Subgroups — #37527180

Order 22976.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040009	C₂	1	2	yes	maximal, complemented
2	#26039996	C₂	1	3	yes	maximal, complemented
3	#26039987	C₂	1	6	yes	maximal, complemented
4	#26039983	C₂	1	3	yes	maximal, complemented
5	#127×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₅₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₃₅₉	C₂²	1	6	yes	—
8	C₂²×Dic₃₅₉	C₂²	1	3	yes	complemented
9	Dic₃₅₉⋊C₄	C₂²	1	6	yes	complemented
10	C₂²⋊Dic₃₅₉	C₂²	1	3	yes	—
11	C₂²⋊Dic₃₅₉	C₂²	1	3	yes	complemented
12	(C₂×D₃₅₉)⋊C₄	C₂²	1	6	yes	—
13	C₂×D₄×C₃₅₉	C₂²	1	3	yes	—
14	C₂×D₄×C₃₅₉	C₂²	1	3	yes	complemented
15	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₃₅₉	—	4	3	no	—
17	C₂²×D₃₅₉	C₂³	1	2	yes	—
18	C₂²⋊D₃₅₉	—	4	6	no	—
19	C₂×Dic₃₅₉	—	4	3	no	—
20	C₂×Dic₃₅₉	C₂³	1	6	yes	—
21	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
22	C₂³×C₃₅₉	C₂³	1	3	yes	—
23	D₄×C₃₅₉	—	4	3	no	—
24	D₄×C₃₅₉	—	4	3	no	—
25	C₂×C₄×C₃₅₉	C₂³	1	3	yes	—
26	C₂²×C₃₅₉	—	2	3	no	—
27	C₂²×C₃₅₉	—	2	6	no	—
28	C₂²×C₃₅₉	—	2	3	no	—
29	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
30	C₂×D₃₅₉	—	2	6	no	—
31	C₂×D₃₅₉	—	4	4	no	—
32	C₄×C₃₅₉	—	2	3	no	—
33	Dic₃₅₉	—	2	6	no	—
34	C₂×C₃₅₉	—	4	1	no	—
35	C₂×C₃₅₉	—	4	3	no	—
36	C₂×C₃₅₉	H(5,𝔽₂)	1	3	yes	—
37	D₃₅₉	—	4	2	no	—
38	C₃₅₉	#560	1	1	yes	characteristic, complemented, Sylow
39	#560	—	359	1	no	maximal, Sylow
40	#127	D₃₅₉	1	1	yes	characteristic, complemented
41	#126	—	359	2	no	—
42	#123	—	359	3	no	—
43	#121	—	359	6	no	—
44	#120	—	359	3	no	—
45	C₂⁴	—	359	1	no	—
46	C₂×D₄	—	359	6	no	—
47	C₂×D₄	C₂×D₃₅₉	1	3	yes	—
48	C₂×D₄	C₂×D₃₅₉	1	3	yes	complemented
49	C₂²×C₄	—	359	3	no	—
50	C₄⋊C₄	—	359	6	no	—
51	C₂²⋊C₄	—	359	3	no	—
52	C₂²⋊C₄	—	359	3	no	—
53	C₂²⋊C₄	—	359	6	no	—
54	C₄²	C₂×D₃₅₉	1	1	yes	characteristic, complemented
55	C₂³	—	359	2	no	—
56	C₂³	—	1436	3	no	—
57	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
58	C₂³	C₂²×D₃₅₉	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	1436	6	no	—
62	C₂×C₄	—	359	6	no	—
63	C₂×C₄	—	1436	3	no	—
64	C₂×C₄	C₂²×D₃₅₉	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	1436	4	no	—
69	C₂²	—	718	6	no	—
70	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	718	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	1436	2	no	—
76	C₂	#26040060	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	#127×C₃₅₉	C₂
hypercenter	#127	D₃₅₉
hypocenter	C₃₅₉	#560
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #26040009	C₂	semidirect
2. #26039996	C₂	semidirect
3. #26039987	C₂	semidirect
4. #26039983	C₂	semidirect
5. #127×C₃₅₉	C₂	semidirect
8. C₂²×Dic₃₅₉	C₂²	semidirect
9. Dic₃₅₉⋊C₄	C₂²	semidirect
11. C₂²⋊Dic₃₅₉	C₂²	semidirect
14. C₂×D₄×C₃₅₉	C₂²	semidirect
15. C₄²×C₃₅₉	C₂²	semidirect
38. C₃₅₉	#560	semidirect
40. #127	D₃₅₉	semidirect
48. C₂×D₄	C₂×D₃₅₉	semidirect
54. C₄²	C₂×D₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
#26040009	2
#26039996	2
#26039987	2
#26039983	2
#127×C₃₅₉	2
#560	359