Subgroups — C₂×C₄²×D₃₄₉ #36374588

Order 22336.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃₄₉	C₂	1	3	yes	maximal
2	C₂×C₄×Dic₃₄₉	C₂	1	3	yes	maximal, complemented
3	C₄²×D₃₄₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄²×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₄₉	C₂²	1	1	yes	characteristic
6	C₂²×Dic₃₄₉	C₂²	1	3	yes	—
7	C₂×C₄×D₃₄₉	C₄	1	24	yes	direct product factor
8	C₂×C₄×D₃₄₉	C₂²	1	12	yes	—
9	C₄×Dic₃₄₉	C₂²	1	12	yes	complemented
10	C₂²×C₄×C₃₄₉	C₂²	1	3	yes	—
11	C₄²×C₃₄₉	C₂²	1	4	yes	complemented
12	C₂²×D₃₄₉	C₂×C₄	1	12	yes	—
13	C₂²×D₃₄₉	C₂³	1	2	yes	—
14	C₂×Dic₃₄₉	C₂×C₄	1	12	yes	complemented
15	C₂×Dic₃₄₉	C₂³	1	6	yes	—
16	C₄×D₃₄₉	C₂×C₄	1	48	yes	direct product factor
17	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₃₄₉	C₂×C₄	1	12	yes	complemented
19	C₂×C₄×C₃₄₉	C₂³	1	6	yes	—
20	C₂²×C₃₄₉	C₂²×C₄	1	6	yes	—
21	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
22	C₂×D₃₄₉	C₄²	1	16	yes	direct product factor
23	C₂×D₃₄₉	C₂²×C₄	1	12	yes	—
24	C₄×C₃₄₉	C₂²×C₄	1	12	yes	complemented
25	Dic₃₄₉	C₂²×C₄	1	12	yes	complemented
26	C₂×C₃₄₉	C₂×C₄²	1	4	yes	complemented
27	C₂×C₃₄₉	C₂³×C₄	1	3	yes	—
28	D₃₄₉	C₂×C₄²	1	8	yes	direct product factor
29	C₃₄₉	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₄²	—	349	1	no	maximal, Sylow
31	C₂³×C₄	—	349	3	no	—
32	C₂×C₄²	—	349	3	no	—
33	C₂×C₄²	—	349	8	no	—
34	C₂×C₄²	D₃₄₉	1	1	yes	characteristic, direct product factor
35	C₂⁴	—	349	1	no	—
36	C₂²×C₄	—	349	3	no	—
37	C₂²×C₄	—	349	24	no	—
38	C₂²×C₄	—	349	12	no	—
39	C₂²×C₄	C₂×D₃₄₉	1	3	yes	—
40	C₄²	—	349	12	no	—
41	C₄²	C₂×D₃₄₉	1	4	yes	direct product factor
42	C₂³	—	349	12	no	—
43	C₂³	—	349	2	no	—
44	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
45	C₂×C₄	—	349	12	no	—
46	C₂×C₄	—	349	6	no	—
47	C₂×C₄	—	349	48	no	—
48	C₂×C₄	C₄×D₃₄₉	1	12	yes	direct product factor
49	C₂×C₄	C₂²×D₃₄₉	1	6	yes	—
50	C₂²	—	349	16	no	—
51	C₂²	—	349	12	no	—
52	C₂²	C₂×C₄×D₃₄₉	1	6	yes	—
53	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
54	C₄	—	349	12	no	—
55	C₄	C₂×C₄×D₃₄₉	1	12	yes	direct product factor
56	C₂	—	349	8	no	—
57	C₂	C₄²×D₃₄₉	1	4	yes	direct product factor
58	C₂	C₂²×C₄×D₃₄₉	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₃₄₉
commutator, hypocenter	C₃₄₉	C₂²×C₄²
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₄²×C₃₄₉	C₂
socle	C₂³×C₃₄₉	C₂³
prime radical	C₂³×D₃₄₉	C₂²
cosocle	C₂²×C₃₄₉	C₂⁴
LCS	G ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₄₉	C₂	semidirect
3. C₄²×D₃₄₉	C₂	direct (N × H)
4. C₂×C₄²×C₃₄₉	C₂	semidirect
7. C₂×C₄×D₃₄₉	C₄	direct (N × H)
9. C₄×Dic₃₄₉	C₂²	semidirect
11. C₄²×C₃₄₉	C₂²	semidirect
14. C₂×Dic₃₄₉	C₂×C₄	semidirect
16. C₄×D₃₄₉	C₂×C₄	direct (N × H)
18. C₂×C₄×C₃₄₉	C₂×C₄	semidirect
22. C₂×D₃₄₉	C₄²	direct (N × H)
24. C₄×C₃₄₉	C₂²×C₄	semidirect
25. Dic₃₄₉	C₂²×C₄	semidirect
26. C₂×C₃₄₉	C₂×C₄²	semidirect
28. D₃₄₉	C₂×C₄²	direct (N × H)
29. C₃₄₉	C₂²×C₄²	semidirect
34. C₂×C₄²	D₃₄₉	direct (N × H)
41. C₄²	C₂×D₃₄₉	direct (N × H)
48. C₂×C₄	C₄×D₃₄₉	direct (N × H)
55. C₄	C₂×C₄×D₃₄₉	direct (N × H)
57. C₂	C₄²×D₃₄₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃₄₉	2
C₂×C₄×Dic₃₄₉	2
C₄²×D₃₄₉	2
C₂×C₄²×C₃₄₉	2
C₂²×C₄²	349

Subgroups — C2×C42×D349 #36374588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₃₄₉ #36374588