Subgroups — #30656691

Order 16448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861601	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#22551549	C₂	1	1	yes	maximal, characteristic
4	#22551503	C₂	1	1	yes	maximal, characteristic, complemented
5	#22551502	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
9	#16861601	C₄	1	4	yes	complemented
10	C₂×#14523339	C₂²	1	1	yes	characteristic
11	C₂×#14523339	C₂²	1	1	yes	characteristic
12	C₈×D₂₅₇	C₄	1	4	yes	complemented
13	#16861597	C₄	1	2	yes	—
14	#16861595	C₄	1	2	yes	—
15	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
19	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₅₇	C₂×C₄	1	2	yes	—
23	C₄×D₂₅₇	C₂×C₄	1	2	yes	—
24	#14523339	C₂×C₄	1	2	yes	—
25	#14523339	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
28	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
29	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
30	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₂₅₇	—	2	2	no	—
32	C₂×D₂₅₇	C₄²	1	2	yes	—
33	C₄×C₂₅₇	—	2	2	no	—
34	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
36	Dic₂₅₇	—	2	2	no	—
37	Dic₂₅₇	C₄²	1	2	yes	—
38	C₂×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₂₅₇	#131	1	2	yes	—
40	D₂₅₇	—	2	2	no	—
41	C₂₅₇	#405	1	1	yes	characteristic, complemented, Sylow
42	#405	—	257	1	no	maximal, Sylow
43	C₂×M₁₆	—	257	1	no	—
44	C₂²×C₈	—	257	1	no	—
45	#117	—	257	1	no	—
46	#97	—	257	1	no	—
47	#97	—	257	1	no	—
48	C₄×C₈	—	257	1	no	—
49	C₄×C₈	D₂₅₇	1	1	yes	characteristic, complemented
50	C₂²×C₄	—	257	1	no	—
51	M₁₆	—	257	4	no	—
52	C₂×C₈	—	257	1	no	—
53	C₂×C₈	—	257	1	no	—
54	C₂×C₈	—	257	4	no	—
55	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
57	C₄⋊C₄	—	257	2	no	—
58	C₂²⋊C₄	—	257	2	no	—
59	C₄²	—	257	1	no	—
60	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
61	C₂³	—	257	1	no	—
62	C₂×C₄	—	257	1	no	—
63	C₂×C₄	—	257	2	no	—
64	C₂×C₄	—	257	2	no	—
65	C₂×C₄	—	257	2	no	—
66	C₂×C₄	C₄×D₂₅₇	1	2	yes	—
67	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
68	C₈	—	257	2	no	—
69	C₈	—	257	2	no	—
70	C₈	C₄×D₂₅₇	1	2	yes	—
71	C₈	C₄×D₂₅₇	1	2	yes	—
72	C₂²	—	257	2	no	—
73	C₂²	—	514	2	no	—
74	C₂²	C₂×C₄×D₂₅₇	1	1	yes	characteristic
75	C₄	—	257	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	514	2	no	—
78	C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
79	C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
80	C₂	—	514	2	no	—
81	C₂	C₄²×D₂₅₇	1	1	yes	characteristic
82	C₂	#22551598	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₅₇
commutator	C₂×C₂₅₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₄×C₈×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂²×C₄
prime radical	C₂²×D₂₅₇	C₂×C₄
cosocle	C₂×C₄×C₂₅₇	C₂³
hypercenter	C₄×C₈	D₂₅₇
hypocenter	C₂₅₇	#405
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #22551503	C₂	semidirect
5. #22551502	C₂	semidirect
6. C₈×Dic₂₅₇	C₂	semidirect
7. C₄×C₈×C₂₅₇	C₂	semidirect
9. #16861601	C₄	semidirect
12. C₈×D₂₅₇	C₄	semidirect
41. C₂₅₇	#405	semidirect
49. C₄×C₈	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861601	2
C₂×C₈×D₂₅₇	2
#22551549	2
#22551503	2
#22551502	2
C₈×Dic₂₅₇	2
C₄×C₈×C₂₅₇	2
#405	257