Subgroups — #16810745

Order 3392.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133224	C₂	1	2	yes	maximal, complemented
2	#13133153	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133144	C₂	1	2	yes	maximal, complemented
4	#13133127	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#41×C₅₃	C₂²	1	2	yes	—
7	Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
8	SD₁₆×C₅₃	C₄	1	2	yes	complemented
9	D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
10	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#11719917	—	2	2	no	—
13	#11719917	C₂²	1	2	yes	—
14	C₂×#88322	—	2	2	no	—
15	C₂×#88322	C₂²	1	1	yes	characteristic
16	Q₈×C₅₃	C₂×C₄	1	2	yes	—
17	D₄×C₅₃	—	2	2	no	—
18	D₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅₃	—	2	2	no	—
20	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₅₃	—	2	1	no	—
22	C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	#88322	—	2	2	no	—
25	#88322	D₄	1	2	yes	—
26	C₂²×C₅₃	—	2	2	no	—
27	C₂²×C₅₃	#41	1	1	yes	characteristic
28	C₄×C₅₃	—	2	2	no	—
29	C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	Dic₅₃	—	4	1	no	—
32	C₂×C₅₃	—	2	1	no	—
33	C₂×C₅₃	—	4	2	no	—
34	C₂×C₅₃	C₄×D₄	1	1	yes	characteristic
35	#444	—	53	1	no	maximal, Sylow
36	C₅₃	#444	1	1	yes	characteristic, complemented, Sylow
37	#135	D₅₃	1	1	yes	characteristic
38	#131	—	53	2	no	—
39	#108	—	53	1	no	—
40	#104	—	53	2	no	—
41	#97	—	53	1	no	—
42	#41	C₂×D₅₃	1	2	yes	—
43	Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₅₃	1	2	yes	complemented
45	D₈	Dic₅₃	1	1	yes	characteristic, complemented
46	M₁₆	—	53	2	no	—
47	M₁₆	—	106	2	no	—
48	C₂×C₈	—	53	1	no	—
49	C₂×C₈	—	106	2	no	—
50	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
51	C₄²	—	53	1	no	—
52	Q₈	C₂×Dic₅₃	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₅₃	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	106	1	no	—
57	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
58	C₈	—	53	2	no	—
59	C₈	—	106	2	no	—
60	C₈	C₂×Dic₅₃	1	1	yes	characteristic
61	C₈	C₂×Dic₅₃	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#11719947	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	212	1	no	—
66	C₄	D₄×D₅₃	1	1	yes	characteristic
67	C₄	C₂²×Dic₅₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₃
Fitting, prime radical	#135×C₅₃	C₂
socle	C₂×C₅₃	C₄×D₄
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	#135	D₅₃
hypocenter	C₅₃	#444
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #13133224	C₂	semidirect
2. #13133153	C₂	semidirect
3. #13133144	C₂	semidirect
4. #13133127	C₂	semidirect
7. Q₁₆×C₅₃	C₄	semidirect
8. SD₁₆×C₅₃	C₄	semidirect
9. D₈×C₅₃	C₄	semidirect
36. C₅₃	#444	semidirect
43. Q₁₆	Dic₅₃	semidirect
44. SD₁₆	Dic₅₃	semidirect
45. D₈	Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133224	2
#13133153	2
#13133144	2
#13133127	2
#135×C₅₃	2
#444	53