Subgroups — C₂×#26013639 #36374688

Order 22336.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₉₈	C₂	1	2	yes	maximal
2	#26013639	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₃₄₉	C₂	1	4	yes	maximal
4	C₂×C₄²×C₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₉₈	—	2	8	no	—
6	C₂×Dic₆₉₈	C₂²	1	8	yes	—
7	C₂²×Dic₃₄₉	C₂²	1	4	yes	—
8	C₄⋊Dic₃₄₉	C₂²	1	16	yes	—
9	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
11	C₄²×C₃₄₉	C₂²	1	4	yes	—
12	Dic₆₉₈	—	2	16	no	—
13	C₂×Dic₃₄₉	—	2	8	no	—
14	C₂×Dic₃₄₉	C₂³	1	8	yes	—
15	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₃₄₉	D₄	1	4	yes	—
17	C₂×C₄×C₃₄₉	Q₈	1	8	yes	complemented
18	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
19	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
20	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
21	C₂²×C₃₄₉	C₂×Q₈	1	4	yes	—
22	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
23	C₄×C₃₄₉	C₂×D₄	1	4	yes	—
24	C₄×C₃₄₉	C₂×Q₈	1	8	yes	complemented
25	Dic₃₄₉	—	2	8	no	—
26	C₂×C₃₄₉	C₄⋊Q₈	1	4	yes	complemented
27	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₃₄₉	C₂²×Q₈	1	2	yes	—
29	C₃₄₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₄⋊Q₈	—	349	1	no	maximal, Sylow
31	C₂²×Q₈	—	349	2	no	—
32	C₄⋊Q₈	—	349	8	no	—
33	C₂×C₄⋊C₄	—	349	4	no	—
34	C₂×C₄²	D₃₄₉	1	1	yes	characteristic
35	C₂×Q₈	—	349	8	no	—
36	C₂×Q₈	—	698	8	no	—
37	C₂²×C₄	—	349	4	no	—
38	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
40	C₄⋊C₄	—	349	16	no	—
41	C₄²	C₂×D₃₄₉	1	4	yes	—
42	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
43	Q₈	—	698	16	no	—
44	C₂×C₄	—	349	8	no	—
45	C₂×C₄	—	698	8	no	—
46	C₂×C₄	D₁₃₉₆	1	4	yes	—
47	C₂×C₄	Dic₆₉₈	1	8	yes	complemented
48	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
49	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
50	C₂²	C₂×D₁₃₉₆	1	2	yes	—
51	C₂²	C₂×Dic₆₉₈	1	4	yes	—
52	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
53	C₄	—	698	8	no	—
54	C₄	C₂×D₁₃₉₆	1	4	yes	—
55	C₄	C₂×Dic₆₉₈	1	8	yes	complemented
56	C₂	#26013639	1	4	yes	direct product factor
57	C₂	C₂²×D₁₃₉₆	1	1	yes	characteristic
58	C₂	C₂²×Dic₆₉₈	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator, cosocle	C₂²×C₃₄₉	C₂⁴
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₄²×C₃₄₉	C₂
socle, prime radical	C₂³×C₃₄₉	C₂³
hypercenter	C₂×C₄²	D₃₄₉
hypocenter	C₃₄₉	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #26013639	C₂	direct (N × H)
17. C₂×C₄×C₃₄₉	Q₈	semidirect
24. C₄×C₃₄₉	C₂×Q₈	semidirect
26. C₂×C₃₄₉	C₄⋊Q₈	semidirect
29. C₃₄₉	C₂×C₄⋊Q₈	semidirect
47. C₂×C₄	Dic₆₉₈	semidirect
55. C₄	C₂×Dic₆₉₈	semidirect
56. C₂	#26013639	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×Dic₆₉₈	2
#26013639	2
C₂×C₄⋊Dic₃₄₉	2
C₂×C₄²×C₃₄₉	2
C₂×C₄⋊Q₈	349

Subgroups — C2×#26013639 #36374688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013639 #36374688