Subgroups — #418×C₂₅₁ #30631237

Order 16064.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#102×C₂₅₁	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₂₅₁	—	2	1	no	—
6	C₂×D₄×C₂₅₁	—	2	1	no	—
7	C₂×D₄×C₂₅₁	—	2	2	no	—
8	C₂×D₄×C₂₅₁	C₄	1	4	yes	complemented
9	C₂×D₄×C₂₅₁	C₂²	1	2	yes	—
10	C₂²×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
11	M₁₆×C₂₅₁	—	2	2	no	—
12	C₂×C₈×C₂₅₁	C₂²	1	2	yes	complemented
13	C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
15	C₄²×C₂₅₁	—	2	1	no	—
16	C₂³×C₂₅₁	—	2	4	no	—
17	C₂³×C₂₅₁	—	2	2	no	—
18	C₂³×C₂₅₁	—	4	4	no	—
19	C₂³×C₂₅₁	D₄	1	1	yes	characteristic
20	D₄×C₂₅₁	—	2	4	no	—
21	D₄×C₂₅₁	—	2	2	no	—
22	D₄×C₂₅₁	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₂₅₁	—	2	2	no	—
24	C₂×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₅₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₁	D₄	1	2	yes	—
27	C₂×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
28	C₈×C₂₅₁	—	2	2	no	—
29	C₂²×C₂₅₁	—	2	2	no	—
30	C₂²×C₂₅₁	—	2	4	no	—
31	C₂²×C₂₅₁	—	4	8	no	—
32	C₂²×C₂₅₁	—	4	4	no	—
33	C₂²×C₂₅₁	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₂₅₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₅₁	—	4	2	no	—
36	C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
37	C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₂₅₁	—	2	2	no	—
40	C₂×C₂₅₁	—	4	4	no	—
41	C₂×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
43	C₂₅₁	#418	1	1	yes	characteristic, direct product factor, Sylow
44	#418	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×D₄	C₂×C₂₅₁	1	1	yes	characteristic
46	C₂×M₁₆	C₂×C₂₅₁	1	1	yes	characteristic, complemented
47	#117	C₂×C₂₅₁	1	1	yes	characteristic, complemented
48	#102	C₂×C₂₅₁	1	4	yes	complemented
49	C₂⁴	—	2	1	no	—
50	C₂×D₄	—	2	1	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	C₄×C₂₅₁	1	4	yes	complemented
53	C₂×D₄	C₂²×C₂₅₁	1	2	yes	—
54	C₂²×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	C₂×C₈	C₂²×C₂₅₁	1	2	yes	complemented
57	C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	complemented
58	C₂²⋊C₄	—	2	1	no	—
59	C₄²	—	2	1	no	—
60	C₂³	—	2	4	no	—
61	C₂³	—	2	2	no	—
62	C₂³	—	4	4	no	—
63	C₂³	D₄×C₂₅₁	1	1	yes	characteristic
64	D₄	—	2	4	no	—
65	D₄	—	2	2	no	—
66	D₄	C₂×C₄×C₂₅₁	1	4	yes	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
69	C₂×C₄	D₄×C₂₅₁	1	1	yes	characteristic
70	C₂×C₄	D₄×C₂₅₁	1	2	yes	—
71	C₂×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	4	8	no	—
76	C₂²	—	4	4	no	—
77	C₂²	C₂²⋊C₄×C₂₅₁	1	2	yes	—
78	C₂²	C₂×D₄×C₂₅₁	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	C₂²⋊C₄×C₂₅₁	1	2	yes	—
81	C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
82	C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	4	no	—
85	C₂	C₂×C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
86	C₂	AGL(1,ℤ/8)×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂×D₄
commutator	C₄	C₂²×C₄×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂³×C₂₅₁
prime radical	C₂²×D₄×C₂₅₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₂²×C₄×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×C₂₅₁	C₂	semidirect
3. #117×C₂₅₁	C₂	semidirect
4. #102×C₂₅₁	C₂	semidirect
8. C₂×D₄×C₂₅₁	C₄	semidirect
12. C₂×C₈×C₂₅₁	C₂²	semidirect
13. C₄⋊C₄×C₂₅₁	C₂²	semidirect
43. C₂₅₁	#418	direct (N × H)
44. #418	C₂₅₁	direct (N × H)
46. C₂×M₁₆	C₂×C₂₅₁	semidirect
47. #117	C₂×C₂₅₁	semidirect
48. #102	C₂×C₂₅₁	semidirect
52. C₂×D₄	C₄×C₂₅₁	semidirect
56. C₂×C₈	C₂²×C₂₅₁	semidirect
57. C₄⋊C₄	C₂²×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×C₂₅₁	2
C₂×M₁₆×C₂₅₁	2
#117×C₂₅₁	2
#102×C₂₅₁	2
#418	251

Subgroups — #418×C251 #30631237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #418×C₂₅₁ #30631237