Group #20291054 — subgroups

Subgroups — D₅₃⋊#364 #20291054

Order 6784.

168 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 74 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	D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#16811546	C₂	1	2	yes	maximal, complemented
3	#16811545	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133292	—	2	2	no	—
6	#13133291	C₂²	1	4	yes	—
7	D₅₃⋊C₁₆	—	2	2	no	—
8	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₄²×D₅₃	C₄	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	—	2	1	no	—
16	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₅₃	—	2	1	no	—
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂³	1	1	yes	characteristic
24	C₈×D₅₃	C₈	1	2	yes	—
25	C₈×D₅₃	C₈	1	2	yes	—
26	C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₈×D₅₃	D₄	1	1	yes	characteristic
28	C₈×D₅₃	Q₈	1	1	yes	characteristic
29	C₈×D₅₃	Q₈	1	1	yes	characteristic
30	#11719910	—	2	4	no	—
31	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	—	2	1	no	—
35	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
36	C₄×D₅₃	—	2	4	no	—
37	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
41	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₈×C₅₃	C₂×C₈	1	2	yes	—
44	#88322	C₂×C₈	1	2	yes	—
45	#88322	C₂×D₄	1	1	yes	characteristic
46	#88322	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	C₂×D₅₃	—	2	2	no	—
49	C₂×D₅₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×D₅₃	C₄⋊C₈	1	1	yes	characteristic
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	Dic₅₃	—	2	2	no	—
55	Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
56	Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×#364	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	D₅₃	—	2	1	no	—
61	D₅₃	#364	1	2	yes	complemented
62	C₂×M₃₂	—	53	2	no	—
63	C₂×C₄×C₈	—	53	1	no	—
64	#364	—	53	2	no	—
65	#364	—	53	2	no	—
66	C₅₃	C₂×#364	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	53	1	no	—
68	C₂²×C₈	—	53	1	no	—
69	C₂×C₄²	—	53	1	no	—
70	M₃₂	—	53	4	no	—
71	M₃₂	—	106	2	no	—
72	C₂×C₁₆	—	106	2	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	—	53	1	no	—
75	C₄×C₈	—	53	1	no	—
76	C₄×C₈	#6157	1	1	yes	characteristic
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	—	106	1	no	—
79	C₂×C₈	—	53	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	1	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	2	no	—
87	C₂×C₈	C₂×#6157	1	1	yes	characteristic
88	C₂×C₈	C₂×#6157	1	1	yes	characteristic
89	C₄²	—	53	1	no	—
90	C₄²	—	106	1	no	—
91	C₄²	C₂×#6157	1	1	yes	characteristic
92	C₁₆	—	106	4	no	—
93	C₂³	—	53	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	106	4	no	—
101	C₂×C₄	—	106	1	no	—
102	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
103	C₈	—	53	1	no	—
104	C₈	—	53	1	no	—
105	C₈	—	53	2	no	—
106	C₈	D₅₃⋊C₈	1	2	yes	—
107	C₈	C₄⋊#6157	1	1	yes	characteristic
108	C₈	C₄⋊#6157	1	1	yes	characteristic
109	C₂²	—	53	1	no	—
110	C₂²	—	53	1	no	—
111	C₂²	—	106	2	no	—
112	C₂²	D₅₃⋊M₁₆	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	53	1	no	—
115	C₄	—	53	1	no	—
116	C₄	—	106	2	no	—
117	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
118	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	53	2	no	—
121	C₂	—	106	1	no	—
122	C₂	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal

#16811546

C₂

yes

maximal, complemented

#16811545

C₂

yes

maximal, complemented

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

#13133292

—

#13133291

C₂²

yes

—

D₅₃⋊C₁₆

—

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₈×D₅₃

C₈

yes

—

C₈×D₅₃

C₈

yes

—

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

#11719910

—

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×C₈

yes

—

#88322

C₂×C₈

yes

—

#88322

C₂×D₄

yes

characteristic

#88322

C₂×Q₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₈

yes

characteristic

C₂×D₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊C₈

yes

characteristic

Dic₅₃

C₄⋊C₈

yes

characteristic

C₂×#364

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

D₅₃

—

D₅₃

#364

yes

complemented

C₂×M₃₂

—

C₂×C₄×C₈

—

#364

—

#364

—

C₅₃

C₂×#364

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

106

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

106

—

101

C₂×C₄

—

106

—

102

C₂×C₄

C₂²×#6157

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

D₅₃⋊C₈

yes

—

107

C₈

C₄⋊#6157

yes

characteristic

108

C₈

C₄⋊#6157

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

106

—

112

C₂²

D₅₃⋊M₁₆

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

106

—

117

C₄

C₂×C₄⋊#6157

yes

characteristic

118

C₄

C₂×D₅₃⋊C₈

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

106

—

122

C₂

D₅₃⋊(C₄⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₈	C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄⋊C₈
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	C₂×#364
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄⋊#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂×C₅₃

C₂×C₄⋊C₈

C₄×C₈

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #16811546	C₂	semidirect
3. #16811545	C₂	semidirect
61. D₅₃	#364	semidirect
66. C₅₃	C₂×#364	semidirect

H ≅ G/N

type

2. #16811546

C₂

semidirect

3. #16811545

C₂

semidirect

61. D₅₃

#364

semidirect

66. C₅₃

C₂×#364

semidirect

Subgroups — D₅₃⋊#364 #20291054

Conjugacy classes of subgroups

Complements

Subgroups — D53⋊#364 #20291054

Conjugacy classes of subgroups

Complements

Subgroups — D₅₃⋊#364 #20291054