Subgroups — #36373948

Order 22336.

58 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	#26013573	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₁₆×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₃₄₉	—	2	1	no	—
5	#19150208	—	2	1	no	—
6	C₂×Q₈×C₃₄₉	—	2	1	no	—
7	Q₁₆×C₃₄₉	—	2	1	no	—
8	Q₁₆×C₃₄₉	C₄	1	2	yes	complemented
9	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₄₉	—	4	1	no	—
11	Q₈×C₃₄₉	—	2	2	no	—
12	Q₈×C₃₄₉	—	4	1	no	—
13	C₂×C₄×C₃₄₉	—	4	1	no	—
14	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
15	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₃₄₉	D₄	1	1	yes	characteristic
17	C₂²×C₃₄₉	D₈	1	1	yes	characteristic
18	C₄×C₃₄₉	—	4	2	no	—
19	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₃₄₉	SD₁₆	1	1	yes	characteristic
21	Dic₃₄₉	—	8	1	no	—
22	C₂×C₃₄₉	#102	1	1	yes	characteristic
23	C₂×C₃₄₉	SD₃₂	1	1	yes	characteristic
24	C₂×C₃₄₉	Q₃₂	1	1	yes	characteristic
25	C₃₄₉	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	349	1	no	maximal, Sylow
27	C₂×Q₁₆	D₃₄₉	1	1	yes	characteristic
28	C₂×C₁₆	—	349	1	no	—
29	#107	—	349	1	no	—
30	C₂×Q₈	—	2	1	no	—
31	Q₁₆	—	2	1	no	—
32	Q₁₆	Dic₃₄₉	1	2	yes	complemented
33	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
34	C₄⋊C₄	—	698	1	no	—
35	C₁₆	—	698	1	no	—
36	Q₈	—	2	2	no	—
37	Q₈	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	1396	1	no	—
40	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
41	C₈	C₂×Dic₃₄₉	1	1	yes	characteristic
42	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
43	C₂²	#19150218	1	1	yes	characteristic
44	C₄	—	4	2	no	—
45	C₄	—	2792	1	no	—
46	C₄	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
47	C₄	#19150221	1	1	yes	characteristic
48	C₂	#26013560	1	1	yes	characteristic
49	C₂	#26013582	1	1	yes	characteristic
50	C₂	#26013585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19150218
commutator	C₈×C₃₄₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₄₉
Fitting	C₂×Q₁₆×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	D₈
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	C₂×Q₁₆	D₃₄₉
hypocenter	C₃₄₉	#358
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
8. Q₁₆×C₃₄₉	C₄	semidirect
25. C₃₄₉	#358	semidirect
32. Q₁₆	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
#26013573	2
C₂×Q₁₆×C₃₄₉	2
#358	349