	2	yes	—
101	C₂×C₄	#11719917	1	2	yes	—
102	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
106	C₈	—	212	4	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
110	C₂²	C₂²×#88322	1	1	yes	characteristic
111	C₂²	C₂×#11719917	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	1	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂²⋊#88322	1	4	yes	—
117	C₄	#13133136	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	C₂×C₂²⋊#88322	1	1	yes	characteristic
121	C₂	#16810597	1	1	yes	characteristic
122	C₂	C₂×#13133136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal

#16810349

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₂²⋊#88322

C₂²

yes

—

C₄×#88322

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

#101

yes

—

#1461

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

D₅₃

yes

characteristic

#431

—

#324

—

C₅₃

#1461

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#88322

yes

complemented

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#88322

yes

—

100

C₂×C₄

C₂×#88322

yes

—

101

C₂×C₄

#11719917

yes

—

102

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₅₃

yes

—

104

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

106

C₈

—

212

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

110

C₂²

C₂²×#88322

yes

characteristic

111

C₂²

C₂×#11719917

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂²⋊#88322

yes

—

117

C₄

#13133136

yes

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊#88322

yes

characteristic

121

C₂

#16810597

yes

characteristic

122

C₂

C₂×#13133136

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1461
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

#1461

LCS

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
16. C₂×Q₈×C₅₃	C₈	semidirect
65. C₅₃	#1461	semidirect
78. C₂×Q₈	#88322	semidirect

H ≅ G/N

type

16. C₂×Q₈×C₅₃

C₈

semidirect

65. C₅₃

#1461

semidirect

78. C₂×Q₈

#88322

semidirect

Subgroups — #20273039

Conjugacy classes of subgroups

Complements