Subgroups — #44550116

Order 32128.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548626	C₂	1	1	yes	maximal, characteristic
2	#30631533	C₂	1	1	yes	maximal, characteristic
3	#30631441	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
7	#22548626	C₄	1	4	yes	complemented
8	C₂₅₁⋊#95	C₄	1	4	yes	—
9	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₅₁	C₄	1	4	yes	complemented
12	C₂²×Dic₂₅₁	C₂×C₄	1	4	yes	—
13	C₂×C₂₅₁⋊C₈	—	2	4	no	—
14	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×Dic₂₅₁	—	2	4	no	—
16	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₅₁	—	2	4	no	—
19	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
20	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
21	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₅₁	—	2	4	no	—
23	C₂×Dic₂₅₁	—	2	8	no	—
24	C₂₅₁⋊C₈	—	2	4	no	—
25	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₁	—	2	4	no	—
27	C₂×C₄×C₂₅₁	C₄²	1	4	yes	—
28	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₅₁	#41	1	4	yes	—
31	C₈×C₂₅₁	—	2	4	no	—
32	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₂₅₁	#117	1	2	yes	—
34	C₂²×C₂₅₁	#131	1	4	yes	—
35	C₄×C₂₅₁	—	2	4	no	—
36	C₄×C₂₅₁	#117	1	2	yes	—
37	C₄×C₂₅₁	#117	1	2	yes	—
38	Dic₂₅₁	—	4	4	no	—
39	C₂×C₂₅₁	#376	1	1	yes	characteristic
40	C₂×C₂₅₁	#405	1	2	yes	—
41	C₂×C₂₅₁	#433	1	4	yes	—
42	C₂₅₁	#1746	1	1	yes	characteristic, complemented, Sylow
43	#1746	—	251	1	no	maximal, Sylow
44	C₂×#97	—	251	1	no	—
45	C₂×C₄×C₈	D₂₅₁	1	1	yes	characteristic
46	#376	—	251	1	no	—
47	#336	—	251	4	no	—
48	C₂²×C₈	—	251	2	no	—
49	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
50	C₂×C₄²	—	251	2	no	—
51	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
52	#97	—	251	4	no	—
53	C₄×C₈	Dic₂₅₁	1	4	yes	complemented
54	#95	—	251	4	no	—
55	C₂²×C₄	—	251	4	no	—
56	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
57	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	502	4	no	—
60	C₂×C₈	—	251	4	no	—
61	C₂×C₈	C₂×Dic₂₅₁	1	4	yes	—
62	C₄²	—	502	4	no	—
63	C₄²	C₄×D₂₅₁	1	2	yes	—
64	C₄²	C₂×Dic₂₅₁	1	2	yes	—
65	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	502	4	no	—
68	C₂×C₄	—	502	8	no	—
69	C₂×C₄	C₄×Dic₂₅₁	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
72	C₂×C₄	#16860810	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	502	4	no	—
75	C₂²	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
76	C₂²	#22548672	1	2	yes	—
77	C₂²	#22548721	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	1004	4	no	—
80	C₄	#22548672	1	2	yes	—
81	C₄	#22548674	1	2	yes	—
82	C₂	#30631533	1	1	yes	characteristic
83	C₂	#30631642	1	2	yes	—
84	C₂	#30631746	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂×C₄²
Fitting	C₂×C₄×C₈×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₁
hypocenter	C₂₅₁	#1746
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #22548626	C₄	semidirect
11. C₄×C₈×C₂₅₁	C₄	semidirect
42. C₂₅₁	#1746	semidirect
53. C₄×C₈	Dic₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548626	2
#30631533	2
#30631441	2
C₂×C₄×C₈×C₂₅₁	2
#1746	251