Group #20290927 — subgroups

Subgroups — #20290927

Order 6784.

184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#16811586	C₂	1	1	yes	maximal, characteristic
3	#16811510	C₂	1	2	yes	maximal, complemented
4	C₁₆×#6157	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₁₆	C₄	1	1	yes	characteristic
7	#13133296	C₄	1	1	yes	characteristic
8	D₅₃⋊C₁₆	C₄	1	2	yes	complemented
9	D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#11719910	C₄	1	2	yes	complemented
12	#13133271	C₄	1	2	yes	—
13	C₈×#6157	C₂²	1	2	yes	—
14	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	#13133157	C₄	1	1	yes	characteristic, complemented
16	#13133156	C₄	1	2	yes	complemented
17	C₁₆×D₅₃	C₂²	1	2	yes	—
18	M₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²⋊#6157	C₈	1	2	yes	—
21	#11719940	C₈	1	2	yes	—
22	C₄⋊#6157	C₈	1	2	yes	—
23	C₄×#6157	C₂×C₄	1	2	yes	—
24	#11719936	C₈	1	2	yes	—
25	D₅₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#88322	C₂×C₄	1	1	yes	characteristic
28	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×D₅₃	C₂³	1	1	yes	characteristic
32	#11719910	C₂×C₄	1	2	yes	—
33	#11719910	C₂×C₄	1	2	yes	—
34	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	#11719908	C₂×C₄	1	2	yes	—
36	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×#6157	C₂×C₈	1	4	yes	—
38	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄×D₅₃	C₄²	1	1	yes	characteristic
41	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
42	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
44	#88324	C₂×C₈	1	4	yes	—
45	C₈×C₅₃	C₄²	1	1	yes	characteristic
46	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	#88322	C₄²	1	1	yes	characteristic
48	#88322	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₂×D₅₃	—	2	2	no	—
51	C₂×D₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×D₅₃	C₂²×C₈	1	1	yes	characteristic
53	#6157	—	2	4	no	—
54	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
55	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
56	Dic₅₃	C₄×C₈	1	1	yes	characteristic
57	Dic₅₃	C₂²×C₈	1	1	yes	characteristic
58	#2108	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
61	D₅₃	—	2	1	no	—
62	D₅₃	#504	1	2	yes	—
63	C₂×M₃₂	—	53	1	no	—
64	C₂²×C₁₆	—	53	1	no	—
65	#405	—	53	1	no	—
66	#346	—	53	2	no	—
67	C₄×C₁₆	—	53	2	no	—
68	C₅₃	#2108	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	53	1	no	—
70	C₂²×C₈	—	53	1	no	—
71	#117	—	53	1	no	—
72	M₃₂	—	53	1	no	—
73	M₃₂	—	53	2	no	—
74	M₃₂	#6157	1	1	yes	characteristic, complemented
75	C₂×C₁₆	—	53	1	no	—
76	C₂×C₁₆	—	53	1	no	—
77	C₂×C₁₆	—	53	2	no	—
78	C₂×C₁₆	—	53	2	no	—
79	C₂×C₁₆	—	53	2	no	—
80	#97	—	53	2	no	—
81	C₄×C₈	—	53	2	no	—
82	C₂²×C₄	—	53	1	no	—
83	M₁₆	—	53	2	no	—
84	M₁₆	—	53	2	no	—
85	C₂×C₈	—	53	2	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	—	53	1	no	—
91	C₂×C₈	C₂×#6157	1	1	yes	characteristic
92	C₄⋊C₄	—	53	2	no	—
93	C₂²⋊C₄	—	53	2	no	—
94	C₄²	—	53	2	no	—
95	C₁₆	—	53	2	no	—
96	C₁₆	—	53	2	no	—
97	C₁₆	—	53	2	no	—
98	C₁₆	C₂×#6157	1	2	yes	—
99	C₂³	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	53	4	no	—
106	C₂×C₄	C₄×#6157	1	1	yes	characteristic
107	C₈	—	53	4	no	—
108	C₈	—	53	1	no	—
109	C₈	—	53	1	no	—
110	C₈	C₄×#6157	1	1	yes	characteristic
111	C₈	C₂²×#6157	1	1	yes	characteristic
112	C₂²	—	53	1	no	—
113	C₂²	—	53	1	no	—
114	C₂²	—	106	2	no	—
115	C₂²	C₈×#6157	1	1	yes	characteristic
116	C₄	—	53	1	no	—
117	C₄	—	53	1	no	—
118	C₄	—	106	4	no	—
119	C₄	C₈×#6157	1	1	yes	characteristic
120	C₄	C₂×C₄×#6157	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	53	2	no	—
123	C₂	—	106	1	no	—
124	C₂	C₂×C₈×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

