Subgroups — C₈₉×#8817 #36386665

Order 22784.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#1834	C₂	1	6	yes	maximal
2	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
4	#384×C₈₉	C₂²	1	4	yes	—
5	#340×C₈₉	—	2	12	no	—
6	C₈²×C₈₉	C₄	1	4	yes	complemented
7	C₂²×C₈×C₈₉	D₄	1	6	yes	—
8	C₂×C₄⋊C₄×C₈₉	—	2	12	no	—
9	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₈₉	C₂×C₄	1	6	yes	—
11	C₄×C₈×C₈₉	Q₈	1	6	yes	—
12	C₂²×C₄×C₈₉	—	4	4	no	—
13	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
14	C₂×C₈×C₈₉	—	2	12	no	—
15	C₂×C₈×C₈₉	C₄⋊C₄	1	12	yes	—
16	C₄⋊C₄×C₈₉	—	4	24	no	—
17	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₈₉	C₂×Q₈	1	3	yes	—
19	C₂³×C₈₉	#127	1	1	yes	characteristic
20	C₂×C₄×C₈₉	—	8	12	no	—
21	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
22	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
23	C₂×C₄×C₈₉	#135	1	12	yes	—
24	C₈×C₈₉	—	2	12	no	—
25	C₂²×C₈₉	#384	1	1	yes	characteristic
26	C₂²×C₈₉	#495	1	6	yes	—
27	C₄×C₈₉	—	16	4	no	—
28	C₄×C₈₉	#427	1	6	yes	—
29	C₄×C₈₉	#499	1	6	yes	—
30	#8817	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₈₉	#1711	1	4	yes	—
32	C₂×C₈₉	#1846	1	3	yes	—
33	#1834	C₂×C₈₉	1	6	yes	—
34	C₂×C₈²	C₂×C₈₉	1	1	yes	characteristic
35	C₈₉	#8817	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₄×C₈	C₂²×C₈₉	1	3	yes	—
37	#384	C₂²×C₈₉	1	4	yes	—
38	#340	—	2	12	no	—
39	C₈²	C₄×C₈₉	1	4	yes	complemented
40	C₂²×C₈	D₄×C₈₉	1	6	yes	—
41	C₂×C₄⋊C₄	—	2	12	no	—
42	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
43	C₄×C₈	C₂×C₄×C₈₉	1	6	yes	—
44	C₄×C₈	Q₈×C₈₉	1	6	yes	—
45	C₂²×C₄	—	4	4	no	—
46	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊C₄×C₈₉	1	12	yes	—
49	C₄⋊C₄	—	4	24	no	—
50	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
51	C₄²	C₂×Q₈×C₈₉	1	3	yes	—
52	C₂³	#127×C₈₉	1	1	yes	characteristic
53	C₂×C₄	—	8	12	no	—
54	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	3	yes	—
55	C₂×C₄	C₄⋊Q₈×C₈₉	1	3	yes	—
56	C₂×C₄	#135×C₈₉	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	#384×C₈₉	1	1	yes	characteristic
59	C₂²	#495×C₈₉	1	6	yes	—
60	C₄	—	16	4	no	—
61	C₄	#427×C₈₉	1	6	yes	—
62	C₄	#499×C₈₉	1	6	yes	—
63	C₂	C₈₉×#1711	1	4	yes	—
64	C₂	C₈₉×#1846	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#127
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₈²×C₈₉	C₄	semidirect
30. #8817	C₈₉	direct (N × H)
35. C₈₉	#8817	direct (N × H)
39. C₈²	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#1834	2
C₂×C₈²×C₈₉	2
#8817	89

Subgroups — C89×#8817 #36386665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#8817 #36386665