Subgroups — #30657784

Order 16448.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551679	C₂	1	2	yes	maximal, complemented
2	#22551662	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551656	C₂	1	2	yes	maximal, complemented
4	#22551646	C₂	1	1	yes	maximal, characteristic, complemented
5	#22551611	C₂	1	1	yes	maximal, characteristic
6	#16861628	C₂²	1	2	yes	—
7	#16861627	—	2	2	no	—
8	#16861626	C₂²	1	1	yes	characteristic
9	C₂×#14523346	—	2	2	no	—
10	C₄×#11759967	C₂²	1	1	yes	characteristic
11	#16861617	C₂²	1	2	yes	—
12	Dic₁₀₂₈	C₄	1	1	yes	characteristic, complemented
13	#16861606	C₄	1	2	yes	complemented
14	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
15	D₈×C₂₅₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11759967	—	2	1	no	—
17	#14523346	—	2	2	no	—
18	#14523346	D₄	1	2	yes	—
19	Dic₅₁₄	C₂×C₄	1	2	yes	—
20	C₂²⋊D₂₅₇	—	2	2	no	—
21	C₂×Dic₂₅₇	—	2	2	no	—
22	C₄×D₂₅₇	C₂³	1	1	yes	characteristic
23	#14523339	C₂×C₄	1	1	yes	characteristic
24	D₄×C₂₅₇	C₂×C₄	1	2	yes	—
25	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₅₇	—	2	2	no	—
27	C₂×D₂₅₇	#41	1	1	yes	characteristic
28	#11759967	—	4	1	no	—
29	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
30	Dic₂₅₇	—	2	2	no	—
31	Dic₂₅₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₂₅₇	—	4	2	no	—
33	C₂×C₂₅₇	C₄×D₄	1	1	yes	characteristic
34	D₂₅₇	—	2	1	no	—
35	C₂₅₇	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	257	1	no	maximal, Sylow
37	#135	—	257	1	no	—
38	#131	—	257	2	no	—
39	#108	—	257	1	no	—
40	#104	—	257	2	no	—
41	#97	—	257	1	no	—
42	#41	—	257	2	no	—
43	Q₁₆	—	257	1	no	—
44	SD₁₆	—	257	2	no	—
45	D₈	#11759967	1	1	yes	characteristic, complemented
46	M₁₆	—	257	2	no	—
47	M₁₆	—	514	2	no	—
48	C₂×C₈	—	257	1	no	—
49	C₂×C₈	—	257	1	no	—
50	C₂×C₈	—	514	2	no	—
51	C₄²	—	257	1	no	—
52	Q₈	—	257	2	no	—
53	D₄	—	514	2	no	—
54	D₄	C₂×#11759967	1	2	yes	—
55	C₂×C₄	—	257	1	no	—
56	C₂×C₄	—	514	2	no	—
57	C₂×C₄	—	514	1	no	—
58	C₈	—	257	2	no	—
59	C₈	—	257	1	no	—
60	C₈	—	514	2	no	—
61	C₈	C₂×#11759967	1	1	yes	characteristic
62	C₂²	—	257	1	no	—
63	C₂²	—	2	2	no	—
64	C₄	—	257	1	no	—
65	C₄	—	514	2	no	—
66	C₄	—	1028	1	no	—
67	C₄	C₂²×#11759967	1	1	yes	characteristic
68	C₂	—	514	1	no	—
69	C₂	—	4	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	D₈×C₂₅₇	C₄
socle	C₂×C₂₅₇	C₄×D₄
prime radical	#22551611	C₂
cosocle	C₄×D₂₅₇	C₂³
hypercenter	D₈	#11759967
hypocenter	C₂₅₇	#444
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #22551679	C₂	semidirect
2. #22551662	C₂	semidirect
3. #22551656	C₂	semidirect
4. #22551646	C₂	semidirect
12. Dic₁₀₂₈	C₄	semidirect
13. #16861606	C₄	semidirect
15. D₈×C₂₅₇	C₄	semidirect
35. C₂₅₇	#444	semidirect
45. D₈	#11759967	semidirect

Maximal subgroups

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

subgroup	index
#22551679	2
#22551662	2
#22551656	2
#22551646	2
#22551611	2
#444	257