Subgroups — #37526555

Order 22976.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₃₅₉	C₂	1	2	yes	maximal, complemented
2	#26039947	C₂	1	1	yes	maximal, characteristic, complemented
3	#26039936	C₂	1	2	yes	maximal, complemented
4	#26039925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₉	—	2	2	no	—
7	C₂×C₃₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
9	C₂²⋊Dic₃₅₉	—	2	2	no	—
10	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₃₅₉	C₂²	1	2	yes	—
12	D₈×C₃₅₉	C₄	1	4	yes	complemented
13	C₂×C₈×C₃₅₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₉	—	2	2	no	—
15	C₂×Dic₃₅₉	—	4	4	no	—
16	C₂×Dic₃₅₉	D₄	1	2	yes	—
17	C₃₅₉⋊C₈	—	2	1	no	—
18	C₂³×C₃₅₉	—	2	2	no	—
19	D₄×C₃₅₉	—	2	2	no	—
20	D₄×C₃₅₉	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
22	C₈×C₃₅₉	C₂×C₄	1	2	yes	complemented
23	C₂²×C₃₅₉	—	2	4	no	—
24	C₂²×C₃₅₉	—	4	2	no	—
25	C₂²×C₃₅₉	—	4	2	no	—
26	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
27	C₄×C₃₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₃₅₉	#41	1	1	yes	characteristic
29	Dic₃₅₉	—	2	2	no	—
30	Dic₃₅₉	—	4	2	no	—
31	C₂×C₃₅₉	—	4	4	no	—
32	C₂×C₃₅₉	C₄×D₄	1	1	yes	characteristic
33	C₂×C₃₅₉	AGL(1,ℤ/8)	1	2	yes	—
34	C₃₅₉	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	359	1	no	maximal, Sylow
36	C₂×D₈	D₃₅₉	1	1	yes	characteristic
37	C₄×D₄	—	359	2	no	—
38	#106	—	359	1	no	—
39	#102	—	359	2	no	—
40	#97	—	359	1	no	—
41	C₂×D₄	C₂×D₃₅₉	1	2	yes	—
42	C₂²×C₄	—	718	2	no	—
43	D₈	Dic₃₅₉	1	4	yes	complemented
44	C₂×C₈	—	359	1	no	—
45	C₂×C₈	C₂×D₃₅₉	1	1	yes	characteristic
46	C₄⋊C₄	—	359	2	no	—
47	C₂²⋊C₄	—	718	2	no	—
48	C₄²	—	359	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₃₅₉	1	4	yes	complemented
52	C₂×C₄	—	359	2	no	—
53	C₂×C₄	—	1436	4	no	—
54	C₂×C₄	—	718	2	no	—
55	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
56	C₈	—	718	1	no	—
57	C₈	C₂×Dic₃₅₉	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₃₅₉	1	1	yes	characteristic
62	C₄	—	1436	2	no	—
63	C₄	—	718	2	no	—
64	C₄	C₂²×Dic₃₅₉	1	1	yes	characteristic
65	C₄	#19153340	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	D₄×Dic₃₅₉	1	1	yes	characteristic
68	C₂	#26040039	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₅₉
commutator	C₄×C₃₅₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting, prime radical	C₂×D₈×C₃₅₉	C₂
socle	C₂²×C₃₅₉	C₂×D₄
cosocle	C₂×C₄×C₃₅₉	C₂³
hypercenter	C₂×D₈	D₃₅₉
hypocenter	C₃₅₉	#442
LCS	G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₃₅₉	C₂	semidirect
2. #26039947	C₂	semidirect
3. #26039936	C₂	semidirect
4. #26039925	C₂	semidirect
12. D₈×C₃₅₉	C₄	semidirect
20. D₄×C₃₅₉	C₂×C₄	semidirect
22. C₈×C₃₅₉	C₂×C₄	semidirect
34. C₃₅₉	#442	semidirect
43. D₈	Dic₃₅₉	semidirect
51. D₄	C₂×Dic₃₅₉	semidirect
57. C₈	C₂×Dic₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₃₅₉	2
#26039947	2
#26039936	2
#26039925	2
C₂×D₈×C₃₅₉	2
#442	359