Subgroups — #36375012

Order 22336.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013718	C₂	1	2	yes	maximal
2	C₂×#19150216	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13063703	—	2	2	no	—
4	C₂²⋊#13063703	—	2	2	no	—
5	C₄×#13063703	C₄	1	4	yes	complemented
6	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
7	#19150216	—	2	2	no	—
8	C₂×#14584095	C₄	1	1	yes	characteristic, complemented
9	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic, complemented
10	C₂×#13063703	—	2	4	no	—
11	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
12	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
13	C₄×D₃₄₉	C₂×C₄	1	2	yes	—
14	C₄×D₃₄₉	D₄	1	2	yes	—
15	#14584095	—	2	1	no	—
16	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₄₉	—	2	1	no	—
18	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₃₄₉	—	4	1	no	—
20	C₂×D₃₄₉	C₂²⋊C₄	1	2	yes	—
21	#13063703	—	4	4	no	—
22	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
23	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₃₄₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₃₄₉	—	2	1	no	—
26	C₂×C₃₄₉	#95	1	1	yes	characteristic
27	D₃₄₉	—	2	2	no	—
28	C₃₄₉	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
29	H(3,ℤ/4)	—	349	1	no	maximal, Sylow
30	C₂×M₁₆	—	349	1	no	—
31	#117	—	349	2	no	—
32	C₂²×C₄	—	349	1	no	—
33	M₁₆	—	698	2	no	—
34	C₂×C₈	—	349	1	no	—
35	C₂×C₈	#13063703	1	1	yes	characteristic, complemented
36	C₄⋊C₄	—	698	2	no	—
37	C₂²⋊C₄	—	698	2	no	—
38	C₄²	—	349	4	no	—
39	C₂³	—	349	1	no	—
40	C₂×C₄	—	349	1	no	—
41	C₂×C₄	—	349	2	no	—
42	C₂×C₄	—	349	2	no	—
43	C₂×C₄	—	698	4	no	—
44	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	698	1	no	—
47	C₂²	—	349	2	no	—
48	C₂²	—	1396	1	no	—
49	C₂²	C₄⋊#13063703	1	1	yes	characteristic
50	C₄	—	349	2	no	—
51	C₄	—	1396	4	no	—
52	C₄	C₄×#13063703	1	1	yes	characteristic
53	C₄	C₂²⋊#13063703	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	698	2	no	—
56	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#13063703
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂×C₄	C₂×#13063703
Fitting	C₂×C₈×C₃₄₉	C₄
socle	C₂×C₃₄₉	#95
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×D₃₄₉	C₂²
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	H(3,ℤ/4)
LCS	G ▷ C₄×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄×#13063703	C₄	semidirect
8. C₂×#14584095	C₄	semidirect
9. C₂×C₈×C₃₄₉	C₄	semidirect
28. C₃₄₉	H(3,ℤ/4)	semidirect
35. C₂×C₈	#13063703	semidirect

Maximal subgroups

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

subgroup	index
#26013718	2
C₂×#19150216	2
H(3,ℤ/4)	349