Subgroups — #36373894

Order 22336.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14584095	C₂	1	1	yes	maximal, characteristic
2	C₄×#14584095	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14584095	—	2	1	no	—
5	C₂×#14584095	C₄	1	1	yes	characteristic
6	C₂×#14584095	C₄	1	1	yes	characteristic
7	C₂×C₈×C₃₄₉	—	2	1	no	—
8	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
9	#14584095	—	2	1	no	—
10	#14584095	—	4	1	no	—
11	#14584095	C₈	1	2	yes	complemented
12	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
14	C₂×C₄×C₃₄₉	Q₈	1	1	yes	characteristic
15	C₈×C₃₄₉	—	4	1	no	—
16	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₃₄₉	—	2	1	no	—
18	C₄×C₃₄₉	C₂×C₈	1	1	yes	characteristic
19	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
20	C₄×C₃₄₉	SD₁₆	1	1	yes	characteristic
21	C₄×C₃₄₉	SD₁₆	1	1	yes	characteristic
22	C₂×C₃₄₉	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₃₄₉	#106	1	1	yes	characteristic
24	C₂×C₃₄₉	#108	1	1	yes	characteristic
25	C₃₄₉	#334	1	1	yes	characteristic, complemented, Sylow
26	#334	—	349	1	no	maximal, Sylow
27	C₄⋊C₈	—	349	1	no	—
28	C₄⋊C₈	D₃₄₉	1	1	yes	characteristic
29	C₄×C₈	—	349	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	349	1	no	—
32	C₂×C₈	—	349	1	no	—
33	C₂×C₈	—	698	1	no	—
34	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
35	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
37	C₂×C₄	Dic₆₉₈	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	349	2	no	—
40	C₈	—	1396	1	no	—
41	C₈	—	698	1	no	—
42	C₂²	#19150212	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₃₄₉	1	1	yes	characteristic
45	C₄	#19150216	1	1	yes	characteristic
46	C₄	#19150220	1	1	yes	characteristic
47	C₄	#19150221	1	1	yes	characteristic
48	C₂	#26013568	1	1	yes	characteristic
49	C₂	#26013570	1	1	yes	characteristic
50	C₂	#26013574	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₄₉
commutator	C₄×C₃₄₉	C₂×C₈
Frattini	C₄²	C₂×D₃₄₉
Fitting	C₄⋊C₈×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₄⋊C₄
cosocle	C₄²×C₃₄₉	C₂²
hypercenter	C₄⋊C₈	D₃₄₉
hypocenter	C₃₄₉	#334
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. #14584095	C₈	semidirect
25. C₃₄₉	#334	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14584095	2
C₄×#14584095	2
C₄⋊C₈×C₃₄₉	2
#334	349