Group #20275558 — subgroups

Subgroups — #20275558

Order 6784.

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

normal subgroups only

This group has 292 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133135	C₂	1	4	yes	maximal
3	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
6	C₂×#11719917	C₂²	1	2	yes	—
7	#13133217	—	2	1	no	—
8	#13133217	—	2	1	no	—
9	#13133217	—	2	2	no	—
10	C₂×C₄⋊Dic₅₃	C₄	1	4	yes	—
11	#13133135	—	2	8	no	—
12	#13133128	—	2	1	no	—
13	C₄×#88322	—	2	1	no	—
14	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
20	C₂×Dic₁₀₆	—	2	4	no	—
21	C₂×Dic₁₀₆	—	2	2	no	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	4	no	—
24	M₁₆×C₅₃	—	2	4	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₄⋊Dic₅₃	—	2	4	no	—
30	C₄⋊Dic₅₃	—	2	4	no	—
31	#11719917	—	2	2	no	—
32	#11719917	D₄	1	4	yes	—
33	C₂×#88322	—	2	2	no	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	2	yes	—
42	C₂×Dic₅₃	—	2	4	no	—
43	C₂×Dic₅₃	—	2	8	no	—
44	Dic₁₀₆	—	4	4	no	—
45	Dic₁₀₆	—	4	2	no	—
46	Dic₁₀₆	—	4	2	no	—
47	C₈×C₅₃	—	4	2	no	—
48	#88322	—	2	4	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₅₃	C₄×D₄	1	2	yes	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	#101	1	4	yes	complemented
54	C₄×C₅₃	#121	1	2	yes	—
55	C₄×C₅₃	#124	1	1	yes	characteristic
56	C₄×C₅₃	#127	1	1	yes	characteristic
57	Dic₅₃	—	4	4	no	—
58	#1966	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	#390	1	1	yes	characteristic
61	C₂×C₅₃	C₂×#101	1	2	yes	—
62	C₂×C₄⋊Q₈	—	53	1	no	—
63	#423	D₅₃	1	1	yes	characteristic
64	C₂×#101	—	53	4	no	—
65	C₄×M₁₆	—	53	1	no	—
66	C₅₃	#1966	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	—	53	2	no	—
68	C₂×M₁₆	—	53	2	no	—
69	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
70	C₄⋊Q₈	—	106	1	no	—
71	C₄⋊Q₈	—	106	1	no	—
72	C₄⋊Q₈	—	106	2	no	—
73	C₂×C₄⋊C₄	—	53	4	no	—
74	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#101	—	106	8	no	—
77	#97	—	106	1	no	—
78	C₄×C₈	—	106	1	no	—
79	C₂×Q₈	—	106	2	no	—
80	C₂×Q₈	—	106	4	no	—
81	C₂×Q₈	—	106	4	no	—
82	C₂×Q₈	—	106	2	no	—
83	C₂²×C₄	—	53	4	no	—
84	C₂²×C₄	C₄×D₅₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	M₁₆	—	53	4	no	—
88	M₁₆	—	106	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	106	2	no	—
91	C₄⋊C₄	—	106	4	no	—
92	C₄⋊C₄	—	106	4	no	—
93	C₄²	D₂₁₂	1	1	yes	characteristic
94	C₄²	D₂₁₂	1	1	yes	characteristic
95	C₄²	C₂²⋊D₅₃	1	2	yes	—
96	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
97	Q₈	—	212	4	no	—
98	Q₈	—	212	2	no	—
99	Q₈	—	212	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	106	4	no	—
102	C₂×C₄	—	106	8	no	—
103	C₂×C₄	#11719921	1	2	yes	—
104	C₂×C₄	#11719921	1	2	yes	—
105	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
106	C₂×C₄	D₄×D₅₃	1	2	yes	—
107	C₂×C₄	#11719949	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
109	C₈	—	4	2	no	—
110	C₈	—	106	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×#11719921	1	1	yes	characteristic
113	C₂²	#13133179	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	212	4	no	—
116	C₄	#13133135	1	4	yes	complemented
117	C₄	#13133192	1	2	yes	—
118	C₄	#13133206	1	1	yes	characteristic
119	C₄	#13133213	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#16810524	1	1	yes	characteristic
122	C₂	C₂×#13133135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

C₂×#13133135

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic

C₅₃×#423

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133217

—

#13133217

—

#13133217

—

C₂×C₄⋊Dic₅₃

C₄

yes

—

#13133135

—

#13133128

—

C₄×#88322

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

#11719917

D₄

yes

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

#101

yes

complemented

C₄×C₅₃

#121

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1966

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

C₂×#101

yes

—

C₂×C₄⋊Q₈

—

#423

D₅₃

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

C₅₃

#1966

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#101

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

D₂₁₂

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

Q₈

—

212

—

Q₈

—

212

—

Q₈

—

212

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

106

—

103

C₂×C₄

#11719921

yes

—

104

C₂×C₄

#11719921

yes

—

105

C₂×C₄

C₂×D₂₁₂

yes

characteristic

106

C₂×C₄

D₄×D₅₃

yes

—

107

C₂×C₄

#11719949

yes

—

108

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

109

C₈

—

110

C₈

—

106

—

111

C₂²

—

112

C₂²

C₂×#11719921

yes

characteristic

113

C₂²

#13133179

yes

—

114

C₄

—

115

C₄

—

212

—

116

C₄

#13133135

yes

complemented

117

C₄

#13133192

yes

—

118

C₄

#13133206

yes

characteristic

119

C₄

#13133213

yes

characteristic

120

C₂

—

121

C₂

#16810524

yes

characteristic

122

C₂

C₂×#13133135

yes

—

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1966
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
53. C₄×C₅₃	#101	semidirect
66. C₅₃	#1966	semidirect
116. C₄	#13133135	semidirect

H ≅ G/N

type

53. C₄×C₅₃

#101

semidirect

66. C₅₃

#1966

semidirect

116. C₄

#13133135

semidirect

Subgroups — #20275558

Conjugacy classes of subgroups

Complements