Subgroups — #44550113

Order 32128.

272 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	C₄²×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#30631441	C₂	1	4	yes	maximal
4	C₂×#97×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₅₁	C₄	1	4	yes	—
7	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
8	#22548626	C₄	1	4	yes	complemented
9	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
11	#97×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	8	no	—
16	C₄×Dic₂₅₁	—	2	4	no	—
17	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₂₅₁	—	2	4	no	—
20	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
21	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
22	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
23	C₂×Dic₂₅₁	—	2	4	no	—
24	C₂×Dic₂₅₁	—	2	4	no	—
25	C₂×Dic₂₅₁	M₁₆	1	8	yes	—
26	C₂₅₁⋊C₈	—	2	4	no	—
27	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₅₁	—	2	4	no	—
29	C₂×C₄×C₂₅₁	C₄²	1	4	yes	—
30	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	#41	1	4	yes	—
33	C₈×C₂₅₁	—	2	4	no	—
34	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
35	C₂²×C₂₅₁	#117	1	2	yes	—
36	C₂²×C₂₅₁	C₂×M₁₆	1	4	yes	—
37	C₄×C₂₅₁	—	2	4	no	—
38	C₄×C₂₅₁	#117	1	2	yes	—
39	C₄×C₂₅₁	#117	1	2	yes	—
40	Dic₂₅₁	—	2	8	no	—
41	C₂×C₂₅₁	#376	1	1	yes	characteristic
42	C₂×C₂₅₁	C₄×M₁₆	1	2	yes	—
43	C₂×C₂₅₁	#432	1	4	yes	—
44	C₂₅₁	#1745	1	1	yes	characteristic, complemented, Sylow
45	#1745	—	251	1	no	maximal, Sylow
46	C₂×#97	—	251	1	no	—
47	C₂×#97	D₂₅₁	1	1	yes	characteristic
48	C₄³	—	251	1	no	—
49	#336	—	251	4	no	—
50	C₂²×C₈	—	251	2	no	—
51	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
52	C₂×C₄²	—	251	4	no	—
53	C₂×C₄²	—	251	2	no	—
54	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
55	#97	—	251	4	no	—
56	#97	Dic₂₅₁	1	4	yes	complemented
57	C₂²×C₄	—	251	4	no	—
58	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
59	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	502	4	no	—
62	C₂×C₈	—	251	4	no	—
63	C₂×C₈	C₂×Dic₂₅₁	1	4	yes	—
64	C₄²	—	502	8	no	—
65	C₄²	—	502	4	no	—
66	C₄²	C₄×D₂₅₁	1	2	yes	—
67	C₄²	C₂×Dic₂₅₁	1	2	yes	—
68	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	502	4	no	—
71	C₂×C₄	—	502	4	no	—
72	C₂×C₄	—	251	8	no	—
73	C₂×C₄	C₄×Dic₂₅₁	1	4	yes	—
74	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
75	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
76	C₂×C₄	#16860810	1	4	yes	—
77	C₈	—	2	4	no	—
78	C₈	—	502	4	no	—
79	C₂²	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
80	C₂²	#22548672	1	2	yes	—
81	C₂²	M₁₆×D₂₅₁	1	4	yes	—
82	C₄	—	2	4	no	—
83	C₄	—	502	8	no	—
84	C₄	#22548672	1	2	yes	—
85	C₄	#22548674	1	2	yes	—
86	C₂	#30631533	1	1	yes	characteristic
87	C₂	M₁₆×Dic₂₅₁	1	2	yes	—
88	C₂	#30631742	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂×C₄²
Fitting	C₂×#97×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#97	D₂₅₁
hypocenter	C₂₅₁	#1745
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
8. #22548626	C₄	semidirect
11. #97×C₂₅₁	C₄	semidirect
44. C₂₅₁	#1745	semidirect
56. #97	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
C₄²×Dic₂₅₁	2
#30631441	2
C₂×#97×C₂₅₁	2
#1745	251