Subgroups — C₂²×#19150234 #36374972

Order 22336.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150234	C₂	1	24	yes	maximal, direct product factor
2	C₂²×D₁₃₉₆	C₂	1	3	yes	maximal, complemented
3	C₂²×C₄×D₃₄₉	C₂	1	3	yes	maximal, complemented
4	C₂²×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₄₉	C₂²	1	3	yes	—
6	#19150234	C₂²	1	64	yes	direct product factor
7	C₂×D₁₃₉₆	C₂²	1	36	yes	complemented
8	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₄₉	C₂²	1	36	yes	complemented
10	C₂×Q₈×C₃₄₉	C₂²	1	12	yes	complemented
11	C₂²×C₄×C₃₄₉	C₂²	1	3	yes	—
12	C₂²×D₃₄₉	—	2	12	no	—
13	C₂²×D₃₄₉	C₂³	1	18	yes	—
14	D₁₃₉₆	C₂³	1	48	yes	complemented
15	C₂×Dic₃₄₉	C₂³	1	6	yes	—
16	C₄×D₃₄₉	C₂³	1	48	yes	complemented
17	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
18	Q₈×C₃₄₉	C₂³	1	16	yes	complemented
19	C₂×C₄×C₃₄₉	C₂³	1	18	yes	—
20	C₂²×C₃₄₉	#41	1	4	yes	complemented
21	C₂²×C₃₄₉	C₂⁴	1	3	yes	—
22	C₂×D₃₄₉	—	2	36	no	—
23	C₂×D₃₄₉	C₂⁴	1	12	yes	—
24	C₄×C₃₄₉	C₂⁴	1	12	yes	—
25	Dic₃₄₉	C₂⁴	1	4	yes	—
26	C₂×C₃₄₉	C₂×#41	1	6	yes	complemented
27	C₂×C₃₄₉	C₂⁵	1	1	yes	characteristic
28	D₃₄₉	—	2	12	no	—
29	C₃₄₉	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂²×#41	—	349	1	no	maximal, Sylow
31	C₂×#41	—	349	24	no	—
32	C₂²×Q₈	D₃₄₉	1	1	yes	characteristic, complemented
33	C₂²×D₄	—	349	3	no	—
34	C₂³×C₄	—	349	3	no	—
35	C₂⁴	—	349	3	no	—
36	#41	—	349	64	no	—
37	C₂×Q₈	C₂×D₃₄₉	1	12	yes	complemented
38	C₂×D₄	—	349	36	no	—
39	C₂²×C₄	—	349	1	no	—
40	C₂²×C₄	—	349	36	no	—
41	C₂²×C₄	C₂×D₃₄₉	1	3	yes	—
42	C₂³	—	349	18	no	—
43	C₂³	—	698	12	no	—
44	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
45	Q₈	C₂²×D₃₄₉	1	16	yes	complemented
46	D₄	—	349	48	no	—
47	C₂×C₄	—	349	6	no	—
48	C₂×C₄	—	349	48	no	—
49	C₂×C₄	C₂²×D₃₄₉	1	18	yes	—
50	C₂²	—	349	12	no	—
51	C₂²	—	698	36	no	—
52	C₂²	#19150234	1	4	yes	direct product factor
53	C₂²	C₂³×D₃₄₉	1	3	yes	—
54	C₄	—	349	4	no	—
55	C₄	C₂³×D₃₄₉	1	12	yes	—
56	C₂	—	698	12	no	—
57	C₂	C₂×#19150234	1	6	yes	direct product factor
58	C₂	C₂⁴×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator, cosocle	C₂×C₃₄₉	C₂⁵
Frattini	C₂	C₂⁴×D₃₄₉
Fitting	C₂²×Q₈×C₃₄₉	C₂
socle	C₂³×C₃₄₉	C₂³
hypercenter	C₂²×Q₈	D₃₄₉
hypocenter	C₃₄₉	C₂²×#41
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19150234	C₂	direct (N × H)
2. C₂²×D₁₃₉₆	C₂	semidirect
3. C₂²×C₄×D₃₄₉	C₂	semidirect
4. C₂²×Q₈×C₃₄₉	C₂	semidirect
6. #19150234	C₂²	direct (N × H)
7. C₂×D₁₃₉₆	C₂²	semidirect
9. C₂×C₄×D₃₄₉	C₂²	semidirect
10. C₂×Q₈×C₃₄₉	C₂²	semidirect
14. D₁₃₉₆	C₂³	semidirect
16. C₄×D₃₄₉	C₂³	semidirect
18. Q₈×C₃₄₉	C₂³	semidirect
20. C₂²×C₃₄₉	#41	semidirect
26. C₂×C₃₄₉	C₂×#41	semidirect
29. C₃₄₉	C₂²×#41	semidirect
32. C₂²×Q₈	D₃₄₉	semidirect
37. C₂×Q₈	C₂×D₃₄₉	semidirect
45. Q₈	C₂²×D₃₄₉	semidirect
52. C₂²	#19150234	direct (N × H)
57. C₂	C₂×#19150234	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19150234	2
C₂²×D₁₃₉₆	2
C₂²×C₄×D₃₄₉	2
C₂²×Q₈×C₃₄₉	2
C₂²×#41	349

Subgroups — C22×#19150234 #36374972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19150234 #36374972