Subgroups — C₂×#26013687 #36374978

Order 22336.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013687	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19150234	C₂	1	4	yes	maximal, complemented
3	C₂×#19150233	C₂	1	6	yes	maximal, complemented
4	C₂×Q₈×D₃₄₉	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150234	C₂²	1	32	yes	complemented
7	#19150233	C₂²	1	48	yes	complemented
8	C₂×Dic₆₉₈	C₂²	1	6	yes	—
9	Q₈×D₃₄₉	C₂²	1	32	yes	complemented
10	C₂×D₁₃₉₆	C₂²	1	6	yes	—
11	C₂×C₂²⋊D₃₄₉	C₂²	1	4	yes	—
12	C₂×C₄×D₃₄₉	C₂²	1	12	yes	—
13	C₂×Q₈×C₃₄₉	C₂²	1	4	yes	—
14	C₂×Q₈×C₃₄₉	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₃₄₉	C₂²	1	3	yes	—
16	C₂²×D₃₄₉	C₂³	1	4	yes	—
17	Dic₆₉₈	C₂³	1	24	yes	—
18	D₁₃₉₆	C₂³	1	24	yes	—
19	C₂²⋊D₃₄₉	C₂³	1	16	yes	—
20	C₂×Dic₃₄₉	C₂³	1	4	yes	—
21	C₄×D₃₄₉	C₂³	1	48	yes	—
22	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
23	Q₈×C₃₄₉	C₂³	1	16	yes	—
24	C₂×C₄×C₃₄₉	C₂³	1	6	yes	—
25	C₂×C₄×C₃₄₉	C₂³	1	12	yes	—
26	C₂²×C₃₄₉	—	2	2	no	—
27	C₂²×C₃₄₉	C₂⁴	1	2	yes	—
28	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
29	C₂×D₃₄₉	—	2	8	no	—
30	C₂×D₃₄₉	C₂⁴	1	8	yes	—
31	C₄×C₃₄₉	C₂⁴	1	12	yes	—
32	Dic₃₄₉	C₂⁴	1	8	yes	—
33	C₂×C₃₄₉	—	2	2	no	—
34	C₂×C₃₄₉	2₋¹⁺⁴	1	2	yes	complemented
35	C₂×C₃₄₉	C₂⁵	1	1	yes	characteristic
36	D₃₄₉	—	2	8	no	—
37	C₃₄₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
38	C₂×2₋¹⁺⁴	—	349	1	no	maximal, Sylow
39	2₋¹⁺⁴	—	349	16	no	—
40	C₂×#41	—	349	4	no	—
41	C₂×#41	—	349	6	no	—
42	C₂²×Q₈	—	349	4	no	—
43	C₂²×Q₈	D₃₄₉	1	1	yes	characteristic, complemented
44	#41	—	349	32	no	—
45	#41	—	349	48	no	—
46	C₂×Q₈	—	349	32	no	—
47	C₂×Q₈	—	349	6	no	—
48	C₂×Q₈	C₂×D₃₄₉	1	4	yes	—
49	C₂×Q₈	C₂×D₃₄₉	1	8	yes	complemented
50	C₂×D₄	—	349	4	no	—
51	C₂×D₄	—	349	6	no	—
52	C₂²×C₄	—	349	12	no	—
53	C₂²×C₄	C₂×D₃₄₉	1	3	yes	—
54	C₂³	—	349	4	no	—
55	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
56	Q₈	—	349	24	no	—
57	Q₈	C₂²×D₃₄₉	1	16	yes	—
58	D₄	—	349	16	no	—
59	D₄	—	349	24	no	—
60	C₂×C₄	—	349	4	no	—
61	C₂×C₄	—	349	48	no	—
62	C₂×C₄	C₂²×D₃₄₉	1	12	yes	—
63	C₂×C₄	C₂²×D₃₄₉	1	6	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	349	8	no	—
66	C₂²	—	698	8	no	—
67	C₂²	C₂³×D₃₄₉	1	2	yes	—
68	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
69	C₄	—	349	8	no	—
70	C₄	C₂³×D₃₄₉	1	12	yes	—
71	C₂	—	2	2	no	—
72	C₂	—	698	8	no	—
73	C₂	#26013687	1	2	yes	direct product factor
74	C₂	C₂⁴×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₃₄₉
commutator, cosocle	C₂×C₃₄₉	C₂⁵
Frattini	C₂	C₂⁴×D₃₄₉
Fitting	C₂²×Q₈×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂⁴
hypercenter	C₂²×Q₈	D₃₄₉
hypocenter	C₃₄₉	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. #26013687	C₂	direct (N × H)
2. C₂×#19150234	C₂	semidirect
3. C₂×#19150233	C₂	semidirect
4. C₂×Q₈×D₃₄₉	C₂	semidirect
5. C₂²×Q₈×C₃₄₉	C₂	semidirect
6. #19150234	C₂²	semidirect
7. #19150233	C₂²	semidirect
9. Q₈×D₃₄₉	C₂²	semidirect
14. C₂×Q₈×C₃₄₉	C₂²	semidirect
34. C₂×C₃₄₉	2₋¹⁺⁴	semidirect
37. C₃₄₉	C₂×2₋¹⁺⁴	semidirect
43. C₂²×Q₈	D₃₄₉	semidirect
49. C₂×Q₈	C₂×D₃₄₉	semidirect
73. C₂	#26013687	direct (N × H)

Maximal subgroups

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

subgroup	index
#26013687	2
C₂×#19150234	2
C₂×#19150233	2
C₂×Q₈×D₃₄₉	2
C₂²×Q₈×C₃₄₉	2
C₂×2₋¹⁺⁴	349

Subgroups — C2×#26013687 #36374978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013687 #36374978