Subgroups — #45662294

Order 32896.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656855	C₂	1	1	yes	maximal, characteristic
2	#30656558	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈²×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22551614	C₄	1	1	yes	characteristic
5	#22551598	—	2	1	no	—
6	#22551532	—	2	1	no	—
7	#22551528	C₄	1	1	yes	characteristic
8	#22551520	—	2	1	no	—
9	C₄×C₈×C₂₅₇	—	2	1	no	—
10	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
11	#16861618	—	2	1	no	—
12	Dic₁₀₂₈	C₈	1	1	yes	characteristic, complemented
13	#16861607	—	2	1	no	—
14	D₂₀₅₆	C₈	1	1	yes	characteristic, complemented
15	#16861602	—	2	1	no	—
16	#16861601	—	4	1	no	—
17	C₈×D₂₅₇	—	4	1	no	—
18	#16861593	—	4	1	no	—
19	C₂×C₈×C₂₅₇	—	2	2	no	—
20	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
22	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
23	Dic₅₁₄	—	2	1	no	—
24	D₁₀₂₈	—	2	1	no	—
25	C₂²⋊D₂₅₇	—	4	1	no	—
26	C₄×D₂₅₇	—	4	1	no	—
27	#14523339	—	4	1	no	—
28	C₂×C₄×C₂₅₇	—	2	1	no	—
29	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₈×C₂₅₇	—	2	4	no	—
31	C₈×C₂₅₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₅₇	M₁₆	1	1	yes	characteristic
33	C₈×C₂₅₇	D₈	1	1	yes	characteristic
34	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₅₇	#104	1	1	yes	characteristic
36	C₂×D₂₅₇	—	4	1	no	—
37	C₄×C₂₅₇	—	2	2	no	—
38	C₄×C₂₅₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₅₇	#102	1	1	yes	characteristic
40	Dic₂₅₇	—	4	1	no	—
41	C₂×C₂₅₇	—	2	1	no	—
42	C₂×C₂₅₇	D₄⋊C₈	1	1	yes	characteristic
43	D₂₅₇	—	8	1	no	—
44	C₂₅₇	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	257	1	no	maximal, Sylow
46	#443	—	257	1	no	—
47	#364	—	257	1	no	—
48	C₈²	D₂₅₇	1	1	yes	characteristic, complemented
49	#135	—	257	1	no	—
50	#131	—	514	1	no	—
51	M₃₂	—	514	1	no	—
52	#108	—	257	1	no	—
53	#104	—	514	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
56	#41	—	514	1	no	—
57	Q₁₆	—	257	1	no	—
58	SD₁₆	—	514	1	no	—
59	D₈	—	257	1	no	—
60	M₁₆	—	514	1	no	—
61	M₁₆	—	1028	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	1028	1	no	—
64	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
65	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
66	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
67	C₁₆	—	1028	1	no	—
68	Q₈	—	514	1	no	—
69	D₄	—	514	1	no	—
70	D₄	—	1028	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	1028	1	no	—
73	C₂×C₄	#16861595	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	1028	1	no	—
76	C₈	C₈×D₂₅₇	1	1	yes	characteristic
77	C₈	#16861601	1	1	yes	characteristic
78	C₈	D₂₀₅₆	1	1	yes	characteristic
79	C₈	#16861607	1	1	yes	characteristic
80	C₂²	—	1028	1	no	—
81	C₂²	#22551520	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	1028	1	no	—
84	C₄	#22551504	1	1	yes	characteristic
85	C₄	#22551514	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2056	1	no	—
88	C₂	#30656466	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₀₅₆
commutator	C₈×C₂₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	C₈²×C₂₅₇	C₂
socle	C₂×C₂₅₇	D₄⋊C₈
prime radical	#22551614	C₄
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	C₈²	D₂₅₇
hypocenter	C₂₅₇	#1335
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #30656558	C₂	semidirect
3. C₈²×C₂₅₇	C₂	semidirect
12. Dic₁₀₂₈	C₈	semidirect
14. D₂₀₅₆	C₈	semidirect
44. C₂₅₇	#1335	semidirect
48. C₈²	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#30656855	2
#30656558	2
C₈²×C₂₅₇	2
#1335	257