Subgroups — #36374256

Order 22336.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₃₄₉	C₂	1	2	yes	maximal, complemented
2	#26013571	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013560	C₂	1	2	yes	maximal, complemented
4	#26013549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄₉	—	2	2	no	—
7	C₂×#14584095	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₄₉	C₂²	1	2	yes	—
9	C₂²⋊Dic₃₄₉	—	2	2	no	—
10	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
12	D₈×C₃₄₉	C₄	1	4	yes	complemented
13	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₄₉	—	2	2	no	—
15	C₂×Dic₃₄₉	—	4	4	no	—
16	C₂×Dic₃₄₉	D₄	1	2	yes	—
17	#14584095	—	2	1	no	—
18	C₂³×C₃₄₉	—	2	2	no	—
19	D₄×C₃₄₉	—	2	2	no	—
20	D₄×C₃₄₉	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
22	C₈×C₃₄₉	C₂×C₄	1	2	yes	complemented
23	C₂²×C₃₄₉	—	2	4	no	—
24	C₂²×C₃₄₉	—	4	2	no	—
25	C₂²×C₃₄₉	—	4	2	no	—
26	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
27	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₃₄₉	#41	1	1	yes	characteristic
29	Dic₃₄₉	—	2	2	no	—
30	Dic₃₄₉	—	4	2	no	—
31	C₂×C₃₄₉	—	4	4	no	—
32	C₂×C₃₄₉	C₄×D₄	1	1	yes	characteristic
33	C₂×C₃₄₉	AGL(1,ℤ/8)	1	2	yes	—
34	C₃₄₉	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	349	1	no	maximal, Sylow
36	C₂×D₈	D₃₄₉	1	1	yes	characteristic
37	C₄×D₄	—	349	2	no	—
38	#106	—	349	1	no	—
39	#102	—	349	2	no	—
40	#97	—	349	1	no	—
41	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
42	C₂²×C₄	—	698	2	no	—
43	D₈	Dic₃₄₉	1	4	yes	complemented
44	C₂×C₈	—	349	1	no	—
45	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
46	C₄⋊C₄	—	349	2	no	—
47	C₂²⋊C₄	—	698	2	no	—
48	C₄²	—	349	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₃₄₉	1	4	yes	complemented
52	C₂×C₄	—	349	2	no	—
53	C₂×C₄	—	1396	4	no	—
54	C₂×C₄	—	698	2	no	—
55	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
56	C₈	—	698	1	no	—
57	C₈	C₂×Dic₃₄₉	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₃₄₉	1	1	yes	characteristic
62	C₄	—	1396	2	no	—
63	C₄	—	698	2	no	—
64	C₄	C₂²×Dic₃₄₉	1	1	yes	characteristic
65	C₄	#19150232	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	D₄×Dic₃₄₉	1	1	yes	characteristic
68	C₂	#26013663	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₄₉
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting, prime radical	C₂×D₈×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₂×D₈	D₃₄₉
hypocenter	C₃₄₉	#442
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₃₄₉	C₂	semidirect
2. #26013571	C₂	semidirect
3. #26013560	C₂	semidirect
4. #26013549	C₂	semidirect
12. D₈×C₃₄₉	C₄	semidirect
20. D₄×C₃₄₉	C₂×C₄	semidirect
22. C₈×C₃₄₉	C₂×C₄	semidirect
34. C₃₄₉	#442	semidirect
43. D₈	Dic₃₄₉	semidirect
51. D₄	C₂×Dic₃₄₉	semidirect
57. C₈	C₂×Dic₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₃₄₉	2
#26013571	2
#26013560	2
#26013549	2
C₂×D₈×C₃₄₉	2
#442	349