Subgroups — C₂×#22551665 #30657791

Order 16448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551665	C₂	1	4	yes	maximal, direct product factor
2	C₂×#16861621	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#16861621	C₂²	1	4	yes	complemented
5	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
6	C₂×#14523339	C₂²	1	1	yes	characteristic
7	C₈×D₂₅₇	C₄	1	2	yes	—
8	C₈×D₂₅₇	C₂²	1	2	yes	—
9	C₂×C₈×C₂₅₇	C₄	1	1	yes	characteristic
10	C₂²×D₂₅₇	C₈	1	1	yes	characteristic
11	C₂×Dic₂₅₇	C₈	1	1	yes	characteristic
12	C₄×D₂₅₇	C₈	1	2	yes	—
13	C₄×D₂₅₇	C₂×C₄	1	2	yes	—
14	#14523339	C₂×C₄	1	1	yes	characteristic
15	#14523339	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
18	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
19	C₂×D₂₅₇	C₁₆	1	4	yes	complemented
20	C₂×D₂₅₇	C₂×C₈	1	2	yes	—
21	C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
22	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
23	Dic₂₅₇	C₂×C₈	1	2	yes	—
24	C₂×C₂₅₇	C₂×C₁₆	1	2	yes	complemented
25	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
26	D₂₅₇	C₂×C₁₆	1	4	yes	complemented
27	C₂₅₇	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂²×C₁₆	—	257	1	no	maximal, Sylow
29	C₂²×C₈	—	257	1	no	—
30	C₂×C₁₆	—	257	2	no	—
31	C₂×C₁₆	—	257	4	no	—
32	C₂²×C₄	—	257	1	no	—
33	C₂×C₈	—	257	1	no	—
34	C₂×C₈	—	257	2	no	—
35	C₂×C₈	—	257	2	no	—
36	C₂×C₈	#11759967	1	1	yes	characteristic
37	C₁₆	—	257	4	no	—
38	C₂³	—	257	1	no	—
39	C₂×C₄	—	257	1	no	—
40	C₂×C₄	—	257	2	no	—
41	C₂×C₄	—	257	2	no	—
42	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
43	C₈	—	257	1	no	—
44	C₈	—	257	1	no	—
45	C₈	C₂×#11759967	1	2	yes	—
46	C₂²	—	257	4	no	—
47	C₂²	—	257	2	no	—
48	C₂²	#16861626	1	1	yes	characteristic
49	C₄	—	257	2	no	—
50	C₄	#16861626	1	1	yes	characteristic
51	C₄	C₂²×#11759967	1	1	yes	characteristic
52	C₂	—	257	4	no	—
53	C₂	#22551665	1	2	yes	direct product factor
54	C₂	C₂×#16861626	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	#11759967
commutator, hypocenter	C₂₅₇	C₂²×C₁₆
Frattini	C₄	C₂²×#11759967
Fitting	C₂×C₈×C₂₅₇	C₄
socle	C₂²×C₂₅₇	C₂×C₈
prime radical	C₂²×D₂₅₇	C₈
cosocle	#14523339	C₂³
LCS	G ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #22551665	C₂	direct (N × H)
2. C₂×#16861621	C₂	semidirect
4. #16861621	C₂²	semidirect
19. C₂×D₂₅₇	C₁₆	semidirect
24. C₂×C₂₅₇	C₂×C₁₆	semidirect
26. D₂₅₇	C₂×C₁₆	semidirect
27. C₂₅₇	C₂²×C₁₆	semidirect
53. C₂	#22551665	direct (N × H)

Maximal subgroups

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

subgroup	index
#22551665	2
C₂×#16861621	2
C₂×C₈×D₂₅₇	2
C₂²×C₁₆	257

Subgroups — C2×#22551665 #30657791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22551665 #30657791