Subgroups — #36448207

Order 22784.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020380	C₂	1	6	yes	maximal
2	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151468	C₂²	1	4	yes	—
4	#19151361	—	2	12	no	—
5	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
6	C₈²×C₈₉	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₈₉	—	2	12	no	—
8	C₂²×C₈×C₈₉	D₄	1	6	yes	—
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×Dic₈₉	—	4	4	no	—
13	C₄⋊Dic₈₉	—	4	24	no	—
14	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
15	C₂×C₈×C₈₉	—	2	12	no	—
16	C₂×C₈×C₈₉	C₄⋊C₄	1	12	yes	—
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	C₂×C₄⋊C₄	1	3	yes	—
21	C₂×C₄×C₈₉	C₄⋊Q₈	1	3	yes	—
22	C₂×C₄×C₈₉	#135	1	12	yes	—
23	C₂×Dic₈₉	—	8	12	no	—
24	C₈×C₈₉	—	2	12	no	—
25	C₂²×C₈₉	#384	1	1	yes	characteristic
26	C₂²×C₈₉	#495	1	6	yes	—
27	C₄×C₈₉	#427	1	6	yes	—
28	C₄×C₈₉	#499	1	6	yes	—
29	Dic₈₉	—	16	4	no	—
30	#8817	—	89	1	no	maximal, Sylow
31	C₂×C₈₉	#1711	1	4	yes	—
32	C₂×C₈₉	#1846	1	3	yes	—
33	#1834	—	89	6	no	—
34	C₂×C₈²	D₈₉	1	1	yes	characteristic
35	C₈₉	#8817	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
37	#384	—	89	4	no	—
38	#340	—	178	12	no	—
39	C₈²	Dic₈₉	1	4	yes	complemented
40	C₂²×C₈	D₃₅₆	1	6	yes	—
41	C₂×C₄⋊C₄	—	178	12	no	—
42	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
43	C₄×C₈	Dic₁₇₈	1	6	yes	—
44	C₄×C₈	C₂×Dic₈₉	1	6	yes	—
45	C₂²×C₄	—	356	4	no	—
46	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊Dic₈₉	1	12	yes	—
49	C₄⋊C₄	—	356	24	no	—
50	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
51	C₄²	C₂×Dic₁₇₈	1	3	yes	—
52	C₂³	#15668009	1	1	yes	characteristic
53	C₂×C₄	—	712	12	no	—
54	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
55	C₂×C₄	#15668012	1	3	yes	—
56	C₂×C₄	#15668034	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	#19151468	1	1	yes	characteristic
59	C₂²	#19151982	1	6	yes	—
60	C₄	—	1424	4	no	—
61	C₄	#19151636	1	6	yes	—
62	C₄	#19152000	1	6	yes	—
63	C₂	#26019864	1	4	yes	—
64	C₂	#26020416	1	3	yes	—

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#8817
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₈₉	C₄	semidirect
35. C₈₉	#8817	semidirect
39. C₈²	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020380	2
C₂×C₈²×C₈₉	2
#8817	89