Subgroups — #45664071

Order 32896.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551503	C₂	1	1	yes	maximal, characteristic
2	#30656583	C₂	1	1	yes	maximal, characteristic
3	#30656491	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14523339	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₅₇	C₂²	1	2	yes	—
7	#22551503	C₄	1	4	yes	complemented
8	#22551499	C₄	1	4	yes	—
9	C₂²×C₈×C₂₅₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₅₇	C₄	1	4	yes	complemented
12	C₂²×Dic₂₅₇	C₂×C₄	1	4	yes	—
13	C₂×#14523339	—	2	4	no	—
14	C₂×#14523339	C₂×C₄	1	4	yes	—
15	C₄×Dic₂₅₇	—	2	4	no	—
16	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₅₇	—	2	4	no	—
19	C₂×C₈×C₂₅₇	C₂×C₄	1	4	yes	—
20	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
21	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₅₇	—	2	4	no	—
23	C₂×Dic₂₅₇	—	2	8	no	—
24	#14523339	—	2	4	no	—
25	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₇	—	2	4	no	—
27	C₂×C₄×C₂₅₇	C₄²	1	4	yes	—
28	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₅₇	#41	1	4	yes	—
31	C₈×C₂₅₇	—	2	4	no	—
32	C₂²×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₂₅₇	#117	1	2	yes	—
34	C₂²×C₂₅₇	#131	1	4	yes	—
35	C₄×C₂₅₇	—	2	4	no	—
36	C₄×C₂₅₇	#117	1	2	yes	—
37	C₄×C₂₅₇	#117	1	2	yes	—
38	Dic₂₅₇	—	4	4	no	—
39	C₂×C₂₅₇	#376	1	1	yes	characteristic
40	C₂×C₂₅₇	#405	1	2	yes	—
41	C₂×C₂₅₇	#433	1	4	yes	—
42	C₂₅₇	#1746	1	1	yes	characteristic, complemented, Sylow
43	#1746	—	257	1	no	maximal, Sylow
44	C₂×#97	—	257	1	no	—
45	C₂×C₄×C₈	D₂₅₇	1	1	yes	characteristic
46	#376	—	257	1	no	—
47	#336	—	257	4	no	—
48	C₂²×C₈	—	257	2	no	—
49	C₂²×C₈	C₂×D₂₅₇	1	2	yes	—
50	C₂×C₄²	—	257	2	no	—
51	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
52	#97	—	257	4	no	—
53	C₄×C₈	Dic₂₅₇	1	4	yes	complemented
54	#95	—	257	4	no	—
55	C₂²×C₄	—	257	4	no	—
56	C₂²×C₄	C₄×D₂₅₇	1	2	yes	—
57	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
58	C₂×C₈	—	257	4	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	514	4	no	—
61	C₂×C₈	C₂×Dic₂₅₇	1	4	yes	—
62	C₄²	—	514	4	no	—
63	C₄²	C₄×D₂₅₇	1	2	yes	—
64	C₄²	C₂×Dic₂₅₇	1	2	yes	—
65	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	514	8	no	—
68	C₂×C₄	—	514	4	no	—
69	C₂×C₄	C₄×Dic₂₅₇	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
72	C₂×C₄	#16861618	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	514	4	no	—
75	C₂²	C₂×C₄×Dic₂₅₇	1	1	yes	characteristic
76	C₂²	#22551549	1	2	yes	—
77	C₂²	#22551598	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	1028	4	no	—
80	C₄	#22551549	1	2	yes	—
81	C₄	#22551551	1	2	yes	—
82	C₂	#30656583	1	1	yes	characteristic
83	C₂	#30656692	1	2	yes	—
84	C₂	#30656796	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₇
commutator	C₂²×C₂₅₇	C₂×C₄²
Fitting	C₂×C₄×C₈×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂²×C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₇
hypocenter	C₂₅₇	#1746
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #22551503	C₄	semidirect
11. C₄×C₈×C₂₅₇	C₄	semidirect
42. C₂₅₇	#1746	semidirect
53. C₄×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×#22551503	2
#30656583	2
#30656491	2
C₂×C₄×C₈×C₂₅₇	2
#1746	257