#16811586

C₂

yes

maximal, characteristic

#16811510

C₂

yes

maximal, complemented

C₁₆×#6157

C₂

yes

maximal, complemented

M₃₂×D₅₃

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

C₄

yes

characteristic

#13133296

C₄

yes

characteristic

D₅₃⋊C₁₆

C₄

yes

complemented

D₅₃⋊C₁₆

C₂²

yes

characteristic

D₅₃⋊C₁₆

C₂²

yes

characteristic

C₂×#11719910

C₄

yes

complemented

#13133271

C₄

yes

—

C₈×#6157

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133157

C₄

yes

characteristic, complemented

#13133156

C₄

yes

complemented

C₁₆×D₅₃

C₂²

yes

—

M₃₂×C₅₃

C₄

yes

characteristic, complemented

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂²⋊#6157

C₈

yes

—

#11719940

C₈

yes

—

C₄⋊#6157

C₈

yes

—

C₄×#6157

C₂×C₄

yes

—

#11719936

C₈

yes

—

D₅₃⋊C₈

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

#11719910

C₂×C₄

yes

—

#11719910

C₂×C₄

yes

—

C₁₆×C₅₃

C₂×C₄

yes

—

#11719908

C₂×C₄

yes

—

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×#6157

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₄²

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₂²×C₄

yes

characteristic

#88324

C₂×C₈

yes

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₄²

yes

characteristic

#88322

C₂²×C₄

yes

characteristic

C₂²×C₅₃

C₄×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄×C₈

yes

characteristic

C₂×D₅₃

C₂²×C₈

yes

characteristic

#6157

—

C₄×C₅₃

C₄×C₈

yes

characteristic

C₄×C₅₃

C₂×C₄²

yes

characteristic

Dic₅₃

C₄×C₈

yes

characteristic

Dic₅₃

C₂²×C₈

yes

characteristic

#2108

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

D₅₃

—

D₅₃

#504

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#346

—

C₄×C₁₆

—

C₅₃

#2108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

M₃₂

#6157

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₂×#6157

yes

—

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄×#6157

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

C₄×#6157

yes

characteristic

111

C₈

C₂²×#6157

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

106

—

115

C₂²

C₈×#6157

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

106

—

119

C₄

C₈×#6157

yes

characteristic

120

C₄

C₂×C₄×#6157

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

106

—

124

C₂

C₂×C₈×#6157

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#6157
commutator, socle	C₂×C₅₃	C₂×C₄×C₈
Fitting	M₃₂×C₅₃	C₄
hypercenter	M₃₂	#6157
hypocenter	C₅₃	#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ M₃₂

subgroup

G/H

center, Frattini

C₈

C₂²×#6157

commutator, socle

C₂×C₅₃

C₂×C₄×C₈

Fitting

M₃₂×C₅₃

C₄

M₃₂

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₈ ◁ M₃₂

N	H ≅ G/N	type
3. #16811510	C₂	semidirect
4. C₁₆×#6157	C₂	semidirect
8. D₅₃⋊C₁₆	C₄	semidirect
11. C₂×#11719910	C₄	semidirect
15. #13133157	C₄	semidirect
16. #13133156	C₄	semidirect
18. M₃₂×C₅₃	C₄	semidirect
68. C₅₃	#2108	semidirect
74. M₃₂	#6157	semidirect

H ≅ G/N

type

3. #16811510

C₂

semidirect

4. C₁₆×#6157

C₂

semidirect

8. D₅₃⋊C₁₆

C₄

semidirect

11. C₂×#11719910

C₄

semidirect

15. #13133157

C₄

semidirect

16. #13133156

C₄

semidirect

18. M₃₂×C₅₃

C₄

semidirect

68. C₅₃

#2108

semidirect

74. M₃₂

#6157

semidirect

Subgroups — #20290927

Conjugacy classes of subgroups

Complements