Subgroups — #36373885

Order 22336.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013631	C₂	1	1	yes	maximal, characteristic
2	#26013550	C₂	1	1	yes	maximal, characteristic
3	#97×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14584095	—	2	1	no	—
5	C₄⋊Dic₃₄₉	C₄	1	1	yes	characteristic
6	C₄⋊Dic₃₄₉	C₄	1	1	yes	characteristic
7	#19150212	—	2	1	no	—
8	#19150212	—	2	1	no	—
9	C₂×C₈×C₃₄₉	—	2	1	no	—
10	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₄₉	—	2	1	no	—
12	C₂×Dic₃₄₉	—	2	1	no	—
13	#14584095	—	2	2	no	—
14	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
17	C₈×C₃₄₉	—	2	2	no	—
18	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₃₄₉	—	2	1	no	—
20	C₄×C₃₄₉	—	2	1	no	—
21	C₄×C₃₄₉	—	2	1	no	—
22	Dic₃₄₉	—	4	1	no	—
23	Dic₃₄₉	—	4	1	no	—
24	C₂×C₃₄₉	#101	1	1	yes	characteristic
25	C₂×C₃₄₉	#104	1	1	yes	characteristic
26	C₂×C₃₄₉	#104	1	1	yes	characteristic
27	C₃₄₉	#330	1	1	yes	characteristic, complemented, Sylow
28	#330	—	349	1	no	maximal, Sylow
29	#125	—	349	1	no	—
30	#97	—	349	1	no	—
31	#97	D₃₄₉	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	698	1	no	—
34	C₄⋊C₄	—	349	1	no	—
35	C₄⋊C₄	—	349	1	no	—
36	C₄⋊C₄	—	698	1	no	—
37	C₄⋊C₄	—	698	1	no	—
38	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
39	C₂×C₄	—	698	1	no	—
40	C₂×C₄	—	698	1	no	—
41	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
42	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
43	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	—	698	2	no	—
46	C₂²	#19150210	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	1396	1	no	—
51	C₄	—	1396	1	no	—
52	C₂	#26013557	1	1	yes	characteristic
53	C₂	#26013565	1	1	yes	characteristic
54	C₂	#26013567	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19150210
commutator	C₂×C₄×C₃₄₉	C₂×C₄
Frattini	C₄²	C₂×D₃₄₉
Fitting	#97×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂²⋊C₄
cosocle	C₄²×C₃₄₉	C₂²
hypercenter	#97	D₃₄₉
hypocenter	C₃₄₉	#330
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
27. C₃₄₉	#330	semidirect

Maximal subgroups

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

subgroup	index
#26013631	2
#26013550	2
#97×C₃₄₉	2
#330	349