Group #20291063 — subgroups

Subgroups — #20291063

Order 6784.

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

normal subgroups only

This group has 144 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×#13133287	C₂	1	1	yes	maximal, characteristic
2	#16811633	C₂	1	1	yes	maximal, characteristic
3	#16811548	C₂	1	2	yes	maximal, complemented
4	#16811547	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	—	2	1	no	—
7	#13133296	—	2	1	no	—
8	#13133287	—	2	1	no	—
9	#13133287	C₂²	1	2	yes	—
10	#13133286	—	2	1	no	—
11	#13133285	C₂²	1	1	yes	characteristic
12	#13133285	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133157	C₄	1	1	yes	characteristic, complemented
15	#13133156	C₂²	1	2	yes	—
16	C₁₆×D₅₃	C₄	1	2	yes	complemented
17	M₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
19	C₂²⋊#6157	—	4	1	no	—
20	C₂×#88324	—	4	1	no	—
21	C₄⋊#6157	—	2	1	no	—
22	C₄⋊#6157	—	2	1	no	—
23	C₄×#6157	—	4	1	no	—
24	#11719936	—	2	2	no	—
25	#11719936	—	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	SD₁₆	1	1	yes	characteristic
35	C₂×#6157	—	4	1	no	—
36	C₂×#6157	—	4	1	no	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
39	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
41	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
42	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
43	#88324	—	4	2	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	#106	1	1	yes	characteristic
49	C₂×D₅₃	—	4	1	no	—
50	C₂×D₅₃	#106	1	1	yes	characteristic
51	C₂×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
52	#6157	—	8	1	no	—
53	#6157	—	8	1	no	—
54	C₄×C₅₃	#106	1	1	yes	characteristic
55	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	Dic₅₃	#106	1	1	yes	characteristic
57	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
58	#2155	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
61	D₅₃	—	2	1	no	—
62	D₅₃	—	2	1	no	—
63	C₂×M₃₂	—	53	1	no	—
64	C₂×#108	—	53	1	no	—
65	#427	—	53	1	no	—
66	#365	—	53	2	no	—
67	#365	—	53	2	no	—
68	C₅₃	#2155	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	106	1	no	—
70	C₂²×C₈	—	53	1	no	—
71	#117	—	106	1	no	—
72	M₃₂	—	53	1	no	—
73	M₃₂	—	53	2	no	—
74	M₃₂	#6157	1	1	yes	characteristic, complemented
75	C₂×C₁₆	—	53	2	no	—
76	#108	—	53	2	no	—
77	#108	—	106	1	no	—
78	#107	—	106	1	no	—
79	#106	—	53	1	no	—
80	#106	—	53	1	no	—
81	C₂²×C₄	—	53	1	no	—
82	M₁₆	—	106	2	no	—
83	M₁₆	—	212	1	no	—
84	C₂×C₈	—	53	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	—	212	1	no	—
90	C₂×C₈	C₂×#6157	1	1	yes	characteristic
91	C₄⋊C₄	—	106	1	no	—
92	C₄⋊C₄	—	106	1	no	—
93	C₂²⋊C₄	—	212	1	no	—
94	C₄²	—	212	1	no	—
95	C₁₆	—	53	2	no	—
96	C₁₆	C₂×#6157	1	2	yes	—
97	C₂³	—	53	1	no	—
98	C₂×C₄	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	—	212	1	no	—
105	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
106	C₈	—	53	1	no	—
107	C₈	—	53	1	no	—
108	C₈	—	212	2	no	—
109	C₈	C₄⋊#6157	1	1	yes	characteristic
110	C₈	C₂²×#6157	1	1	yes	characteristic
111	C₂²	—	53	1	no	—
112	C₂²	—	53	1	no	—
113	C₂²	—	212	1	no	—
114	C₂²	#13133285	1	1	yes	characteristic
115	C₄	—	53	1	no	—
116	C₄	—	53	1	no	—
117	C₄	—	424	1	no	—
118	C₄	—	424	1	no	—
119	C₄	#13133285	1	1	yes	characteristic
120	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	106	1	no	—
123	C₂	—	106	1	no	—
124	C₂	C₂×#13133285	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133287

C₂

yes

maximal, characteristic

#16811633

C₂

yes

maximal, characteristic

#16811548

C₂

yes

maximal, complemented

#16811547

C₂

yes

maximal, complemented

M₃₂×D₅₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#13133286

—

#13133285

C₂²

yes

characteristic

#13133285

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133157

C₄

yes

characteristic, complemented

#13133156

C₂²

yes

—

C₁₆×D₅₃

C₄

yes

complemented

M₃₂×C₅₃

C₄

yes

characteristic, complemented

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂²⋊#6157

—

—

—

—

—

—

—

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

SD₁₆

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

#88324

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

#88322

C₂×Q₈

yes

characteristic

C₂²×C₅₃

#106

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#106

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

characteristic

#6157

—

#6157

—

C₄×C₅₃

#106

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

Dic₅₃

#106

yes

characteristic

Dic₅₃

C₂×SD₁₆

yes

characteristic

#2155

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#106

yes

characteristic

D₅₃

—

D₅₃

—

C₂×M₃₂

—

—

—

—

—

C₅₃

#2155

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

#117

—

106

—

M₃₂

—

M₃₂

—

M₃₂

#6157

yes

characteristic, complemented

C₂×C₁₆

—

#108

—

#108

—

106

—

#107

—

106

—

#106

—

#106

—

C₂²×C₄

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

212

—

C₁₆

—

C₁₆

C₂×#6157

yes

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

212

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₄⋊#6157

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

212

—

109

C₈

C₄⋊#6157

yes

characteristic

110

C₈

C₂²×#6157

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

212

—

114

C₂²

#13133285

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

424

—

118

C₄

—

424

—

119

C₄

#13133285

yes

characteristic

120

C₄

C₂×C₄⋊#6157

yes

characteristic

121

C₂

—

122

C₂

—

106

—

123

C₂

—

106

—

124

C₂

C₂×#13133285

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13133285
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	M₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₂×#106
hypercenter	M₃₂	#6157
hypocenter	C₅₃	#2155
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₂

C₂×#13133285

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. #16811548	C₂	semidirect
4. #16811547	C₂	semidirect
14. #13133157	C₄	semidirect
16. C₁₆×D₅₃	C₄	semidirect
17. M₃₂×C₅₃	C₄	semidirect
68. C₅₃	#2155	semidirect
74. M₃₂	#6157	semidirect

H ≅ G/N

type

3. #16811548

C₂

semidirect

4. #16811547

C₂

semidirect

14. #13133157

C₄

semidirect

16. C₁₆×D₅₃

C₄

semidirect

17. M₃₂×C₅₃

C₄

semidirect

68. C₅₃

#2155

semidirect

74. M₃₂

#6157

semidirect

Subgroups — #20291063

Conjugacy classes of subgroups

Complements