Subgroups — #36442108

Order 22784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667918	C₂	1	1	yes	maximal, characteristic
2	#26018328	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667918	C₄	1	1	yes	characteristic
5	C₂×#15667918	C₄	1	1	yes	characteristic
6	#19151419	—	2	1	no	—
7	#364×C₈₉	—	2	1	no	—
8	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	#15667952	—	4	1	no	—
10	#15667918	—	2	1	no	—
11	#15667918	C₈	1	2	yes	—
12	M₃₂×C₈₉	—	4	1	no	—
13	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
16	#13083549	—	8	1	no	—
17	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
19	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
20	C₁₆×C₈₉	—	2	1	no	—
21	C₁₆×C₈₉	—	8	1	no	—
22	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
26	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
27	C₈×C₈₉	#108	1	1	yes	characteristic
28	C₈×C₈₉	D₁₆	1	1	yes	characteristic
29	C₈×C₈₉	Q₃₂	1	1	yes	characteristic
30	C₂²×C₈₉	#368	1	1	yes	characteristic
31	C₄×C₈₉	—	2	2	no	—
32	C₄×C₈₉	#335	1	1	yes	characteristic
33	C₄×C₈₉	#366	1	1	yes	characteristic
34	#7462	—	89	1	no	maximal, Sylow
35	C₂×C₈₉	—	2	1	no	—
36	C₂×C₈₉	#1371	1	1	yes	characteristic
37	C₄×C₃₂	—	89	1	no	—
38	#1373	—	89	1	no	—
39	#1373	D₈₉	1	1	yes	characteristic
40	C₈₉	#7462	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃₂	—	89	1	no	—
42	C₂×C₃₂	—	89	1	no	—
43	#364	—	2	1	no	—
44	#364	—	178	1	no	—
45	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
46	M₃₂	—	4	1	no	—
47	M₃₂	—	356	1	no	—
48	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
49	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
50	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
51	C₃₂	—	89	2	no	—
52	C₃₂	—	178	1	no	—
53	C₂×C₈	#13083553	1	1	yes	characteristic
54	C₂×C₈	#13083559	1	1	yes	characteristic
55	C₄²	#13083564	1	1	yes	characteristic
56	C₁₆	—	2	1	no	—
57	C₁₆	—	8	1	no	—
58	C₁₆	—	712	1	no	—
59	C₁₆	C₈×D₈₉	1	1	yes	characteristic
60	C₁₆	#13083557	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#15667945	1	1	yes	characteristic
63	C₈	#15667941	1	1	yes	characteristic
64	C₈	#15667947	1	1	yes	characteristic
65	C₈	#15667953	1	1	yes	characteristic
66	C₈	#15667958	1	1	yes	characteristic
67	C₂²	#19151427	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	#19151350	1	1	yes	characteristic
70	C₄	#19151423	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#26018323	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667953
commutator	C₁₆×C₈₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×#1373	C₂
socle	C₂×C₈₉	#1371
prime radical	C₂²×C₈₉	#368
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	#1373	D₈₉
hypocenter	C₈₉	#7462
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
40. C₈₉	#7462	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667918	2
#26018328	2
C₈₉×#1373	2
#7462	89