Subgroups — #16810873

Order 3392.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133188	C₂	1	2	yes	maximal, complemented
3	#13133149	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133138	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₅₃	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₅₃	—	2	2	no	—
8	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	D₈×C₅₃	—	2	2	no	—
10	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₅₃	—	2	2	no	—
12	C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×#88322	C₂²	1	1	yes	characteristic
14	#11719912	—	2	2	no	—
15	C₂³×C₅₃	—	2	2	no	—
16	C₂²×D₅₃	D₄	1	1	yes	characteristic
17	D₄×C₅₃	—	2	2	no	—
18	D₄×C₅₃	—	2	2	no	—
19	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²⋊D₅₃	—	4	4	no	—
21	C₂×Dic₅₃	—	2	2	no	—
22	C₂×Dic₅₃	D₄	1	1	yes	characteristic
23	C₄×D₅₃	—	2	1	no	—
24	C₄×D₅₃	—	2	1	no	—
25	C₈×C₅₃	D₄	1	2	yes	complemented
26	#88322	—	2	1	no	—
27	C₂²×C₅₃	—	4	2	no	—
28	C₂²×C₅₃	—	4	2	no	—
29	C₂²×C₅₃	—	4	2	no	—
30	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×D₅₃	—	2	1	no	—
32	C₂×D₅₃	—	2	2	no	—
33	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅₃	#41	1	1	yes	characteristic
35	Dic₅₃	—	2	1	no	—
36	Dic₅₃	—	4	2	no	—
37	C₂×C₅₃	—	8	2	no	—
38	C₂×C₅₃	#121	1	1	yes	characteristic
39	C₂×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
40	D₅₃	—	4	1	no	—
41	#469	—	53	1	no	maximal, Sylow
42	C₅₃	#469	1	1	yes	characteristic, complemented, Sylow
43	C₂×D₈	D₅₃	1	1	yes	characteristic, complemented
44	C₂×M₁₆	—	53	1	no	—
45	#121	—	53	2	no	—
46	#106	—	53	1	no	—
47	#102	—	53	2	no	—
48	C₂×D₄	—	106	2	no	—
49	C₂×D₄	C₂×D₅₃	1	2	yes	—
50	C₂²×C₄	—	53	1	no	—
51	D₈	—	2	2	no	—
52	M₁₆	—	106	2	no	—
53	C₂×C₈	—	53	1	no	—
54	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
55	C₄⋊C₄	—	53	2	no	—
56	C₂²⋊C₄	—	106	2	no	—
57	C₂³	—	2	2	no	—
58	C₂³	—	53	1	no	—
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	212	4	no	—
62	C₂×C₄	—	53	1	no	—
63	C₂×C₄	—	106	1	no	—
64	C₂×C₄	—	106	1	no	—
65	C₂×C₄	—	106	2	no	—
66	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
67	C₈	—	106	1	no	—
68	C₈	C₂²⋊D₅₃	1	2	yes	complemented
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	106	1	no	—
73	C₂²	—	106	2	no	—
74	C₂²	D₄×D₅₃	1	1	yes	characteristic
75	C₄	—	106	1	no	—
76	C₄	—	212	2	no	—
77	C₄	#11719947	1	1	yes	characteristic
78	C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	C₂	—	8	2	no	—
80	C₂	—	212	1	no	—
81	C₂	#13133188	1	1	yes	characteristic
82	C₂	#13133241	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₂×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
hypercenter	C₂×D₈	D₅₃
hypocenter	C₅₃	#469
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#11719912	C₂	semidirect
2. #13133188	C₂	semidirect
3. #13133149	C₂	semidirect
4. #13133138	C₂	semidirect
5. C₂×D₈×C₅₃	C₂	semidirect
25. C₈×C₅₃	D₄	semidirect
42. C₅₃	#469	semidirect
43. C₂×D₈	D₅₃	semidirect
68. C₈	C₂²⋊D₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11719912	2
#13133188	2
#13133149	2
#13133138	2
C₂×D₈×C₅₃	2
#469	53