Subgroups — C₂×#22551649 #30657706

Order 16448.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523346	C₂	1	2	yes	maximal, complemented
2	#22551649	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×D₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523346	—	2	4	no	—
5	C₂×#14523346	C₂²	1	4	yes	complemented
6	C₂³×D₂₅₇	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₅₇	—	2	4	no	—
9	C₂×C₄×D₂₅₇	C₄	1	2	yes	—
10	C₂×C₄×D₂₅₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₅₇	C₄	1	1	yes	characteristic
12	#14523346	—	2	4	no	—
13	C₂²×D₂₅₇	—	2	2	no	—
14	C₂²×D₂₅₇	—	2	2	no	—
15	C₂²×D₂₅₇	C₈	1	4	yes	complemented
16	C₂²×D₂₅₇	C₂×C₄	1	2	yes	—
17	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₅₇	D₄	1	4	yes	—
19	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
20	C₄×D₂₅₇	—	2	4	no	—
21	C₄×D₂₅₇	—	2	4	no	—
22	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₇	—	2	2	no	—
24	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
25	C₂²×C₂₅₇	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₂₅₇	C₂×C₈	1	2	yes	—
27	C₂²×C₂₅₇	M₁₆	1	2	yes	—
28	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₂₅₇	—	2	8	no	—
30	C₂×D₂₅₇	—	2	2	no	—
31	C₂×D₂₅₇	—	2	2	no	—
32	C₂×D₂₅₇	C₂×C₈	1	4	yes	complemented
33	C₄×C₂₅₇	—	2	2	no	—
34	Dic₂₅₇	C₂²⋊C₄	1	2	yes	—
35	Dic₂₅₇	C₂×D₄	1	2	yes	—
36	C₂×C₂₅₇	C₂²⋊C₈	1	4	yes	complemented
37	C₂×C₂₅₇	C₂×C₂²⋊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	—	2	4	no	—
41	C₂₅₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₂²⋊C₈	—	257	1	no	maximal, Sylow
43	C₂³×C₄	—	257	1	no	—
44	C₂²×C₈	—	257	2	no	—
45	C₂²⋊C₈	—	257	4	no	—
46	C₂⁴	—	257	1	no	—
47	C₂²×C₄	—	257	1	no	—
48	C₂²×C₄	—	257	2	no	—
49	C₂²×C₄	—	257	2	no	—
50	C₂²×C₄	—	514	4	no	—
51	C₂²×C₄	#11759967	1	1	yes	characteristic
52	C₂×C₈	—	257	4	no	—
53	C₂×C₈	—	514	4	no	—
54	C₂³	—	257	4	no	—
55	C₂³	—	257	2	no	—
56	C₂³	—	514	2	no	—
57	C₂³	—	514	2	no	—
58	C₂³	C₂×#11759967	1	1	yes	characteristic
59	C₂×C₄	—	257	4	no	—
60	C₂×C₄	—	257	1	no	—
61	C₂×C₄	—	257	1	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	514	4	no	—
64	C₂×C₄	—	514	4	no	—
65	C₂×C₄	C₂×#11759967	1	2	yes	—
66	C₈	—	514	4	no	—
67	C₂²	—	257	4	no	—
68	C₂²	—	514	8	no	—
69	C₂²	—	514	2	no	—
70	C₂²	—	514	2	no	—
71	C₂²	C₂²⋊#11759967	1	2	yes	—
72	C₂²	#16861626	1	2	yes	—
73	C₂²	#16861628	1	2	yes	—
74	C₂²	C₂²×#11759967	1	1	yes	characteristic
75	C₄	—	257	2	no	—
76	C₄	—	257	2	no	—
77	C₄	—	2	2	no	—
78	C₂	—	514	4	no	—
79	C₂	#22551649	1	4	yes	direct product factor
80	C₂	C₂×C₂²⋊#11759967	1	1	yes	characteristic
81	C₂	C₂×#16861626	1	1	yes	characteristic
82	C₂	C₂×#16861628	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523346	C₂	semidirect
2. #22551649	C₂	direct (N × H)
5. C₂×#14523346	C₂²	semidirect
15. C₂²×D₂₅₇	C₈	semidirect
32. C₂×D₂₅₇	C₂×C₈	semidirect
36. C₂×C₂₅₇	C₂²⋊C₈	semidirect
41. C₂₅₇	C₂×C₂²⋊C₈	semidirect
79. C₂	#22551649	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523346	2
#22551649	2
C₂²×C₄×D₂₅₇	2
C₂×C₂²⋊C₈	257

Subgroups — C2×#22551649 #30657706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22551649 #30657706