Subgroups — #413×C₂₅₇ #30656282

Order 16448.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₂₅₇	C₂	1	2	yes	maximal, complemented
3	#101×C₂₅₇	C₂	1	2	yes	maximal, complemented
4	#100×C₂₅₇	C₂	1	2	yes	maximal, complemented
5	#41×C₂₅₇	—	2	2	no	—
6	#41×C₂₅₇	—	2	2	no	—
7	C₂×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₂₅₇	C₄	1	2	yes	—
9	C₂×D₄×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₅₇	C₄	1	2	yes	—
11	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
12	M₁₆×C₂₅₇	—	2	2	no	—
13	M₁₆×C₂₅₇	C₂²	1	4	yes	—
14	C₂×C₈×C₂₅₇	—	2	2	no	—
15	C₂³×C₂₅₇	C₂×C₄	1	2	yes	—
16	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	Q₈×C₂₅₇	—	2	2	no	—
18	D₄×C₂₅₇	—	2	4	no	—
19	D₄×C₂₅₇	—	2	2	no	—
20	C₂×C₄×C₂₅₇	—	2	2	no	—
21	C₂×C₄×C₂₅₇	—	2	2	no	—
22	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
25	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
26	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
27	C₈×C₂₅₇	—	2	4	no	—
28	C₂²×C₂₅₇	—	2	2	no	—
29	C₂²×C₂₅₇	—	4	1	no	—
30	C₂²×C₂₅₇	—	4	2	no	—
31	C₂²×C₂₅₇	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₂₅₇	—	2	2	no	—
34	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₂₅₇	C₂×D₄	1	2	yes	—
37	C₂×C₂₅₇	—	2	1	no	—
38	C₂×C₂₅₇	—	2	2	no	—
39	C₂×C₂₅₇	—	4	2	no	—
40	C₂×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂₅₇	#413	1	1	yes	characteristic, direct product factor, Sylow
42	#413	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×#41	C₂×C₂₅₇	1	1	yes	characteristic
44	C₂×M₁₆	C₂×C₂₅₇	1	2	yes	complemented
45	#101	C₂×C₂₅₇	1	2	yes	complemented
46	#100	C₂×C₂₅₇	1	2	yes	complemented
47	#41	—	2	2	no	—
48	#41	—	2	2	no	—
49	C₂×Q₈	C₂²×C₂₅₇	1	1	yes	characteristic
50	C₂×D₄	C₄×C₂₅₇	1	2	yes	—
51	C₂×D₄	C₂²×C₂₅₇	1	1	yes	characteristic
52	C₂²×C₄	C₄×C₂₅₇	1	2	yes	—
53	C₂²×C₄	C₂²×C₂₅₇	1	1	yes	characteristic
54	M₁₆	—	2	2	no	—
55	M₁₆	C₂²×C₂₅₇	1	4	yes	—
56	C₂×C₈	—	2	2	no	—
57	C₂³	C₂×C₄×C₂₅₇	1	2	yes	—
58	C₂³	C₂×C₄×C₂₅₇	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	D₄	—	2	4	no	—
61	D₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	C₂×C₄×C₂₅₇	1	2	yes	—
65	C₂×C₄	C₂×C₄×C₂₅₇	1	1	yes	characteristic
66	C₂×C₄	D₄×C₂₅₇	1	2	yes	—
67	C₂×C₄	D₄×C₂₅₇	1	2	yes	—
68	C₂×C₄	C₂³×C₂₅₇	1	1	yes	characteristic
69	C₈	—	2	4	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	2	no	—
73	C₂²	C₂²⋊C₄×C₂₅₇	1	2	yes	—
74	C₂²	C₂²×C₄×C₂₅₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	C₂²⋊C₄×C₂₅₇	1	1	yes	characteristic
77	C₄	C₂²⋊C₄×C₂₅₇	1	1	yes	characteristic
78	C₄	C₂×D₄×C₂₅₇	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	2	2	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊C₄×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₅₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₂₅₇
Frattini, cosocle	C₂×C₄	C₂³×C₂₅₇
socle	C₂×C₂₅₇	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₂₅₇	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₅₇ ◁ C₂²×C₄×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×C₂₅₇	C₂	semidirect
3. #101×C₂₅₇	C₂	semidirect
4. #100×C₂₅₇	C₂	semidirect
41. C₂₅₇	#413	direct (N × H)
42. #413	C₂₅₇	direct (N × H)
44. C₂×M₁₆	C₂×C₂₅₇	semidirect
45. #101	C₂×C₂₅₇	semidirect
46. #100	C₂×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₂₅₇	2
C₂×M₁₆×C₂₅₇	2
#101×C₂₅₇	2
#100×C₂₅₇	2
#413	257

Subgroups — #413×C257 #30656282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #413×C₂₅₇ #30656282