Subgroups — #37588104

Order 24512.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃₈₃⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₃₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₈₃⋊C₈	—	2	2	no	—
4	C₂×C₈×C₃₈₃	C₄	1	1	yes	characteristic
5	C₂×C₈×C₃₈₃	C₄	1	1	yes	characteristic
6	C₄²×C₃₈₃	C₂²	1	1	yes	characteristic
7	C₃₈₃⋊C₈	—	4	2	no	—
8	C₂×C₄×C₃₈₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₃₈₃	D₄	1	1	yes	characteristic
10	C₂×C₄×C₃₈₃	Q₈	1	1	yes	characteristic
11	C₈×C₃₈₃	—	2	1	no	—
12	C₈×C₃₈₃	C₈	1	2	yes	complemented
13	C₂²×C₃₈₃	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₃₈₃	—	2	1	no	—
15	C₄×C₃₈₃	C₂×C₈	1	1	yes	characteristic
16	C₄×C₃₈₃	M₁₆	1	1	yes	characteristic
17	C₄×C₃₈₃	D₈	1	1	yes	characteristic
18	C₄×C₃₈₃	Q₁₆	1	1	yes	characteristic
19	C₂×C₃₈₃	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₃₈₃	#107	1	1	yes	characteristic
21	C₂×C₃₈₃	#108	1	1	yes	characteristic
22	C₃₈₃	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	383	1	no	maximal, Sylow
24	C₄⋊C₈	—	383	2	no	—
25	C₄×C₈	D₃₈₃	1	1	yes	characteristic
26	C₂×C₈	—	766	2	no	—
27	C₂×C₈	Dic₃₈₃	1	1	yes	characteristic
28	C₂×C₈	Dic₃₈₃	1	1	yes	characteristic
29	C₄²	C₂×D₃₈₃	1	1	yes	characteristic
30	C₂×C₄	C₂×Dic₃₈₃	1	1	yes	characteristic
31	C₂×C₄	D₁₅₃₂	1	1	yes	characteristic
32	C₂×C₄	Dic₇₆₆	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	1532	2	no	—
35	C₈	C₃₈₃⋊C₈	1	2	yes	complemented
36	C₂²	C₄⋊Dic₃₈₃	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	C₂×C₃₈₃⋊C₈	1	1	yes	characteristic
39	C₄	#20259441	1	1	yes	characteristic
40	C₄	D₃₀₆₄	1	1	yes	characteristic
41	C₄	Dic₁₅₃₂	1	1	yes	characteristic
42	C₂	C₄⋊(C₃₈₃⋊C₈)	1	1	yes	characteristic
43	C₂	#27133775	1	1	yes	characteristic
44	C₂	#27133777	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₅₃₂
commutator	C₄×C₃₈₃	C₂×C₈
Frattini	C₄²	C₂×D₃₈₃
Fitting	C₄×C₈×C₃₈₃	C₂
socle, prime radical	C₂²×C₃₈₃	C₄⋊C₄
cosocle	C₄²×C₃₈₃	C₂²
hypercenter	C₄×C₈	D₃₈₃
hypocenter	C₃₈₃	#335
LCS	G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₃₈₃	C₈	semidirect
22. C₃₈₃	#335	semidirect
35. C₈	C₃₈₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃₈₃⋊C₈)	2
C₄×C₈×C₃₈₃	2
#335	383