Subgroups — #37526840

Order 22976.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040014	C₂	1	2	yes	maximal
2	#26039947	C₂	1	1	yes	maximal, characteristic
3	#26039946	C₂	1	2	yes	maximal
4	C₈×Dic₃₅₉	C₂	1	1	yes	maximal, characteristic
5	#106×C₃₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₈	—	2	2	no	—
7	C₂×C₃₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
9	Dic₃₅₉⋊C₄	—	2	2	no	—
10	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₃₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₅₉	C₂²	1	2	yes	—
13	Dic₇₁₈	—	4	4	no	—
14	C₂×Dic₃₅₉	—	2	2	no	—
15	C₂×Dic₃₅₉	D₄	1	2	yes	—
16	C₃₅₉⋊C₈	Q₈	1	2	yes	complemented
17	C₂×C₄×C₃₅₉	—	2	2	no	—
18	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
19	C₈×C₃₅₉	Q₈	1	2	yes	complemented
20	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₃₅₉	—	4	2	no	—
22	C₄×C₃₅₉	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₃₅₉	C₂×Q₈	1	1	yes	characteristic
24	Dic₃₅₉	—	4	2	no	—
25	Dic₃₅₉	SD₁₆	1	4	yes	—
26	C₂×C₃₅₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₃₅₉	C₂×SD₁₆	1	2	yes	—
28	C₃₅₉	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	359	1	no	maximal, Sylow
30	C₄⋊Q₈	—	359	2	no	—
31	#106	—	359	1	no	—
32	#106	—	359	2	no	—
33	#106	D₃₅₉	1	1	yes	characteristic
34	C₄×C₈	—	359	1	no	—
35	C₂×Q₈	—	718	2	no	—
36	C₂×C₈	—	359	1	no	—
37	C₂×C₈	C₂×D₃₅₉	1	1	yes	characteristic
38	C₄⋊C₄	—	359	2	no	—
39	C₄⋊C₄	—	718	2	no	—
40	C₄⋊C₄	C₂×D₃₅₉	1	2	yes	—
41	C₄²	—	359	1	no	—
42	Q₈	—	1436	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	359	2	no	—
45	C₂×C₄	—	718	2	no	—
46	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
47	C₈	—	359	2	no	—
48	C₈	Dic₇₁₈	1	2	yes	complemented
49	C₂²	D₄×D₃₅₉	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	359	4	no	—
52	C₄	—	1436	2	no	—
53	C₄	Q₈×D₃₅₉	1	1	yes	characteristic
54	C₄	C₂×Dic₇₁₈	1	1	yes	characteristic
55	C₂	#26040014	1	1	yes	characteristic
56	C₂	SD₁₆×D₃₅₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₅₉
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	#106×C₃₅₉	C₂
socle, prime radical	C₂²×C₃₅₉	C₂×D₄
hypercenter	#106	D₃₅₉
hypocenter	C₃₅₉	#498
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
16. C₃₅₉⋊C₈	Q₈	semidirect
19. C₈×C₃₅₉	Q₈	semidirect
28. C₃₅₉	#498	semidirect
48. C₈	Dic₇₁₈	semidirect

Maximal subgroups

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

subgroup	index
#26040014	2
#26039947	2
#26039946	2
C₈×Dic₃₅₉	2
#106×C₃₅₉	2
#498	359