Subgroups — #28416839

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	#21418469	C₂	1	1	yes	maximal, characteristic
3	#21418468	C₂	1	2	yes	maximal
4	C₈×Dic₂₃₃	C₂	1	1	yes	maximal, characteristic
5	#107×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	1	yes	characteristic
16	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
17	#14272338	Q₈	1	2	yes	complemented
18	C₂×C₄×C₂₃₃	—	2	2	no	—
19	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
20	C₈×C₂₃₃	Q₈	1	2	yes	complemented
21	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₃₃	—	4	2	no	—
23	C₄×C₂₃₃	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₂₃₃	C₂×Q₈	1	1	yes	characteristic
25	Dic₂₃₃	—	4	2	no	—
26	Dic₂₃₃	D₈	1	2	yes	—
27	Dic₂₃₃	Q₁₆	1	2	yes	—
28	C₂×C₂₃₃	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₂₃₃	C₂×D₈	1	1	yes	characteristic
30	C₂×C₂₃₃	C₂×Q₁₆	1	1	yes	characteristic
31	C₂₃₃	#500	1	1	yes	characteristic, complemented, Sylow
32	#500	—	233	1	no	maximal, Sylow
33	C₄⋊Q₈	—	233	2	no	—
34	#107	—	233	1	no	—
35	#107	—	233	2	no	—
36	#107	D₂₃₃	1	1	yes	characteristic
37	C₄×C₈	—	233	1	no	—
38	C₂×Q₈	—	466	2	no	—
39	C₂×C₈	—	233	1	no	—
40	C₂×C₈	C₂×D₂₃₃	1	1	yes	characteristic
41	C₄⋊C₄	—	466	2	no	—
42	C₄⋊C₄	—	233	2	no	—
43	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
44	C₄²	—	233	1	no	—
45	Q₈	—	932	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	466	2	no	—
48	C₂×C₄	—	233	1	no	—
49	C₂×C₄	—	233	1	no	—
50	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
51	C₈	—	233	2	no	—
52	C₈	Dic₄₆₆	1	2	yes	complemented
53	C₂²	D₄×D₂₃₃	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	932	2	no	—
56	C₄	—	233	2	no	—
57	C₄	—	233	2	no	—
58	C₄	Q₈×D₂₃₃	1	1	yes	characteristic
59	C₄	C₂×Dic₄₆₆	1	1	yes	characteristic
60	C₂	#21418534	1	1	yes	characteristic
61	C₂	D₈×D₂₃₃	1	1	yes	characteristic
62	C₂	Q₁₆×D₂₃₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₃
commutator, cosocle	C₂×C₄×C₂₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting	#107×C₂₃₃	C₂
socle, prime radical	C₂²×C₂₃₃	C₂×D₄
hypercenter	#107	D₂₃₃
hypocenter	C₂₃₃	#500
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
17. #14272338	Q₈	semidirect
20. C₈×C₂₃₃	Q₈	semidirect
31. C₂₃₃	#500	semidirect
52. 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
#21418469	2
#21418468	2
C₈×Dic₂₃₃	2
#107×C₂₃₃	2
#500	233