Subgroups — #36373962

Order 22336.

50 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	—	2	1	no	—
7	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
8	C₁₆×C₃₄₉	—	2	1	no	—
9	C₂×Dic₃₄₉	C₈	1	1	yes	characteristic
10	C₂×Dic₃₄₉	C₈	1	1	yes	characteristic
11	#14584095	—	2	1	no	—
12	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
13	C₈×C₃₄₉	D₄	1	1	yes	characteristic
14	C₈×C₃₄₉	Q₈	1	1	yes	characteristic
15	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
16	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
18	Dic₃₄₉	—	2	1	no	—
19	Dic₃₄₉	C₁₆	1	2	yes	complemented
20	C₂×C₃₄₉	C₄⋊C₈	1	1	yes	characteristic
21	C₂×C₃₄₉	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₃₄₉	M₃₂	1	1	yes	characteristic
23	C₃₄₉	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
24	C₄⋊C₁₆	—	349	1	no	maximal, Sylow
25	C₂×C₁₆	—	349	1	no	—
26	C₂×C₁₆	D₃₄₉	1	1	yes	characteristic
27	C₄×C₈	—	349	1	no	—
28	C₂×C₈	—	349	1	no	—
29	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
30	C₄²	—	349	1	no	—
31	C₁₆	—	2	1	no	—
32	C₁₆	—	698	1	no	—
33	C₂×C₄	—	349	1	no	—
34	C₂×C₄	—	349	1	no	—
35	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
36	C₈	—	698	1	no	—
37	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
38	C₈	Dic₆₉₈	1	1	yes	characteristic
39	C₂²	C₈×D₃₄₉	1	1	yes	characteristic
40	C₄	—	349	2	no	—
41	C₄	—	698	1	no	—
42	C₄	#19150212	1	1	yes	characteristic
43	C₄	#19150216	1	1	yes	characteristic
44	C₂	#26013568	1	1	yes	characteristic
45	C₂	C₁₆×D₃₄₉	1	1	yes	characteristic
46	C₂	#26013578	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃₄₉
commutator	C₂×C₃₄₉	C₂×C₁₆
Fitting	C₂×C₁₆×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂×C₈
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	C₂×C₁₆	D₃₄₉
hypocenter	C₃₄₉	C₄⋊C₁₆
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
19. Dic₃₄₉	C₁₆	semidirect
23. C₃₄₉	C₄⋊C₁₆	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
C₄⋊C₁₆	349