Subgroups — #16810739

Order 3392.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133239	C₂	1	1	yes	maximal, characteristic
2	#13133225	C₂	1	2	yes	maximal, complemented
3	#13133143	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719945	C₂²	1	2	yes	—
7	M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×#88322	C₂²	1	1	yes	characteristic
11	Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
12	D₄₂₄	C₄	1	1	yes	characteristic, complemented
13	#11719913	C₄	1	2	yes	complemented
14	#11719912	—	2	2	no	—
15	C₈×D₅₃	—	2	2	no	—
16	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²⋊D₅₃	—	2	2	no	—
18	C₂×Dic₅₃	—	2	1	no	—
19	D₂₁₂	C₂×C₄	1	2	yes	—
20	C₄×D₅₃	—	2	2	no	—
21	Dic₁₀₆	C₂×C₄	1	2	yes	—
22	C₈×C₅₃	—	2	2	no	—
23	C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#88322	D₄	1	1	yes	characteristic
26	#88322	D₄	1	1	yes	characteristic
27	C₂²×C₅₃	#41	1	1	yes	characteristic
28	C₂×D₅₃	—	2	2	no	—
29	C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	Dic₅₃	—	2	2	no	—
32	Dic₅₃	—	2	2	no	—
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	C₄×D₄	1	1	yes	characteristic
35	D₅₃	—	4	2	no	—
36	#443	—	53	1	no	maximal, Sylow
37	C₅₃	#443	1	1	yes	characteristic, complemented, Sylow
38	#135	—	53	1	no	—
39	#131	—	53	2	no	—
40	#108	D₅₃	1	1	yes	characteristic, complemented
41	#104	—	53	2	no	—
42	C₄×C₈	—	53	1	no	—
43	#41	—	53	2	no	—
44	Q₁₆	—	53	1	no	—
45	SD₁₆	—	53	2	no	—
46	D₈	—	53	1	no	—
47	M₁₆	—	106	2	no	—
48	M₁₆	C₂×D₅₃	1	2	yes	—
49	C₂×C₈	—	53	1	no	—
50	C₂×C₈	—	106	2	no	—
51	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
52	C₄²	—	53	1	no	—
53	Q₈	—	53	2	no	—
54	D₄	—	53	2	no	—
55	D₄	—	106	2	no	—
56	C₂×C₄	—	106	1	no	—
57	C₂×C₄	—	106	2	no	—
58	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	53	1	no	—
61	C₈	—	53	1	no	—
62	C₈	C₄×D₅₃	1	1	yes	characteristic
63	C₈	C₄×D₅₃	1	1	yes	characteristic
64	C₂²	—	106	2	no	—
65	C₂²	#11719949	1	1	yes	characteristic
66	C₄	—	106	2	no	—
67	C₄	—	106	2	no	—
68	C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
69	C₄	D₄×D₅₃	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	212	2	no	—
72	C₂	#13133179	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	#108×C₅₃	C₂
socle	C₂×C₅₃	C₄×D₄
prime radical	#13133239	C₂
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	#108	D₅₃
hypocenter	C₅₃	#443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #13133225	C₂	semidirect
3. #13133143	C₂	semidirect
4. C₈×Dic₅₃	C₂	semidirect
5. #108×C₅₃	C₂	semidirect
11. Dic₂₁₂	C₄	semidirect
12. D₄₂₄	C₄	semidirect
13. #11719913	C₄	semidirect
37. C₅₃	#443	semidirect
40. #108	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133239	2
#13133225	2
#13133143	2
C₈×Dic₅₃	2
#108×C₅₃	2
#443	53