Subgroups — #28416546

Order 14912.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₈₆₄	C₂	1	1	yes	maximal, characteristic
2	#21418497	C₂	1	2	yes	maximal, complemented
3	#21418455	C₂	1	2	yes	maximal, complemented
4	#21418445	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₃₂	C₂²	1	2	yes	—
7	C₂×C₄×D₂₃₃	—	2	2	no	—
8	D₁₈₆₄	C₄	1	4	yes	complemented
9	C₂×#14272338	C₂²	1	1	yes	characteristic
10	#16836571	—	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	C₂²×D₂₃₃	—	2	2	no	—
15	D₉₃₂	—	2	2	no	—
16	D₉₃₂	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₂₃₃	D₄	1	2	yes	—
18	C₄×D₂₃₃	—	4	4	no	—
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	complemented
23	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₃₃	—	2	4	no	—
25	C₂×D₂₃₃	—	4	2	no	—
26	C₂×D₂₃₃	—	4	2	no	—
27	C₄×C₂₃₃	—	4	2	no	—
28	C₄×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₃₃	#41	1	1	yes	characteristic
30	Dic₂₃₃	—	2	2	no	—
31	C₂×C₂₃₃	C₄×D₄	1	1	yes	characteristic
32	C₂×C₂₃₃	AGL(1,ℤ/8)	1	2	yes	—
33	D₂₃₃	—	4	4	no	—
34	C₂₃₃	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	233	1	no	maximal, Sylow
36	C₂×D₈	—	233	1	no	—
37	C₄×D₄	—	233	2	no	—
38	#106	D₂₃₃	1	1	yes	characteristic, complemented
39	#102	—	233	2	no	—
40	#97	—	233	1	no	—
41	C₂×D₄	—	233	2	no	—
42	C₂²×C₄	—	466	2	no	—
43	D₈	—	233	4	no	—
44	C₂×C₈	—	233	1	no	—
45	C₂×C₈	C₂×D₂₃₃	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
47	C₂²⋊C₄	—	466	2	no	—
48	C₄²	—	233	1	no	—
49	C₂³	—	466	2	no	—
50	D₄	—	466	2	no	—
51	D₄	—	233	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	932	4	no	—
54	C₂×C₄	—	233	2	no	—
55	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
56	C₈	—	466	1	no	—
57	C₈	C₄×D₂₃₃	1	2	yes	complemented
58	C₂²	—	932	2	no	—
59	C₂²	—	932	2	no	—
60	C₂²	—	466	4	no	—
61	C₂²	D₄×D₂₃₃	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	466	2	no	—
64	C₄	C₂×C₄×D₂₃₃	1	1	yes	characteristic
65	C₄	#16836595	1	1	yes	characteristic
66	C₂	—	932	4	no	—
67	C₂	#21418497	1	1	yes	characteristic
68	C₂	#21418562	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃₃
commutator	C₄×C₂₃₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting	#106×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×D₄
prime radical	C₂×D₁₈₆₄	C₂
cosocle	C₂×C₄×C₂₃₃	C₂³
hypercenter	#106	D₂₃₃
hypocenter	C₂₃₃	#442
LCS	G ▷ 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
2. #21418497	C₂	semidirect
3. #21418455	C₂	semidirect
4. #21418445	C₂	semidirect
5. #106×C₂₃₃	C₂	semidirect
8. D₁₈₆₄	C₄	semidirect
16. D₉₃₂	C₂×C₄	semidirect
22. C₈×C₂₃₃	C₂×C₄	semidirect
34. C₂₃₃	#442	semidirect
38. #106	D₂₃₃	semidirect
57. C₈	C₄×D₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₈₆₄	2
#21418497	2
#21418455	2
#21418445	2
#106×C₂₃₃	2
#442	233