Subgroups — #28417164

Order 14912.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418535	C₂	1	3	yes	maximal
2	#21418526	C₂	1	6	yes	maximal
3	Q₈×Dic₂₃₃	C₂	1	2	yes	maximal
4	C₄×Dic₄₆₆	C₂	1	3	yes	maximal
5	C₄×Q₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₆₆	C₂²	1	3	yes	—
7	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₃₃	C₂²	1	6	yes	—
9	C₄⋊Dic₂₃₃	C₂²	1	6	yes	—
10	#16836573	C₂²	1	6	yes	—
11	C₄×Dic₂₃₃	C₂²	1	6	yes	—
12	C₂×Q₈×C₂₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₃₃	C₂²	1	3	yes	—
14	C₄²×C₂₃₃	C₂²	1	3	yes	—
15	Dic₄₆₆	—	2	6	no	—
16	C₂×Dic₂₃₃	C₂³	1	2	yes	—
17	C₂×Dic₂₃₃	C₂³	1	6	yes	—
18	Q₈×C₂₃₃	Q₈	1	4	yes	complemented
19	C₂×C₄×C₂₃₃	C₂³	1	3	yes	—
20	C₂×C₄×C₂₃₃	C₂³	1	3	yes	—
21	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
22	C₂²×C₂₃₃	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₃₃	—	2	3	no	—
24	C₄×C₂₃₃	C₂×Q₈	1	6	yes	—
25	C₄×C₂₃₃	#41	1	2	yes	—
26	Dic₂₃₃	—	2	6	no	—
27	Dic₂₃₃	—	2	2	no	—
28	C₂×C₂₃₃	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₂₃₃	C₂×#41	1	1	yes	characteristic
30	C₂×C₂₃₃	2₋¹⁺⁴	1	1	yes	characteristic
31	C₂₃₃	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	233	1	no	maximal, Sylow
33	C₄⋊Q₈	—	233	3	no	—
34	#125	—	233	6	no	—
35	C₄×Q₈	—	233	2	no	—
36	C₄×Q₈	—	233	3	no	—
37	C₄×Q₈	D₂₃₃	1	1	yes	characteristic
38	C₂×Q₈	—	233	3	no	—
39	C₂×Q₈	C₂×D₂₃₃	1	1	yes	characteristic
40	C₄⋊C₄	—	233	1	no	—
41	C₄⋊C₄	—	233	6	no	—
42	C₄⋊C₄	—	233	6	no	—
43	C₄⋊C₄	—	233	6	no	—
44	C₄⋊C₄	C₂×D₂₃₃	1	3	yes	—
45	C₄²	—	233	6	no	—
46	C₄²	C₂×D₂₃₃	1	3	yes	—
47	Q₈	—	466	6	no	—
48	Q₈	Dic₄₆₆	1	4	yes	complemented
49	C₂×C₄	—	233	2	no	—
50	C₂×C₄	—	233	6	no	—
51	C₂×C₄	C₂²×D₂₃₃	1	3	yes	—
52	C₂×C₄	C₂²×D₂₃₃	1	3	yes	—
53	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
54	C₂²	C₂³×D₂₃₃	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	466	6	no	—
57	C₄	—	466	2	no	—
58	C₄	C₂×Dic₄₆₆	1	6	yes	—
59	C₄	#16836595	1	2	yes	—
60	C₂	C₂²×Dic₄₆₆	1	1	yes	characteristic
61	C₂	C₂×#16836595	1	1	yes	characteristic
62	C₂	#21418582	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₃
commutator, socle, prime radical, cosocle	C₂²×C₂₃₃	C₂⁴
Fitting	C₄×Q₈×C₂₃₃	C₂
hypercenter	C₄×Q₈	D₂₃₃
hypocenter	C₂₃₃	#557
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₂₃₃	Q₈	semidirect
31. C₂₃₃	#557	semidirect
48. Q₈	Dic₄₆₆	semidirect

Maximal subgroups

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

subgroup	index
#21418535	2
#21418526	2
Q₈×Dic₂₃₃	2
C₄×Dic₄₆₆	2
C₄×Q₈×C₂₃₃	2
#557	233