Subgroups — #49093491

Order 35968.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31823895	C₂	1	1	yes	maximal, characteristic
2	#31823291	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22582901	—	2	1	no	—
5	Dic₂₂₄₈	C₄	1	1	yes	characteristic, complemented
6	#22582824	—	2	1	no	—
7	D₄₄₉₆	C₄	1	1	yes	characteristic, complemented
8	#22582813	—	2	1	no	—
9	C₂×C₁₆×C₂₈₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₈₁	—	2	1	no	—
11	#17985138	—	4	1	no	—
12	Dic₁₁₂₄	—	2	1	no	—
13	#17985127	—	4	1	no	—
14	D₂₂₄₈	—	2	1	no	—
15	C₄⋊Dic₂₈₁	—	4	1	no	—
16	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
17	C₁₆×C₂₈₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₈₁	D₄	1	1	yes	characteristic
19	Dic₅₆₂	—	4	1	no	—
20	D₁₁₂₄	—	4	1	no	—
21	C₂²⋊D₂₈₁	—	8	1	no	—
22	C₂×Dic₂₈₁	—	8	1	no	—
23	C₄×D₂₈₁	—	8	1	no	—
24	C₂×C₄×C₂₈₁	D₈	1	1	yes	characteristic
25	C₈×C₂₈₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₈₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₈₁	D₁₆	1	1	yes	characteristic
28	C₂×D₂₈₁	—	8	1	no	—
29	C₄×C₂₈₁	#102	1	1	yes	characteristic
30	C₄×C₂₈₁	SD₃₂	1	1	yes	characteristic
31	Dic₂₈₁	—	8	1	no	—
32	Dic₂₈₁	—	16	1	no	—
33	C₂×C₂₈₁	—	2	1	no	—
34	C₂×C₂₈₁	#357	1	1	yes	characteristic
35	D₂₈₁	—	16	1	no	—
36	C₂₈₁	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	281	1	no	maximal, Sylow
38	#508	—	281	1	no	—
39	M₆₄	D₂₈₁	1	1	yes	characteristic, complemented
40	#367	—	281	1	no	—
41	#135	—	562	1	no	—
42	Q₃₂	—	281	1	no	—
43	SD₃₂	—	562	1	no	—
44	D₁₆	—	281	1	no	—
45	C₂×C₁₆	C₂×D₂₈₁	1	1	yes	characteristic
46	#107	—	562	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	1124	1	no	—
49	Q₁₆	—	562	1	no	—
50	SD₁₆	—	1124	1	no	—
51	D₈	—	562	1	no	—
52	C₂×C₈	D₁₁₂₄	1	1	yes	characteristic
53	C₄⋊C₄	—	1124	1	no	—
54	C₁₆	C₄×D₂₈₁	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₂₈₁	1	1	yes	characteristic
56	Q₈	—	1124	1	no	—
57	D₄	—	1124	1	no	—
58	D₄	—	2248	1	no	—
59	C₂×C₄	—	2248	1	no	—
60	C₂×C₄	—	2248	1	no	—
61	C₂×C₄	D₂₂₄₈	1	1	yes	characteristic
62	C₈	#17985115	1	1	yes	characteristic
63	C₈	#17985127	1	1	yes	characteristic
64	C₂²	—	2248	1	no	—
65	C₂²	D₄₄₉₆	1	1	yes	characteristic
66	C₄	—	4496	1	no	—
67	C₄	—	2248	1	no	—
68	C₄	#22582801	1	1	yes	characteristic
69	C₄	#22582824	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	4496	1	no	—
72	C₂	#31823265	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31823265
commutator	C₁₆×C₂₈₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₈₁
Fitting	M₆₄×C₂₈₁	C₂
socle	C₂×C₂₈₁	#357
prime radical	#31823895	C₂
cosocle	C₂×C₁₆×C₂₈₁	C₂²
hypercenter	M₆₄	D₂₈₁
hypocenter	C₂₈₁	#1418
LCS	G ▷ C₁₆×C₂₈₁ ▷ C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #31823291	C₂	semidirect
3. M₆₄×C₂₈₁	C₂	semidirect
5. Dic₂₂₄₈	C₄	semidirect
7. D₄₄₉₆	C₄	semidirect
36. C₂₈₁	#1418	semidirect
39. M₆₄	D₂₈₁	semidirect

Maximal subgroups

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

subgroup	index
#31823895	2
#31823291	2
M₆₄×C₂₈₁	2
#1418	281