Subgroups — #30657613

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861628	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#16861628	—	2	2	no	—
4	#16861628	C₄	1	4	yes	—
5	C₂×#14523346	—	2	2	no	—
6	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
7	C₂×#14523339	C₄	1	1	yes	characteristic
8	C₈×D₂₅₇	—	2	2	no	—
9	C₂×C₈×C₂₅₇	C₄	1	1	yes	characteristic
10	#14523346	—	2	4	no	—
11	C₂²×D₂₅₇	Q₈	1	1	yes	characteristic
12	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
13	C₄×D₂₅₇	C₂×C₄	1	2	yes	—
14	C₄×D₂₅₇	D₄	1	2	yes	—
15	#14523339	—	2	1	no	—
16	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₅₇	—	2	1	no	—
18	C₂²×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₂₅₇	—	2	1	no	—
20	C₂×D₂₅₇	—	2	1	no	—
21	C₂×D₂₅₇	C₄⋊C₄	1	2	yes	—
22	C₄×C₂₅₇	C₄²	1	1	yes	characteristic
23	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₂₅₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₂₅₇	#95	1	1	yes	characteristic
26	C₂×C₂₅₇	#108	1	1	yes	characteristic
27	C₂×C₂₅₇	#108	1	1	yes	characteristic
28	D₂₅₇	—	2	2	no	—
29	C₂₅₇	#341	1	1	yes	characteristic, complemented, Sylow
30	#341	—	257	1	no	maximal, Sylow
31	C₂×M₁₆	—	257	2	no	—
32	C₂²×C₈	—	257	1	no	—
33	C₂²×C₄	—	257	1	no	—
34	M₁₆	—	257	4	no	—
35	M₁₆	—	514	2	no	—
36	C₂×C₈	—	257	1	no	—
37	C₂×C₈	—	514	2	no	—
38	C₂×C₈	—	514	2	no	—
39	C₂×C₈	#11759967	1	1	yes	characteristic
40	C₂³	—	257	1	no	—
41	C₂×C₄	—	257	1	no	—
42	C₂×C₄	—	257	2	no	—
43	C₂×C₄	—	257	2	no	—
44	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	514	4	no	—
47	C₈	—	514	1	no	—
48	C₂²	—	257	2	no	—
49	C₂²	—	514	1	no	—
50	C₂²	—	514	1	no	—
51	C₂²	C₄⋊#11759967	1	1	yes	characteristic
52	C₄	—	257	2	no	—
53	C₄	C₄×#11759967	1	1	yes	characteristic
54	C₄	C₂²⋊#11759967	1	1	yes	characteristic
55	C₂	—	514	2	no	—
56	C₂	(C₂×#11759967)⋊C₄	1	1	yes	characteristic
57	C₂	#22551662	1	1	yes	characteristic
58	C₂	#22551663	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11759967
commutator	C₄×C₂₅₇	C₄²
Frattini	C₂×C₄	C₂×#11759967
Fitting	C₂×C₈×C₂₅₇	C₄
socle	C₂²×C₂₅₇	C₄⋊C₄
prime radical	C₂²×D₂₅₇	Q₈
cosocle	C₂×C₄×D₂₅₇	C₂²
hypercenter	C₂×C₈	#11759967
hypocenter	C₂₅₇	#341
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
29. 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×#16861628	2
C₂×C₈×D₂₅₇	2
#341	257