Subgroups — #36375065

Order 22336.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150236	C₂	1	2	yes	maximal
2	C₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	#19150236	—	2	2	no	—
4	C₂×#14584095	C₂²	1	1	yes	characteristic
5	C₄×Dic₃₄₉	C₄	1	1	yes	characteristic
6	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic
7	C₂×Dic₃₄₉	C₈	1	1	yes	characteristic
8	C₂×Dic₃₄₉	C₈	1	1	yes	characteristic
9	#14584095	D₄	1	1	yes	characteristic
10	#14584095	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
12	C₈×C₃₄₉	—	2	1	no	—
13	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
14	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
15	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
16	Dic₃₄₉	—	2	1	no	—
17	Dic₃₄₉	C₁₆	1	2	yes	complemented
18	C₂×C₃₄₉	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₃₄₉	C₂×C₁₆	1	1	yes	characteristic
20	C₂×C₃₄₉	M₃₂	1	1	yes	characteristic
21	C₃₄₉	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₄⋊C₁₆	—	349	1	no	maximal, Sylow
23	C₂×C₁₆	—	349	2	no	—
24	C₄×C₈	—	349	1	no	—
25	C₂×C₈	—	349	1	no	—
26	C₂×C₈	#13063703	1	1	yes	characteristic
27	C₄²	—	349	1	no	—
28	C₁₆	—	698	2	no	—
29	C₂×C₄	—	349	1	no	—
30	C₂×C₄	—	349	1	no	—
31	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	349	1	no	—
34	C₈	—	349	1	no	—
35	C₂²	D₃₄₉⋊C₈	1	1	yes	characteristic
36	C₄	—	349	2	no	—
37	C₄	—	698	1	no	—
38	C₄	C₄⋊#13063703	1	1	yes	characteristic
39	C₄	#19150242	1	1	yes	characteristic
40	C₂	#26013706	1	1	yes	characteristic
41	C₂	D₃₄₉⋊C₁₆	1	1	yes	characteristic
42	C₂	#26013714	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#13063703
commutator	C₂×C₃₄₉	C₂×C₁₆
Fitting	C₂×C₈×C₃₄₉	C₄
socle, prime radical	C₂²×C₃₄₉	C₂×C₈
cosocle	C₂×#14584095	C₂²
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	C₄⋊C₁₆
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
17. Dic₃₄₉	C₁₆	semidirect
21. 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×#19150236	2
C₈×Dic₃₄₉	2
C₄⋊C₁₆	349