Subgroups — #30657793

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551667	C₂	1	2	yes	maximal, complemented
2	#22551666	C₂	1	2	yes	maximal, complemented
3	#22551665	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
5	#16861621	C₂²	1	4	yes	complemented
6	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
7	C₂×#14523339	C₂²	1	1	yes	characteristic
8	C₈×D₂₅₇	C₄	1	1	yes	characteristic
9	C₈×D₂₅₇	C₄	1	1	yes	characteristic
10	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
11	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₅₇	C₄	1	1	yes	characteristic
13	C₂²×D₂₅₇	C₈	1	1	yes	characteristic
14	C₂×Dic₂₅₇	C₈	1	1	yes	characteristic
15	C₄×D₂₅₇	C₈	1	1	yes	characteristic
16	C₄×D₂₅₇	C₈	1	1	yes	characteristic
17	C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
19	#14523339	C₂×C₄	1	1	yes	characteristic
20	#14523339	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
22	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
25	C₂×D₂₅₇	—	2	2	no	—
26	C₂×D₂₅₇	C₂×C₈	1	1	yes	characteristic
27	C₂×D₂₅₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
29	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
30	Dic₂₅₇	C₂×C₈	1	1	yes	characteristic
31	Dic₂₅₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₂₅₇	—	2	1	no	—
33	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
34	D₂₅₇	—	2	1	no	—
35	D₂₅₇	M₃₂	1	2	yes	complemented
36	C₂₅₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
37	C₂×M₃₂	—	257	1	no	maximal, Sylow
38	C₂²×C₈	—	257	1	no	—
39	M₃₂	—	257	2	no	—
40	M₃₂	—	257	2	no	—
41	C₂×C₁₆	—	257	2	no	—
42	C₂²×C₄	—	257	1	no	—
43	C₂×C₈	—	257	1	no	—
44	C₂×C₈	—	257	1	no	—
45	C₂×C₈	—	257	1	no	—
46	C₂×C₈	—	257	1	no	—
47	C₂×C₈	—	257	1	no	—
48	C₂×C₈	#11759967	1	1	yes	characteristic
49	C₁₆	—	257	4	no	—
50	C₂³	—	257	1	no	—
51	C₂×C₄	—	257	1	no	—
52	C₂×C₄	—	257	1	no	—
53	C₂×C₄	—	257	1	no	—
54	C₂×C₄	—	257	1	no	—
55	C₂×C₄	—	257	1	no	—
56	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
57	C₈	—	257	1	no	—
58	C₈	—	257	1	no	—
59	C₈	C₂×#11759967	1	1	yes	characteristic
60	C₈	C₂×#11759967	1	1	yes	characteristic
61	C₂²	—	257	1	no	—
62	C₂²	—	257	1	no	—
63	C₂²	—	514	2	no	—
64	C₂²	#16861626	1	1	yes	characteristic
65	C₄	—	257	1	no	—
66	C₄	—	257	1	no	—
67	C₄	#16861626	1	1	yes	characteristic
68	C₄	C₂²×#11759967	1	1	yes	characteristic
69	C₂	—	257	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	514	1	no	—
72	C₂	C₂×#16861626	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#11759967
commutator, socle	C₂×C₂₅₇	C₂²×C₈
Frattini	C₄	C₂²×#11759967
Fitting	C₂×C₈×C₂₅₇	C₄
prime radical	C₂²×D₂₅₇	C₈
cosocle	#14523339	C₂³
hypercenter	C₂×C₈	#11759967
hypocenter	C₂₅₇	C₂×M₃₂
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #22551667	C₂	semidirect
2. #22551666	C₂	semidirect
3. #22551665	C₂	semidirect
5. #16861621	C₂²	semidirect
35. D₂₅₇	M₃₂	semidirect
36. C₂₅₇	C₂×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#22551667	2
#22551666	2
#22551665	2
C₂×C₈×D₂₅₇	2
C₂×M₃₂	257