Subgroups — #30657742

Order 16448.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861628	C₂	1	2	yes	maximal
2	#22551659	C₂	1	4	yes	maximal, complemented
3	C₄²×D₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#16861628	—	2	4	no	—
5	C₂×#14523346	C₂²	1	4	yes	—
6	C₂×C₄×D₂₅₇	C₄	1	2	yes	—
7	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₅₇	C₄	1	1	yes	characteristic
9	C₄×Dic₂₅₇	C₂²	1	2	yes	—
10	C₄²×C₂₅₇	C₄	1	1	yes	characteristic
11	#14523346	—	2	4	no	—
12	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
14	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
15	C₄×D₂₅₇	—	2	4	no	—
16	C₄×D₂₅₇	D₄	1	2	yes	—
17	C₄×D₂₅₇	Q₈	1	2	yes	—
18	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₂₅₇	—	2	2	no	—
22	C₂×D₂₅₇	C₄⋊C₄	1	2	yes	—
23	C₄×C₂₅₇	—	2	2	no	—
24	C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
25	Dic₂₅₇	M₁₆	1	4	yes	complemented
26	Dic₂₅₇	C₂×D₄	1	1	yes	characteristic
27	Dic₂₅₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₂₅₇	C₂×M₁₆	1	2	yes	—
30	D₂₅₇	—	2	2	no	—
31	C₂₅₇	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	257	1	no	maximal, Sylow
33	C₂×M₁₆	—	257	2	no	—
34	C₂×C₄²	—	257	1	no	—
35	C₄⋊C₈	—	257	4	no	—
36	C₂²×C₄	—	257	1	no	—
37	C₂²×C₄	—	257	2	no	—
38	M₁₆	—	514	4	no	—
39	C₂×C₈	—	257	4	no	—
40	C₄²	—	257	2	no	—
41	C₄²	—	257	1	no	—
42	C₄²	#11759967	1	1	yes	characteristic
43	C₂³	—	257	1	no	—
44	C₂×C₄	—	257	1	no	—
45	C₂×C₄	—	257	2	no	—
46	C₂×C₄	—	257	2	no	—
47	C₂×C₄	—	257	2	no	—
48	C₂×C₄	—	514	4	no	—
49	C₂×C₄	C₂×#11759967	1	2	yes	—
50	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
51	C₈	—	514	4	no	—
52	C₂²	—	257	2	no	—
53	C₂²	—	514	2	no	—
54	C₂²	C₂²×#11759967	1	1	yes	characteristic
55	C₄	—	257	1	no	—
56	C₄	—	257	1	no	—
57	C₄	—	257	4	no	—
58	C₄	—	2	2	no	—
59	C₄	C₄⋊#11759967	1	2	yes	—
60	C₂	—	514	2	no	—
61	C₂	C₂×C₄⋊#11759967	1	1	yes	characteristic
62	C₂	D₂₅₇⋊M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11759967
commutator, socle	C₂²×C₂₅₇	C₂²×C₄
Fitting	C₄²×C₂₅₇	C₄
prime radical	C₂²×D₂₅₇	C₂×C₄
cosocle	C₂×Dic₂₅₇	C₂³
hypercenter	C₄²	#11759967
hypocenter	C₂₅₇	#423
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #22551659	C₂	semidirect
25. Dic₂₅₇	M₁₆	semidirect
31. C₂₅₇	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861628	2
#22551659	2
C₄²×D₂₅₇	2
#423	257