Subgroups — #37526196

Order 22976.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃₅₉⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₅₉⋊C₈	—	2	2	no	—
4	C₂×C₈×C₃₅₉	C₄	1	1	yes	characteristic
5	C₂×C₈×C₃₅₉	C₄	1	1	yes	characteristic
6	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
7	C₃₅₉⋊C₈	—	4	2	no	—
8	C₂×C₄×C₃₅₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
10	C₂×C₄×C₃₅₉	Q₈	1	1	yes	characteristic
11	C₈×C₃₅₉	—	2	1	no	—
12	C₈×C₃₅₉	C₈	1	2	yes	complemented
13	C₂²×C₃₅₉	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₃₅₉	—	2	1	no	—
15	C₄×C₃₅₉	C₂×C₈	1	1	yes	characteristic
16	C₄×C₃₅₉	M₁₆	1	1	yes	characteristic
17	C₄×C₃₅₉	D₈	1	1	yes	characteristic
18	C₄×C₃₅₉	Q₁₆	1	1	yes	characteristic
19	C₂×C₃₅₉	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₃₅₉	#107	1	1	yes	characteristic
21	C₂×C₃₅₉	#108	1	1	yes	characteristic
22	C₃₅₉	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	359	1	no	maximal, Sylow
24	C₄⋊C₈	—	359	2	no	—
25	C₄×C₈	D₃₅₉	1	1	yes	characteristic
26	C₂×C₈	—	718	2	no	—
27	C₂×C₈	Dic₃₅₉	1	1	yes	characteristic
28	C₂×C₈	Dic₃₅₉	1	1	yes	characteristic
29	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
30	C₂×C₄	C₂×Dic₃₅₉	1	1	yes	characteristic
31	C₂×C₄	D₁₄₃₆	1	1	yes	characteristic
32	C₂×C₄	Dic₇₁₈	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	1436	2	no	—
35	C₈	C₃₅₉⋊C₈	1	2	yes	complemented
36	C₂²	C₄⋊Dic₃₅₉	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	C₂×C₃₅₉⋊C₈	1	1	yes	characteristic
39	C₄	#19153325	1	1	yes	characteristic
40	C₄	D₂₈₇₂	1	1	yes	characteristic
41	C₄	Dic₁₄₃₆	1	1	yes	characteristic
42	C₂	C₄⋊(C₃₅₉⋊C₈)	1	1	yes	characteristic
43	C₂	#26039949	1	1	yes	characteristic
44	C₂	#26039951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₄₃₆
commutator	C₄×C₃₅₉	C₂×C₈
Frattini	C₄²	C₂×D₃₅₉
Fitting	C₄×C₈×C₃₅₉	C₂
socle, prime radical	C₂²×C₃₅₉	C₄⋊C₄
cosocle	C₄²×C₃₅₉	C₂²
hypercenter	C₄×C₈	D₃₅₉
hypocenter	C₃₅₉	#335
LCS	G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₃₅₉	C₈	semidirect
22. C₃₅₉	#335	semidirect
35. C₈	C₃₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃₅₉⋊C₈)	2
C₄×C₈×C₃₅₉	2
#335	359