Subgroups — #37526847

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	#26039949	C₂	1	1	yes	maximal, characteristic
3	#26039948	C₂	1	2	yes	maximal
4	C₈×Dic₃₅₉	C₂	1	1	yes	maximal, characteristic
5	#107×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	1	yes	characteristic
16	C₂×Dic₃₅₉	D₄	1	1	yes	characteristic
17	C₃₅₉⋊C₈	Q₈	1	2	yes	complemented
18	C₂×C₄×C₃₅₉	—	2	2	no	—
19	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
20	C₈×C₃₅₉	Q₈	1	2	yes	complemented
21	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₃₅₉	—	4	2	no	—
23	C₄×C₃₅₉	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₃₅₉	C₂×Q₈	1	1	yes	characteristic
25	Dic₃₅₉	—	4	2	no	—
26	Dic₃₅₉	D₈	1	2	yes	—
27	Dic₃₅₉	Q₁₆	1	2	yes	—
28	C₂×C₃₅₉	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₃₅₉	C₂×D₈	1	1	yes	characteristic
30	C₂×C₃₅₉	C₂×Q₁₆	1	1	yes	characteristic
31	C₃₅₉	#500	1	1	yes	characteristic, complemented, Sylow
32	#500	—	359	1	no	maximal, Sylow
33	C₄⋊Q₈	—	359	2	no	—
34	#107	—	359	1	no	—
35	#107	—	359	2	no	—
36	#107	D₃₅₉	1	1	yes	characteristic
37	C₄×C₈	—	359	1	no	—
38	C₂×Q₈	—	718	2	no	—
39	C₂×C₈	—	359	1	no	—
40	C₂×C₈	C₂×D₃₅₉	1	1	yes	characteristic
41	C₄⋊C₄	—	359	2	no	—
42	C₄⋊C₄	—	718	2	no	—
43	C₄⋊C₄	C₂×D₃₅₉	1	2	yes	—
44	C₄²	—	359	1	no	—
45	Q₈	—	1436	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	359	1	no	—
48	C₂×C₄	—	359	1	no	—
49	C₂×C₄	—	718	2	no	—
50	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
51	C₈	—	359	2	no	—
52	C₈	Dic₇₁₈	1	2	yes	complemented
53	C₂²	D₄×D₃₅₉	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	359	2	no	—
56	C₄	—	359	2	no	—
57	C₄	—	1436	2	no	—
58	C₄	Q₈×D₃₅₉	1	1	yes	characteristic
59	C₄	C₂×Dic₇₁₈	1	1	yes	characteristic
60	C₂	#26040014	1	1	yes	characteristic
61	C₂	D₈×D₃₅₉	1	1	yes	characteristic
62	C₂	Q₁₆×D₃₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₉
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	#107×C₃₅₉	C₂
socle, prime radical	C₂²×C₃₅₉	C₂×D₄
hypercenter	#107	D₃₅₉
hypocenter	C₃₅₉	#500
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
17. C₃₅₉⋊C₈	Q₈	semidirect
20. C₈×C₃₅₉	Q₈	semidirect
31. C₃₅₉	#500	semidirect
52. 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
#26039949	2
#26039948	2
C₈×Dic₃₅₉	2
#107×C₃₅₉	2
#500	359