Subgroups — C₂×#26013608 #36374630

Order 22336.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₃₄₉	C₂	1	3	yes	maximal, complemented
2	#26013608	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊Dic₃₄₉	C₂	1	3	yes	maximal, complemented
4	C₂⁵×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₄₉	C₂²	1	1	yes	characteristic, complemented
6	C₂×C₂²⋊D₃₄₉	—	2	12	no	—
7	C₂×C₂²⋊D₃₄₉	C₂²	1	12	yes	complemented
8	C₂²×Dic₃₄₉	C₂²	1	3	yes	complemented
9	C₂²⋊Dic₃₄₉	C₂²	1	12	yes	complemented
10	C₂⁴×C₃₄₉	—	2	12	no	—
11	C₂⁴×C₃₄₉	C₂²	1	3	yes	—
12	C₂⁴×C₃₄₉	C₂²	1	4	yes	complemented
13	C₂²×D₃₄₉	—	2	6	no	—
14	C₂²×D₃₄₉	C₂³	1	2	yes	complemented
15	C₂²⋊D₃₄₉	—	2	24	no	—
16	C₂×Dic₃₄₉	—	2	6	no	—
17	C₂×Dic₃₄₉	C₂³	1	6	yes	complemented
18	C₂³×C₃₄₉	—	2	32	no	—
19	C₂³×C₃₄₉	—	2	12	no	—
20	C₂³×C₃₄₉	—	2	24	no	—
21	C₂³×C₃₄₉	D₄	1	12	yes	complemented
22	C₂³×C₃₄₉	C₂³	1	6	yes	—
23	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
24	C₂²×C₃₄₉	—	2	32	no	—
25	C₂²×C₃₄₉	—	2	24	no	—
26	C₂²×C₃₄₉	—	2	12	no	—
27	C₂²×C₃₄₉	C₂×D₄	1	6	yes	—
28	C₂²×C₃₄₉	C₂×D₄	1	12	yes	complemented
29	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
30	C₂×D₃₄₉	—	2	6	no	—
31	C₂×D₃₄₉	—	4	4	no	—
32	Dic₃₄₉	—	2	6	no	—
33	C₂×C₃₄₉	—	2	12	no	—
34	C₂×C₃₄₉	#120	1	4	yes	complemented
35	C₂×C₃₄₉	C₂²×D₄	1	3	yes	—
36	D₃₄₉	—	4	2	no	—
37	C₃₄₉	C₂×#120	1	1	yes	characteristic, complemented, Sylow
38	C₂×#120	—	349	1	no	maximal, Sylow
39	C₂⁵	D₃₄₉	1	1	yes	characteristic, complemented
40	C₂²×D₄	—	349	3	no	—
41	#120	—	349	8	no	—
42	C₂×C₂²⋊C₄	—	349	3	no	—
43	C₂⁴	—	2	12	no	—
44	C₂⁴	—	349	1	no	—
45	C₂⁴	C₂×D₃₄₉	1	3	yes	—
46	C₂⁴	C₂×D₃₄₉	1	4	yes	complemented
47	C₂×D₄	—	349	12	no	—
48	C₂×D₄	—	698	12	no	—
49	C₂²×C₄	—	349	3	no	—
50	C₂²⋊C₄	—	349	12	no	—
51	C₂³	—	2	32	no	—
52	C₂³	—	2	12	no	—
53	C₂³	—	2	24	no	—
54	C₂³	—	349	2	no	—
55	C₂³	—	698	6	no	—
56	C₂³	C₂²⋊D₃₄₉	1	12	yes	complemented
57	C₂³	C₂²×D₃₄₉	1	6	yes	—
58	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
59	D₄	—	698	24	no	—
60	C₂×C₄	—	349	6	no	—
61	C₂×C₄	—	698	6	no	—
62	C₂²	—	2	32	no	—
63	C₂²	—	2	24	no	—
64	C₂²	—	2	12	no	—
65	C₂²	—	1396	4	no	—
66	C₂²	—	698	6	no	—
67	C₂²	C₂×C₂²⋊D₃₄₉	1	6	yes	—
68	C₂²	C₂×C₂²⋊D₃₄₉	1	12	yes	complemented
69	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
70	C₄	—	698	6	no	—
71	C₂	—	2	12	no	—
72	C₂	—	1396	2	no	—
73	C₂	#26013608	1	4	yes	direct product factor
74	C₂	C₂²×C₂²⋊D₃₄₉	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator, cosocle	C₂²×C₃₄₉	C₂⁴
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂⁵×C₃₄₉	C₂
socle	C₂³×C₃₄₉	C₂³
hypercenter	C₂⁵	D₃₄₉
hypocenter	C₃₄₉	C₂×#120
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₃₄₉	C₂	semidirect
2. #26013608	C₂	direct (N × H)
3. C₂×C₂²⋊Dic₃₄₉	C₂	semidirect
4. C₂⁵×C₃₄₉	C₂	semidirect
5. C₂³×D₃₄₉	C₂²	semidirect
7. C₂×C₂²⋊D₃₄₉	C₂²	semidirect
8. C₂²×Dic₃₄₉	C₂²	semidirect
9. C₂²⋊Dic₃₄₉	C₂²	semidirect
12. C₂⁴×C₃₄₉	C₂²	semidirect
14. C₂²×D₃₄₉	C₂³	semidirect
17. C₂×Dic₃₄₉	C₂³	semidirect
21. C₂³×C₃₄₉	D₄	semidirect
28. C₂²×C₃₄₉	C₂×D₄	semidirect
34. C₂×C₃₄₉	#120	semidirect
37. C₃₄₉	C₂×#120	semidirect
39. C₂⁵	D₃₄₉	semidirect
46. C₂⁴	C₂×D₃₄₉	semidirect
56. C₂³	C₂²⋊D₃₄₉	semidirect
68. C₂²	C₂×C₂²⋊D₃₄₉	semidirect
73. C₂	#26013608	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×C₂²⋊D₃₄₉	2
#26013608	2
C₂×C₂²⋊Dic₃₄₉	2
C₂⁵×C₃₄₉	2
C₂×#120	349

Subgroups — C2×#26013608 #36374630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013608 #36374630