Subgroups — #496×C₂₅₇ #30656365

Order 16448.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×C₂₅₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×C₂₅₇	C₂	1	2	yes	maximal, complemented
3	#127×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#124×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₂₅₇	—	2	2	no	—
8	C₂×D₄×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×D₄×C₂₅₇	C₂²	1	2	yes	—
10	SD₁₆×C₂₅₇	—	2	4	no	—
11	D₈×C₂₅₇	—	2	4	no	—
12	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₂₅₇	—	2	2	no	—
14	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂³×C₂₅₇	—	2	1	no	—
16	C₂³×C₂₅₇	—	2	2	no	—
17	Q₈×C₂₅₇	—	2	2	no	—
18	D₄×C₂₅₇	—	2	2	no	—
19	D₄×C₂₅₇	—	2	2	no	—
20	D₄×C₂₅₇	—	2	2	no	—
21	D₄×C₂₅₇	—	4	4	no	—
22	C₂×C₄×C₂₅₇	—	2	1	no	—
23	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
24	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₈×C₂₅₇	D₄	1	4	yes	complemented
26	C₂²×C₂₅₇	—	4	1	no	—
27	C₂²×C₂₅₇	—	4	2	no	—
28	C₂²×C₂₅₇	—	4	4	no	—
29	C₂²×C₂₅₇	—	4	2	no	—
30	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₅₇	—	2	2	no	—
32	C₄×C₂₅₇	—	4	1	no	—
33	C₄×C₂₅₇	C₂×D₄	1	2	yes	—
34	C₂×C₂₅₇	—	8	2	no	—
35	C₂×C₂₅₇	—	8	1	no	—
36	C₂×C₂₅₇	#127	1	1	yes	characteristic
37	C₂×C₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
38	C₂₅₇	#496	1	1	yes	characteristic, direct product factor, Sylow
39	#496	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×SD₁₆	C₂×C₂₅₇	1	2	yes	complemented
41	C₂×D₈	C₂×C₂₅₇	1	2	yes	complemented
42	#127	C₂×C₂₅₇	1	1	yes	characteristic, complemented
43	#124	C₂×C₂₅₇	1	1	yes	characteristic, complemented
44	#97	C₂×C₂₅₇	1	1	yes	characteristic, complemented
45	C₂×Q₈	C₂²×C₂₅₇	1	1	yes	characteristic
46	C₂×D₄	—	2	2	no	—
47	C₂×D₄	C₂²×C₂₅₇	1	1	yes	characteristic
48	C₂×D₄	C₂²×C₂₅₇	1	2	yes	—
49	SD₁₆	—	2	4	no	—
50	D₈	—	2	4	no	—
51	C₂×C₈	C₂²×C₂₅₇	1	2	yes	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂²×C₂₅₇	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	2	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	4	4	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	D₄×C₂₅₇	1	2	yes	—
63	C₂×C₄	C₂³×C₂₅₇	1	1	yes	characteristic
64	C₈	D₄×C₂₅₇	1	4	yes	complemented
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	4	4	no	—
68	C₂²	—	4	2	no	—
69	C₂²	C₂×D₄×C₂₅₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	C₂×D₄×C₂₅₇	1	2	yes	—
73	C₂	—	8	2	no	—
74	C₂	—	8	1	no	—
75	C₂	#127×C₂₅₇	1	1	yes	characteristic
76	C₂	AGL(1,ℤ/8)×C₂₅₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₅₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₇ ◁ C₄²×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×SD₁₆×C₂₅₇	C₂	semidirect
2. C₂×D₈×C₂₅₇	C₂	semidirect
3. #127×C₂₅₇	C₂	semidirect
4. #124×C₂₅₇	C₂	semidirect
5. #97×C₂₅₇	C₂	semidirect
25. C₈×C₂₅₇	D₄	semidirect
38. C₂₅₇	#496	direct (N × H)
39. #496	C₂₅₇	direct (N × H)
40. C₂×SD₁₆	C₂×C₂₅₇	semidirect
41. C₂×D₈	C₂×C₂₅₇	semidirect
42. #127	C₂×C₂₅₇	semidirect
43. #124	C₂×C₂₅₇	semidirect
44. #97	C₂×C₂₅₇	semidirect
64. C₈	D₄×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×C₂₅₇	2
C₂×D₈×C₂₅₇	2
#127×C₂₅₇	2
#124×C₂₅₇	2
#97×C₂₅₇	2
#496	257

Subgroups — #496×C257 #30656365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #496×C₂₅₇ #30656365