Subgroups — #36375007

Order 22336.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14584102	C₂	1	2	yes	maximal
2	C₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14584102	—	2	2	no	—
4	C₂×#14584095	C₄	1	1	yes	characteristic
5	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic
7	#14584102	—	4	2	no	—
8	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
9	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
10	#14584095	—	2	1	no	—
11	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
12	C₈×C₃₄₉	C₈	1	2	yes	complemented
13	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₃₄₉	C₂×C₈	1	1	yes	characteristic
15	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
16	Dic₃₄₉	—	2	1	no	—
17	Dic₃₄₉	D₈	1	1	yes	characteristic
18	Dic₃₄₉	Q₁₆	1	1	yes	characteristic
19	C₂×C₃₄₉	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₃₄₉	#107	1	1	yes	characteristic
21	C₂×C₃₄₉	#108	1	1	yes	characteristic
22	C₃₄₉	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	349	1	no	maximal, Sylow
24	C₄⋊C₈	—	349	2	no	—
25	C₄×C₈	—	349	1	no	—
26	C₂×C₈	—	349	1	no	—
27	C₂×C₈	—	698	2	no	—
28	C₂×C₈	#13063703	1	1	yes	characteristic
29	C₄²	—	349	1	no	—
30	C₂×C₄	—	349	1	no	—
31	C₂×C₄	—	349	1	no	—
32	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
33	C₈	—	1396	2	no	—
34	C₈	—	698	1	no	—
35	C₈	#14584102	1	2	yes	complemented
36	C₂²	C₄⋊#13063703	1	1	yes	characteristic
37	C₄	—	349	1	no	—
38	C₄	—	349	1	no	—
39	C₄	—	698	1	no	—
40	C₄	C₂×#14584102	1	1	yes	characteristic
41	C₄	#19150243	1	1	yes	characteristic
42	C₂	C₄⋊#14584102	1	1	yes	characteristic
43	C₂	#26013708	1	1	yes	characteristic
44	C₂	#26013710	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#13063703
commutator	C₄×C₃₄₉	C₂×C₈
Frattini	C₂×C₄	C₂×#13063703
Fitting	C₂×C₈×C₃₄₉	C₄
socle, prime radical	C₂²×C₃₄₉	C₄⋊C₄
cosocle	C₄×Dic₃₄₉	C₂²
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	#335
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₃₄₉	C₈	semidirect
22. C₃₄₉	#335	semidirect
35. C₈	#14584102	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14584102	2
C₈×Dic₃₄₉	2
#335	349