Subgroups — #36442320

Order 22784.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#26018463	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1425	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₁₂	—	2	1	no	—
5	Dic₁₄₂₄	—	2	1	no	—
6	Dic₁₄₂₄	C₄	1	2	yes	—
7	#19151317	—	2	1	no	—
8	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
9	#368×C₈₉	—	2	1	no	—
10	C₂×Dic₃₅₆	—	4	1	no	—
11	Dic₇₁₂	—	2	2	no	—
12	Dic₇₁₂	—	4	1	no	—
13	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
14	#108×C₈₉	—	4	1	no	—
15	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₈₉	D₄	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	8	1	no	—
18	Dic₃₅₆	—	4	2	no	—
19	Dic₃₅₆	—	8	1	no	—
20	M₁₆×C₈₉	—	8	1	no	—
21	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
22	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₈₉	SD₁₆	1	1	yes	characteristic
24	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
25	C₂×Dic₈₉	—	16	1	no	—
26	Dic₁₇₈	—	8	2	no	—
27	Dic₁₇₈	—	16	1	no	—
28	C₈×C₈₉	—	16	1	no	—
29	C₈×C₈₉	#102	1	1	yes	characteristic
30	C₈×C₈₉	SD₃₂	1	1	yes	characteristic
31	C₂²×C₈₉	SD₆₄	1	1	yes	characteristic
32	C₄×C₈₉	#357	1	1	yes	characteristic
33	C₄×C₈₉	D₃₂	1	1	yes	characteristic
34	Dic₈₉	—	16	2	no	—
35	#7542	—	89	1	no	maximal, Sylow
36	C₂×C₈₉	—	2	1	no	—
37	C₂×C₈₉	#1415	1	1	yes	characteristic
38	C₂×Q₆₄	—	89	1	no	—
39	M₁₂₈	—	89	1	no	—
40	#1425	D₈₉	1	1	yes	characteristic
41	C₈₉	#7542	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₃₂	—	178	1	no	—
43	Q₆₄	—	89	2	no	—
44	Q₆₄	—	178	1	no	—
45	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
46	#368	—	2	1	no	—
47	C₆₄	—	178	1	no	—
48	C₂×Q₁₆	—	356	1	no	—
49	Q₃₂	—	356	1	no	—
50	Q₃₂	—	178	2	no	—
51	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
52	#108	—	4	1	no	—
53	C₃₂	C₄×D₈₉	1	1	yes	characteristic
54	C₃₂	D₃₅₆	1	1	yes	characteristic
55	C₂×Q₈	—	712	1	no	—
56	Q₁₆	—	356	2	no	—
57	Q₁₆	—	712	1	no	—
58	M₁₆	—	8	1	no	—
59	C₂×C₈	#13083559	1	1	yes	characteristic
60	C₁₆	#13083551	1	1	yes	characteristic
61	C₁₆	#13083561	1	1	yes	characteristic
62	Q₈	—	1424	1	no	—
63	Q₈	—	712	2	no	—
64	C₂×C₄	—	1424	1	no	—
65	C₂×C₄	#15667953	1	1	yes	characteristic
66	C₈	—	16	1	no	—
67	C₈	#15667932	1	1	yes	characteristic
68	C₈	#15667955	1	1	yes	characteristic
69	C₂²	#19151435	1	1	yes	characteristic
70	C₄	—	1424	2	no	—
71	C₄	#19151398	1	1	yes	characteristic
72	C₄	#19151433	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#26018429	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26018429
commutator	C₃₂×C₈₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₈₉×#1425	C₂
socle	C₂×C₈₉	#1415
prime radical	C₂²×C₈₉	SD₆₄
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	#1425	D₈₉
hypocenter	C₈₉	#7542
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1425

Complements

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

N	H ≅ G/N	type
41. C₈₉	#7542	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₄₂₄	2
#26018463	2
C₈₉×#1425	2
#7542	89