Group #20277114 — subgroups

Subgroups — #384×D₅₃ #20277114

Order 6784.

888 conjugacy classes of subgroups; only the 365 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 365 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×C₄⋊C₄×D₅₃	C₂	1	6	yes	maximal
2	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810492	C₂	1	6	yes	maximal, complemented
5	C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
7	C₂²×C₄×D₅₃	C₂²	1	3	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
9	C₂×#11719919	C₂²	1	12	yes	—
10	C₄⋊C₄×D₅₃	—	2	24	no	—
11	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
12	C₄²×D₅₃	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×D₅₃	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×Dic₅₃	C₂³	1	4	yes	—
19	C₂²×Dic₅₃	C₂³	1	3	yes	—
20	C₂×C₄×D₅₃	—	2	24	no	—
21	C₂×C₄×D₅₃	C₂×C₄	1	12	yes	—
22	C₂×C₄×D₅₃	D₄	1	12	yes	—
23	C₂×C₄×D₅₃	Q₈	1	12	yes	—
24	C₄⋊C₄×C₅₃	—	2	12	no	—
25	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
26	C₄⋊Dic₅₃	—	2	12	no	—
27	#11719919	—	2	24	no	—
28	C₄×Dic₅₃	C₂×C₄	1	12	yes	complemented
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
31	C₂²×D₅₃	C₂×D₄	1	6	yes	—
32	C₂²×D₅₃	C₂×Q₈	1	6	yes	—
33	C₂×C₄×C₅₃	—	2	12	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
37	C₂×Dic₅₃	—	2	12	no	—
38	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
39	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
40	C₂×Dic₅₃	C₂×Q₈	1	6	yes	—
41	C₄×D₅₃	—	4	16	no	—
42	C₄×D₅₃	C₄⋊C₄	1	48	yes	complemented
43	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
44	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
45	C₂²×C₅₃	C₂²×Q₈	1	3	yes	—
46	C₂×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
47	C₂×D₅₃	#127	1	4	yes	—
48	C₂×D₅₃	C₄⋊Q₈	1	12	yes	—
49	C₄×C₅₃	—	4	4	no	—
50	C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	complemented
51	Dic₅₃	—	4	4	no	—
52	Dic₅₃	C₂×C₄⋊C₄	1	12	yes	complemented
53	C₂×#384	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
55	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
56	C₂×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
57	D₅₃	#384	1	8	yes	direct product factor
58	C₂²×C₄⋊C₄	—	53	6	no	—
59	C₂²×C₄²	—	53	1	no	—
60	#384	—	53	6	no	—
61	#384	—	53	1	no	—
62	#384	D₅₃	1	1	yes	characteristic, direct product factor
63	C₅₃	C₂×#384	1	1	yes	characteristic, complemented, Sylow
64	C₂³×C₄	—	53	3	no	—
65	C₂³×C₄	—	53	4	no	—
66	C₂×C₄⋊C₄	—	53	12	no	—
67	C₂×C₄⋊C₄	—	53	6	no	—
68	C₂×C₄⋊C₄	—	106	24	no	—
69	C₂×C₄⋊C₄	C₂×D₅₃	1	6	yes	—
70	C₂×C₄²	—	53	3	no	—
71	C₂×C₄²	—	53	8	no	—
72	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
73	C₂⁴	—	53	1	no	—
74	C₂²×C₄	—	53	3	no	—
75	C₂²×C₄	—	53	12	no	—
76	C₂²×C₄	—	53	12	no	—
77	C₂²×C₄	—	53	12	no	—
78	C₂²×C₄	—	53	4	no	—
79	C₂²×C₄	—	106	24	no	—
80	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
81	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
82	C₄⋊C₄	—	2	12	no	—
83	C₄⋊C₄	—	106	24	no	—
84	C₄⋊C₄	—	106	12	no	—
85	C₄²	—	53	12	no	—
86	C₄²	C₄×D₅₃	1	4	yes	complemented
87	C₂³	—	53	2	no	—
88	C₂³	—	53	6	no	—
89	C₂³	—	53	6	no	—
90	C₂³	C₂³×D₅₃	1	1	yes	characteristic
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	53	6	no	—
93	C₂×C₄	—	53	6	no	—
94	C₂×C₄	—	53	6	no	—
95	C₂×C₄	—	53	48	no	—
96	C₂×C₄	—	106	12	no	—
97	C₂×C₄	—	212	16	no	—
98	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
99	C₂×C₄	D₄×D₅₃	1	6	yes	—
100	C₂×C₄	Q₈×D₅₃	1	6	yes	—
101	C₂²	—	53	12	no	—
102	C₂²	—	53	4	no	—
103	C₂²	—	53	12	no	—
104	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
105	C₂²	C₂×D₄×D₅₃	1	3	yes	—
106	C₂²	C₂×Q₈×D₅₃	1	3	yes	—
107	C₄	—	4	4	no	—
108	C₄	—	53	12	no	—
109	C₄	—	212	4	no	—
110	C₄	C₄⋊C₄×D₅₃	1	12	yes	complemented
111	C₂	—	53	8	no	—
112	C₂	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
113	C₂	#127×D₅₃	1	1	yes	characteristic
114	C₂	C₄⋊Q₈×D₅₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×D₅₃

