Subgroups — #30657777

Order 16448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11759967	C₂	1	2	yes	maximal
2	#22551660	C₂	1	1	yes	maximal, characteristic
3	#22551655	C₂	1	2	yes	maximal
4	#22551646	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₂₅₇	C₂	1	1	yes	maximal, characteristic
6	#16861626	C₂²	1	1	yes	characteristic
7	C₄⋊#11759967	—	2	2	no	—
8	C₄⋊#11759967	C₂²	1	2	yes	—
9	C₄×#11759967	—	2	2	no	—
10	C₄×#11759967	C₂²	1	1	yes	characteristic
11	Q₈×D₂₅₇	C₂²	1	2	yes	—
12	Dic₁₀₂₈	C₄	1	1	yes	characteristic, complemented
13	#16861608	C₄	1	2	yes	complemented
14	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₂₅₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11759967	—	2	2	no	—
17	C₂×#11759967	D₄	1	2	yes	—
18	#14523346	—	2	1	no	—
19	Dic₅₁₄	—	2	2	no	—
20	Dic₅₁₄	C₂×C₄	1	2	yes	—
21	C₄×D₂₅₇	—	2	2	no	—
22	C₄×D₂₅₇	C₂³	1	1	yes	characteristic
23	#14523339	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₂₅₇	C₂×C₄	1	2	yes	—
25	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₂₅₇	C₂×D₄	1	1	yes	characteristic
27	#11759967	—	2	2	no	—
28	#11759967	—	4	2	no	—
29	C₄×C₂₅₇	—	2	2	no	—
30	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₂₅₇	—	2	2	no	—
32	Dic₂₅₇	#41	1	1	yes	characteristic
33	C₂×C₂₅₇	C₄×D₄	1	1	yes	characteristic
34	D₂₅₇	#137	1	2	yes	—
35	C₂₅₇	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	257	1	no	maximal, Sylow
37	C₂×Q₁₆	—	257	1	no	—
38	C₄×Q₈	—	257	2	no	—
39	#106	—	257	1	no	—
40	#103	—	257	2	no	—
41	#97	—	257	1	no	—
42	C₂×Q₈	—	257	2	no	—
43	Q₁₆	—	257	2	no	—
44	Q₁₆	—	257	1	no	—
45	Q₁₆	#11759967	1	1	yes	characteristic, complemented
46	C₂×C₈	—	257	1	no	—
47	C₂×C₈	—	257	1	no	—
48	C₄⋊C₄	—	257	2	no	—
49	C₄⋊C₄	—	514	2	no	—
50	C₄²	—	257	1	no	—
51	C₄²	—	514	2	no	—
52	Q₈	—	257	2	no	—
53	Q₈	—	514	2	no	—
54	Q₈	C₂×#11759967	1	2	yes	—
55	C₂×C₄	—	257	1	no	—
56	C₂×C₄	—	257	2	no	—
57	C₂×C₄	—	514	2	no	—
58	C₂×C₄	—	514	2	no	—
59	C₈	—	257	1	no	—
60	C₈	—	514	1	no	—
61	C₈	C₂×#11759967	1	1	yes	characteristic
62	C₂²	—	257	1	no	—
63	C₄	—	257	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	514	2	no	—
66	C₄	—	514	2	no	—
67	C₄	—	1028	2	no	—
68	C₄	C₂²×#11759967	1	1	yes	characteristic
69	C₂	—	257	2	no	—
70	C₂	D₄×#11759967	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11759967
commutator	C₄×C₂₅₇	C₂²×C₄
Frattini	C₄	C₂²×#11759967
Fitting	Q₁₆×C₂₅₇	C₄
socle	C₂×C₂₅₇	C₄×D₄
prime radical	C₂×D₂₅₇	C₂×D₄
cosocle	C₄×D₂₅₇	C₂³
hypercenter	Q₁₆	#11759967
hypocenter	C₂₅₇	#441
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₁₀₂₈	C₄	semidirect
13. #16861608	C₄	semidirect
15. Q₁₆×C₂₅₇	C₄	semidirect
35. C₂₅₇	#441	semidirect
45. Q₁₆	#11759967	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11759967	2
#22551660	2
#22551655	2
#22551646	2
Q₁₆×D₂₅₇	2
#441	257