Subgroups — #37526354

Order 22976.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₃₅₉)⋊C₄	C₂	1	3	yes	maximal
2	C₃₅₉⋊#95	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₅₉	C₂²	1	1	yes	characteristic
5	C₂²×Dic₃₅₉	C₂²	1	3	yes	—
6	(C₂×D₃₅₉)⋊C₄	—	2	6	no	—
7	C₂×Q₈×C₃₅₉	—	2	6	no	—
8	C₂²×C₄×C₃₅₉	C₂²	1	3	yes	—
9	C₂²×D₃₅₉	—	2	3	no	—
10	C₂²×D₃₅₉	—	2	3	no	—
11	C₂²×D₃₅₉	Q₈	1	2	yes	complemented
12	C₂×Dic₃₅₉	—	2	3	no	—
13	C₂×Dic₃₅₉	—	2	6	no	—
14	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
15	Q₈×C₃₅₉	—	2	8	no	—
16	C₂×C₄×C₃₅₉	—	2	6	no	—
17	C₂×C₄×C₃₅₉	D₄	1	6	yes	—
18	C₂²×C₃₅₉	C₂×D₄	1	3	yes	—
19	C₂²×C₃₅₉	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₃₅₉	#41	1	3	yes	—
21	C₂×D₃₅₉	—	2	6	no	—
22	C₂×D₃₅₉	—	4	1	no	—
23	C₂×D₃₅₉	—	4	3	no	—
24	C₄×C₃₅₉	—	2	6	no	—
25	Dic₃₅₉	—	4	3	no	—
26	C₂×C₃₅₉	#120	1	1	yes	characteristic
27	C₂×C₃₅₉	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₃₅₉	#124	1	3	yes	—
29	D₃₅₉	—	4	2	no	—
30	C₃₅₉	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	359	1	no	maximal, Sylow
32	C₂²×Q₈	D₃₅₉	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	359	3	no	—
34	#95	—	359	3	no	—
35	C₂⁴	—	359	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	359	3	no	—
38	C₂²×C₄	C₂×D₃₅₉	1	3	yes	—
39	C₂²⋊C₄	—	718	6	no	—
40	C₂³	—	359	2	no	—
41	C₂³	—	718	3	no	—
42	C₂³	—	718	3	no	—
43	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	718	3	no	—
47	C₂×C₄	—	718	6	no	—
48	C₂×C₄	C₂²⋊D₃₅₉	1	6	yes	—
49	C₂²	—	1436	1	no	—
50	C₂²	—	1436	3	no	—
51	C₂²	—	718	6	no	—
52	C₂²	C₂×C₂²⋊D₃₅₉	1	3	yes	—
53	C₂²	Q₈×D₃₅₉	1	1	yes	characteristic
54	C₂²	#19153342	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	1436	3	no	—
57	C₂	—	1436	2	no	—
58	C₂	#26039984	1	1	yes	characteristic
59	C₂	#26039991	1	3	yes	—
60	C₂	#26040004	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₅₉
commutator, socle, cosocle	C₂³×C₃₅₉	C₂³
Fitting	C₂²×Q₈×C₃₅₉	C₂
prime radical	C₂³×D₃₅₉	C₂²
hypercenter	C₂²×Q₈	D₃₅₉
hypocenter	C₃₅₉	#393
LCS	G ▷ C₂³×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃₅₉⋊#95	C₂	semidirect
3. C₂²×Q₈×C₃₅₉	C₂	semidirect
11. C₂²×D₃₅₉	Q₈	semidirect
30. C₃₅₉	#393	semidirect
32. C₂²×Q₈	D₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₃₅₉)⋊C₄	2
C₃₅₉⋊#95	2
C₂²×Q₈×C₃₅₉	2
#393	359