Subgroups — #16810547

Order 3392.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719919	C₂	1	3	yes	maximal, complemented
2	C₂×#11719921	C₂	1	3	yes	maximal
3	#95×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅₃	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₅₃	C₂²	1	3	yes	—
6	C₂²×Dic₅₃	C₂²	1	3	yes	—
7	#11719921	—	2	6	no	—
8	#11719919	—	2	6	no	—
9	C₂³×C₅₃	C₂³	1	1	yes	characteristic
10	C₂²×D₅₃	—	2	3	no	—
11	C₂²×D₅₃	—	2	3	no	—
12	C₂²×D₅₃	Q₈	1	2	yes	—
13	C₂×C₄×C₅₃	—	2	3	no	—
14	C₂×C₄×C₅₃	—	2	3	no	—
15	C₂×C₄×C₅₃	—	2	3	no	—
16	C₂×Dic₅₃	—	2	3	no	—
17	C₂×Dic₅₃	—	2	3	no	—
18	C₂×Dic₅₃	D₄	1	6	yes	—
19	C₂²×C₅₃	C₂×D₄	1	3	yes	—
20	C₂²×C₅₃	C₂×Q₈	1	1	yes	characteristic
21	C₂²×C₅₃	#41	1	3	yes	—
22	C₂×D₅₃	—	2	6	no	—
23	C₂×D₅₃	—	4	3	no	—
24	C₂×D₅₃	—	4	1	no	—
25	C₄×C₅₃	—	4	3	no	—
26	Dic₅₃	—	2	6	no	—
27	C₂×C₅₃	#121	1	3	yes	—
28	C₂×C₅₃	C₂²⋊Q₈	1	3	yes	—
29	C₂×C₅₃	#126	1	1	yes	characteristic
30	D₅₃	—	4	2	no	—
31	#396	—	53	1	no	maximal, Sylow
32	C₅₃	#396	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₄⋊C₄	—	53	3	no	—
34	C₂×C₂²⋊C₄	—	53	3	no	—
35	#95	D₅₃	1	1	yes	characteristic, complemented
36	C₂⁴	—	53	1	no	—
37	C₂²×C₄	—	53	3	no	—
38	C₂²×C₄	C₂×D₅₃	1	3	yes	—
39	C₄⋊C₄	—	106	6	no	—
40	C₂²⋊C₄	—	106	6	no	—
41	C₂³	—	53	2	no	—
42	C₂³	—	106	3	no	—
43	C₂³	—	106	3	no	—
44	C₂³	C₂²×D₅₃	1	1	yes	characteristic
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	—	53	6	no	—
49	C₂×C₄	—	106	3	no	—
50	C₂×C₄	—	106	3	no	—
51	C₂²	—	106	6	no	—
52	C₂²	—	212	3	no	—
53	C₂²	—	212	1	no	—
54	C₂²	#11719945	1	3	yes	—
55	C₂²	D₄×D₅₃	1	3	yes	—
56	C₂²	Q₈×D₅₃	1	1	yes	characteristic
57	C₄	—	4	3	no	—
58	C₄	—	106	6	no	—
59	C₂	—	212	2	no	—
60	C₂	#13133190	1	3	yes	—
61	C₂	#13133195	1	3	yes	—
62	C₂	#13133211	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂³
Fitting	#95×C₅₃	C₂
prime radical	C₂³×D₅₃	C₂²
hypercenter	#95	D₅₃
hypocenter	C₅₃	#396
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂×#11719919	C₂	semidirect
3. #95×C₅₃	C₂	semidirect
32. C₅₃	#396	semidirect
35. #95	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719919	2
C₂×#11719921	2
#95×C₅₃	2
#396	53