Subgroups — C₁₆×#11719910 #28298085

Order 13568.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719910	C₂	1	2	yes	maximal
2	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719910	C₄	1	4	yes	—
4	C₄×#11719908	C₄	1	1	yes	characteristic
5	C₈×#88322	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719910	C₈	1	8	yes	—
8	C₂×#11719908	C₈	1	2	yes	—
9	C₄×#88322	C₂×C₄	1	2	yes	—
10	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
11	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₅₃	C₄²	1	1	yes	characteristic
15	C₂×#88322	C₂×C₈	1	4	yes	—
16	#11719910	C₁₆	1	16	yes	direct product factor
17	C₁₆×C₅₃	C₁₆	1	4	yes	complemented
18	#11719908	C₁₆	1	4	yes	complemented
19	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
20	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
21	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
22	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
23	#88322	C₂×C₁₆	1	8	yes	—
24	C₁₆²	—	53	1	no	maximal, Sylow
25	C₂²×C₅₃	C₈²	1	1	yes	characteristic
26	C₄×C₅₃	C₄×C₁₆	1	4	yes	—
27	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
28	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
29	C₈×C₁₆	—	53	2	no	—
30	C₈×C₁₆	—	53	1	no	—
31	C₂×C₅₃	C₈×C₁₆	1	2	yes	—
32	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
33	C₄×C₁₆	—	53	4	no	—
34	C₄×C₁₆	—	53	1	no	—
35	C₄×C₁₆	#6157	1	1	yes	characteristic
36	C₈²	—	53	1	no	—
37	C₅₃	C₁₆²	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	—	53	8	no	—
39	C₂×C₁₆	—	53	2	no	—
40	C₂×C₁₆	#88324	1	2	yes	—
41	C₄×C₈	—	53	2	no	—
42	C₄×C₈	C₂×#6157	1	1	yes	characteristic
43	C₂×C₈	—	53	4	no	—
44	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
45	C₂×C₈	C₂×#88324	1	1	yes	characteristic
46	C₄²	C₄×#6157	1	1	yes	characteristic
47	C₁₆	—	53	16	no	—
48	C₁₆	—	53	4	no	—
49	C₁₆	#11719910	1	4	yes	direct product factor
50	C₂×C₄	C₈×#6157	1	2	yes	—
51	C₂×C₄	C₄×#88324	1	1	yes	characteristic
52	C₈	—	53	8	no	—
53	C₈	C₂×#11719910	1	2	yes	—
54	C₈	D₅₃⋊C₁₆	1	2	yes	—
55	C₂²	C₈×#88324	1	1	yes	characteristic
56	C₄	C₁₆×#6157	1	4	yes	—
57	C₄	C₄×#11719910	1	1	yes	characteristic
58	C₄	#16811509	1	1	yes	characteristic
59	C₂	C₁₆×#88324	1	2	yes	—
60	C₂	C₈×#11719910	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₁₆	#6157
commutator, hypocenter	C₅₃	C₁₆²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₈²
cosocle	C₈×#88322	C₂²
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
16. #11719910	C₁₆	direct (N × H)
17. C₁₆×C₅₃	C₁₆	semidirect
18. #11719908	C₁₆	semidirect
37. C₅₃	C₁₆²	semidirect
49. C₁₆	#11719910	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#11719910	2
C₁₆×#88322	2
C₁₆²	53

Subgroups — C16×#11719910 #28298085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#11719910 #28298085