Subgroups — C₂×#13133127 #16810571

Order 3392.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133127	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×Dic₅₃	C₄	1	2	yes	—
7	C₂×C₈×C₅₃	C₄	1	4	yes	complemented
8	C₂×C₈×C₅₃	C₂²	1	2	yes	—
9	C₄×Dic₅₃	C₄	1	2	yes	—
10	C₄×Dic₅₃	C₂²	1	2	yes	—
11	C₂×#88322	C₄	1	4	yes	complemented
12	C₂×#88322	C₂²	1	2	yes	—
13	C₂³×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₅₃	—	2	4	no	—
19	C₂×Dic₅₃	C₂×C₄	1	4	yes	—
20	C₈×C₅₃	C₂×C₄	1	4	yes	complemented
21	#88322	C₂×C₄	1	4	yes	complemented
22	C₂²×C₅₃	C₄²	1	2	yes	—
23	C₂²×C₅₃	M₁₆	1	4	yes	—
24	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₅₃	C₄²	1	2	yes	—
26	C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	Dic₅₃	—	2	4	no	—
29	C₂×C₅₃	#97	1	4	yes	complemented
30	C₂×C₅₃	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₅₃	C₂×M₁₆	1	2	yes	—
32	C₂×#97	—	53	1	no	maximal, Sylow
33	C₅₃	C₂×#97	1	1	yes	characteristic, complemented, Sylow
34	C₂²×C₈	—	53	1	no	—
35	C₂²×C₈	D₅₃	1	1	yes	characteristic
36	C₂×C₄²	—	53	1	no	—
37	#97	—	53	4	no	—
38	C₂²×C₄	—	53	2	no	—
39	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
40	C₂×C₈	—	53	4	no	—
41	C₂×C₈	—	53	2	no	—
42	C₂×C₈	Dic₅₃	1	4	yes	complemented
43	C₂×C₈	C₂×D₅₃	1	2	yes	—
44	C₄²	—	53	2	no	—
45	C₄²	—	53	2	no	—
46	C₂³	C₄×D₅₃	1	1	yes	characteristic
47	C₂×C₄	—	53	4	no	—
48	C₂×C₄	—	106	4	no	—
49	C₂×C₄	C₄×D₅₃	1	2	yes	—
50	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
51	C₂×C₄	C₂×Dic₅₃	1	2	yes	—
52	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
53	C₈	—	53	4	no	—
54	C₈	C₂×Dic₅₃	1	4	yes	complemented
55	C₂²	#11719912	1	4	yes	—
56	C₂²	C₄×Dic₅₃	1	2	yes	—
57	C₂²	C₂×C₄×D₅₃	1	1	yes	characteristic
58	C₄	—	106	4	no	—
59	C₄	C₄×Dic₅₃	1	2	yes	—
60	C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
61	C₄	C₂²×Dic₅₃	1	1	yes	characteristic
62	C₂	#13133127	1	4	yes	direct product factor
63	C₂	C₂×C₄×Dic₅₃	1	1	yes	characteristic
64	C₂	C₂×#11719912	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₂²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	C₂²×C₈	D₅₃
hypocenter	C₅₃	C₂×#97
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #13133127	C₂	direct (N × H)
7. C₂×C₈×C₅₃	C₄	semidirect
11. C₂×#88322	C₄	semidirect
20. C₈×C₅₃	C₂×C₄	semidirect
21. #88322	C₂×C₄	semidirect
29. C₂×C₅₃	#97	semidirect
33. C₅₃	C₂×#97	semidirect
42. C₂×C₈	Dic₅₃	semidirect
54. C₈	C₂×Dic₅₃	semidirect
62. C₂	#13133127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88322	2
C₂×C₄×Dic₅₃	2
#13133127	2
C₂²×C₈×C₅₃	2
C₂×#97	53

Subgroups — C2×#13133127 #16810571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13133127 #16810571