Subgroups — #30657103

Order 16448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551589	C₂	1	1	yes	maximal, characteristic
2	#22551512	C₂	1	4	yes	maximal, complemented
3	#22551503	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₂₈	—	2	2	no	—
6	C₂×D₁₀₂₈	C₂²	1	2	yes	—
7	C₂×#14523339	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₅₇	—	2	2	no	—
9	C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
10	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
11	C₂²×D₂₅₇	—	2	2	no	—
12	D₁₀₂₈	—	2	4	no	—
13	D₁₀₂₈	—	4	4	no	—
14	#14523339	—	2	2	no	—
15	C₂×C₄×C₂₅₇	—	2	2	no	—
16	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
17	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
18	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
19	C₂×D₂₅₇	—	4	2	no	—
20	C₂×D₂₅₇	—	4	4	no	—
21	C₄×C₂₅₇	—	2	2	no	—
22	C₄×C₂₅₇	—	4	2	no	—
23	C₄×C₂₅₇	#41	1	2	yes	—
24	C₂×C₂₅₇	#124	1	1	yes	characteristic
25	C₂×C₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
26	D₂₅₇	—	8	2	no	—
27	C₂₅₇	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	257	1	no	maximal, Sylow
29	#127	—	257	1	no	—
30	#125	D₂₅₇	1	1	yes	characteristic, complemented
31	#102	—	257	4	no	—
32	#97	—	257	1	no	—
33	C₂×D₄	—	257	2	no	—
34	C₂×D₄	—	514	2	no	—
35	C₂×C₈	—	257	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
38	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
39	C₂³	—	514	2	no	—
40	D₄	—	514	4	no	—
41	D₄	—	1028	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
44	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
45	C₈	—	514	2	no	—
46	C₂²	—	1028	2	no	—
47	C₂²	—	1028	4	no	—
48	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#16861619	1	2	yes	—
52	C₂	—	2056	2	no	—
53	C₂	#22551581	1	1	yes	characteristic
54	C₂	#22551617	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	#125×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	#22551589	C₂
hypercenter	#125	D₂₅₇
hypocenter	C₂₅₇	#490
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #22551512	C₂	semidirect
3. #22551503	C₂	semidirect
4. #125×C₂₅₇	C₂	semidirect
27. C₂₅₇	#490	semidirect
30. #125	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551589	2
#22551512	2
#22551503	2
#125×C₂₅₇	2
#490	257