Subgroups — #36442102

Order 22784.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018450	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1422	C₂	1	1	yes	maximal, characteristic
4	#19151432	—	2	1	no	—
5	C₄×#13083549	C₄	1	1	yes	characteristic
6	C₈×#10623494	C₄	1	1	yes	characteristic
7	M₆₄×C₈₉	—	2	1	no	—
8	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083549	C₈	1	1	yes	characteristic
10	C₂×#13083549	C₈	1	1	yes	characteristic
11	C₄×#10623494	—	2	1	no	—
12	#15667918	—	4	1	no	—
13	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₈₉	—	4	1	no	—
17	C₂×#10623494	—	2	2	no	—
18	#13083549	—	2	1	no	—
19	#13083549	C₁₆	1	2	yes	—
20	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
22	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
23	C₁₆×C₈₉	—	2	1	no	—
24	C₁₆×C₈₉	D₈	1	1	yes	characteristic
25	C₁₆×C₈₉	Q₁₆	1	1	yes	characteristic
26	C₂×C₄×C₈₉	—	2	1	no	—
27	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₈₉	#107	1	1	yes	characteristic
29	C₈×C₈₉	#108	1	1	yes	characteristic
30	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₈₉	M₃₂	1	1	yes	characteristic
32	#10623494	—	2	4	no	—
33	C₂²×C₈₉	#364	1	1	yes	characteristic
34	C₄×C₈₉	—	2	2	no	—
35	C₄×C₈₉	#335	1	1	yes	characteristic
36	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
37	#7459	—	89	1	no	maximal, Sylow
38	C₂×C₈₉	—	2	1	no	—
39	C₂×C₈₉	#1370	1	1	yes	characteristic
40	#1422	—	89	1	no	—
41	#1422	D₈₉	1	1	yes	characteristic
42	C₈×C₁₆	—	89	1	no	—
43	C₈₉	#7459	1	1	yes	characteristic, complemented, Sylow
44	M₆₄	—	2	1	no	—
45	M₆₄	—	178	1	no	—
46	C₄×C₁₆	—	89	1	no	—
47	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
48	C₈²	—	89	1	no	—
49	C₂×C₁₆	—	89	1	no	—
50	C₂×C₁₆	—	89	1	no	—
51	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
52	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
53	C₄×C₈	—	178	1	no	—
54	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
55	C₃₂	—	4	1	no	—
56	C₃₂	—	356	1	no	—
57	C₂×C₈	—	178	2	no	—
58	C₂×C₈	#13083553	1	1	yes	characteristic
59	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
60	C₄²	#13083557	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	89	2	no	—
63	C₁₆	—	178	1	no	—
64	C₁₆	#13083559	1	1	yes	characteristic
65	C₁₆	#13083564	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#15667941	1	1	yes	characteristic
68	C₈	—	178	4	no	—
69	C₈	#15667945	1	1	yes	characteristic
70	C₈	#15667947	1	1	yes	characteristic
71	C₈	C₁₆×D₈₉	1	1	yes	characteristic
72	C₈	#15667951	1	1	yes	characteristic
73	C₂²	#19151418	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#19151350	1	1	yes	characteristic
76	C₄	#19151416	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#26018321	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#13083559
commutator	C₈×C₈₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×#1422	C₂
socle	C₂×C₈₉	#1370
prime radical	C₂²×C₈₉	#364
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	#1422	D₈₉
hypocenter	C₈₉	#7459
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
43. C₈₉	#7459	semidirect

Maximal subgroups

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

subgroup	index
#26018450	2
C₁₆×#10623494	2
C₈₉×#1422	2
#7459	89