Subgroups — #36469172

Order 22784.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022534	C₂	1	4	yes	maximal
2	#26019870	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19152003	—	2	2	no	—
5	#19152003	C₂²	1	4	yes	—
6	#19151978	—	2	2	no	—
7	#19151426	—	2	8	no	—
8	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
9	C₈²×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₅₆	—	4	4	no	—
11	#15668012	—	2	4	no	—
12	#15667946	—	2	8	no	—
13	#15667946	—	4	4	no	—
14	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
15	C₄×C₈×C₈₉	D₄	1	2	yes	—
16	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	4	4	no	—
18	Dic₃₅₆	—	8	8	no	—
19	C₄⋊Dic₈₉	—	4	4	no	—
20	C₄⋊Dic₈₉	—	4	4	no	—
21	C₂×C₈×C₈₉	D₈	1	4	yes	—
22	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
23	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
24	C₁₆×C₈₉	Q₁₆	1	8	yes	complemented
25	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
26	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
27	C₂×Dic₈₉	—	8	4	no	—
28	Dic₁₇₈	—	8	8	no	—
29	C₈×C₈₉	SD₃₂	1	8	yes	—
30	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
31	C₂²×C₈₉	#493	1	1	yes	characteristic
32	C₄×C₈₉	#494	1	2	yes	—
33	C₄×C₈₉	C₂×SD₃₂	1	4	yes	—
34	Dic₈₉	—	16	4	no	—
35	#12523	—	89	1	no	maximal, Sylow
36	C₂×C₈₉	#1713	1	1	yes	characteristic
37	C₂×C₈₉	#2248	1	2	yes	—
38	#2254	—	89	4	no	—
39	#1713	—	89	2	no	—
40	C₈×C₁₆	D₈₉	1	1	yes	characteristic
41	C₈₉	#12523	1	1	yes	characteristic, complemented, Sylow
42	#500	—	89	4	no	—
43	#500	—	178	2	no	—
44	#494	—	178	2	no	—
45	#367	—	178	8	no	—
46	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
47	C₈²	C₂×D₈₉	1	1	yes	characteristic
48	C₂×Q₁₆	—	356	4	no	—
49	C₄⋊Q₈	—	178	4	no	—
50	C₂×C₁₆	D₃₅₆	1	4	yes	—
51	#107	—	356	4	no	—
52	#107	—	178	8	no	—
53	C₄×C₈	D₃₅₆	1	2	yes	—
54	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
55	C₂×Q₈	—	356	4	no	—
56	Q₁₆	—	712	8	no	—
57	C₂×C₈	D₇₁₂	1	4	yes	—
58	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
59	C₄⋊C₄	—	356	4	no	—
60	C₄⋊C₄	—	356	4	no	—
61	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
62	C₁₆	Dic₃₅₆	1	8	yes	complemented
63	Q₈	—	712	8	no	—
64	C₂×C₄	—	712	4	no	—
65	C₂×C₄	#15668009	1	1	yes	characteristic
66	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
67	C₈	#15667957	1	8	yes	—
68	C₈	C₂×Dic₃₅₆	1	4	yes	—
69	C₂²	#19151975	1	1	yes	characteristic
70	C₄	—	1424	4	no	—
71	C₄	#19151978	1	2	yes	—
72	C₄	C₂×#15667957	1	4	yes	—
73	C₂	#26019870	1	1	yes	characteristic
74	C₂	#26022508	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#12523
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₈₉	Q₁₆	semidirect
41. C₈₉	#12523	semidirect
62. C₁₆	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26022534	2
#26019870	2
C₈×C₁₆×C₈₉	2
#12523	89