Subgroups — C₂×#26013573 #36374178

Order 22336.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₃₄₉	C₂	1	2	yes	maximal
2	#26013573	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₄₉	—	2	2	no	—
5	C₄⋊Dic₃₄₉	—	2	2	no	—
6	C₄⋊Dic₃₄₉	C₂²	1	4	yes	—
7	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₃₄₉	C₄	1	4	yes	complemented
9	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
10	C₂×Dic₃₄₉	—	2	4	no	—
11	C₂×Dic₃₄₉	—	4	4	no	—
12	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
13	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
15	C₂×C₄×C₃₄₉	Q₈	1	2	yes	—
16	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
17	C₈×C₃₄₉	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃₄₉	C₄⋊C₄	1	2	yes	—
19	C₂²×C₃₄₉	D₈	1	2	yes	—
20	C₂²×C₃₄₉	Q₁₆	1	2	yes	—
21	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₃₄₉	C₄⋊C₄	1	2	yes	—
23	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
25	Dic₃₄₉	—	4	4	no	—
26	C₂×C₃₄₉	#107	1	4	yes	complemented
27	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃₄₉	C₂×D₈	1	1	yes	characteristic
29	C₂×C₃₄₉	C₂×Q₁₆	1	1	yes	characteristic
30	C₃₄₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
31	C₂×#107	—	349	1	no	maximal, Sylow
32	C₂²×C₈	D₃₄₉	1	1	yes	characteristic
33	C₂×C₄⋊C₄	—	349	2	no	—
34	#107	—	349	4	no	—
35	C₂²×C₄	—	698	2	no	—
36	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
37	C₂×C₈	Dic₃₄₉	1	4	yes	complemented
38	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
39	C₄⋊C₄	—	349	4	no	—
40	C₄⋊C₄	—	698	2	no	—
41	C₂³	D₁₃₉₆	1	1	yes	characteristic
42	C₂×C₄	—	1396	4	no	—
43	C₂×C₄	—	698	4	no	—
44	C₂×C₄	C₂×Dic₃₄₉	1	2	yes	—
45	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
46	C₂×C₄	Dic₆₉₈	1	2	yes	—
47	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
48	C₈	C₂×Dic₃₄₉	1	4	yes	complemented
49	C₂²	C₄⋊Dic₃₄₉	1	2	yes	—
50	C₂²	D₂₇₉₂	1	2	yes	—
51	C₂²	Dic₁₃₉₆	1	2	yes	—
52	C₂²	C₂×D₁₃₉₆	1	1	yes	characteristic
53	C₄	—	1396	4	no	—
54	C₄	C₄⋊Dic₃₄₉	1	2	yes	—
55	C₄	C₂²×Dic₃₄₉	1	1	yes	characteristic
56	C₄	C₂×Dic₆₉₈	1	1	yes	characteristic
57	C₂	#26013573	1	4	yes	direct product factor
58	C₂	C₂×C₄⋊Dic₃₄₉	1	1	yes	characteristic
59	C₂	C₂×D₂₇₉₂	1	1	yes	characteristic
60	C₂	C₂×Dic₁₃₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₃₉₆
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₂²×C₈×C₃₄₉	C₂
socle, prime radical	C₂³×C₃₄₉	D₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₂²×C₈	D₃₄₉
hypocenter	C₃₄₉	C₂×#107
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #26013573	C₂	direct (N × H)
8. C₂×C₈×C₃₄₉	C₄	semidirect
17. C₈×C₃₄₉	C₂×C₄	semidirect
26. C₂×C₃₄₉	#107	semidirect
30. C₃₄₉	C₂×#107	semidirect
37. C₂×C₈	Dic₃₄₉	semidirect
48. C₈	C₂×Dic₃₄₉	semidirect
57. C₂	#26013573	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×C₄⋊Dic₃₄₉	2
#26013573	2
C₂²×C₈×C₃₄₉	2
C₂×#107	349

Subgroups — C2×#26013573 #36374178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013573 #36374178