Subgroups — #36373901

Order 22336.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013598	C₂	1	2	yes	maximal
2	C₂×M₁₆×C₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₃₄₉	—	2	2	no	—
4	C₂²⋊Dic₃₄₉	—	2	2	no	—
5	C₄×Dic₃₄₉	C₄	1	4	yes	complemented
6	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
7	M₁₆×C₃₄₉	—	2	2	no	—
8	C₂×C₈×C₃₄₉	C₄	1	2	yes	complemented
9	C₂×Dic₃₄₉	—	2	4	no	—
10	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
11	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
14	C₂×C₄×C₃₄₉	Q₈	1	1	yes	characteristic
15	C₈×C₃₄₉	—	2	2	no	—
16	C₂²×C₃₄₉	—	4	1	no	—
17	C₂²×C₃₄₉	C₂²⋊C₄	1	2	yes	—
18	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
20	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₄₉	C₄⋊C₄	1	2	yes	—
22	Dic₃₄₉	—	4	4	no	—
23	C₂×C₃₄₉	—	2	2	no	—
24	C₂×C₃₄₉	—	2	1	no	—
25	C₂×C₃₄₉	#95	1	1	yes	characteristic
26	C₃₄₉	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
27	H(3,ℤ/4)	—	349	1	no	maximal, Sylow
28	C₂×M₁₆	D₃₄₉	1	1	yes	characteristic
29	#117	—	349	2	no	—
30	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	C₂×C₈	Dic₃₄₉	1	2	yes	complemented
33	C₄⋊C₄	—	698	2	no	—
34	C₂²⋊C₄	—	698	2	no	—
35	C₄²	—	349	4	no	—
36	C₂³	D₁₃₉₆	1	1	yes	characteristic
37	C₂×C₄	—	698	4	no	—
38	C₂×C₄	C₄×D₃₄₉	1	2	yes	—
39	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
40	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
41	C₂×C₄	Dic₆₉₈	1	1	yes	characteristic
42	C₈	—	2	2	no	—
43	C₂²	—	4	1	no	—
44	C₂²	#19150210	1	2	yes	—
45	C₂²	C₄⋊Dic₃₄₉	1	1	yes	characteristic
46	C₄	—	1396	4	no	—
47	C₄	C₄×Dic₃₄₉	1	1	yes	characteristic
48	C₄	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
49	C₄	#19150212	1	2	yes	—
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	#26013546	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₃₄₉
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂²×C₄	C₂×D₃₄₉
Fitting	C₂×M₁₆×C₃₄₉	C₂
socle	C₂×C₃₄₉	#95
prime radical	C₂³×C₃₄₉	D₄
cosocle	C₂²×C₄×C₃₄₉	C₂²
hypercenter	C₂×M₁₆	D₃₄₉
hypocenter	C₃₄₉	H(3,ℤ/4)
LCS	G ▷ C₄×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄×Dic₃₄₉	C₄	semidirect
8. C₂×C₈×C₃₄₉	C₄	semidirect
26. C₃₄₉	H(3,ℤ/4)	semidirect
32. C₂×C₈	Dic₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013598	2
C₂×M₁₆×C₃₄₉	2
H(3,ℤ/4)	349