Subgroups — #2241×C₂₅₇ #45660783

Order 32896.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	#494×C₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#358×C₂₅₇	C₂	1	4	yes	maximal
4	C₄×C₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₂₅₇	—	2	1	no	—
6	C₂×Q₁₆×C₂₅₇	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₅₇	—	2	1	no	—
8	C₄⋊Q₈×C₂₅₇	—	2	1	no	—
9	C₂×C₁₆×C₂₅₇	C₂²	1	2	yes	—
10	#107×C₂₅₇	—	2	1	no	—
11	#107×C₂₅₇	C₂²	1	2	yes	—
12	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₂₅₇	—	2	2	no	—
14	C₂×Q₈×C₂₅₇	—	4	1	no	—
15	Q₁₆×C₂₅₇	—	2	4	no	—
16	Q₁₆×C₂₅₇	—	4	2	no	—
17	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₅₇	—	2	2	no	—
20	C₄⋊C₄×C₂₅₇	—	4	1	no	—
21	C₄⋊C₄×C₂₅₇	—	4	1	no	—
22	C₄⋊C₄×C₂₅₇	—	4	1	no	—
23	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
24	C₁₆×C₂₅₇	—	2	2	no	—
25	Q₈×C₂₅₇	—	4	2	no	—
26	Q₈×C₂₅₇	—	4	2	no	—
27	Q₈×C₂₅₇	—	8	2	no	—
28	C₂×C₄×C₂₅₇	—	4	2	no	—
29	C₂×C₄×C₂₅₇	—	4	2	no	—
30	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₅₇	SD₁₆	1	2	yes	—
34	C₈×C₂₅₇	#41	1	2	yes	—
35	C₂²×C₂₅₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₅₇	—	8	2	no	—
37	C₄×C₂₅₇	—	8	2	no	—
38	C₄×C₂₅₇	SD₃₂	1	2	yes	—
39	C₄×C₂₅₇	Q₃₂	1	2	yes	—
40	C₄×C₂₅₇	#124	1	1	yes	characteristic
41	C₄×C₂₅₇	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₂₅₇	#486	1	1	yes	characteristic
43	C₂×C₂₅₇	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₂₅₇	C₂×Q₃₂	1	1	yes	characteristic
45	C₂₅₇	#2241	1	1	yes	characteristic, direct product factor, Sylow
46	#2241	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	#500	C₂×C₂₅₇	1	1	yes	characteristic
48	#494	C₂×C₂₅₇	1	1	yes	characteristic
49	#358	C₂×C₂₅₇	1	4	yes	—
50	C₄×C₁₆	C₂×C₂₅₇	1	1	yes	characteristic
51	C₂×Q₁₆	—	2	1	no	—
52	C₂×Q₁₆	C₂²×C₂₅₇	1	2	yes	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₂×C₁₆	C₂²×C₂₅₇	1	2	yes	—
56	#107	—	2	1	no	—
57	#107	C₂²×C₂₅₇	1	2	yes	—
58	C₄×C₈	C₂²×C₂₅₇	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	C₂×C₈	D₄×C₂₅₇	1	1	yes	characteristic
64	C₂×C₈	C₂³×C₂₅₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄²	D₄×C₂₅₇	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	D₈×C₂₅₇	1	1	yes	characteristic
77	C₂×C₄	D₈×C₂₅₇	1	1	yes	characteristic
78	C₂×C₄	C₂×D₄×C₂₅₇	1	1	yes	characteristic
79	C₈	SD₁₆×C₂₅₇	1	2	yes	—
80	C₈	#41×C₂₅₇	1	2	yes	—
81	C₂²	C₂×D₈×C₂₅₇	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	8	2	no	—
84	C₄	SD₃₂×C₂₅₇	1	2	yes	—
85	C₄	Q₃₂×C₂₅₇	1	2	yes	—
86	C₄	#124×C₂₅₇	1	1	yes	characteristic
87	C₄	C₂×SD₁₆×C₂₅₇	1	1	yes	characteristic
88	C₂	#486×C₂₅₇	1	1	yes	characteristic
89	C₂	C₂×SD₃₂×C₂₅₇	1	1	yes	characteristic
90	C₂	C₂×Q₃₂×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₅₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₅₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₇ ◁ C₄²×C₂₅₇ ◁ C₄×C₈×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
45. C₂₅₇	#2241	direct (N × H)
46. #2241	C₂₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#500×C₂₅₇	2
#494×C₂₅₇	2
#358×C₂₅₇	2
C₄×C₁₆×C₂₅₇	2
#2241	257

Subgroups — #2241×C257 #45660783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2241×C₂₅₇ #45660783