Subgroups — #36374096

Order 22336.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150216	C₂	1	2	yes	maximal
2	C₄²×D₃₄₉	C₂	1	1	yes	maximal, characteristic
3	#26013550	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₃₄₉	C₂	1	2	yes	maximal, complemented
5	#97×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₄₉	C₄	1	2	yes	—
7	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
8	#19150216	C₄	1	8	yes	complemented
9	C₂×#14584095	C₂²	1	2	yes	—
10	C₄×Dic₃₄₉	C₄	1	2	yes	—
11	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
13	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
14	C₂²×D₃₄₉	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
17	C₄×D₃₄₉	—	2	4	no	—
18	C₄×D₃₄₉	C₂×C₄	1	4	yes	—
19	#14584095	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
22	C₈×C₃₄₉	C₂×C₄	1	4	yes	complemented
23	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₃₄₉	—	2	2	no	—
25	C₂×D₃₄₉	C₄²	1	2	yes	—
26	C₄×C₃₄₉	—	2	2	no	—
27	C₄×C₃₄₉	C₂²×C₄	1	2	yes	—
28	Dic₃₄₉	C₄²	1	2	yes	—
29	Dic₃₄₉	M₁₆	1	4	yes	complemented
30	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₃₄₉	C₂×M₁₆	1	2	yes	—
32	D₃₄₉	—	2	2	no	—
33	C₃₄₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×M₁₆	—	349	1	no	maximal, Sylow
35	C₂×M₁₆	—	349	2	no	—
36	C₂×C₄²	—	349	1	no	—
37	#97	—	349	1	no	—
38	#97	D₃₄₉	1	1	yes	characteristic, complemented
39	C₄×C₈	—	349	2	no	—
40	C₂²×C₄	—	349	1	no	—
41	C₂²×C₄	—	349	2	no	—
42	M₁₆	—	349	8	no	—
43	C₂×C₈	—	349	2	no	—
44	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
45	C₄²	—	349	2	no	—
46	C₄²	—	349	1	no	—
47	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
48	C₂³	—	349	1	no	—
49	C₂×C₄	—	349	1	no	—
50	C₂×C₄	—	349	2	no	—
51	C₂×C₄	—	349	4	no	—
52	C₂×C₄	—	698	4	no	—
53	C₂×C₄	C₄×D₃₄₉	1	2	yes	—
54	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
55	C₈	—	349	4	no	—
56	C₈	C₄×D₃₄₉	1	4	yes	complemented
57	C₂²	—	349	2	no	—
58	C₂²	—	698	2	no	—
59	C₂²	C₂×C₄×D₃₄₉	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	349	2	no	—
62	C₄	—	349	4	no	—
63	C₄	C₂×C₄×D₃₄₉	1	2	yes	—
64	C₂	—	698	2	no	—
65	C₂	C₄²×D₃₄₉	1	1	yes	characteristic
66	C₂	M₁₆×D₃₄₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₄₉
commutator	C₂×C₃₄₉	C₂×C₄²
Fitting	#97×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂²×C₄
prime radical	C₂²×D₃₄₉	C₂×C₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	#97	D₃₄₉
hypocenter	C₃₄₉	C₄×M₁₆
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #26013550	C₂	semidirect
4. C₈×Dic₃₄₉	C₂	semidirect
5. #97×C₃₄₉	C₂	semidirect
8. #19150216	C₄	semidirect
19. #14584095	C₂×C₄	semidirect
22. C₈×C₃₄₉	C₂×C₄	semidirect
29. Dic₃₄₉	M₁₆	semidirect
33. C₃₄₉	C₄×M₁₆	semidirect
38. #97	D₃₄₉	semidirect
56. C₈	C₄×D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150216	2
C₄²×D₃₄₉	2
#26013550	2
C₈×Dic₃₄₉	2
#97×C₃₄₉	2
C₄×M₁₆	349