Group #20272842 — subgroups

Subgroups — #20272842

Order 6784.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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×#13133136	C₂	1	2	yes	maximal
2	C₅₃×#391	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719917	—	2	2	no	—
4	#13133136	—	2	2	no	—
5	#13133136	C₄	1	4	yes	complemented
6	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	#95×C₅₃	—	2	2	no	—
10	C₂×Q₈×C₅₃	—	4	1	no	—
11	C₂×Q₈×C₅₃	—	4	1	no	—
12	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	Q₈	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	2	yes	—
19	C₄⋊C₄×C₅₃	—	4	1	no	—
20	C₄²×C₅₃	—	4	1	no	—
21	#11719917	—	2	4	no	—
22	#11719917	—	4	2	no	—
23	C₂×#88322	—	4	2	no	—
24	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	Q₈×C₅₃	—	4	2	no	—
26	Q₈×C₅₃	—	4	1	no	—
27	Q₈×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	#88322	—	4	4	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	#95	1	1	yes	characteristic
45	C₂²×C₅₃	#99	1	1	yes	characteristic
46	C₂²×C₅₃	#99	1	1	yes	characteristic
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	—	8	1	no	—
53	#1394	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	#342	1	1	yes	characteristic
57	C₂×C₅₃	#355	1	1	yes	characteristic
58	C₂×C₅₃	#356	1	1	yes	characteristic
59	C₂×#101	—	53	2	no	—
60	#391	D₅₃	1	1	yes	characteristic
61	C₅₃	#1394	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂×M₁₆	—	106	2	no	—
64	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
65	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
66	#101	—	53	4	no	—
67	#101	—	106	2	no	—
68	#95	—	2	2	no	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	4	1	no	—
71	C₂×Q₈	Dic₁₀₆	1	1	yes	characteristic
72	C₂×Q₈	C₄×D₅₃	1	2	yes	—
73	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
78	M₁₆	—	106	4	no	—
79	M₁₆	—	212	2	no	—
80	C₂×C₈	—	212	2	no	—
81	C₄⋊C₄	—	4	1	no	—
82	C₄²	—	4	1	no	—
83	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
84	Q₈	—	4	1	no	—
85	Q₈	—	4	2	no	—
86	Q₈	—	4	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
97	C₂×C₄	#11719919	1	2	yes	—
98	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	#11719921	1	2	yes	—
100	C₈	—	212	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	#13133124	1	1	yes	characteristic
104	C₂²	#13133132	1	1	yes	characteristic
105	C₂²	#13133132	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	2	no	—
111	C₄	—	8	1	no	—
112	C₂	—	2	1	no	—
113	C₂	—	2	1	no	—
114	C₂	#16810390	1	1	yes	characteristic
115	C₂	#16810420	1	1	yes	characteristic
116	C₂	#16810422	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133136

C₂

yes

maximal

C₅₃×#391

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133136

—

#13133136

C₄

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1394

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₅₃

#1394

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₄²

Dic₅₃

yes

characteristic

#101

—

#101

—

106

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₀₆

yes

characteristic

C₂×Q₈

C₄×D₅₃

yes

—

C₂×Q₈

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

#11719921

yes

—

100

C₈

—

212

—

101

C₂²

—

102

C₂²

—

103

C₂²

#13133124

yes

characteristic

104

C₂²

#13133132

yes

characteristic

105

C₂²

#13133132

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₂

—

113

C₂

—

114

C₂

#16810390

yes

characteristic

115

C₂

#16810420

yes

characteristic

116

C₂

#16810422

yes

characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂²×Q₈	C₂×D₅₃
Fitting	C₅₃×#391	C₂
socle	C₂²×C₅₃	#99
hypercenter	#391	D₅₃
hypocenter	C₅₃	#1394
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #391

subgroup

G/H

center

C₂²

#13133132

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂²×Q₈

C₂×D₅₃

Fitting

C₅₃×#391

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #391

N	H ≅ G/N	type
5. #13133136	C₄	semidirect
61. C₅₃	#1394	semidirect

H ≅ G/N

type

5. #13133136

C₄

semidirect

61. C₅₃

#1394

semidirect

Subgroups — #20272842

Conjugacy classes of subgroups

Complements