Subgroups — #30657781

Order 16448.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551680	C₂	1	2	yes	maximal, complemented
2	#22551663	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551657	C₂	1	2	yes	maximal, complemented
4	C₈×#11759967	C₂	1	1	yes	maximal, characteristic, complemented
5	#22551615	C₂	1	1	yes	maximal, characteristic
6	#16861628	—	2	2	no	—
7	#16861628	C₂²	1	2	yes	—
8	#16861626	—	2	2	no	—
9	#16861626	C₂²	1	1	yes	characteristic
10	C₄×#11759967	C₂²	1	1	yes	characteristic
11	#16861619	C₂²	1	2	yes	—
12	#16861605	C₄	1	2	yes	complemented
13	D₂₀₅₆	C₄	1	1	yes	characteristic, complemented
14	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₂₅₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11759967	—	2	1	no	—
17	#14523346	—	2	2	no	—
18	#14523346	D₄	1	1	yes	characteristic
19	#14523346	D₄	1	1	yes	characteristic
20	D₁₀₂₈	—	2	2	no	—
21	D₁₀₂₈	C₂×C₄	1	2	yes	—
22	C₄×D₂₅₇	—	2	2	no	—
23	C₄×D₂₅₇	C₂³	1	1	yes	characteristic
24	#14523339	C₂×C₄	1	1	yes	characteristic
25	Q₈×C₂₅₇	C₂×C₄	1	2	yes	—
26	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂×D₂₅₇	—	2	2	no	—
28	C₂×D₂₅₇	#41	1	1	yes	characteristic
29	#11759967	—	2	2	no	—
30	C₄×C₂₅₇	—	2	2	no	—
31	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
32	Dic₂₅₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₂₅₇	C₄×D₄	1	1	yes	characteristic
34	D₂₅₇	—	2	1	no	—
35	D₂₅₇	—	4	2	no	—
36	C₂₅₇	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	257	1	no	maximal, Sylow
38	#135	—	257	1	no	—
39	#131	—	257	2	no	—
40	#108	—	257	1	no	—
41	#104	—	257	2	no	—
42	C₄×C₈	—	257	1	no	—
43	#41	—	257	2	no	—
44	Q₁₆	#11759967	1	1	yes	characteristic, complemented
45	SD₁₆	—	257	2	no	—
46	D₈	—	257	1	no	—
47	M₁₆	—	257	2	no	—
48	M₁₆	—	514	2	no	—
49	C₂×C₈	—	257	1	no	—
50	C₂×C₈	—	257	1	no	—
51	C₂×C₈	—	514	2	no	—
52	C₄²	—	257	1	no	—
53	Q₈	C₂×#11759967	1	2	yes	—
54	D₄	—	257	2	no	—
55	D₄	—	514	2	no	—
56	C₂×C₄	—	257	1	no	—
57	C₂×C₄	—	514	2	no	—
58	C₂×C₄	—	514	1	no	—
59	C₈	—	257	1	no	—
60	C₈	—	257	1	no	—
61	C₈	—	257	1	no	—
62	C₈	—	514	2	no	—
63	C₈	C₂×#11759967	1	1	yes	characteristic
64	C₂²	—	257	1	no	—
65	C₂²	—	514	2	no	—
66	C₄	—	257	1	no	—
67	C₄	—	2	2	no	—
68	C₄	—	514	2	no	—
69	C₄	C₂²×#11759967	1	1	yes	characteristic
70	C₂	—	514	1	no	—
71	C₂	—	1028	2	no	—
72	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	#22551615	C₂
cosocle	C₄×D₂₅₇	C₂³
hypercenter	Q₁₆	#11759967
hypocenter	C₂₅₇	#443
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
1. #22551680	C₂	semidirect
2. #22551663	C₂	semidirect
3. #22551657	C₂	semidirect
4. C₈×#11759967	C₂	semidirect
12. #16861605	C₄	semidirect
13. D₂₀₅₆	C₄	semidirect
15. Q₁₆×C₂₅₇	C₄	semidirect
36. C₂₅₇	#443	semidirect
44. Q₁₆	#11759967	semidirect

Maximal subgroups

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

subgroup	index
#22551680	2
#22551663	2
#22551657	2
C₈×#11759967	2
#22551615	2
#443	257