Subgroups — #37526807

Order 22976.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040007	C₂	1	1	yes	maximal, characteristic, complemented
2	#26039936	C₂	1	4	yes	maximal, complemented
3	#26039926	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₃₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₅₉⋊C₈	C₂²	1	2	yes	—
6	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
7	Dic₃₅₉⋊C₄	—	2	2	no	—
8	C₂×D₄×C₃₅₉	—	2	2	no	—
9	C₂×D₄×C₃₅₉	C₂²	1	2	yes	—
10	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₅₉	—	2	2	no	—
12	C₃₅₉⋊C₈	—	2	2	no	—
13	C₂³×C₃₅₉	—	2	2	no	—
14	D₄×C₃₅₉	—	2	4	no	—
15	D₄×C₃₅₉	—	4	4	no	—
16	C₂×C₄×C₃₅₉	D₄	1	2	yes	—
17	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
18	C₂²×C₃₅₉	—	4	2	no	—
19	C₂²×C₃₅₉	—	4	4	no	—
20	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₃₅₉	—	2	2	no	—
22	C₄×C₃₅₉	#41	1	2	yes	—
23	Dic₃₅₉	—	4	2	no	—
24	C₂×C₃₅₉	—	8	2	no	—
25	C₂×C₃₅₉	#124	1	1	yes	characteristic
26	C₂×C₃₅₉	AGL(1,ℤ/8)	1	2	yes	—
27	C₃₅₉	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	359	1	no	maximal, Sylow
29	#127	D₃₅₉	1	1	yes	characteristic
30	#125	—	359	1	no	—
31	#102	—	359	4	no	—
32	#97	—	359	1	no	—
33	C₂×D₄	—	2	2	no	—
34	C₂×D₄	C₂×D₃₅₉	1	2	yes	—
35	C₂×C₈	—	359	2	no	—
36	C₄⋊C₄	—	359	2	no	—
37	C₄⋊C₄	—	718	2	no	—
38	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
39	C₂³	—	2	2	no	—
40	D₄	—	2	4	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	718	2	no	—
43	C₂×C₄	C₂²⋊D₃₅₉	1	2	yes	—
44	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
45	C₈	—	718	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	4	no	—
48	C₂²	C₂×C₂²⋊D₃₅₉	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	1436	2	no	—
51	C₄	#19153340	1	2	yes	—
52	C₂	—	8	2	no	—
53	C₂	#26040002	1	1	yes	characteristic
54	C₂	#26040041	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₅₉
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting, prime radical	#127×C₃₅₉	C₂
socle	C₂²×C₃₅₉	C₂×D₄
hypercenter	#127	D₃₅₉
hypocenter	C₃₅₉	#490
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #26040007	C₂	semidirect
2. #26039936	C₂	semidirect
3. #26039926	C₂	semidirect
27. C₃₅₉	#490	semidirect

Maximal subgroups

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

subgroup	index
#26040007	2
#26039936	2
#26039926	2
#127×C₃₅₉	2
#490	359