Subgroups — #16810941

Order 3392.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133191	C₂	1	1	yes	maximal, characteristic
2	#13133148	C₂	1	2	yes	maximal, complemented
3	#13133137	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₅₃	—	2	1	no	—
7	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₅₃	—	2	1	no	—
9	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	#11719921	—	2	1	no	—
13	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×#88322	C₂²	1	2	yes	—
15	C₂³×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×D₅₃	—	2	1	no	—
17	C₂×C₄×C₅₃	—	2	2	no	—
18	C₂×C₄×C₅₃	—	2	2	no	—
19	C₂×C₄×C₅₃	—	4	2	no	—
20	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²⋊D₅₃	—	4	2	no	—
23	C₂×Dic₅₃	—	2	1	no	—
24	D₂₁₂	—	2	2	no	—
25	#88322	—	2	2	no	—
26	C₂²×C₅₃	—	2	1	no	—
27	C₂²×C₅₃	—	2	1	no	—
28	C₂²×C₅₃	SD₁₆	1	2	yes	—
29	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₅₃	—	4	1	no	—
31	C₂×D₅₃	—	4	1	no	—
32	C₂×D₅₃	—	4	1	no	—
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	—	4	2	no	—
35	C₄×C₅₃	#41	1	2	yes	—
36	Dic₅₃	—	4	1	no	—
37	C₂×C₅₃	—	2	2	no	—
38	C₂×C₅₃	#123	1	1	yes	characteristic
39	C₂×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
41	D₅₃	—	8	1	no	—
42	#481	—	53	1	no	maximal, Sylow
43	C₅₃	#481	1	1	yes	characteristic, complemented, Sylow
44	#121	—	53	1	no	—
45	C₂×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
46	#106	—	53	2	no	—
47	#102	—	53	2	no	—
48	C₂²⋊C₈	—	53	1	no	—
49	C₂×D₄	—	53	1	no	—
50	C₂×D₄	—	106	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
53	C₂×C₈	—	53	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	53	1	no	—
56	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
57	C₂²⋊C₄	—	106	1	no	—
58	C₂³	—	106	1	no	—
59	C₂³	C₂²⋊D₅₃	1	1	yes	characteristic
60	D₄	—	106	2	no	—
61	D₄	—	212	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	106	1	no	—
66	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
68	C₈	—	106	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	212	1	no	—
72	C₂²	—	212	1	no	—
73	C₂²	—	212	1	no	—
74	C₂²	#11719924	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	212	1	no	—
79	C₄	#11719945	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	424	1	no	—
82	C₂	#13133200	1	1	yes	characteristic
83	C₂	C₂×#11719924	1	1	yes	characteristic
84	C₂	#13133243	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊C₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₄
prime radical	#13133191	C₂
hypercenter	C₂×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#481
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #13133148	C₂	semidirect
3. #13133137	C₂	semidirect
4. C₂²⋊#88322	C₂	semidirect
5. C₂×C₄⋊C₄×C₅₃	C₂	semidirect
43. C₅₃	#481	semidirect
45. C₂×C₄⋊C₄	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133191	2
#13133148	2
#13133137	2
C₂²⋊#88322	2
C₂×C₄⋊C₄×C₅₃	2
#481	53