Group #20288117 — subgroups

Subgroups — #20288117

Order 6784.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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	#16811252	C₂	1	12	yes	maximal, complemented
2	#16811249	C₂	1	3	yes	maximal
3	#16811175	C₂	1	12	yes	maximal, complemented
4	C₂×#13133173	C₂	1	3	yes	maximal
5	C₅₃×#543	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
7	#13133216	C₂²	1	12	yes	complemented
8	#13133215	C₂²	1	12	yes	complemented
9	#13133207	C₂²	1	24	yes	complemented
10	#13133196	C₂²	1	12	yes	—
11	#13133194	C₂²	1	24	yes	—
12	Q₈×Dic₅₃	C₂²	1	4	yes	complemented
13	#13133182	C₂²	1	12	yes	complemented
14	#13133173	C₂²	1	12	yes	—
15	#13133173	C₂²	1	12	yes	—
16	C₂×#11719919	C₂²	1	6	yes	—
17	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
18	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
19	C₂²⋊Q₈×C₅₃	C₂²	1	12	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
21	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
24	C₂²×Dic₅₃	Q₈	1	8	yes	complemented
25	C₂²×Dic₅₃	C₂³	1	6	yes	—
26	C₂²×Dic₅₃	C₂³	1	3	yes	—
27	C₂²×Dic₅₃	C₂³	1	3	yes	—
28	C₂×Dic₁₀₆	C₂³	1	12	yes	—
29	C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
30	C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
31	C₂²⋊Dic₅₃	C₂³	1	12	yes	—
32	C₄⋊Dic₅₃	C₂³	1	12	yes	—
33	#11719919	C₂³	1	24	yes	—
34	#11719919	C₂³	1	24	yes	—
35	C₄×Dic₅₃	C₂³	1	12	yes	—
36	C₄×Dic₅₃	C₂³	1	12	yes	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
39	C₂³×C₅₃	C₂⁴	1	3	yes	—
40	Q₈×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	—	4	6	no	—
42	C₂×C₄×C₅₃	C₂⁴	1	12	yes	—
43	C₂×Dic₅₃	—	4	6	no	—
44	C₂×Dic₅₃	C₂×Q₈	1	12	yes	—
45	C₂×Dic₅₃	C₂×Q₈	1	12	yes	—
46	C₂×Dic₅₃	C₂⁴	1	12	yes	—
47	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂⁴	1	3	yes	—
49	Dic₁₀₆	—	4	12	no	—
50	C₂²×C₅₃	—	2	6	no	—
51	C₂²×C₅₃	—	4	4	no	—
52	C₂²×C₅₃	C₂²×Q₈	1	6	yes	—
53	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
54	C₄×C₅₃	—	2	12	no	—
55	Dic₅₃	—	2	12	no	—
56	Dic₅₃	C₂²×Q₈	1	2	yes	—
57	Dic₅₃	C₂²×Q₈	1	6	yes	—
58	#3458	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	6	no	—
60	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
61	C₂×C₅₃	#585	1	2	yes	—
62	#544	—	53	12	no	—
63	#543	—	53	3	no	—
64	#543	D₅₃	1	1	yes	characteristic
65	#533	—	53	12	no	—
66	C₂×#117	—	53	3	no	—
67	C₅₃	#3458	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₄	—	53	1	no	—
69	C₄⋊Q₈	—	53	12	no	—
70	C₄⋊Q₈	—	53	12	no	—
71	#125	—	53	24	no	—
72	C₂²⋊Q₈	—	53	12	no	—
73	C₂²⋊Q₈	—	53	24	no	—
74	C₂²⋊Q₈	C₂×D₅₃	1	12	yes	—
75	C₄×Q₈	—	53	4	no	—
76	C₄×Q₈	—	53	12	no	—
77	#117	—	53	12	no	—
78	#117	—	53	12	no	—
79	C₂×C₄⋊C₄	—	53	6	no	—
80	C₂×C₂²⋊C₄	—	53	3	no	—
81	C₂×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
82	C₂×C₄²	—	53	6	no	—
83	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×Q₈	—	53	12	no	—
85	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
86	C₂²×C₄	—	53	8	no	—
87	C₂²×C₄	—	53	3	no	—
88	C₂²×C₄	—	53	3	no	—
89	C₂²×C₄	—	53	6	no	—
90	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
91	C₄⋊C₄	—	53	24	no	—
92	C₄⋊C₄	—	53	24	no	—
93	C₄⋊C₄	—	53	12	no	—
94	C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
95	C₂²⋊C₄	—	53	12	no	—
96	C₂²⋊C₄	C₂²×D₅₃	1	12	yes	—
97	C₄²	—	53	12	no	—
98	C₄²	—	53	12	no	—
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₅₃	1	4	yes	—
101	C₂³	C₂³×D₅₃	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	212	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	53	12	no	—
106	C₂×C₄	—	53	12	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	53	3	no	—
109	C₂×C₄	—	53	12	no	—
110	C₂×C₄	—	212	6	no	—
111	C₂×C₄	C₂³×D₅₃	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₅₃	1	6	yes	—
115	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	53	2	no	—
118	C₄	—	53	6	no	—
119	C₄	—	106	12	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
122	C₂	#16811458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811252

C₂

yes

maximal, complemented

#16811249

C₂

yes

maximal

#16811175

C₂

yes

maximal, complemented

C₂×#13133173

C₂

yes

maximal

C₅₃×#543

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

#13133216

C₂²

yes

complemented

#13133215

C₂²

yes

complemented

#13133207

C₂²

yes

complemented

#13133196

C₂²

yes

—

#13133194

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

complemented

#13133182

C₂²

yes

complemented

#13133173

C₂²

yes

—

#13133173

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂²⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

Q₈

yes

complemented

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

C₂²×Q₈

yes

—

Dic₅₃

C₂²×Q₈

yes

—

#3458

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

#585

yes

—

#544

—

#543

—

#543

D₅₃

yes

characteristic

#533

—

C₂×#117

—

C₅₃

#3458

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₅₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₅₃

yes

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

Q₈×D₅₃

yes

—

101

C₂³

C₂³×D₅₃

yes

—

102

Q₈

—

103

Q₈

—

212

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

212

—

111

C₂×C₄

C₂³×D₅₃

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₅₃

yes

—

115

C₂²

C₂⁴×D₅₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

106

—

120

C₂

—

121

C₂

C₂²×Q₈×D₅₃

yes

characteristic

122

C₂

#16811458

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#543	C₂
hypercenter	#543	D₅₃
hypocenter	C₅₃	#3458
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₅₃

commutator, socle

C₂²×C₅₃

C₂⁵

Fitting

C₅₃×#543

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
1. #16811252	C₂	semidirect
3. #16811175	C₂	semidirect
7. #13133216	C₂²	semidirect
8. #13133215	C₂²	semidirect
9. #13133207	C₂²	semidirect
12. Q₈×Dic₅₃	C₂²	semidirect
13. #13133182	C₂²	semidirect
24. C₂²×Dic₅₃	Q₈	semidirect
67. C₅₃	#3458	semidirect

H ≅ G/N

type

1. #16811252

C₂

semidirect

3. #16811175

C₂

semidirect

7. #13133216

C₂²

semidirect

8. #13133215

C₂²

semidirect

9. #13133207

C₂²

semidirect

12. Q₈×Dic₅₃

C₂²

semidirect

13. #13133182

C₂²

semidirect

24. C₂²×Dic₅₃

Q₈

semidirect

67. C₅₃

#3458

semidirect

Subgroups — #20288117

Conjugacy classes of subgroups

Complements