Subgroups — #30656644

Order 16448.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#22551499	C₂	1	4	yes	maximal
4	C₂²×Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₅₇	C₂²	1	2	yes	—
6	C₂²×Dic₂₅₇	C₂²	1	2	yes	—
7	C₄⋊Dic₂₅₇	—	2	2	no	—
8	C₄×Dic₂₅₇	—	2	2	no	—
9	C₂×Q₈×C₂₅₇	—	2	2	no	—
10	C₂×Q₈×C₂₅₇	—	2	2	no	—
11	C₂×Q₈×C₂₅₇	C₄	1	4	yes	complemented
12	C₂²×C₄×C₂₅₇	C₂²	1	2	yes	—
13	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅₇	—	2	2	no	—
15	C₂×Dic₂₅₇	—	2	2	no	—
16	C₂×Dic₂₅₇	—	2	2	no	—
17	C₂×Dic₂₅₇	—	2	4	no	—
18	C₂×Dic₂₅₇	Q₈	1	4	yes	complemented
19	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
20	Q₈×C₂₅₇	—	2	8	no	—
21	C₂×C₄×C₂₅₇	—	2	2	no	—
22	C₂×C₄×C₂₅₇	—	2	2	no	—
23	C₂×C₄×C₂₅₇	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
26	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
27	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₂₅₇	C₂×Q₈	1	2	yes	—
31	C₂²×C₂₅₇	#41	1	2	yes	—
32	C₄×C₂₅₇	—	2	4	no	—
33	C₄×C₂₅₇	C₂²⋊C₄	1	4	yes	—
34	Dic₂₅₇	—	2	4	no	—
35	Dic₂₅₇	—	4	2	no	—
36	C₂×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₅₇	C₄×Q₈	1	2	yes	—
38	C₂×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₂₅₇	#124	1	1	yes	characteristic
40	C₂×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
41	C₂₅₇	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	257	1	no	maximal, Sylow
43	C₂²×Q₈	D₂₅₇	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	257	1	no	—
45	C₂×C₄²	—	257	1	no	—
46	#95	—	257	4	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	Dic₂₅₇	1	4	yes	complemented
50	C₂²×C₄	—	257	2	no	—
51	C₂²×C₄	—	257	2	no	—
52	C₂²×C₄	C₂×D₂₅₇	1	2	yes	—
53	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
54	C₄⋊C₄	—	514	2	no	—
55	C₄²	—	514	2	no	—
56	C₂³	C₂²×D₂₅₇	1	1	yes	characteristic
57	Q₈	—	2	8	no	—
58	C₂×C₄	—	257	4	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	514	2	no	—
62	C₂×C₄	—	514	2	no	—
63	C₂×C₄	—	514	2	no	—
64	C₂×C₄	—	514	4	no	—
65	C₂×C₄	C₂×Dic₂₅₇	1	4	yes	—
66	C₂×C₄	C₂×Dic₂₅₇	1	2	yes	—
67	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
68	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
69	C₂²	C₂²×Dic₂₅₇	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
71	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
72	C₂²	Q₈×D₂₅₇	1	2	yes	—
73	C₂²	#16861619	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	514	4	no	—
76	C₄	—	1028	2	no	—
77	C₄	C₂²⋊Dic₂₅₇	1	4	yes	—
78	C₂	C₂×C₂²⋊Dic₂₅₇	1	1	yes	characteristic
79	C₂	Q₈×Dic₂₅₇	1	2	yes	—
80	C₂	#22551568	1	2	yes	—
81	C₂	#22551581	1	1	yes	characteristic
82	C₂	#22551590	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₅₇
commutator	C₂²×C₂₅₇	C₂²×C₄
Fitting	C₂²×Q₈×C₂₅₇	C₂
socle, prime radical, cosocle	C₂³×C₂₅₇	C₂³
hypercenter	C₂²×Q₈	D₂₅₇
hypocenter	C₂₅₇	#391
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
11. C₂×Q₈×C₂₅₇	C₄	semidirect
18. C₂×Dic₂₅₇	Q₈	semidirect
41. C₂₅₇	#391	semidirect
49. C₂×Q₈	Dic₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₅₇	2
C₂×C₄×Dic₂₅₇	2
#22551499	2
C₂²×Q₈×C₂₅₇	2
#391	257