Subgroups — C₈₉×#7475 #36385323

Order 22784.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2158	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
4	#427×C₈₉	—	2	2	no	—
5	M₆₄×C₈₉	—	2	2	no	—
6	C₂×C₃₂×C₈₉	C₄	1	2	yes	complemented
7	#367×C₈₉	C₄	1	4	yes	complemented
8	#366×C₈₉	—	2	2	no	—
9	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
10	#117×C₈₉	—	4	2	no	—
11	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
14	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
15	#107×C₈₉	—	2	4	no	—
16	#106×C₈₉	—	4	2	no	—
17	C₃₂×C₈₉	—	2	2	no	—
18	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
19	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
22	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
23	C₄⋊C₄×C₈₉	—	4	4	no	—
24	C₂²⋊C₄×C₈₉	—	8	2	no	—
25	C₄²×C₈₉	—	8	2	no	—
26	C₁₆×C₈₉	C₄²	1	1	yes	characteristic
27	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
29	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₈₉	—	8	4	no	—
31	C₂×C₄×C₈₉	#102	1	2	yes	—
32	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
33	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
34	C₂×C₄×C₈₉	Q₃₂	1	1	yes	characteristic
35	C₈×C₈₉	#95	1	1	yes	characteristic
36	C₈×C₈₉	#103	1	2	yes	—
37	C₈×C₈₉	#106	1	1	yes	characteristic
38	C₂²×C₈₉	—	4	1	no	—
39	C₂²×C₈₉	#357	1	2	yes	—
40	C₂²×C₈₉	#366	1	1	yes	characteristic
41	C₄×C₈₉	—	16	4	no	—
42	C₄×C₈₉	#340	1	1	yes	characteristic
43	C₄×C₈₉	#358	1	2	yes	—
44	C₄×C₈₉	#367	1	1	yes	characteristic
45	#7475	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₈₉	—	2	2	no	—
47	C₂×C₈₉	—	2	1	no	—
48	C₂×C₈₉	#1380	1	1	yes	characteristic
49	C₂×M₆₄	C₂×C₈₉	1	1	yes	characteristic
50	#2158	C₂×C₈₉	1	2	yes	—
51	C₈₉	#7475	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
53	#427	—	2	2	no	—
54	M₆₄	—	2	2	no	—
55	C₂×C₃₂	C₄×C₈₉	1	2	yes	complemented
56	#367	C₄×C₈₉	1	4	yes	complemented
57	#366	—	2	2	no	—
58	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
59	#117	—	4	2	no	—
60	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
61	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
62	C₂×C₁₆	D₄×C₈₉	1	2	yes	—
63	C₂×C₁₆	Q₈×C₈₉	1	1	yes	characteristic
64	#107	—	2	4	no	—
65	#106	—	4	2	no	—
66	C₃₂	—	2	2	no	—
67	C₂²×C₄	D₈×C₈₉	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
69	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
70	C₂×C₈	SD₁₆×C₈₉	1	2	yes	—
71	C₂×C₈	Q₁₆×C₈₉	1	1	yes	characteristic
72	C₄⋊C₄	—	4	4	no	—
73	C₂²⋊C₄	—	8	2	no	—
74	C₄²	—	8	2	no	—
75	C₁₆	C₄²×C₈₉	1	1	yes	characteristic
76	C₁₆	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
77	C₁₆	C₄⋊C₄×C₈₉	1	2	yes	—
78	C₂³	D₁₆×C₈₉	1	1	yes	characteristic
79	C₂×C₄	—	8	4	no	—
80	C₂×C₄	#102×C₈₉	1	2	yes	—
81	C₂×C₄	#107×C₈₉	1	1	yes	characteristic
82	C₂×C₄	SD₃₂×C₈₉	1	2	yes	—
83	C₂×C₄	Q₃₂×C₈₉	1	1	yes	characteristic
84	C₈	#95×C₈₉	1	1	yes	characteristic
85	C₈	#103×C₈₉	1	2	yes	—
86	C₈	#106×C₈₉	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	#357×C₈₉	1	2	yes	—
89	C₂²	#366×C₈₉	1	1	yes	characteristic
90	C₄	—	16	4	no	—
91	C₄	#340×C₈₉	1	1	yes	characteristic
92	C₄	#358×C₈₉	1	2	yes	—
93	C₄	#367×C₈₉	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	C₈₉×#1380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	#367
commutator	C₁₆	C₄²×C₈₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₈₉
socle	C₂×C₈₉	#1380
prime radical	C₂³×C₈₉	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₈×C₈₉ ◁ C₂²×C₁₆×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₃₂×C₈₉	C₄	semidirect
7. #367×C₈₉	C₄	semidirect
45. #7475	C₈₉	direct (N × H)
51. C₈₉	#7475	direct (N × H)
55. C₂×C₃₂	C₄×C₈₉	semidirect
56. #367	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄×C₈₉	2
C₈₉×#2158	2
#7475	89

Subgroups — C89×#7475 #36385323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7475 #36385323