Subgroups — #30656692

Order 16448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861602	C₂	1	1	yes	maximal, characteristic
2	#22551551	C₂	1	1	yes	maximal, characteristic
3	#22551502	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₅₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	#16861602	C₄	1	4	yes	complemented
7	C₂×#14523339	C₂²	1	2	yes	—
8	C₄⋊Dic₂₅₇	C₄	1	2	yes	—
9	C₂²⋊Dic₂₅₇	C₄	1	2	yes	—
10	C₄×Dic₂₅₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₅₇	C₄	1	4	yes	complemented
13	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₇	C₂×C₄	1	4	yes	—
16	#14523339	C₂×C₄	1	4	yes	—
17	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
22	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
23	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
24	C₂²×C₂₅₇	—	2	2	no	—
25	C₂²×C₂₅₇	C₄²	1	2	yes	—
26	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₂₅₇	C₄²	1	2	yes	—
28	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
30	Dic₂₅₇	—	2	4	no	—
31	C₂×C₂₅₇	—	2	2	no	—
32	C₂×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₂₅₇	#131	1	2	yes	—
34	C₂₅₇	#405	1	1	yes	characteristic, complemented, Sylow
35	#405	—	257	1	no	maximal, Sylow
36	C₂×M₁₆	—	257	1	no	—
37	C₂²×C₈	D₂₅₇	1	1	yes	characteristic
38	#117	—	257	1	no	—
39	#97	—	257	2	no	—
40	C₄×C₈	—	257	2	no	—
41	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
42	M₁₆	—	257	4	no	—
43	C₂×C₈	—	257	2	no	—
44	C₂×C₈	Dic₂₅₇	1	4	yes	complemented
45	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
46	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
47	C₄⋊C₄	—	257	2	no	—
48	C₂²⋊C₄	—	257	2	no	—
49	C₄²	—	257	2	no	—
50	C₂³	C₄×D₂₅₇	1	1	yes	characteristic
51	C₂×C₄	—	257	4	no	—
52	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
53	C₂×C₄	C₄×D₂₅₇	1	2	yes	—
54	C₂×C₄	C₂×Dic₂₅₇	1	2	yes	—
55	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
56	C₈	—	257	4	no	—
57	C₈	C₂×Dic₂₅₇	1	2	yes	—
58	C₈	C₂×Dic₂₅₇	1	2	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₂₅₇	1	2	yes	—
61	C₂²	C₂×C₄×D₂₅₇	1	1	yes	characteristic
62	C₄	—	514	4	no	—
63	C₄	C₄×Dic₂₅₇	1	2	yes	—
64	C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
65	C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄×Dic₂₅₇	1	1	yes	characteristic
68	C₂	#22551598	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #22551502	C₂	semidirect
4. C₈×Dic₂₅₇	C₂	semidirect
6. #16861602	C₄	semidirect
12. C₂×C₈×C₂₅₇	C₄	semidirect
34. C₂₅₇	#405	semidirect
44. C₂×C₈	Dic₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861602	2
#22551551	2
#22551502	2
C₈×Dic₂₅₇	2
C₂²×C₈×C₂₅₇	2
#405	257