Subgroups — #37527186

Order 22976.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040002	C₂	1	3	yes	maximal, complemented
2	#26040001	C₂	1	6	yes	maximal, complemented
3	#26039984	C₂	1	2	yes	maximal, complemented
4	#26039983	C₂	1	3	yes	maximal, complemented
5	#120×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₅₉	C₂²	1	1	yes	characteristic, complemented
7	C₂×Dic₇₁₈	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊D₃₅₉	C₂²	1	6	yes	—
9	C₂²⋊Dic₃₅₉	C₂²	1	3	yes	complemented
10	C₂²⋊Dic₃₅₉	C₂²	1	6	yes	complemented
11	(C₂×D₃₅₉)⋊C₄	C₂²	1	6	yes	complemented
12	C₄×Dic₃₅₉	C₂²	1	3	yes	complemented
13	C₂⁴×C₃₅₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₄×C₃₅₉	C₂²	1	3	yes	—
15	C₂²⋊C₄×C₃₅₉	C₂²	1	3	yes	complemented
16	C₂²×D₃₅₉	—	4	3	no	—
17	C₂²×D₃₅₉	C₂³	1	2	yes	—
18	Dic₇₁₈	—	4	3	no	—
19	C₂²⋊D₃₅₉	—	4	6	no	—
20	C₂×Dic₃₅₉	C₂³	1	6	yes	—
21	C₂³×C₃₅₉	—	4	3	no	—
22	C₂³×C₃₅₉	C₂³	1	3	yes	—
23	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
24	D₄×C₃₅₉	—	4	3	no	—
25	C₂×C₄×C₃₅₉	C₂³	1	3	yes	—
26	C₂²×C₃₅₉	—	2	3	no	—
27	C₂²×C₃₅₉	—	2	3	no	—
28	C₂²×C₃₅₉	—	2	6	no	—
29	C₂²×C₃₅₉	—	4	4	no	—
30	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₅₉	—	2	6	no	—
32	C₂×D₃₅₉	—	4	4	no	—
33	C₄×C₃₅₉	—	2	3	no	—
34	Dic₃₅₉	—	2	6	no	—
35	C₂×C₃₅₉	—	4	1	no	—
36	C₂×C₃₅₉	—	4	3	no	—
37	C₂×C₃₅₉	H(5,𝔽₂)	1	3	yes	—
38	D₃₅₉	—	4	2	no	—
39	C₃₅₉	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
40	H(3,𝔽₄)	—	359	1	no	maximal, Sylow
41	#124	—	359	3	no	—
42	#124	—	359	6	no	—
43	#120	—	359	2	no	—
44	#120	—	359	3	no	—
45	#120	D₃₅₉	1	1	yes	characteristic, complemented
46	C₂⁴	—	359	1	no	—
47	C₂⁴	C₂×D₃₅₉	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	359	3	no	—
49	C₂×D₄	—	359	6	no	—
50	C₂×D₄	C₂×D₃₅₉	1	3	yes	—
51	C₂²⋊C₄	—	359	3	no	—
52	C₂²⋊C₄	—	359	6	no	—
53	C₂²⋊C₄	—	359	6	no	—
54	C₂²⋊C₄	C₂×D₃₅₉	1	3	yes	complemented
55	C₄²	—	359	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	359	2	no	—
58	C₂³	—	1436	3	no	—
59	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
60	C₂³	C₂²×D₃₅₉	1	3	yes	—
61	Q₈	—	1436	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	1436	6	no	—
64	C₂×C₄	—	359	6	no	—
65	C₂×C₄	C₂²×D₃₅₉	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	1436	4	no	—
71	C₂²	—	718	6	no	—
72	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	718	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	1436	2	no	—
78	C₂	#26040060	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	#120×C₃₅₉	C₂
hypercenter	#120	D₃₅₉
hypocenter	C₃₅₉	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. #26040002	C₂	semidirect
2. #26040001	C₂	semidirect
3. #26039984	C₂	semidirect
4. #26039983	C₂	semidirect
5. #120×C₃₅₉	C₂	semidirect
6. C₂³×D₃₅₉	C₂²	semidirect
7. C₂×Dic₇₁₈	C₂²	semidirect
9. C₂²⋊Dic₃₅₉	C₂²	semidirect
10. C₂²⋊Dic₃₅₉	C₂²	semidirect
11. (C₂×D₃₅₉)⋊C₄	C₂²	semidirect
12. C₄×Dic₃₅₉	C₂²	semidirect
13. C₂⁴×C₃₅₉	C₂²	semidirect
15. C₂²⋊C₄×C₃₅₉	C₂²	semidirect
39. C₃₅₉	H(3,𝔽₄)	semidirect
45. #120	D₃₅₉	semidirect
47. C₂⁴	C₂×D₃₅₉	semidirect
54. C₂²⋊C₄	C₂×D₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
#26040002	2
#26040001	2
#26039984	2
#26039983	2
#120×C₃₅₉	2
H(3,𝔽₄)	359