Subgroups — #37526249

Order 22976.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040037	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19153341	—	2	1	no	—
5	Dic₁₄₃₆	C₄	1	1	yes	characteristic
6	#19153330	—	2	1	no	—
7	D₂₈₇₂	C₄	1	1	yes	characteristic
8	C₃₅₉⋊C₁₆	—	2	1	no	—
9	M₁₆×C₃₅₉	—	2	1	no	—
10	C₂×C₈×C₃₅₉	C₂²	1	1	yes	characteristic
11	Dic₇₁₈	—	2	1	no	—
12	D₁₄₃₆	—	2	1	no	—
13	C₂²⋊D₃₅₉	—	4	1	no	—
14	C₄×D₃₅₉	—	4	1	no	—
15	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
16	C₈×C₃₅₉	—	4	1	no	—
17	C₈×C₃₅₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₃₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₃₅₉	SD₁₆	1	1	yes	characteristic
20	C₂×D₃₅₉	—	4	1	no	—
21	C₄×C₃₅₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₅₉	D₈	1	1	yes	characteristic
23	Dic₃₅₉	—	4	1	no	—
24	C₂×C₃₅₉	—	2	1	no	—
25	C₂×C₃₅₉	#102	1	1	yes	characteristic
26	D₃₅₉	—	8	1	no	—
27	C₃₅₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	359	1	no	maximal, Sylow
29	#135	—	359	1	no	—
30	C₂×C₁₆	—	359	1	no	—
31	#108	D₃₅₉	1	1	yes	characteristic, complemented
32	#41	—	718	1	no	—
33	Q₁₆	—	359	1	no	—
34	SD₁₆	—	718	1	no	—
35	D₈	—	359	1	no	—
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂×D₃₅₉	1	1	yes	characteristic
38	C₁₆	—	718	1	no	—
39	Q₈	—	718	1	no	—
40	D₄	—	1436	1	no	—
41	D₄	—	718	1	no	—
42	C₂×C₄	—	1436	1	no	—
43	C₂×C₄	C₂²⋊D₃₅₉	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₄×D₃₅₉	1	1	yes	characteristic
46	C₈	D₁₄₃₆	1	1	yes	characteristic
47	C₂²	—	1436	1	no	—
48	C₂²	#19153328	1	1	yes	characteristic
49	C₄	—	1436	1	no	—
50	C₄	(C₂×D₃₅₉)⋊C₄	1	1	yes	characteristic
51	C₄	#19153326	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	2872	1	no	—
54	C₂	#26039935	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19153326
commutator	C₈×C₃₅₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₅₉
Fitting	#108×C₃₅₉	C₂
socle	C₂×C₃₅₉	#102
prime radical	#26040037	C₂
cosocle	C₂×C₈×C₃₅₉	C₂²
hypercenter	#108	D₃₅₉
hypocenter	C₃₅₉	#359
LCS	G ▷ C₈×C₃₅₉ ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₅₉⋊C₁₆	C₂	semidirect
3. #108×C₃₅₉	C₂	semidirect
27. C₃₅₉	#359	semidirect
31. #108	D₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
#26040037	2
C₂×C₃₅₉⋊C₁₆	2
#108×C₃₅₉	2
#359	359