Subgroups — #28416529

Order 14912.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₉₃₂	C₂	1	1	yes	maximal, characteristic
2	#21418500	C₂	1	2	yes	maximal
3	#21418458	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	C₂²	1	2	yes	—
7	Dic₉₃₂	C₄	1	4	yes	complemented
8	C₂×#14272338	C₂²	1	1	yes	characteristic
9	#16836573	—	2	2	no	—
10	C₄×Dic₂₃₃	—	2	2	no	—
11	C₄×Dic₂₃₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
14	Dic₄₆₆	—	2	2	no	—
15	Dic₄₆₆	C₂×C₄	1	4	yes	—
16	C₂×Dic₂₃₃	—	2	2	no	—
17	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
18	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
19	#14272338	—	2	1	no	—
20	C₂×C₄×C₂₃₃	—	2	2	no	—
21	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
22	C₈×C₂₃₃	C₂×C₄	1	2	yes	—
23	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₃₃	—	4	2	no	—
25	C₄×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₂₃₃	#41	1	1	yes	characteristic
27	Dic₂₃₃	—	2	4	no	—
28	Dic₂₃₃	—	2	1	no	—
29	Dic₂₃₃	Q₁₆	1	2	yes	complemented
30	C₂×C₂₃₃	C₄×D₄	1	1	yes	characteristic
31	C₂×C₂₃₃	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₂₃₃	#135	1	1	yes	characteristic
33	C₂₃₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	233	1	no	maximal, Sylow
35	C₂×Q₁₆	—	233	1	no	—
36	C₄×Q₈	—	233	2	no	—
37	#107	D₂₃₃	1	1	yes	characteristic
38	#103	—	233	2	no	—
39	C₄×C₈	—	233	1	no	—
40	C₂×Q₈	—	233	2	no	—
41	Q₁₆	—	233	4	no	—
42	C₂×C₈	—	233	1	no	—
43	C₂×C₈	C₂×D₂₃₃	1	1	yes	characteristic
44	C₄⋊C₄	—	466	2	no	—
45	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
46	C₄²	—	466	2	no	—
47	C₄²	—	233	1	no	—
48	Q₈	—	466	2	no	—
49	Q₈	—	233	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	466	2	no	—
52	C₂×C₄	—	233	1	no	—
53	C₂×C₄	—	233	1	no	—
54	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
55	C₈	—	466	1	no	—
56	C₈	C₄×D₂₃₃	1	2	yes	—
57	C₂²	D₄×D₂₃₃	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	466	4	no	—
60	C₄	—	466	1	no	—
61	C₄	—	233	2	no	—
62	C₄	C₂×C₄×D₂₃₃	1	1	yes	characteristic
63	C₄	#16836595	1	1	yes	characteristic
64	C₂	#21418497	1	1	yes	characteristic
65	C₂	Q₁₆×D₂₃₃	1	1	yes	characteristic
66	C₂	#21418554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₃
commutator	C₄×C₂₃₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting	#107×C₂₃₃	C₂
socle, prime radical	C₂²×C₂₃₃	C₂×D₄
cosocle	C₂×C₄×C₂₃₃	C₂³
hypercenter	#107	D₂₃₃
hypocenter	C₂₃₃	C₄×Q₁₆
LCS	G ▷ 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
7. Dic₉₃₂	C₄	semidirect
29. Dic₂₃₃	Q₁₆	semidirect
33. C₂₃₃	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₉₃₂	2
#21418500	2
#21418458	2
C₈×Dic₂₃₃	2
#107×C₂₃₃	2
C₄×Q₁₆	233