Subgroups — C₂×#22551532 #30657152

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551532	C₂	1	4	yes	maximal, direct product factor
2	C₂×#16861593	C₂	1	2	yes	maximal, complemented
3	C₂²×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#16861593	C₂²	1	4	yes	complemented
5	C₂²×C₄×C₂₅₇	C₄	1	1	yes	characteristic
6	C₂×C₈×C₂₅₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₂₅₇	C₄	1	2	yes	—
8	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
9	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂³×C₂₅₇	C₈	1	1	yes	characteristic
11	C₂×C₄×C₂₅₇	C₈	1	2	yes	—
12	C₂×C₄×C₂₅₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
15	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
16	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₅₇	C₂³	1	1	yes	characteristic
18	C₂²×C₂₅₇	—	2	2	no	—
19	C₂²×C₂₅₇	C₂×C₈	1	2	yes	—
20	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
21	C₄×C₂₅₇	C₂×C₈	1	2	yes	—
22	C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₂₅₇	—	2	2	no	—
25	C₂×C₂₅₇	M₃₂	1	2	yes	complemented
26	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
27	C₂₅₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₃₂	—	257	1	no	maximal, Sylow
29	C₂²×C₈	D₂₅₇	1	1	yes	characteristic
30	M₃₂	—	257	4	no	—
31	C₂×C₁₆	—	257	2	no	—
32	C₂²×C₄	Dic₂₅₇	1	1	yes	characteristic
33	C₂×C₈	Dic₂₅₇	1	1	yes	characteristic
34	C₂×C₈	Dic₂₅₇	1	2	yes	—
35	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
36	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
37	C₁₆	—	257	4	no	—
38	C₂³	#14523339	1	1	yes	characteristic
39	C₂×C₄	#14523339	1	2	yes	—
40	C₂×C₄	#14523339	1	1	yes	characteristic
41	C₂×C₄	C₂×Dic₂₅₇	1	2	yes	—
42	C₂×C₄	C₂×Dic₂₅₇	1	1	yes	characteristic
43	C₈	C₂×Dic₂₅₇	1	2	yes	—
44	C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
45	C₈	C₂²×D₂₅₇	1	1	yes	characteristic
46	C₂²	—	2	2	no	—
47	C₂²	C₂×#14523339	1	2	yes	—
48	C₂²	C₂×#14523339	1	1	yes	characteristic
49	C₄	C₂×#14523339	1	2	yes	—
50	C₄	C₂×#14523339	1	1	yes	characteristic
51	C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	#22551532	1	2	yes	direct product factor
54	C₂	C₂²×#14523339	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #22551532	C₂	direct (N × H)
2. C₂×#16861593	C₂	semidirect
4. #16861593	C₂²	semidirect
25. C₂×C₂₅₇	M₃₂	semidirect
27. C₂₅₇	C₂×M₃₂	semidirect
53. C₂	#22551532	direct (N × H)

Maximal subgroups

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

subgroup	index
#22551532	2
C₂×#16861593	2
C₂²×C₈×C₂₅₇	2
C₂×M₃₂	257

Subgroups — C2×#22551532 #30657152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22551532 #30657152