Subgroups — #30656765

Order 16448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523339	C₂	1	1	yes	maximal, characteristic
2	#22551551	C₂	1	1	yes	maximal, characteristic
3	#22551521	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523339	—	2	2	no	—
6	C₂×#14523339	C₂²	1	2	yes	—
7	C₄⋊Dic₂₅₇	C₄	1	2	yes	—
8	C₂²⋊Dic₂₅₇	C₄	1	2	yes	—
9	C₄×Dic₂₅₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₅₇	—	2	2	no	—
12	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
13	C₂×Dic₂₅₇	C₂×C₄	1	4	yes	—
14	#14523339	—	2	2	no	—
15	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
18	C₂×C₄×C₂₅₇	Q₈	1	2	yes	—
19	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
20	C₈×C₂₅₇	—	2	2	no	—
21	C₂²×C₂₅₇	—	2	2	no	—
22	C₂²×C₂₅₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
25	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
27	Dic₂₅₇	—	2	4	no	—
28	C₂×C₂₅₇	—	2	2	no	—
29	C₂×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₂₅₇	#131	1	2	yes	—
31	C₂₅₇	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	257	1	no	maximal, Sylow
33	C₂×M₁₆	D₂₅₇	1	1	yes	characteristic
34	C₂²×C₈	—	257	1	no	—
35	#117	—	257	1	no	—
36	C₄⋊C₈	—	257	4	no	—
37	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	C₂×C₈	—	257	2	no	—
40	C₂×C₈	—	514	2	no	—
41	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
42	C₄⋊C₄	—	257	2	no	—
43	C₂²⋊C₄	—	257	2	no	—
44	C₄²	—	257	2	no	—
45	C₂³	C₄×D₂₅₇	1	1	yes	characteristic
46	C₂×C₄	—	257	4	no	—
47	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
49	C₂×C₄	Dic₅₁₄	1	2	yes	—
50	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	514	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	#16861597	1	2	yes	—
55	C₂²	C₂×C₄×D₂₅₇	1	1	yes	characteristic
56	C₄	—	514	4	no	—
57	C₄	#16861597	1	2	yes	—
58	C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
59	C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×#16861597	1	1	yes	characteristic
62	C₂	#22551600	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₅₇
commutator, socle	C₂²×C₂₅₇	C₂²×C₄
Fitting	C₂×M₁₆×C₂₅₇	C₂
prime radical	C₂³×C₂₅₇	C₂×C₄
cosocle	C₂×C₄×C₂₅₇	C₂³
hypercenter	C₂×M₁₆	D₂₅₇
hypocenter	C₂₅₇	#424
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22551521	C₂	semidirect
31. C₂₅₇	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523339	2
#22551551	2
#22551521	2
C₂×M₁₆×C₂₅₇	2
#424	257