Subgroups — #30657638

Order 16448.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551652	C₂	1	1	yes	maximal, characteristic, complemented
2	#22551650	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551562	C₂	1	1	yes	maximal, characteristic
4	#16861627	—	2	1	no	—
5	C₂²⋊#11759967	—	2	1	no	—
6	C₂³×D₂₅₇	C₄	1	1	yes	characteristic, complemented
7	C₂×D₁₀₂₈	—	2	1	no	—
8	C₂×C₂²⋊D₂₅₇	C₂²	1	1	yes	characteristic
9	#16861595	—	2	1	no	—
10	C₂²⋊C₄×C₂₅₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#11759967	—	4	1	no	—
12	#14523346	—	4	1	no	—
13	C₂²×D₂₅₇	—	2	1	no	—
14	C₂²×D₂₅₇	—	4	2	no	—
15	C₂²×D₂₅₇	—	4	1	no	—
16	C₂²×D₂₅₇	D₄	1	1	yes	characteristic
17	D₁₀₂₈	—	4	2	no	—
18	C₂²⋊D₂₅₇	—	4	1	no	—
19	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
20	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	—	2	1	no	—
22	C₂²×C₂₅₇	—	2	1	no	—
23	C₂²×C₂₅₇	—	4	1	no	—
24	C₂²×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₂₅₇	—	2	2	no	—
26	C₂×D₂₅₇	—	2	1	no	—
27	C₂×D₂₅₇	—	4	1	no	—
28	C₂×D₂₅₇	—	4	2	no	—
29	C₂×D₂₅₇	—	4	1	no	—
30	C₂×D₂₅₇	—	4	1	no	—
31	C₂×D₂₅₇	—	4	2	no	—
32	#11759967	—	8	1	no	—
33	C₄×C₂₅₇	—	4	1	no	—
34	Dic₂₅₇	—	2	1	no	—
35	C₂×C₂₅₇	—	2	1	no	—
36	C₂×C₂₅₇	—	4	1	no	—
37	C₂×C₂₅₇	#99	1	1	yes	characteristic
38	D₂₅₇	—	4	1	no	—
39	D₂₅₇	—	4	2	no	—
40	C₂₅₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	257	1	no	maximal, Sylow
42	#120	—	257	1	no	—
43	#100	—	257	1	no	—
44	#99	—	257	1	no	—
45	C₂⁴	—	257	1	no	—
46	C₂×D₄	—	257	1	no	—
47	C₂×D₄	—	514	1	no	—
48	M₁₆	—	514	1	no	—
49	C₂²⋊C₄	—	514	1	no	—
50	C₂²⋊C₄	—	514	1	no	—
51	C₂²⋊C₄	#11759967	1	1	yes	characteristic, complemented
52	C₂³	—	257	1	no	—
53	C₂³	—	514	1	no	—
54	C₂³	—	1028	1	no	—
55	C₂³	—	1028	2	no	—
56	C₂³	C₂×#11759967	1	1	yes	characteristic
57	D₄	—	1028	1	no	—
58	D₄	—	1028	2	no	—
59	C₂×C₄	—	257	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	1028	1	no	—
62	C₈	—	1028	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	514	2	no	—
65	C₂²	—	514	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	1028	1	no	—
68	C₂²	—	1028	1	no	—
69	C₂²	—	1028	2	no	—
70	C₂²	—	1028	2	no	—
71	C₂²	—	1028	1	no	—
72	C₂²	C₂²⋊#11759967	1	1	yes	characteristic
73	C₄	—	514	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	2056	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	1028	1	no	—
79	C₂	—	1028	2	no	—
80	C₂	#22551651	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22551651
commutator	C₂³×C₂₅₇	C₂×C₄
Frattini	C₂³	C₂×#11759967
Fitting	C₂²⋊C₄×C₂₅₇	C₄
socle	C₂×C₂₅₇	#99
prime radical	#22551562	C₂
cosocle	C₂×C₂²⋊D₂₅₇	C₂²
hypercenter	C₂²⋊C₄	#11759967
hypocenter	C₂₅₇	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #22551652	C₂	semidirect
2. #22551650	C₂	semidirect
6. C₂³×D₂₅₇	C₄	semidirect
10. C₂²⋊C₄×C₂₅₇	C₄	semidirect
40. C₂₅₇	AΣL(1,𝔽₁₆)	semidirect
51. C₂²⋊C₄	#11759967	semidirect

Maximal subgroups

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

subgroup	index
#22551652	2
#22551650	2
#22551562	2
AΣL(1,𝔽₁₆)	257