Subgroups — #36441044

Order 22784.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018170	C₂	1	2	yes	maximal
2	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄×#13083549	C₄	1	4	yes	—
4	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
6	C₈²×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×#13083549	—	2	4	no	—
8	C₂×C₁₆×C₈₉	C₈	1	2	yes	—
9	C₂×C₁₆×C₈₉	C₈	1	2	yes	—
10	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
11	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
12	#13083549	—	4	4	no	—
13	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
14	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
15	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
16	C₄²×C₈₉	C₄²	1	1	yes	characteristic
17	C₁₆×C₈₉	—	2	2	no	—
18	C₁₆×C₈₉	C₁₆	1	4	yes	complemented
19	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
20	C₂×C₄×C₈₉	#97	1	2	yes	—
21	C₈×C₈₉	—	2	4	no	—
22	C₈×C₈₉	C₂×C₁₆	1	2	yes	—
23	C₈×C₈₉	M₃₂	1	2	yes	—
24	C₂²×C₈₉	#322	1	1	yes	characteristic
25	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
26	C₄×C₈₉	#346	1	1	yes	characteristic
27	C₄×C₈₉	#347	1	4	yes	—
28	#7053	—	89	1	no	maximal, Sylow
29	C₂×C₈₉	#1312	1	1	yes	characteristic
30	C₂×C₈₉	#1313	1	2	yes	—
31	#1312	—	89	2	no	—
32	#1311	D₈₉	1	1	yes	characteristic
33	C₈₉	#7053	1	1	yes	characteristic, complemented, Sylow
34	C₄×C₁₆	—	89	4	no	—
35	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
36	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
37	C₈²	C₂×D₈₉	1	1	yes	characteristic
38	C₂×C₁₆	—	178	4	no	—
39	C₂×C₁₆	#10623494	1	2	yes	—
40	C₂×C₁₆	#10623494	1	2	yes	—
41	C₄×C₈	C₄×D₈₉	1	2	yes	—
42	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
43	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
44	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
45	C₂×C₈	#13083557	1	4	yes	—
46	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
47	C₁₆	—	2	2	no	—
48	C₁₆	—	356	4	no	—
49	C₁₆	#13083549	1	4	yes	complemented
50	C₂×C₄	C₄×#10623494	1	1	yes	characteristic
51	C₂×C₄	#15667922	1	2	yes	—
52	C₈	—	2	4	no	—
53	C₈	C₂×#13083549	1	2	yes	—
54	C₈	#15667952	1	2	yes	—
55	C₂²	#19151319	1	1	yes	characteristic
56	C₄	C₄×#13083549	1	1	yes	characteristic
57	C₄	#19151374	1	1	yes	characteristic
58	C₄	#19151375	1	4	yes	—
59	C₂	#26018170	1	1	yes	characteristic
60	C₂	#26018172	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₈₉
commutator	C₄×C₈₉	C₄×C₁₆
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1311	C₂
socle, prime radical	C₂²×C₈₉	#322
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#7053
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #1311

Complements

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

N	H ≅ G/N	type
18. C₁₆×C₈₉	C₁₆	semidirect
33. C₈₉	#7053	semidirect
49. C₁₆	#13083549	semidirect

Maximal subgroups

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

subgroup	index
#26018170	2
C₈₉×#1311	2
#7053	89