Subgroups — #37526848

Order 22976.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040015	C₂	1	2	yes	maximal
2	#26039949	C₂	1	4	yes	maximal
3	C₄×C₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₁₈	—	2	2	no	—
5	C₄⋊Dic₃₅₉	—	2	2	no	—
6	C₄⋊Dic₃₅₉	C₂²	1	4	yes	—
7	C₂×C₈×C₃₅₉	C₂²	1	2	yes	—
8	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
9	Dic₇₁₈	—	4	4	no	—
10	C₂×Dic₃₅₉	—	2	4	no	—
11	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
12	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
13	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
14	C₈×C₃₅₉	Q₈	1	4	yes	complemented
15	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
16	C₄×C₃₅₉	D₈	1	2	yes	—
17	C₄×C₃₅₉	Q₁₆	1	2	yes	—
18	C₄×C₃₅₉	C₂×Q₈	1	2	yes	—
19	Dic₃₅₉	—	4	4	no	—
20	C₂×C₃₅₉	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₃₅₉	C₂×D₈	1	1	yes	characteristic
22	C₂×C₃₅₉	C₂×Q₁₆	1	1	yes	characteristic
23	C₃₅₉	#500	1	1	yes	characteristic, complemented, Sylow
24	#500	—	359	1	no	maximal, Sylow
25	C₄⋊Q₈	—	359	2	no	—
26	#107	—	359	4	no	—
27	C₄×C₈	D₃₅₉	1	1	yes	characteristic
28	C₂×Q₈	—	718	2	no	—
29	C₂×C₈	C₂×D₃₅₉	1	2	yes	—
30	C₄⋊C₄	—	359	4	no	—
31	C₄⋊C₄	—	718	2	no	—
32	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
33	Q₈	—	1436	4	no	—
34	C₂×C₄	—	718	4	no	—
35	C₂×C₄	D₁₄₃₆	1	1	yes	characteristic
36	C₂×C₄	D₁₄₃₆	1	1	yes	characteristic
37	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
38	C₈	Dic₇₁₈	1	4	yes	complemented
39	C₂²	C₂×D₁₄₃₆	1	1	yes	characteristic
40	C₄	—	1436	4	no	—
41	C₄	D₂₈₇₂	1	2	yes	—
42	C₄	Dic₁₄₃₆	1	2	yes	—
43	C₄	C₂×Dic₇₁₈	1	2	yes	—
44	C₂	#26040015	1	1	yes	characteristic
45	C₂	C₂×D₂₈₇₂	1	1	yes	characteristic
46	C₂	C₂×Dic₁₄₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₄₃₆
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	C₄×C₈×C₃₅₉	C₂
socle, prime radical	C₂²×C₃₅₉	C₂×D₄
hypercenter	C₄×C₈	D₃₅₉
hypocenter	C₃₅₉	#500
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₈×C₃₅₉	Q₈	semidirect
23. C₃₅₉	#500	semidirect
38. C₈	Dic₇₁₈	semidirect

Maximal subgroups

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

subgroup	index
#26040015	2
#26039949	2
C₄×C₈×C₃₅₉	2
#500	359