Subgroups — #36448162

Order 22784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018124	C₂	1	3	yes	maximal
2	#26018117	C₂	1	3	yes	maximal
3	C₈₉×#1455	C₂	1	1	yes	maximal, characteristic
4	#19151655	C₂²	1	3	yes	—
5	#19151466	C₂²	1	1	yes	characteristic
6	C₄×M₁₆×C₈₉	C₂²	1	3	yes	—
7	#322×C₈₉	C₄	1	3	yes	complemented
8	C₈²×C₈₉	C₄	1	1	yes	characteristic, complemented
9	C₄⋊#10623494	—	2	3	no	—
10	C₂²⋊#10623494	—	2	3	no	—
11	#15667923	—	2	3	no	—
12	#15667919	—	2	3	no	—
13	C₂×M₁₆×C₈₉	—	2	3	no	—
14	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
15	#97×C₈₉	—	2	3	no	—
16	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
17	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
18	C₂²×Dic₈₉	—	4	1	no	—
19	C₂×#10623494	—	4	3	no	—
20	C₂²×C₄×C₈₉	#41	1	3	yes	—
21	M₁₆×C₈₉	—	4	6	no	—
22	C₂×C₈×C₈₉	—	2	3	no	—
23	C₂×C₈×C₈₉	—	2	3	no	—
24	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
25	C₄²×C₈₉	#41	1	3	yes	—
26	C₂³×C₈₉	#126	1	1	yes	characteristic
27	C₂×C₄×C₈₉	—	4	3	no	—
28	C₂×C₄×C₈₉	#117	1	3	yes	—
29	C₂×C₄×C₈₉	#126	1	3	yes	—
30	C₂×Dic₈₉	—	8	3	no	—
31	C₈×C₈₉	—	2	6	no	—
32	C₈×C₈₉	—	4	3	no	—
33	#10623494	—	8	3	no	—
34	C₂²×C₈₉	—	2	3	no	—
35	C₂²×C₈₉	#383	1	1	yes	characteristic
36	C₄×C₈₉	—	2	3	no	—
37	C₄×C₈₉	—	2	3	no	—
38	Dic₈₉	—	16	1	no	—
39	#8804	—	89	1	no	maximal, Sylow
40	C₂×C₈₉	—	4	1	no	—
41	C₂×C₈₉	#1841	1	3	yes	—
42	#1455	D₈₉	1	1	yes	characteristic
43	#1291	—	89	3	no	—
44	#1288	—	89	3	no	—
45	C₈₉	#8804	1	1	yes	characteristic, complemented, Sylow
46	#432	—	89	3	no	—
47	C₄×M₁₆	C₂×D₈₉	1	3	yes	—
48	#383	—	89	1	no	—
49	#322	Dic₈₉	1	3	yes	complemented
50	C₈²	Dic₈₉	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
53	C₄⋊C₈	—	178	3	no	—
54	C₂²⋊C₈	—	178	3	no	—
55	#97	—	2	3	no	—
56	#97	—	178	3	no	—
57	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
58	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
59	#95	—	178	3	no	—
60	C₂²×C₄	—	356	1	no	—
61	C₂²×C₄	#13083574	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	356	3	no	—
66	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
67	C₄²	#13083574	1	3	yes	—
68	C₂³	#15668006	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	712	3	no	—
71	C₂×C₄	#15667971	1	3	yes	—
72	C₂×C₄	#15668006	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	712	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#19151466	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	1424	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#26020401	1	3	yes	—

	subgroup	G/H
center	C₂²	#19151466
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1455	C₂
socle	C₂²×C₈₉	#383
prime radical	C₂³×C₈₉	#126
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1455	D₈₉
hypocenter	C₈₉	#8804
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
7. #322×C₈₉	C₄	semidirect
8. C₈²×C₈₉	C₄	semidirect
45. C₈₉	#8804	semidirect
49. #322	Dic₈₉	semidirect
50. C₈²	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26018124	2
#26018117	2
C₈₉×#1455	2
#8804	89