Subgroups — #16810973

Order 3392.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133217	C₂	1	1	yes	maximal, characteristic
2	#13133141	C₂	1	4	yes	maximal
3	C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₅₃	C₂²	1	2	yes	—
6	C₂×Dic₁₀₆	—	2	1	no	—
7	C₄⋊C₄×C₅₃	—	2	1	no	—
8	C₄⋊C₄×C₅₃	—	2	1	no	—
9	C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₅₃	—	2	1	no	—
11	C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×#88322	C₂²	1	2	yes	—
13	Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄×C₅₃	—	2	2	no	—
15	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₅₃	—	2	2	no	—
19	Dic₁₀₆	—	4	2	no	—
20	#88322	—	2	2	no	—
21	C₂²×C₅₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₅₃	—	4	2	no	—
23	C₄×C₅₃	SD₁₆	1	2	yes	—
24	C₄×C₅₃	Q₁₆	1	2	yes	—
25	C₄×C₅₃	#41	1	2	yes	—
26	Dic₅₃	—	4	2	no	—
27	C₂×C₅₃	#124	1	1	yes	characteristic
28	C₂×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
30	#487	—	53	1	no	maximal, Sylow
31	C₅₃	#487	1	1	yes	characteristic, complemented, Sylow
32	C₄⋊Q₈	—	53	1	no	—
33	C₄⋊Q₈	D₅₃	1	1	yes	characteristic
34	#103	—	53	4	no	—
35	C₄×C₈	—	53	1	no	—
36	C₂×Q₈	—	106	1	no	—
37	C₂×Q₈	C₂×D₅₃	1	2	yes	—
38	C₂×C₈	—	53	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	53	2	no	—
42	C₄⋊C₄	—	106	1	no	—
43	C₄²	C₂×D₅₃	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	212	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	106	2	no	—
48	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
51	C₈	—	106	2	no	—
52	C₂²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	212	2	no	—
55	C₄	#11719924	1	2	yes	—
56	C₄	#11719925	1	2	yes	—
57	C₄	#11719947	1	2	yes	—
58	C₂	#13133204	1	1	yes	characteristic
59	C₂	C₂×#11719924	1	1	yes	characteristic
60	C₂	C₂×#11719925	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#487
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
31. C₅₃	#487	semidirect

Maximal subgroups

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

subgroup	index
#13133217	2
#13133141	2
C₄×#88322	2
C₄⋊Q₈×C₅₃	2
#487	53