Subgroups — #30656501

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861602	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523339	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#16861602	—	2	1	no	—
5	#16861602	C₄	1	2	yes	—
6	C₂×#14523339	—	2	1	no	—
7	C₂×#14523339	—	2	1	no	—
8	C₂×#14523339	—	2	1	no	—
9	C₂×#14523339	C₄	1	2	yes	—
10	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₅₇	—	2	1	no	—
12	M₁₆×C₂₅₇	C₄	1	2	yes	—
13	C₂×C₈×C₂₅₇	—	2	1	no	—
14	#14523339	—	2	2	no	—
15	#14523339	—	2	2	no	—
16	C₂³×C₂₅₇	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
23	C₈×C₂₅₇	—	2	2	no	—
24	C₂²×C₂₅₇	—	2	2	no	—
25	C₂²×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₂₅₇	C₄²	1	1	yes	characteristic
29	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₂₅₇	—	2	1	no	—
33	C₂×C₂₅₇	—	2	1	no	—
34	C₂×C₂₅₇	#95	1	1	yes	characteristic
35	C₂×C₂₅₇	#108	1	2	yes	—
36	C₂₅₇	#341	1	1	yes	characteristic, complemented, Sylow
37	#341	—	257	1	no	maximal, Sylow
38	C₂×M₁₆	—	257	1	no	—
39	C₂×M₁₆	D₂₅₇	1	1	yes	characteristic
40	C₂²×C₈	—	257	1	no	—
41	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
42	M₁₆	—	257	2	no	—
43	M₁₆	—	2	1	no	—
44	M₁₆	—	514	1	no	—
45	M₁₆	Dic₂₅₇	1	2	yes	—
46	C₂×C₈	—	257	2	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	514	1	no	—
49	C₂×C₈	—	514	1	no	—
50	C₂×C₈	—	514	1	no	—
51	C₂³	Dic₅₁₄	1	1	yes	characteristic
52	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
53	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
54	C₂×C₄	C₂×Dic₂₅₇	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
57	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	514	2	no	—
60	C₈	—	514	2	no	—
61	C₂²	—	2	2	no	—
62	C₂²	#16861597	1	1	yes	characteristic
63	C₂²	#16861597	1	1	yes	characteristic
64	C₂²	C₄⋊Dic₂₅₇	1	1	yes	characteristic
65	C₄	C₄×Dic₂₅₇	1	1	yes	characteristic
66	C₄	#16861595	1	1	yes	characteristic
67	C₄	#16861595	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₂₅₇	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	#22551499	1	1	yes	characteristic
72	C₂	#22551527	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₅₇
commutator	C₄×C₂₅₇	C₄²
Frattini	C₂²×C₄	C₂×D₂₅₇
Fitting	C₂×M₁₆×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₄⋊C₄
prime radical	C₂³×C₂₅₇	Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂²
hypercenter	C₂×M₁₆	D₂₅₇
hypocenter	C₂₅₇	#341
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
36. C₂₅₇	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861602	2
C₂²×#14523339	2
C₂×M₁₆×C₂₅₇	2
#341	257