Subgroups — #392×C₂₅₇ #30656261

Order 16448.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₂₅₇	C₂	1	3	yes	maximal, complemented
2	C₂×C₂²⋊C₄×C₂₅₇	C₂	1	3	yes	maximal, complemented
3	#95×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₂₅₇	C₂²	1	3	yes	—
5	C₂×D₄×C₂₅₇	—	2	12	no	—
6	C₂×D₄×C₂₅₇	—	2	6	no	—
7	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₅₇	C₂²	1	3	yes	—
9	C₂²⋊C₄×C₂₅₇	—	2	6	no	—
10	C₂³×C₂₅₇	—	2	3	no	—
11	C₂³×C₂₅₇	—	2	6	no	—
12	C₂³×C₂₅₇	—	2	6	no	—
13	C₂³×C₂₅₇	—	2	3	no	—
14	C₂³×C₂₅₇	D₄	1	6	yes	complemented
15	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
16	D₄×C₂₅₇	—	2	24	no	—
17	C₂×C₄×C₂₅₇	—	2	6	no	—
18	C₂×C₄×C₂₅₇	—	2	3	no	—
19	C₂×C₄×C₂₅₇	D₄	1	6	yes	complemented
20	C₂²×C₂₅₇	—	2	12	no	—
21	C₂²×C₂₅₇	—	2	6	no	—
22	C₂²×C₂₅₇	—	4	3	no	—
23	C₂²×C₂₅₇	—	4	6	no	—
24	C₂²×C₂₅₇	—	4	3	no	—
25	C₂²×C₂₅₇	C₂×D₄	1	3	yes	—
26	C₂²×C₂₅₇	C₂×D₄	1	3	yes	—
27	C₂²×C₂₅₇	#41	1	1	yes	characteristic
28	C₄×C₂₅₇	—	2	6	no	—
29	C₄×C₂₅₇	—	4	1	no	—
30	C₂×C₂₅₇	—	4	6	no	—
31	C₂×C₂₅₇	#120	1	3	yes	—
32	C₂×C₂₅₇	#121	1	3	yes	—
33	C₂×C₂₅₇	#127	1	1	yes	characteristic
34	C₂₅₇	#392	1	1	yes	characteristic, direct product factor, Sylow
35	#392	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×D₄	C₂×C₂₅₇	1	3	yes	complemented
37	C₂×C₂²⋊C₄	C₂×C₂₅₇	1	3	yes	complemented
38	#95	C₂×C₂₅₇	1	1	yes	characteristic, complemented
39	C₂⁴	C₂²×C₂₅₇	1	3	yes	—
40	C₂×D₄	—	2	12	no	—
41	C₂×D₄	—	2	6	no	—
42	C₂²×C₄	C₂²×C₂₅₇	1	1	yes	characteristic
43	C₂²×C₄	C₂²×C₂₅₇	1	3	yes	—
44	C₂²⋊C₄	—	2	6	no	—
45	C₂³	—	2	3	no	—
46	C₂³	—	2	6	no	—
47	C₂³	—	2	6	no	—
48	C₂³	—	2	3	no	—
49	C₂³	D₄×C₂₅₇	1	6	yes	complemented
50	C₂³	C₂³×C₂₅₇	1	1	yes	characteristic
51	D₄	—	2	24	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	2	3	no	—
54	C₂×C₄	D₄×C₂₅₇	1	6	yes	complemented
55	C₂²	—	2	12	no	—
56	C₂²	—	2	6	no	—
57	C₂²	—	4	3	no	—
58	C₂²	—	4	6	no	—
59	C₂²	—	4	3	no	—
60	C₂²	C₂×D₄×C₂₅₇	1	3	yes	—
61	C₂²	C₂×D₄×C₂₅₇	1	3	yes	—
62	C₂²	#41×C₂₅₇	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	4	1	no	—
65	C₂	—	4	6	no	—
66	C₂	#120×C₂₅₇	1	3	yes	—
67	C₂	#121×C₂₅₇	1	3	yes	—
68	C₂	#127×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₅₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₂₅₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×C₂₅₇	C₂	semidirect
2. C₂×C₂²⋊C₄×C₂₅₇	C₂	semidirect
3. #95×C₂₅₇	C₂	semidirect
14. C₂³×C₂₅₇	D₄	semidirect
19. C₂×C₄×C₂₅₇	D₄	semidirect
34. C₂₅₇	#392	direct (N × H)
35. #392	C₂₅₇	direct (N × H)
36. C₂²×D₄	C₂×C₂₅₇	semidirect
37. C₂×C₂²⋊C₄	C₂×C₂₅₇	semidirect
38. #95	C₂×C₂₅₇	semidirect
49. C₂³	D₄×C₂₅₇	semidirect
54. C₂×C₄	D₄×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×C₂₅₇	2
C₂×C₂²⋊C₄×C₂₅₇	2
#95×C₂₅₇	2
#392	257

Subgroups — #392×C257 #30656261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #392×C₂₅₇ #30656261