Subgroups — #106×D₃₄₉ #36374173

Order 22336.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₃₄₉	C₂	1	2	yes	maximal
3	#26013571	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013570	C₂	1	2	yes	maximal, complemented
5	#106×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₄₉	—	2	2	no	—
7	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
8	C₂×#14584095	C₂²	1	1	yes	characteristic
9	C₈×D₃₄₉	C₄	1	4	yes	complemented
10	C₄⋊Dic₃₄₉	C₂²	1	2	yes	—
11	#19150212	—	2	2	no	—
12	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
14	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
15	C₂×Dic₃₄₉	—	2	2	no	—
16	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
17	C₄×D₃₄₉	—	4	4	no	—
18	C₄×D₃₄₉	C₂×C₄	1	2	yes	—
19	C₄×D₃₄₉	Q₈	1	2	yes	—
20	#14584095	C₂×C₄	1	2	yes	complemented
21	C₂×C₄×C₃₄₉	—	2	2	no	—
22	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
23	C₈×C₃₄₉	C₂×C₄	1	2	yes	complemented
24	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
25	C₂×D₃₄₉	C₄⋊C₄	1	2	yes	—
26	C₂×D₃₄₉	SD₁₆	1	4	yes	—
27	C₄×C₃₄₉	—	4	2	no	—
28	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
30	Dic₃₄₉	—	4	2	no	—
31	Dic₃₄₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃₄₉	C₂×SD₁₆	1	2	yes	—
34	D₃₄₉	#106	1	4	yes	direct product factor
35	C₃₄₉	C₂×#106	1	1	yes	characteristic, complemented, Sylow
36	C₂×#106	—	349	1	no	maximal, Sylow
37	C₂²×C₈	—	349	1	no	—
38	C₂×C₄⋊C₄	—	349	2	no	—
39	#106	—	349	1	no	—
40	#106	—	349	2	no	—
41	#106	D₃₄₉	1	1	yes	characteristic, direct product factor
42	C₂²×C₄	—	349	1	no	—
43	C₂²×C₄	—	698	2	no	—
44	C₂×C₈	—	349	1	no	—
45	C₂×C₈	—	349	4	no	—
46	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
47	C₄⋊C₄	—	349	2	no	—
48	C₄⋊C₄	—	698	2	no	—
49	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
50	C₂³	—	349	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	349	1	no	—
53	C₂×C₄	—	349	2	no	—
54	C₂×C₄	—	349	2	no	—
55	C₂×C₄	—	1396	4	no	—
56	C₂×C₄	—	698	2	no	—
57	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
58	C₈	—	349	2	no	—
59	C₈	C₄×D₃₄₉	1	2	yes	complemented
60	C₂²	—	349	2	no	—
61	C₂²	—	349	4	no	—
62	C₂²	D₄×D₃₄₉	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	—	349	2	no	—
65	C₄	—	1396	2	no	—
66	C₄	C₂×C₄×D₃₄₉	1	1	yes	characteristic
67	C₄	Q₈×D₃₄₉	1	1	yes	characteristic
68	C₂	—	349	4	no	—
69	C₂	C₄⋊C₄×D₃₄₉	1	1	yes	characteristic
70	C₂	SD₁₆×D₃₄₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₄₉
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#106×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	#106	D₃₄₉
hypocenter	C₃₄₉	C₂×#106
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #26013571	C₂	semidirect
4. #26013570	C₂	semidirect
5. #106×C₃₄₉	C₂	semidirect
9. C₈×D₃₄₉	C₄	semidirect
20. #14584095	C₂×C₄	semidirect
23. C₈×C₃₄₉	C₂×C₄	semidirect
34. D₃₄₉	#106	direct (N × H)
35. C₃₄₉	C₂×#106	semidirect
41. #106	D₃₄₉	direct (N × H)
59. C₈	C₄×D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₄₉	2
C₄⋊C₄×D₃₄₉	2
#26013571	2
#26013570	2
#106×C₃₄₉	2
C₂×#106	349

Subgroups — #106×D349 #36374173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×D₃₄₉ #36374173