Group #20289554 — subgroups

Subgroups — #20289554

Order 6784.

152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 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	#16811640	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133272	C₂	1	1	yes	maximal, characteristic
3	#16810673	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88324	C₄	1	2	yes	—
5	#13133284	—	2	1	no	—
6	C₂²⋊#88324	C₄	1	2	yes	—
7	#13133272	—	2	1	no	—
8	#13133272	—	2	1	no	—
9	C₄×#88324	—	2	1	no	—
10	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	#13133146	—	2	1	no	—
12	C₂²⋊#88322	C₄	1	1	yes	characteristic
13	C₈×Dic₅₃	—	2	1	no	—
14	C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#88324	—	2	2	no	—
19	C₂×#88324	—	2	2	no	—
20	C₂×#88324	—	2	2	no	—
21	C₂×#88324	C₈	1	4	yes	complemented
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₄×Dic₅₃	D₄	1	1	yes	characteristic
24	C₄×Dic₅₃	D₄	1	1	yes	characteristic
25	C₄×Dic₅₃	D₄	1	1	yes	characteristic
26	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
27	C₂×#88322	—	2	1	no	—
28	C₂³×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	—	4	1	no	—
33	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
38	C₂×Dic₅₃	M₁₆	1	2	yes	—
39	#88324	—	2	4	no	—
40	#88324	—	4	2	no	—
41	C₈×C₅₃	—	4	1	no	—
42	#88322	—	4	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	#95	1	1	yes	characteristic
46	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂²×C₅₃	#97	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	Dic₅₃	—	2	1	no	—
51	Dic₅₃	—	2	1	no	—
52	Dic₅₃	C₂²⋊C₈	1	2	yes	—
53	Dic₅₃	C₄⋊C₈	1	2	yes	—
54	#1281	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	#336	1	1	yes	characteristic
58	C₂×C₅₃	#344	1	1	yes	characteristic
59	C₂×C₅₃	#344	1	1	yes	characteristic
60	#431	—	53	1	no	—
61	#431	—	53	1	no	—
62	C₂×#97	—	53	1	no	—
63	C₅₃	#1281	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	53	2	no	—
65	C₂×C₄²	—	53	1	no	—
66	C₄⋊C₈	—	106	1	no	—
67	C₄⋊C₈	—	106	1	no	—
68	C₂²⋊C₈	—	53	2	no	—
69	C₂²⋊C₈	—	53	1	no	—
70	C₂²⋊C₈	#6157	1	1	yes	characteristic
71	#97	—	106	1	no	—
72	#97	—	106	1	no	—
73	C₄×C₈	—	106	1	no	—
74	C₄×C₈	—	106	1	no	—
75	C₂²×C₄	—	53	1	no	—
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	53	4	no	—
80	C₂×C₈	—	106	2	no	—
81	C₂×C₈	—	106	2	no	—
82	C₂×C₈	—	106	2	no	—
83	C₂×C₈	—	106	1	no	—
84	C₄²	—	53	1	no	—
85	C₄²	—	53	1	no	—
86	C₄²	—	53	1	no	—
87	C₄²	—	53	1	no	—
88	C₂³	C₄×#6157	1	1	yes	characteristic
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	53	1	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	2	no	—
93	C₂×C₄	—	53	2	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	212	1	no	—
97	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
99	C₈	—	4	1	no	—
100	C₈	—	106	4	no	—
101	C₈	—	212	1	no	—
102	C₈	—	212	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	(C₂×#6157)⋊C₄	1	1	yes	characteristic
106	C₂²	C₈×#6157	1	1	yes	characteristic
107	C₂²	#13133271	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	53	2	no	—
111	C₄	—	53	2	no	—
112	C₄	—	106	1	no	—
113	C₄	—	106	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	(C₂×#6157)⋊C₈	1	1	yes	characteristic
117	C₂	#16811504	1	1	yes	characteristic
118	C₂	#16811506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811640

C₂

yes

maximal, characteristic

C₂×#13133272

C₂

yes

maximal, characteristic

#16810673

C₂

yes

maximal, characteristic

C₂²×#88324

C₄

yes

—

#13133284

—

C₂²⋊#88324

C₄

yes

—

#13133272

—

#13133272

—

C₄×#88324

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

—

C₂²⋊#88322

C₄

yes

characteristic

C₈×Dic₅₃

—

C₂²⋊C₈×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

—

—

C₈

yes

complemented

C₂×C₈×C₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

M₁₆

yes

—

#88324

—

#88324

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

C₄×C₈

yes

characteristic

C₂²×C₅₃

#97

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊C₈

yes

—

Dic₅₃

C₄⋊C₈

yes

—

#1281

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₂×#97

—

C₅₃

#1281

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

#6157

yes

characteristic

#97

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₄×#6157

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₄⋊#6157

yes

characteristic

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

100

C₈

—

106

—

101

C₈

—

212

—

102

C₈

—

212

—

103

C₂²

—

104

C₂²

—

105

C₂²

(C₂×#6157)⋊C₄

yes

characteristic

106

C₂²

C₈×#6157

yes

characteristic

107

C₂²

#13133271

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

106

—

113

C₄

—

106

—

114

C₂

—

115

C₂

—

116

C₂

(C₂×#6157)⋊C₈

yes

characteristic

117

C₂

#16811504

yes

characteristic

118

C₂

#16811506

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#6157)⋊C₄
commutator	C₂²×C₅₃	C₄×C₈
Frattini	C₂²×C₄	C₂×#6157
Fitting	C₂²⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	#95
hypercenter	C₂²⋊C₈	#6157
hypocenter	C₅₃	#1281
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

(C₂×#6157)⋊C₄

commutator

C₂²×C₅₃

C₄×C₈

Frattini

C₂²×C₄

C₂×#6157

Fitting

C₂²⋊C₈×C₅₃

C₄

C₂²×C₅₃

C₂²⋊C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
21. C₂×#88324	C₈	semidirect
63. C₅₃	#1281	semidirect

H ≅ G/N

type

21. C₂×#88324

C₈

semidirect

63. C₅₃

#1281

semidirect

Subgroups — #20289554

Conjugacy classes of subgroups

Complements