Subgroups — #28416832

Order 14912.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418534	C₂	1	2	yes	maximal
2	#21418467	C₂	1	1	yes	maximal, characteristic
3	#21418466	C₂	1	2	yes	maximal
4	C₈×Dic₂₃₃	C₂	1	1	yes	maximal, characteristic
5	#106×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₆₆	—	2	2	no	—
7	C₂×#14272338	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₃₃	C₂²	1	2	yes	—
9	#16836573	—	2	2	no	—
10	C₄×Dic₂₃₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₃₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
13	Dic₄₆₆	—	4	4	no	—
14	C₂×Dic₂₃₃	—	2	2	no	—
15	C₂×Dic₂₃₃	D₄	1	2	yes	—
16	#14272338	Q₈	1	2	yes	complemented
17	C₂×C₄×C₂₃₃	—	2	2	no	—
18	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
19	C₈×C₂₃₃	Q₈	1	2	yes	complemented
20	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₃₃	—	4	2	no	—
22	C₄×C₂₃₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₂₃₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₂₃₃	—	4	2	no	—
25	Dic₂₃₃	SD₁₆	1	4	yes	—
26	C₂×C₂₃₃	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₂₃₃	C₂×SD₁₆	1	2	yes	—
28	C₂₃₃	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	233	1	no	maximal, Sylow
30	C₄⋊Q₈	—	233	2	no	—
31	#106	—	233	1	no	—
32	#106	—	233	2	no	—
33	#106	D₂₃₃	1	1	yes	characteristic
34	C₄×C₈	—	233	1	no	—
35	C₂×Q₈	—	466	2	no	—
36	C₂×C₈	—	233	1	no	—
37	C₂×C₈	C₂×D₂₃₃	1	1	yes	characteristic
38	C₄⋊C₄	—	466	2	no	—
39	C₄⋊C₄	—	233	2	no	—
40	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
41	C₄²	—	233	1	no	—
42	Q₈	—	932	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	466	2	no	—
45	C₂×C₄	—	233	2	no	—
46	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
47	C₈	—	233	2	no	—
48	C₈	Dic₄₆₆	1	2	yes	complemented
49	C₂²	D₄×D₂₃₃	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	932	2	no	—
52	C₄	—	233	4	no	—
53	C₄	Q₈×D₂₃₃	1	1	yes	characteristic
54	C₄	C₂×Dic₄₆₆	1	1	yes	characteristic
55	C₂	#21418534	1	1	yes	characteristic
56	C₂	SD₁₆×D₂₃₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₃
commutator, cosocle	C₂×C₄×C₂₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting	#106×C₂₃₃	C₂
socle, prime radical	C₂²×C₂₃₃	C₂×D₄
hypercenter	#106	D₂₃₃
hypocenter	C₂₃₃	#498
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
16. #14272338	Q₈	semidirect
19. C₈×C₂₃₃	Q₈	semidirect
28. C₂₃₃	#498	semidirect
48. C₈	Dic₄₆₆	semidirect

Maximal subgroups

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

subgroup	index
#21418534	2
#21418467	2
#21418466	2
C₈×Dic₂₃₃	2
#106×C₂₃₃	2
#498	233