Subgroups — C₂³×#15667918 #36712531

Order 22784.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667918	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667918	C₂²	1	28	yes	direct product factor
4	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₈₉	C₄	1	7	yes	—
6	C₂²×C₁₆×C₈₉	C₂²	1	7	yes	—
7	#15667918	C₂³	1	8	yes	direct product factor
8	C₂³×C₄×C₈₉	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₈₉	C₈	1	7	yes	—
10	C₂²×C₈×C₈₉	C₂×C₄	1	7	yes	—
11	C₂×C₁₆×C₈₉	C₂×C₄	1	21	yes	—
12	C₂×C₁₆×C₈₉	C₂³	1	7	yes	—
13	C₂⁴×C₈₉	C₁₆	1	1	yes	characteristic
14	C₂²×C₄×C₈₉	C₁₆	1	7	yes	—
15	C₂²×C₄×C₈₉	C₂×C₈	1	7	yes	—
16	C₂×C₈×C₈₉	C₂×C₈	1	21	yes	—
17	C₂×C₈×C₈₉	C₂²×C₄	1	7	yes	—
18	C₁₆×C₈₉	C₂²×C₄	1	7	yes	—
19	C₁₆×C₈₉	C₂⁴	1	1	yes	characteristic
20	C₂³×C₈₉	C₃₂	1	8	yes	complemented
21	C₂³×C₈₉	C₂×C₁₆	1	7	yes	—
22	C₂×C₄×C₈₉	C₂×C₁₆	1	21	yes	—
23	C₂×C₄×C₈₉	C₂²×C₈	1	7	yes	—
24	C₈×C₈₉	C₂²×C₈	1	7	yes	—
25	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
26	C₂²×C₈₉	C₂×C₃₂	1	28	yes	complemented
27	C₂²×C₈₉	C₂²×C₁₆	1	7	yes	—
28	C₄×C₈₉	C₂²×C₁₆	1	7	yes	—
29	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
30	C₂³×C₃₂	—	89	1	no	maximal, Sylow
31	C₂×C₈₉	C₂²×C₃₂	1	14	yes	complemented
32	C₂×C₈₉	C₂³×C₁₆	1	1	yes	characteristic
33	C₂³×C₁₆	D₈₉	1	1	yes	characteristic
34	C₂²×C₃₂	—	89	14	no	—
35	C₈₉	C₂³×C₃₂	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
37	C₂²×C₁₆	Dic₈₉	1	7	yes	—
38	C₂²×C₁₆	C₂×D₈₉	1	7	yes	—
39	C₂×C₃₂	—	89	28	no	—
40	C₂³×C₄	#10623494	1	1	yes	characteristic
41	C₂²×C₈	#10623494	1	7	yes	—
42	C₂²×C₈	C₂×Dic₈₉	1	7	yes	—
43	C₂×C₁₆	C₂×Dic₈₉	1	21	yes	—
44	C₂×C₁₆	C₂²×D₈₉	1	7	yes	—
45	C₃₂	—	89	8	no	—
46	C₂⁴	#13083549	1	1	yes	characteristic
47	C₂²×C₄	#13083549	1	7	yes	—
48	C₂²×C₄	C₂×#10623494	1	7	yes	—
49	C₂×C₈	C₂×#10623494	1	21	yes	—
50	C₂×C₈	C₂²×Dic₈₉	1	7	yes	—
51	C₁₆	C₂²×Dic₈₉	1	7	yes	—
52	C₁₆	C₂³×D₈₉	1	1	yes	characteristic
53	C₂³	#15667918	1	8	yes	direct product factor
54	C₂³	C₂×#13083549	1	7	yes	—
55	C₂×C₄	C₂×#13083549	1	21	yes	—
56	C₂×C₄	C₂²×#10623494	1	7	yes	—
57	C₈	C₂²×#10623494	1	7	yes	—
58	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
59	C₂²	C₂×#15667918	1	28	yes	direct product factor
60	C₂²	C₂²×#13083549	1	7	yes	—
61	C₄	C₂²×#13083549	1	7	yes	—
62	C₄	C₂³×#10623494	1	1	yes	characteristic
63	C₂	C₂²×#15667918	1	14	yes	direct product factor
64	C₂	C₂³×#13083549	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₁₆	D₈₉
commutator, hypocenter	C₈₉	C₂³×C₃₂
Frattini	C₁₆	C₂³×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₁₆
cosocle	C₁₆×C₈₉	C₂⁴
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667918	C₂	direct (N × H)
3. C₂×#15667918	C₂²	direct (N × H)
7. #15667918	C₂³	direct (N × H)
20. C₂³×C₈₉	C₃₂	semidirect
26. C₂²×C₈₉	C₂×C₃₂	semidirect
31. C₂×C₈₉	C₂²×C₃₂	semidirect
35. C₈₉	C₂³×C₃₂	semidirect
53. C₂³	#15667918	direct (N × H)
59. C₂²	C₂×#15667918	direct (N × H)
63. C₂	C₂²×#15667918	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×#15667918	2
C₂³×C₁₆×C₈₉	2
C₂³×C₃₂	89

Subgroups — C23×#15667918 #36712531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15667918 #36712531