Group #20291071 — subgroups

Subgroups — #20291071

Order 6784.

140 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 62 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×#13133288	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
3	#16811552	C₂	1	2	yes	maximal, complemented
4	#16811551	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	—	2	1	no	—
7	D₅₃⋊M₁₆	—	2	1	no	—
8	#13133288	—	2	1	no	—
9	#13133288	C₂²	1	1	yes	characteristic
10	#13133288	C₂²	1	2	yes	—
11	#13133288	C₂²	1	1	yes	characteristic
12	#13133287	—	2	1	no	—
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133157	C₄	1	1	yes	characteristic
15	#13133156	C₄	1	2	yes	—
16	C₁₆×D₅₃	C₂²	1	2	yes	—
17	M₃₂×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
19	#11719940	—	4	1	no	—
20	C₂×#88324	—	4	1	no	—
21	#11719936	—	2	1	no	—
22	#11719936	—	2	1	no	—
23	#11719936	—	2	2	no	—
24	#11719936	—	4	1	no	—
25	D₅₃⋊C₈	—	4	1	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#88322	Q₈	1	1	yes	characteristic
28	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×D₅₃	D₄	1	1	yes	characteristic
30	C₈×D₅₃	Q₈	1	1	yes	characteristic
31	C₈×D₅₃	C₂³	1	1	yes	characteristic
32	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	#11719908	C₂×C₄	1	2	yes	—
34	C₂²×D₅₃	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₅₃	D₈	1	1	yes	characteristic
37	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₅₃	D₈	1	1	yes	characteristic
39	C₄×D₅₃	Q₁₆	1	1	yes	characteristic
40	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
41	#88324	—	4	2	no	—
42	#88324	—	4	1	no	—
43	#88324	—	4	1	no	—
44	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	#88322	C₄⋊C₄	1	1	yes	characteristic
47	#88322	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₅₃	#107	1	1	yes	characteristic
49	C₂×D₅₃	—	4	1	no	—
50	C₂×D₅₃	#107	1	1	yes	characteristic
51	C₂×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₅₃	#107	1	1	yes	characteristic
53	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	Dic₅₃	#107	1	1	yes	characteristic
55	Dic₅₃	C₂×D₈	1	1	yes	characteristic
56	#2161	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	1	no	—
58	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
59	D₅₃	—	2	1	no	—
60	D₅₃	—	2	1	no	—
61	C₂×M₃₂	—	53	1	no	—
62	C₂×#108	—	53	1	no	—
63	C₂×#108	—	53	1	no	—
64	#368	—	53	2	no	—
65	#368	—	53	2	no	—
66	C₅₃	#2161	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	106	1	no	—
68	C₂×M₁₆	—	106	1	no	—
69	C₂²×C₈	—	53	1	no	—
70	M₃₂	—	53	1	no	—
71	M₃₂	—	53	2	no	—
72	M₃₂	#6157	1	1	yes	characteristic
73	C₂×C₁₆	—	53	2	no	—
74	#108	—	53	1	no	—
75	#108	—	53	2	no	—
76	#108	—	53	1	no	—
77	#108	—	106	1	no	—
78	#108	—	106	1	no	—
79	C₂²×C₄	—	53	1	no	—
80	M₁₆	—	106	1	no	—
81	M₁₆	—	106	1	no	—
82	M₁₆	—	106	2	no	—
83	M₁₆	—	212	1	no	—
84	M₁₆	—	212	1	no	—
85	C₂×C₈	—	53	1	no	—
86	C₂×C₈	—	53	1	no	—
87	C₂×C₈	—	53	1	no	—
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	212	1	no	—
91	C₂×C₈	—	212	1	no	—
92	C₂×C₈	C₂×#6157	1	1	yes	characteristic
93	C₁₆	—	53	2	no	—
94	C₁₆	C₂×#6157	1	2	yes	—
95	C₂³	—	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	—	53	1	no	—
101	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
102	C₈	—	53	1	no	—
103	C₈	—	53	1	no	—
104	C₈	—	212	2	no	—
105	C₈	—	212	1	no	—
106	C₈	—	212	1	no	—
107	C₈	C₄⋊#6157	1	1	yes	characteristic
108	C₈	C₂²×#6157	1	1	yes	characteristic
109	C₂²	—	53	1	no	—
110	C₂²	—	53	1	no	—
111	C₂²	—	212	1	no	—
112	C₂²	#13133286	1	1	yes	characteristic
113	C₄	—	53	1	no	—
114	C₄	—	53	1	no	—
115	C₄	#13133286	1	1	yes	characteristic
116	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	106	1	no	—
119	C₂	—	106	1	no	—
120	C₂	C₂×#13133286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133288

C₂

yes

maximal, characteristic

D₅₃⋊#108

C₂

yes

maximal, characteristic

#16811552

C₂

yes

maximal, complemented

#16811551

C₂

yes

maximal, complemented

M₃₂×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

—

D₅₃⋊M₁₆

—

#13133288

—

#13133288

C₂²

yes

characteristic

#13133288

C₂²

yes

—

#13133288

C₂²

yes

characteristic

#13133287

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133157

C₄

yes

characteristic

#13133156

C₄

yes

—

C₁₆×D₅₃

C₂²

yes

—

M₃₂×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

—

—

—

—

—

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

Q₈

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

—

#11719908

C₂×C₄

yes

—

C₂²×D₅₃

Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

D₈

yes

characteristic

C₄×D₅₃

Q₁₆

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

—

—

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

#88322

C₂×Q₈

yes

characteristic

C₂²×C₅₃

#107

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#107

yes

characteristic

C₂×D₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

#107

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Dic₅₃

#107

yes

characteristic

Dic₅₃

C₂×D₈

yes

characteristic

#2161

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#107

yes

characteristic

D₅₃

—

D₅₃

—

C₂×M₃₂

—

—

—

—

—

C₅₃

#2161

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

#6157

yes

characteristic

C₂×C₁₆

—

—

—

—

—

106

—

#108

—

106

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₁₆

—

C₁₆

C₂×#6157

yes

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊#6157

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

212

—

105

C₈

—

212

—

106

C₈

—

212

—

107

C₈

C₄⋊#6157

yes

characteristic

108

C₈

C₂²×#6157

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

212

—

112

C₂²

#13133286

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

#13133286

yes

characteristic

116

C₄

C₂×C₄⋊#6157

yes

characteristic

117

C₂

—

118

C₂

—

106

—

119

C₂

—

106

—

120

C₂

C₂×#13133286

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13133286
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	M₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₂×#107
hypercenter	M₃₂	#6157
hypocenter	C₅₃	#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₂

C₂×#13133286

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

M₃₂×C₅₃

C₄

C₂×C₅₃

M₃₂

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

N	H ≅ G/N	type
3. #16811552	C₂	semidirect
4. #16811551	C₂	semidirect
66. C₅₃	#2161	semidirect

H ≅ G/N

type

3. #16811552

C₂

semidirect

4. #16811551

C₂

semidirect

66. C₅₃

#2161

semidirect

Subgroups — #20291071

Conjugacy classes of subgroups

Complements