Subgroups — #16811682

Order 3392.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133292	C₂	1	2	yes	maximal, complemented
2	#13133291	C₂	1	1	yes	maximal, characteristic, complemented
3	D₅₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133225	C₂	1	1	yes	maximal, characteristic
6	#11719945	C₄	1	1	yes	characteristic
7	M₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#88322	C₂²	1	1	yes	characteristic
9	#11719912	C₄	1	2	yes	—
10	C₈×D₅₃	C₂²	1	2	yes	—
11	#11719910	C₂²	1	1	yes	characteristic
12	#11719910	C₂²	1	1	yes	characteristic
13	#11719910	C₂²	1	2	yes	—
14	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²⋊D₅₃	C₈	1	2	yes	—
16	D₂₁₂	C₈	1	1	yes	characteristic
17	C₄×D₅₃	C₂×C₄	1	2	yes	—
18	Dic₁₀₆	C₈	1	1	yes	characteristic
19	C₈×C₅₃	C₂×C₄	1	2	yes	—
20	#88322	C₂×C₄	1	1	yes	characteristic
21	#88322	C₂³	1	1	yes	characteristic
22	C₂²×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×D₅₃	C₂×C₈	1	2	yes	—
24	C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	Dic₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₅₃	—	2	1	no	—
28	C₂×C₅₃	C₂²×C₈	1	1	yes	characteristic
29	D₅₃	—	2	2	no	—
30	#504	—	53	1	no	maximal, Sylow
31	C₅₃	#504	1	1	yes	characteristic, complemented, Sylow
32	#131	—	53	1	no	—
33	M₃₂	—	53	1	no	—
34	M₃₂	—	53	2	no	—
35	C₂×C₁₆	—	53	1	no	—
36	C₂×C₁₆	—	53	2	no	—
37	#41	—	53	1	no	—
38	M₁₆	—	53	2	no	—
39	M₁₆	#6157	1	1	yes	characteristic
40	C₂×C₈	—	53	1	no	—
41	C₂×C₈	—	53	2	no	—
42	C₁₆	—	53	1	no	—
43	C₁₆	—	53	1	no	—
44	C₁₆	—	53	2	no	—
45	Q₈	—	53	1	no	—
46	D₄	—	53	2	no	—
47	D₄	—	53	1	no	—
48	C₂×C₄	—	53	2	no	—
49	C₂×C₄	C₂×#6157	1	1	yes	characteristic
50	C₈	—	53	1	no	—
51	C₈	—	53	1	no	—
52	C₈	C₂×#6157	1	2	yes	—
53	C₂²	—	53	2	no	—
54	C₂²	D₅₃⋊C₈	1	1	yes	characteristic
55	C₄	—	53	2	no	—
56	C₄	D₅₃⋊C₈	1	1	yes	characteristic
57	C₄	C₂²×#6157	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	106	2	no	—
60	C₂	C₂×D₅₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#6157
commutator, socle	C₂×C₅₃	C₂²×C₈
Fitting	M₁₆×C₅₃	C₄
prime radical	#11719945	C₄
cosocle	#88322	C₂³
hypercenter	M₁₆	#6157
hypocenter	C₅₃	#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #13133292	C₂	semidirect
2. #13133291	C₂	semidirect
3. D₅₃⋊C₁₆	C₂	semidirect
4. C₂×#11719910	C₂	semidirect
31. C₅₃	#504	semidirect

Maximal subgroups

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

subgroup	index
#13133292	2
#13133291	2
D₅₃⋊C₁₆	2
C₂×#11719910	2
#13133225	2
#504	53