Subgroups — #393×C₂₅₇ #30656262

Order 16448.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊C₄×C₂₅₇	C₂	1	3	yes	maximal
3	#95×C₂₅₇	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₂₅₇	C₂²	1	1	yes	characteristic
5	C₂×Q₈×C₂₅₇	—	2	6	no	—
6	C₂²×C₄×C₂₅₇	C₂²	1	3	yes	—
7	C₂²×C₄×C₂₅₇	C₂²	1	3	yes	—
8	C₂²⋊C₄×C₂₅₇	—	2	6	no	—
9	C₂³×C₂₅₇	—	2	3	no	—
10	C₂³×C₂₅₇	—	2	3	no	—
11	C₂³×C₂₅₇	Q₈	1	2	yes	complemented
12	C₂³×C₂₅₇	C₂³	1	1	yes	characteristic
13	Q₈×C₂₅₇	—	2	8	no	—
14	C₂×C₄×C₂₅₇	—	2	6	no	—
15	C₂×C₄×C₂₅₇	—	2	3	no	—
16	C₂×C₄×C₂₅₇	—	2	6	no	—
17	C₂×C₄×C₂₅₇	D₄	1	6	yes	—
18	C₂²×C₂₅₇	—	2	6	no	—
19	C₂²×C₂₅₇	—	4	1	no	—
20	C₂²×C₂₅₇	—	4	3	no	—
21	C₂²×C₂₅₇	C₂×D₄	1	3	yes	—
22	C₂²×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
23	C₂²×C₂₅₇	#41	1	3	yes	—
24	C₄×C₂₅₇	—	2	6	no	—
25	C₄×C₂₅₇	—	4	3	no	—
26	C₂×C₂₅₇	—	4	2	no	—
27	C₂×C₂₅₇	#120	1	1	yes	characteristic
28	C₂×C₂₅₇	C₂²⋊Q₈	1	3	yes	—
29	C₂×C₂₅₇	#124	1	3	yes	—
30	C₂₅₇	#393	1	1	yes	characteristic, direct product factor, Sylow
31	#393	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×Q₈	C₂×C₂₅₇	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	C₂×C₂₅₇	1	3	yes	—
34	#95	C₂×C₂₅₇	1	3	yes	complemented
35	C₂⁴	C₂²×C₂₅₇	1	1	yes	characteristic
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	C₂²×C₂₅₇	1	3	yes	—
38	C₂²×C₄	C₂²×C₂₅₇	1	3	yes	—
39	C₂²⋊C₄	—	2	6	no	—
40	C₂³	—	2	3	no	—
41	C₂³	—	2	3	no	—
42	C₂³	Q₈×C₂₅₇	1	2	yes	complemented
43	C₂³	C₂³×C₂₅₇	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	6	no	—
48	C₂×C₄	D₄×C₂₅₇	1	6	yes	—
49	C₂²	—	2	6	no	—
50	C₂²	—	4	1	no	—
51	C₂²	—	4	3	no	—
52	C₂²	C₂×D₄×C₂₅₇	1	3	yes	—
53	C₂²	C₂×Q₈×C₂₅₇	1	1	yes	characteristic
54	C₂²	#41×C₂₅₇	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	4	3	no	—
57	C₂	—	4	2	no	—
58	C₂	#120×C₂₅₇	1	1	yes	characteristic
59	C₂	C₂²⋊Q₈×C₂₅₇	1	3	yes	—
60	C₂	#124×C₂₅₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₅₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₂₅₇
prime radical	C₂⁴×C₂₅₇	C₂²
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×Q₈×C₂₅₇	C₂	semidirect
3. #95×C₂₅₇	C₂	semidirect
11. C₂³×C₂₅₇	Q₈	semidirect
30. C₂₅₇	#393	direct (N × H)
31. #393	C₂₅₇	direct (N × H)
32. C₂²×Q₈	C₂×C₂₅₇	semidirect
34. #95	C₂×C₂₅₇	semidirect
42. C₂³	Q₈×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₅₇	2
C₂×C₂²⋊C₄×C₂₅₇	2
#95×C₂₅₇	2
#393	257

Subgroups — #393×C257 #30656262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #393×C₂₅₇ #30656262