C₂

yes

maximal

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic, complemented

#16810492

C₂

yes

maximal, complemented

C₅₃×#384

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₄⋊C₄×D₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄²×D₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

complemented

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

complemented

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

C₂²×C₄

yes

—

C₂²×D₅₃

C₂×D₄

yes

—

C₂²×D₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₄⋊C₄

yes

complemented

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

—

C₂×D₅₃

#127

yes

—

C₂×D₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

complemented

Dic₅₃

—

Dic₅₃

C₂×C₄⋊C₄

yes

complemented

C₂×#384

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄⋊C₄

yes

—

C₂×C₅₃

C₂×#127

yes

characteristic

C₂×C₅₃

C₂×C₄⋊Q₈

yes

—

D₅₃

#384

yes

direct product factor

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

#384

—

#384

—

#384

D₅₃

yes

characteristic, direct product factor

C₅₃

C₂×#384

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

C₄×D₅₃

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

D₄×D₅₃

yes

—

100

C₂×C₄

Q₈×D₅₃

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂²×C₄×D₅₃

yes

characteristic

105

C₂²

C₂×D₄×D₅₃

yes

—

106

C₂²

C₂×Q₈×D₅₃

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

212

—

110

C₄

C₄⋊C₄×D₅₃

yes

complemented

111

C₂

—

112

C₂

C₂×C₄⋊C₄×D₅₃

yes

—

113

C₂

#127×D₅₃

yes

characteristic

114

C₂

C₄⋊Q₈×D₅₃

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#384	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	#384	D₅₃
hypocenter	C₅₃	C₂×#384
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#384

C₂

socle

C₂³×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
3. #16810493	C₂	semidirect
4. #16810492	C₂	semidirect
5. C₅₃×#384	C₂	semidirect
12. C₄²×D₅₃	C₄	semidirect
25. C₄²×C₅₃	C₂×C₄	semidirect
28. C₄×Dic₅₃	C₂×C₄	semidirect
42. C₄×D₅₃	C₄⋊C₄	semidirect
50. C₄×C₅₃	C₂×C₄⋊C₄	semidirect
52. Dic₅₃	C₂×C₄⋊C₄	semidirect
57. D₅₃	#384	direct (N × H)
62. #384	D₅₃	direct (N × H)
63. C₅₃	C₂×#384	semidirect
86. C₄²	C₄×D₅₃	semidirect
110. C₄	C₄⋊C₄×D₅₃	semidirect

H ≅ G/N

type

3. #16810493

C₂

semidirect

4. #16810492

C₂

semidirect

5. C₅₃×#384

C₂

semidirect

12. C₄²×D₅₃

C₄

semidirect

25. C₄²×C₅₃

C₂×C₄

semidirect

28. C₄×Dic₅₃

C₂×C₄

semidirect

42. C₄×D₅₃

C₄⋊C₄

semidirect

50. C₄×C₅₃

C₂×C₄⋊C₄

semidirect

52. Dic₅₃

C₂×C₄⋊C₄

semidirect

57. D₅₃

#384

direct (N × H)

62. #384

D₅₃

direct (N × H)

63. C₅₃

C₂×#384

semidirect

86. C₄²

C₄×D₅₃

semidirect

110. C₄

C₄⋊C₄×D₅₃

semidirect

Subgroups — #384×D₅₃ #20277114

Conjugacy classes of subgroups

Complements

Subgroups — #384×D53 #20277114

Conjugacy classes of subgroups

Complements

Subgroups — #384×D₅₃ #20277114