Subgroups — #45666205

Order 32896.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657169	C₂	1	1	yes	maximal, characteristic
2	#30656853	C₂	1	2	yes	maximal, complemented
3	#30656543	C₂	1	2	yes	maximal, complemented
4	#30656512	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#22551614	C₂²	1	2	yes	—
7	#22551600	—	2	2	no	—
8	Dic₂₀₅₆	C₄	1	1	yes	characteristic, complemented
9	#22551537	C₄	1	2	yes	complemented
10	D₄₁₁₂	C₄	1	1	yes	characteristic, complemented
11	C₂×#16861593	C₂²	1	1	yes	characteristic
12	#22551518	—	2	2	no	—
13	C₈×Dic₂₅₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
15	#108×C₂₅₇	C₂²	1	2	yes	—
16	#16861618	—	2	2	no	—
17	Dic₁₀₂₈	C₂×C₄	1	2	yes	—
18	#16861607	—	2	2	no	—
19	D₂₀₅₆	C₂×C₄	1	2	yes	—
20	#16861601	—	4	2	no	—
21	C₂×#14523339	D₄	1	1	yes	characteristic
22	C₈×D₂₅₇	—	4	2	no	—
23	C₄×Dic₂₅₇	D₄	1	1	yes	characteristic
24	#16861593	—	2	1	no	—
25	M₁₆×C₂₅₇	—	2	2	no	—
26	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₂₅₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₂₅₇	C₂×C₄	1	1	yes	characteristic
29	Dic₅₁₄	—	2	2	no	—
30	D₁₀₂₈	—	2	2	no	—
31	C₂²⋊D₂₅₇	—	4	2	no	—
32	C₂×Dic₂₅₇	—	2	1	no	—
33	C₄×D₂₅₇	—	4	2	no	—
34	#14523339	D₈	1	2	yes	—
35	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
36	C₈×C₂₅₇	—	4	2	no	—
37	C₈×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₂₅₇	#41	1	1	yes	characteristic
39	C₂²×C₂₅₇	#135	1	1	yes	characteristic
40	C₂×D₂₅₇	—	4	2	no	—
41	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
42	C₄×C₂₅₇	C₂×D₈	1	1	yes	characteristic
43	Dic₂₅₇	—	4	1	no	—
44	Dic₂₅₇	—	4	2	no	—
45	C₂×C₂₅₇	—	2	1	no	—
46	C₂×C₂₅₇	C₄×D₈	1	1	yes	characteristic
47	D₂₅₇	—	8	2	no	—
48	C₂₅₇	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	257	1	no	maximal, Sylow
50	#508	—	257	1	no	—
51	#443	—	257	2	no	—
52	#368	D₂₅₇	1	1	yes	characteristic, complemented
53	#359	—	257	2	no	—
54	#346	—	257	1	no	—
55	#135	—	257	2	no	—
56	#131	—	514	2	no	—
57	Q₃₂	—	257	1	no	—
58	SD₃₂	—	257	2	no	—
59	D₁₆	—	257	1	no	—
60	C₂×C₁₆	—	257	1	no	—
61	C₂×C₁₆	C₂×D₂₅₇	1	1	yes	characteristic
62	#108	C₂×D₂₅₇	1	2	yes	—
63	#104	—	514	2	no	—
64	C₄×C₈	—	257	1	no	—
65	#41	—	514	2	no	—
66	Q₁₆	—	257	2	no	—
67	SD₁₆	—	514	2	no	—
68	D₈	—	257	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	1028	2	no	—
71	C₂×C₈	—	257	1	no	—
72	C₂×C₈	—	1028	2	no	—
73	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
74	C₄²	—	257	1	no	—
75	C₁₆	—	514	1	no	—
76	C₁₆	C₄×D₂₅₇	1	1	yes	characteristic
77	C₁₆	C₄×D₂₅₇	1	1	yes	characteristic
78	Q₈	—	514	2	no	—
79	D₄	—	514	2	no	—
80	D₄	—	1028	2	no	—
81	C₂×C₄	—	514	1	no	—
82	C₂×C₄	—	1028	2	no	—
83	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
84	C₈	—	257	2	no	—
85	C₈	—	4	2	no	—
86	C₈	C₂×C₄×D₂₅₇	1	1	yes	characteristic
87	C₈	#16861619	1	1	yes	characteristic
88	C₂²	—	1028	2	no	—
89	C₂²	#22551615	1	1	yes	characteristic
90	C₄	—	1028	1	no	—
91	C₄	—	1028	2	no	—
92	C₄	#22551554	1	1	yes	characteristic
93	C₄	D₈×D₂₅₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2056	2	no	—
96	C₂	#30656819	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₂₅₇
commutator	C₈×C₂₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₇
Fitting	#368×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₄×D₈
prime radical	#30657169	C₂
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	#368	D₂₅₇
hypocenter	C₂₅₇	#2179
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #30656853	C₂	semidirect
3. #30656543	C₂	semidirect
4. #30656512	C₂	semidirect
5. #368×C₂₅₇	C₂	semidirect
8. Dic₂₀₅₆	C₄	semidirect
9. #22551537	C₄	semidirect
10. D₄₁₁₂	C₄	semidirect
48. C₂₅₇	#2179	semidirect
52. #368	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#30657169	2
#30656853	2
#30656543	2
#30656512	2
#368×C₂₅₇	2
#2179	257