Subgroups — #36373919

Order 22336.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150208	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14584095	C₄	1	1	yes	characteristic
5	C₄×Dic₃₄₉	C₄	1	1	yes	characteristic
6	#19150208	C₄	1	2	yes	complemented
7	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
8	C₁₆×C₃₄₉	C₄	1	2	yes	complemented
9	C₂×Dic₃₄₉	C₈	1	2	yes	—
10	#14584095	C₈	1	2	yes	—
11	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
12	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
13	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
15	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
16	C₄×C₃₄₉	C₂×C₈	1	1	yes	characteristic
17	Dic₃₄₉	—	2	2	no	—
18	C₂×C₃₄₉	C₄×C₈	1	1	yes	characteristic
19	C₂×C₃₄₉	M₃₂	1	2	yes	—
20	C₃₄₉	#346	1	1	yes	characteristic, complemented, Sylow
21	#346	—	349	1	no	maximal, Sylow
22	C₂×C₁₆	—	349	1	no	—
23	C₂×C₁₆	D₃₄₉	1	1	yes	characteristic
24	C₄×C₈	—	349	1	no	—
25	C₂×C₈	—	349	1	no	—
26	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
27	C₄²	—	349	1	no	—
28	C₁₆	—	349	2	no	—
29	C₁₆	Dic₃₄₉	1	2	yes	complemented
30	C₂×C₄	—	349	2	no	—
31	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
32	C₈	—	349	2	no	—
33	C₈	C₄×D₃₄₉	1	1	yes	characteristic
34	C₈	C₂×Dic₃₄₉	1	1	yes	characteristic
35	C₂²	C₈×D₃₄₉	1	1	yes	characteristic
36	C₄	—	698	2	no	—
37	C₄	C₄×Dic₃₄₉	1	1	yes	characteristic
38	C₄	C₈×D₃₄₉	1	1	yes	characteristic
39	C₂	C₈×Dic₃₄₉	1	1	yes	characteristic
40	C₂	#26013578	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃₄₉
commutator	C₂×C₃₄₉	C₄×C₈
Fitting	C₂×C₁₆×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂×C₈
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	C₂×C₁₆	D₃₄₉
hypocenter	C₃₄₉	#346
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
6. #19150208	C₄	semidirect
8. C₁₆×C₃₄₉	C₄	semidirect
20. C₃₄₉	#346	semidirect
29. C₁₆	Dic₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150208	2
C₈×Dic₃₄₉	2
C₂×C₁₆×C₃₄₉	2
#346	349