Subgroups — #107×D₃₄₉ #36374176

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	#26013573	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013572	C₂	1	2	yes	maximal, complemented
5	#107×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	D₈	1	2	yes	—
27	C₂×D₃₄₉	Q₁₆	1	2	yes	—
28	C₄×C₃₄₉	—	4	2	no	—
29	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
31	Dic₃₄₉	—	4	2	no	—
32	Dic₃₄₉	C₄⋊C₄	1	2	yes	—
33	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃₄₉	C₂×D₈	1	1	yes	characteristic
35	C₂×C₃₄₉	C₂×Q₁₆	1	1	yes	characteristic
36	D₃₄₉	#107	1	4	yes	direct product factor
37	C₃₄₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
38	C₂×#107	—	349	1	no	maximal, Sylow
39	C₂²×C₈	—	349	1	no	—
40	C₂×C₄⋊C₄	—	349	2	no	—
41	#107	—	349	1	no	—
42	#107	—	349	2	no	—
43	#107	D₃₄₉	1	1	yes	characteristic, direct product factor
44	C₂²×C₄	—	349	1	no	—
45	C₂²×C₄	—	698	2	no	—
46	C₂×C₈	—	349	1	no	—
47	C₂×C₈	—	349	4	no	—
48	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
49	C₄⋊C₄	—	349	2	no	—
50	C₄⋊C₄	—	698	2	no	—
51	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
52	C₂³	—	349	1	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	349	1	no	—
55	C₂×C₄	—	349	2	no	—
56	C₂×C₄	—	349	2	no	—
57	C₂×C₄	—	1396	4	no	—
58	C₂×C₄	—	698	2	no	—
59	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
60	C₈	—	349	2	no	—
61	C₈	C₄×D₃₄₉	1	2	yes	complemented
62	C₂²	—	349	2	no	—
63	C₂²	—	349	2	no	—
64	C₂²	—	349	2	no	—
65	C₂²	D₄×D₃₄₉	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	349	2	no	—
68	C₄	—	1396	2	no	—
69	C₄	C₂×C₄×D₃₄₉	1	1	yes	characteristic
70	C₄	Q₈×D₃₄₉	1	1	yes	characteristic
71	C₂	—	349	4	no	—
72	C₂	C₄⋊C₄×D₃₄₉	1	1	yes	characteristic
73	C₂	D₈×D₃₄₉	1	1	yes	characteristic
74	C₂	Q₁₆×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₄₉
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#107×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	#107	D₃₄₉
hypocenter	C₃₄₉	C₂×#107
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
3. #26013573	C₂	semidirect
4. #26013572	C₂	semidirect
5. #107×C₃₄₉	C₂	semidirect
9. C₈×D₃₄₉	C₄	semidirect
20. #14584095	C₂×C₄	semidirect
23. C₈×C₃₄₉	C₂×C₄	semidirect
36. D₃₄₉	#107	direct (N × H)
37. C₃₄₉	C₂×#107	semidirect
43. #107	D₃₄₉	direct (N × H)
61. 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
#26013573	2
#26013572	2
#107×C₃₄₉	2
C₂×#107	349

Subgroups — #107×D349 #36374176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #107×D₃₄₉ #36374176