Subgroups — #30657033

Order 16448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551592	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₁₄	C₂	1	1	yes	maximal, characteristic
3	#22551526	C₂	1	2	yes	maximal
4	#22551517	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₁₄	—	2	1	no	—
7	C₂×Dic₅₁₄	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₅₇	—	2	1	no	—
9	C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
11	#16861597	—	2	1	no	—
12	C₄×Dic₂₅₇	—	2	1	no	—
13	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
14	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
15	Dic₅₁₄	—	2	1	no	—
16	Dic₅₁₄	—	4	2	no	—
17	Dic₅₁₄	Q₈	1	2	yes	complemented
18	C₂×Dic₂₅₇	—	2	1	no	—
19	C₂×Dic₂₅₇	—	2	1	no	—
20	C₂×Dic₂₅₇	—	2	2	no	—
21	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
24	C₈×C₂₅₇	—	2	2	no	—
25	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₅₇	—	2	1	no	—
27	C₄×C₂₅₇	Q₁₆	1	2	yes	—
28	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₅₇	#41	1	1	yes	characteristic
30	Dic₂₅₇	—	2	2	no	—
31	Dic₂₅₇	—	4	1	no	—
32	Dic₂₅₇	—	4	2	no	—
33	C₂×C₂₅₇	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₂₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂₅₇	#477	1	1	yes	characteristic, complemented, Sylow
37	#477	—	257	1	no	maximal, Sylow
38	C₄⋊Q₈	—	257	1	no	—
39	C₄×Q₈	—	257	1	no	—
40	#107	—	257	2	no	—
41	C₄⋊C₈	D₂₅₇	1	1	yes	characteristic
42	#103	—	257	2	no	—
43	C₂×Q₈	—	257	1	no	—
44	C₂×Q₈	—	514	1	no	—
45	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
46	C₄⋊C₄	—	257	2	no	—
47	C₄⋊C₄	—	257	1	no	—
48	C₄⋊C₄	—	514	1	no	—
49	C₄⋊C₄	—	514	1	no	—
50	C₄²	—	514	1	no	—
51	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
52	Q₈	—	257	2	no	—
53	Q₈	—	514	1	no	—
54	Q₈	—	1028	2	no	—
55	C₂×C₄	—	514	1	no	—
56	C₂×C₄	—	514	1	no	—
57	C₂×C₄	—	514	2	no	—
58	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
59	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂²	C₂×D₁₀₂₈	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	514	2	no	—
65	C₄	—	1028	2	no	—
66	C₄	—	1028	1	no	—
67	C₄	Dic₁₀₂₈	1	2	yes	—
68	C₄	Q₈×D₂₅₇	1	1	yes	characteristic
69	C₄	#16861617	1	1	yes	characteristic
70	C₂	#22551572	1	1	yes	characteristic
71	C₂	C₂×Dic₁₀₂₈	1	1	yes	characteristic
72	C₂	#22551620	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₂₈
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₄⋊C₈×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₂×D₄
hypercenter	C₄⋊C₈	D₂₅₇
hypocenter	C₂₅₇	#477
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. Dic₅₁₄	Q₈	semidirect
36. C₂₅₇	#477	semidirect

Maximal subgroups

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

subgroup	index
#22551592	2
C₄×Dic₅₁₄	2
#22551526	2
#22551517	2
C₄⋊C₈×C₂₅₇	2
#477	257