Subgroups — #340×C₂₅₇ #30656209

Order 16448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×C₂₅₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₂₅₇	—	2	2	no	—
4	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
5	C₂×C₈×C₂₅₇	—	2	2	no	—
6	C₂×C₈×C₂₅₇	C₄	1	2	yes	complemented
7	C₄⋊C₄×C₂₅₇	—	2	2	no	—
8	C₄⋊C₄×C₂₅₇	C₄	1	4	yes	complemented
9	C₂³×C₂₅₇	D₄	1	1	yes	characteristic
10	C₂×C₄×C₂₅₇	—	2	4	no	—
11	C₂×C₄×C₂₅₇	—	4	2	no	—
12	C₂×C₄×C₂₅₇	—	4	2	no	—
13	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
16	C₂×C₄×C₂₅₇	Q₈	1	1	yes	characteristic
17	C₈×C₂₅₇	—	2	2	no	—
18	C₂²×C₂₅₇	C₂²⋊C₄	1	2	yes	—
19	C₂²×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₂₅₇	D₈	1	1	yes	characteristic
21	C₂²×C₂₅₇	SD₁₆	1	2	yes	—
22	C₂²×C₂₅₇	Q₁₆	1	1	yes	characteristic
23	C₄×C₂₅₇	—	4	4	no	—
24	C₄×C₂₅₇	C₄²	1	1	yes	characteristic
25	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₂₅₇	#95	1	1	yes	characteristic
28	C₂×C₂₅₇	#102	1	2	yes	—
29	C₂×C₂₅₇	#103	1	2	yes	—
30	C₂×C₂₅₇	#106	1	1	yes	characteristic
31	C₂×C₂₅₇	#107	1	1	yes	characteristic
32	C₂₅₇	#340	1	1	yes	characteristic, direct product factor, Sylow
33	#340	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₈	C₂×C₂₅₇	1	1	yes	characteristic
35	C₂×C₄⋊C₄	C₂×C₂₅₇	1	2	yes	—
36	C₂²×C₄	—	2	2	no	—
37	C₂²×C₄	C₂²×C₂₅₇	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	C₄×C₂₅₇	1	2	yes	complemented
40	C₄⋊C₄	—	2	2	no	—
41	C₄⋊C₄	C₄×C₂₅₇	1	4	yes	complemented
42	C₂³	D₄×C₂₅₇	1	1	yes	characteristic
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	—	4	2	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	C₂×C₄×C₂₅₇	1	1	yes	characteristic
47	C₂×C₄	C₂×C₄×C₂₅₇	1	2	yes	—
48	C₂×C₄	D₄×C₂₅₇	1	2	yes	—
49	C₂×C₄	Q₈×C₂₅₇	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₂²	C₂²⋊C₄×C₂₅₇	1	2	yes	—
52	C₂²	C₄⋊C₄×C₂₅₇	1	1	yes	characteristic
53	C₂²	D₈×C₂₅₇	1	1	yes	characteristic
54	C₂²	SD₁₆×C₂₅₇	1	2	yes	—
55	C₂²	Q₁₆×C₂₅₇	1	1	yes	characteristic
56	C₄	—	4	4	no	—
57	C₄	C₄²×C₂₅₇	1	1	yes	characteristic
58	C₄	C₂²⋊C₄×C₂₅₇	1	1	yes	characteristic
59	C₄	C₄⋊C₄×C₂₅₇	1	2	yes	—
60	C₂	#95×C₂₅₇	1	1	yes	characteristic
61	C₂	#102×C₂₅₇	1	2	yes	—
62	C₂	#103×C₂₅₇	1	2	yes	—
63	C₂	#106×C₂₅₇	1	1	yes	characteristic
64	C₂	#107×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₅₇	D₄
commutator	C₄	C₄²×C₂₅₇
Frattini, cosocle	C₂²×C₄	C₂²×C₂₅₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₇ ◁ C₂²×C₄×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₈×C₂₅₇	C₄	semidirect
8. C₄⋊C₄×C₂₅₇	C₄	semidirect
32. C₂₅₇	#340	direct (N × H)
33. #340	C₂₅₇	direct (N × H)
39. C₂×C₈	C₄×C₂₅₇	semidirect
41. C₄⋊C₄	C₄×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×C₂₅₇	2
C₂×C₄⋊C₄×C₂₅₇	2
#340	257

Subgroups — #340×C257 #30656209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #340×C₂₅₇ #30656209