Subgroups — #26018327

Order 11392.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151419	C₂	1	1	yes	maximal, characteristic
2	C₄×#13083549	C₂	1	1	yes	maximal, characteristic
3	#364×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#15667952	—	2	1	no	—
5	C₂×#13083549	C₄	1	1	yes	characteristic
6	C₂×#13083549	C₄	1	1	yes	characteristic
7	M₃₂×C₈₉	—	2	1	no	—
8	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	#13083549	—	2	1	no	—
10	#13083549	—	4	1	no	—
11	#13083549	C₈	1	2	yes	—
12	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
14	C₄²×C₈₉	Q₈	1	1	yes	characteristic
15	C₁₆×C₈₉	—	4	1	no	—
16	C₂×C₄×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
19	C₈×C₈₉	M₁₆	1	1	yes	characteristic
20	C₈×C₈₉	D₈	1	1	yes	characteristic
21	C₈×C₈₉	Q₁₆	1	1	yes	characteristic
22	C₂²×C₈₉	#108	1	1	yes	characteristic
23	C₄×C₈₉	—	2	2	no	—
24	C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₈₉	#107	1	1	yes	characteristic
26	C₂×C₈₉	—	2	1	no	—
27	C₂×C₈₉	#335	1	1	yes	characteristic
28	#1373	—	89	1	no	maximal, Sylow
29	C₈₉	#1373	1	1	yes	characteristic, complemented, Sylow
30	#364	—	89	1	no	—
31	#364	D₈₉	1	1	yes	characteristic
32	C₄×C₁₆	—	89	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	178	1	no	—
35	C₂×C₁₆	—	89	1	no	—
36	C₂×C₁₆	—	89	1	no	—
37	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
38	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
40	C₄²	Dic₁₇₈	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	89	2	no	—
43	C₁₆	—	356	1	no	—
44	C₁₆	—	178	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	#13083553	1	1	yes	characteristic
47	C₈	C₈×D₈₉	1	1	yes	characteristic
48	C₈	#13083557	1	1	yes	characteristic
49	C₈	#13083559	1	1	yes	characteristic
50	C₈	#13083564	1	1	yes	characteristic
51	C₂²	#15667947	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	#15667941	1	1	yes	characteristic
54	C₄	#15667945	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#19151350	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13083559
commutator	C₈×C₈₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₈₉
Fitting	#364×C₈₉	C₂
socle	C₂×C₈₉	#335
prime radical	C₂²×C₈₉	#108
cosocle	C₄×C₈×C₈₉	C₂²
hypercenter	#364	D₈₉
hypocenter	C₈₉	#1373
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
29. C₈₉	#1373	semidirect

Maximal subgroups

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

subgroup	index
#19151419	2
C₄×#13083549	2
#364×C₈₉	2
#1373	89