Subgroups — #30657738

Order 16448.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861626	C₂	1	2	yes	maximal
2	#22551659	C₂	1	2	yes	maximal, complemented
3	C₄⋊#14523346	C₂	1	2	yes	maximal, complemented
4	C₄²×D₂₅₇	C₂	1	1	yes	maximal, characteristic
5	#16861626	—	2	4	no	—
6	C₂×#14523346	C₂²	1	4	yes	—
7	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
9	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₅₇	C₄	1	1	yes	characteristic
11	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
13	C₄²×C₂₅₇	C₄	1	1	yes	characteristic
14	#14523346	—	2	4	no	—
15	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
19	C₄×D₂₅₇	—	2	2	no	—
20	C₄×D₂₅₇	C₈	1	4	yes	complemented
21	C₄×D₂₅₇	D₄	1	2	yes	—
22	C₄×D₂₅₇	Q₈	1	2	yes	—
23	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₅₇	C₄⋊C₄	1	2	yes	—
28	C₂×D₂₅₇	C₂×C₈	1	2	yes	—
29	C₂×D₂₅₇	M₁₆	1	2	yes	—
30	C₄×C₂₅₇	—	2	1	no	—
31	C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
32	C₄×C₂₅₇	C₂×C₈	1	2	yes	complemented
33	Dic₂₅₇	—	2	1	no	—
34	Dic₂₅₇	C₂×C₈	1	2	yes	complemented
35	Dic₂₅₇	C₂×D₄	1	1	yes	characteristic
36	Dic₂₅₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
40	D₂₅₇	C₄⋊C₈	1	4	yes	complemented
41	C₂₅₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄⋊C₈	—	257	1	no	maximal, Sylow
43	C₂²×C₈	—	257	2	no	—
44	C₂×C₄²	—	257	1	no	—
45	C₄⋊C₈	—	257	2	no	—
46	C₄⋊C₈	—	257	2	no	—
47	C₂²×C₄	—	257	1	no	—
48	C₂²×C₄	—	257	1	no	—
49	C₂²×C₄	—	257	1	no	—
50	C₂×C₈	—	257	4	no	—
51	C₂×C₈	—	514	4	no	—
52	C₄²	—	257	1	no	—
53	C₄²	—	257	1	no	—
54	C₄²	—	257	1	no	—
55	C₄²	#11759967	1	1	yes	characteristic
56	C₂³	—	257	1	no	—
57	C₂×C₄	—	257	1	no	—
58	C₂×C₄	—	257	1	no	—
59	C₂×C₄	—	257	2	no	—
60	C₂×C₄	—	257	2	no	—
61	C₂×C₄	—	257	4	no	—
62	C₂×C₄	—	257	1	no	—
63	C₂×C₄	—	514	2	no	—
64	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
65	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
66	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
67	C₈	—	514	4	no	—
68	C₂²	—	257	2	no	—
69	C₂²	—	257	2	no	—
70	C₂²	—	257	2	no	—
71	C₂²	C₂²×#11759967	1	1	yes	characteristic
72	C₄	—	257	1	no	—
73	C₄	—	257	1	no	—
74	C₄	—	257	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	514	1	no	—
77	C₄	C₄⋊#11759967	1	2	yes	—
78	C₄	#16861626	1	2	yes	complemented
79	C₂	—	257	4	no	—
80	C₂	C₂×C₄⋊#11759967	1	1	yes	characteristic
81	C₂	C₂×#16861626	1	1	yes	characteristic
82	C₂	D₂₅₇⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11759967
commutator	C₂×C₂₅₇	C₂²×C₈
Frattini	C₂²	C₂²×#11759967
Fitting	C₄²×C₂₅₇	C₄
socle	C₂²×C₂₅₇	C₂²×C₄
prime radical	C₂²×D₂₅₇	C₂×C₄
cosocle	C₂×Dic₂₅₇	C₂³
hypercenter	C₄²	#11759967
hypocenter	C₂₅₇	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #22551659	C₂	semidirect
3. C₄⋊#14523346	C₂	semidirect
20. C₄×D₂₅₇	C₈	semidirect
32. C₄×C₂₅₇	C₂×C₈	semidirect
34. Dic₂₅₇	C₂×C₈	semidirect
40. D₂₅₇	C₄⋊C₈	semidirect
41. C₂₅₇	C₂×C₄⋊C₈	semidirect
78. C₄	#16861626	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861626	2
#22551659	2
C₄⋊#14523346	2
C₄²×D₂₅₇	2
C₂×C₄⋊C₈	257