Subgroups — #37526205

Order 22976.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₃₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₃₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₅₉	—	2	1	no	—
5	C₂×C₃₅₉⋊C₈	—	2	1	no	—
6	C₂×C₃₅₉⋊C₈	—	2	1	no	—
7	C₂×C₃₅₉⋊C₈	C₄	1	2	yes	complemented
8	C₄⋊Dic₃₅₉	—	2	1	no	—
9	C₄⋊Dic₃₅₉	C₄	1	2	yes	complemented
10	C₂²×C₄×C₃₅₉	—	2	1	no	—
11	C₂²×C₄×C₃₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₅₉	—	2	1	no	—
13	C₄⋊C₄×C₃₅₉	C₄	1	2	yes	complemented
14	C₂×Dic₃₅₉	—	2	2	no	—
15	C₂×Dic₃₅₉	—	4	1	no	—
16	C₂×Dic₃₅₉	—	4	1	no	—
17	C₃₅₉⋊C₈	—	2	2	no	—
18	C₂³×C₃₅₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₅₉	—	2	2	no	—
20	C₂×C₄×C₃₅₉	—	4	1	no	—
21	C₂×C₄×C₃₅₉	—	4	1	no	—
22	C₂×C₄×C₃₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₅₉	Q₈	1	1	yes	characteristic
28	C₂²×C₃₅₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₃₅₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₃₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₃₅₉	D₈	1	1	yes	characteristic
32	C₂²×C₃₅₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₃₅₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₃₅₉	Q₁₆	1	1	yes	characteristic
35	C₄×C₃₅₉	—	4	2	no	—
36	C₄×C₃₅₉	C₄²	1	1	yes	characteristic
37	C₄×C₃₅₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₃₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₃₅₉	C₄⋊C₄	1	1	yes	characteristic
40	Dic₃₅₉	—	4	2	no	—
41	C₂×C₃₅₉	#95	1	1	yes	characteristic
42	C₂×C₃₅₉	#102	1	1	yes	characteristic
43	C₂×C₃₅₉	#102	1	1	yes	characteristic
44	C₂×C₃₅₉	#103	1	1	yes	characteristic
45	C₂×C₃₅₉	#103	1	1	yes	characteristic
46	C₂×C₃₅₉	#106	1	1	yes	characteristic
47	C₂×C₃₅₉	#107	1	1	yes	characteristic
48	C₃₅₉	#340	1	1	yes	characteristic, complemented, Sylow
49	#340	—	359	1	no	maximal, Sylow
50	C₂²×C₈	—	359	1	no	—
51	C₂×C₄⋊C₄	—	359	1	no	—
52	C₂×C₄⋊C₄	D₃₅₉	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	718	1	no	—
55	C₂²×C₄	C₂×D₃₅₉	1	1	yes	characteristic
56	C₂×C₈	—	359	2	no	—
57	C₂×C₈	—	718	1	no	—
58	C₂×C₈	—	718	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	359	2	no	—
61	C₄⋊C₄	—	718	1	no	—
62	C₄⋊C₄	Dic₃₅₉	1	2	yes	complemented
63	C₂³	C₂²⋊D₃₅₉	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	1436	1	no	—
68	C₂×C₄	—	1436	1	no	—
69	C₂×C₄	—	718	2	no	—
70	C₂×C₄	C₄×D₃₅₉	1	1	yes	characteristic
71	C₂×C₄	C₄×D₃₅₉	1	1	yes	characteristic
72	C₂×C₄	C₂×Dic₃₅₉	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₃₅₉	1	1	yes	characteristic
74	C₂×C₄	D₁₄₃₆	1	1	yes	characteristic
75	C₂×C₄	Dic₇₁₈	1	1	yes	characteristic
76	C₈	—	718	2	no	—
77	C₂²	(C₂×D₃₅₉)⋊C₄	1	1	yes	characteristic
78	C₂²	C₂²⋊Dic₃₅₉	1	1	yes	characteristic
79	C₂²	Dic₃₅₉⋊C₄	1	1	yes	characteristic
80	C₂²	#19153326	1	1	yes	characteristic
81	C₂²	#19153328	1	1	yes	characteristic
82	C₂²	#19153329	1	1	yes	characteristic
83	C₂²	#19153331	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	1436	2	no	—
86	C₄	C₄×Dic₃₅₉	1	1	yes	characteristic
87	C₄	(C₂×D₃₅₉)⋊C₄	1	1	yes	characteristic
88	C₄	Dic₃₅₉⋊C₄	1	1	yes	characteristic
89	C₄	C₄⋊Dic₃₅₉	1	1	yes	characteristic
90	C₂	C₃₅₉⋊#95	1	1	yes	characteristic
91	C₂	#26039935	1	1	yes	characteristic
92	C₂	#26039936	1	1	yes	characteristic
93	C₂	#26039938	1	1	yes	characteristic
94	C₂	#26039939	1	1	yes	characteristic
95	C₂	#26039946	1	1	yes	characteristic
96	C₂	#26039948	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₅₉
commutator	C₄×C₃₅₉	C₄²
Frattini	C₂²×C₄	C₂×D₃₅₉
Fitting	C₂×C₄⋊C₄×C₃₅₉	C₂
socle, prime radical	C₂³×C₃₅₉	D₄
cosocle	C₂²×C₄×C₃₅₉	C₂²
hypercenter	C₂×C₄⋊C₄	D₃₅₉
hypocenter	C₃₅₉	#340
LCS	G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂×C₃₅₉⋊C₈	C₄	semidirect
9. C₄⋊Dic₃₅₉	C₄	semidirect
13. C₄⋊C₄×C₃₅₉	C₄	semidirect
48. C₃₅₉	#340	semidirect
62. C₄⋊C₄	Dic₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₅₉⋊C₈	2
C₂×C₄⋊Dic₃₅₉	2
C₂×C₄⋊C₄×C₃₅₉	2
#340	359