Subgroups — #30632300

Order 16064.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22548715	C₂	1	1	yes	maximal, characteristic, complemented
2	#22548710	C₂	1	4	yes	maximal
3	#22548706	C₂	1	4	yes	maximal, complemented
4	#22548695	C₂	1	2	yes	maximal
5	#22548680	C₂	1	2	yes	maximal, complemented
6	#22548672	C₂	1	1	yes	maximal, characteristic
7	#125×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₀₂	C₂²	1	2	yes	—
9	C₂×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
11	C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
12	Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
13	Dic₂₅₁⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₂₅₁)⋊C₄	C₂²	1	2	yes	—
15	(C₂×D₂₅₁)⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₅₁	C₂²	1	4	yes	—
18	C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₅₁	C₂²	1	4	yes	—
20	C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂²×D₂₅₁	C₂³	1	1	yes	characteristic
22	Dic₅₀₂	—	2	4	no	—
23	C₂×Dic₂₅₁	C₂³	1	2	yes	—
24	C₂×Dic₂₅₁	C₂³	1	4	yes	—
25	C₂×Dic₂₅₁	C₂³	1	1	yes	characteristic
26	C₄×D₂₅₁	—	2	2	no	—
27	C₂×C₄×C₂₅₁	C₂³	1	2	yes	—
28	C₂×C₄×C₂₅₁	C₂³	1	4	yes	—
29	C₂×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₂²×C₂₅₁	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₅₁	—	2	1	no	—
32	C₂×D₂₅₁	—	2	2	no	—
33	C₄×C₂₅₁	—	2	2	no	—
34	C₄×C₂₅₁	—	2	4	no	—
35	C₄×C₂₅₁	#41	1	2	yes	—
36	Dic₂₅₁	—	2	1	no	—
37	Dic₂₅₁	—	2	4	no	—
38	Dic₂₅₁	—	2	2	no	—
39	C₂×C₂₅₁	C₂×#41	1	1	yes	characteristic
40	C₂×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
41	D₂₅₁	—	4	1	no	—
42	C₂₅₁	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	251	1	no	maximal, Sylow
44	C₄⋊Q₈	—	251	1	no	—
45	#126	—	251	4	no	—
46	#125	—	251	4	no	—
47	#125	D₂₅₁	1	1	yes	characteristic, complemented
48	C₂²⋊Q₈	—	251	2	no	—
49	C₄×Q₈	—	251	2	no	—
50	#117	—	251	1	no	—
51	C₂×Q₈	—	251	2	no	—
52	C₂²×C₄	—	251	1	no	—
53	C₄⋊C₄	—	251	2	no	—
54	C₄⋊C₄	—	251	4	no	—
55	C₄⋊C₄	—	251	4	no	—
56	C₄⋊C₄	—	251	4	no	—
57	C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
58	C₄⋊C₄	C₂×D₂₅₁	1	4	yes	—
59	C₂²⋊C₄	—	251	2	no	—
60	C₂²⋊C₄	—	251	4	no	—
61	C₄²	—	251	1	no	—
62	C₄²	—	251	4	no	—
63	C₄²	C₂×D₂₅₁	1	1	yes	characteristic
64	C₂³	—	251	1	no	—
65	Q₈	—	502	4	no	—
66	C₂×C₄	—	502	2	no	—
67	C₂×C₄	—	251	2	no	—
68	C₂×C₄	—	251	4	no	—
69	C₂×C₄	—	251	1	no	—
70	C₂×C₄	C₂²×D₂₅₁	1	4	yes	—
71	C₂×C₄	C₂²×D₂₅₁	1	2	yes	—
72	C₂×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
73	C₂²	—	502	1	no	—
74	C₂²	—	502	2	no	—
75	C₂²	C₂³×D₂₅₁	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	502	1	no	—
79	C₄	—	502	4	no	—
80	C₄	—	502	2	no	—
81	C₄	#16860811	1	2	yes	—
82	C₂	—	1004	1	no	—
83	C₂	C₂×#16860811	1	1	yes	characteristic
84	C₂	#22548762	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁴
Fitting	#125×C₂₅₁	C₂
prime radical	C₂²×D₂₅₁	C₂³
hypercenter	#125	D₂₅₁
hypocenter	C₂₅₁	#541
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22548715	C₂	semidirect
3. #22548706	C₂	semidirect
5. #22548680	C₂	semidirect
7. #125×C₂₅₁	C₂	semidirect
42. C₂₅₁	#541	semidirect
47. #125	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#22548715	2
#22548710	2
#22548706	2
#22548695	2
#22548680	2
#22548672	2
#125×C₂₅₁	2
#541	251