Subgroups — C₂×#26013637 #36374686

Order 22336.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic
2	#26013637	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19150212	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₉₈	—	2	4	no	—
7	C₂×Dic₆₉₈	C₂²	1	4	yes	—
8	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
9	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
10	#19150212	C₂²	1	16	yes	—
11	C₄×Dic₃₄₉	C₂²	1	4	yes	—
12	C₂×Q₈×C₃₄₉	—	2	4	no	—
13	C₂×Q₈×C₃₄₉	C₂²	1	4	yes	—
14	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
16	Dic₆₉₈	—	2	8	no	—
17	C₂×Dic₃₄₉	—	2	4	no	—
18	C₂×Dic₃₄₉	Q₈	1	8	yes	complemented
19	C₂×Dic₃₄₉	C₂³	1	4	yes	—
20	C₂×Dic₃₄₉	C₂³	1	4	yes	—
21	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
22	Q₈×C₃₄₉	—	2	8	no	—
23	C₂×C₄×C₃₄₉	—	2	4	no	—
24	C₂×C₄×C₃₄₉	D₄	1	4	yes	—
25	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
26	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
27	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
28	C₂²×C₃₄₉	C₂×Q₈	1	4	yes	—
29	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
30	C₄×C₃₄₉	—	2	4	no	—
31	C₄×C₃₄₉	C₂×D₄	1	4	yes	—
32	Dic₃₄₉	—	2	4	no	—
33	Dic₃₄₉	C₂×Q₈	1	8	yes	complemented
34	C₂×C₃₄₉	C₄⋊Q₈	1	4	yes	complemented
35	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
36	C₂×C₃₄₉	C₂²×Q₈	1	2	yes	—
37	C₃₄₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄⋊Q₈	—	349	1	no	maximal, Sylow
39	C₂²×Q₈	—	349	1	no	—
40	C₂²×Q₈	D₃₄₉	1	1	yes	characteristic
41	C₄⋊Q₈	—	349	8	no	—
42	C₂×C₄⋊C₄	—	349	4	no	—
43	C₂×C₄²	—	349	1	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	349	4	no	—
46	C₂×Q₈	—	698	4	no	—
47	C₂×Q₈	C₂×D₃₄₉	1	4	yes	—
48	C₂²×C₄	—	349	2	no	—
49	C₂²×C₄	—	349	2	no	—
50	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
51	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
52	C₄⋊C₄	—	349	16	no	—
53	C₄²	—	349	4	no	—
54	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
55	Q₈	—	2	8	no	—
56	Q₈	—	698	8	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	349	8	no	—
59	C₂×C₄	—	349	4	no	—
60	C₂×C₄	—	349	4	no	—
61	C₂×C₄	—	698	4	no	—
62	C₂×C₄	C₂²⋊D₃₄₉	1	4	yes	—
63	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
64	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₃₄₉	1	2	yes	—
66	C₂²	Q₈×D₃₄₉	1	4	yes	—
67	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	349	8	no	—
70	C₄	—	698	4	no	—
71	C₄	C₂×C₂²⋊D₃₄₉	1	4	yes	—
72	C₂	#26013637	1	4	yes	direct product factor
73	C₂	C₂²×C₂²⋊D₃₄₉	1	1	yes	characteristic
74	C₂	C₂×Q₈×D₃₄₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #26013637	C₂	direct (N × H)
18. C₂×Dic₃₄₉	Q₈	semidirect
33. Dic₃₄₉	C₂×Q₈	semidirect
34. C₂×C₃₄₉	C₄⋊Q₈	semidirect
37. C₃₄₉	C₂×C₄⋊Q₈	semidirect
72. C₂	#26013637	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
#26013637	2
C₂×#19150212	2
C₂×C₄×Dic₃₄₉	2
C₂²×Q₈×C₃₄₉	2
C₂×C₄⋊Q₈	349

Subgroups — C2×#26013637 #36374686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013637 #36374686