Subgroups — #44558618

Order 32128.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631599	C₂	1	12	yes	maximal, complemented
2	#30631543	C₂	1	2	yes	maximal
3	#379×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₅₀₂	C₂²	1	4	yes	complemented
5	C₂×C₂²⋊Dic₂₅₁	C₂²	1	12	yes	—
6	C₂₅₁⋊#95	C₂²	1	12	yes	complemented
7	C₂⁵×C₂₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	6	yes	—
9	C₂×Dic₅₀₂	—	4	12	no	—
10	C₂²×Dic₂₅₁	C₂³	1	8	yes	—
11	C₂²⋊Dic₂₅₁	—	4	12	no	—
12	C₂⁴×C₂₅₁	—	4	6	no	—
13	C₂⁴×C₂₅₁	Q₈	1	4	yes	complemented
14	C₂⁴×C₂₅₁	C₂³	1	3	yes	—
15	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
16	C₂²⋊C₄×C₂₅₁	—	4	6	no	—
17	Dic₅₀₂	—	4	16	no	—
18	C₂×Dic₂₅₁	—	2	24	no	—
19	C₂³×C₂₅₁	—	2	6	no	—
20	C₂³×C₂₅₁	—	2	12	no	—
21	C₂³×C₂₅₁	—	4	12	no	—
22	C₂³×C₂₅₁	—	4	16	no	—
23	C₂³×C₂₅₁	C₂×Q₈	1	6	yes	—
24	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
25	C₂×C₄×C₂₅₁	—	2	12	no	—
26	C₂²×C₂₅₁	—	2	6	no	—
27	C₂²×C₂₅₁	—	2	12	no	—
28	C₂²×C₂₅₁	—	4	16	no	—
29	C₂²×C₂₅₁	—	4	12	no	—
30	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
31	C₂²×C₂₅₁	H(5,𝔽₂)	1	6	yes	—
32	C₄×C₂₅₁	—	4	4	no	—
33	Dic₂₅₁	—	4	8	no	—
34	C₂×C₂₅₁	—	4	6	no	—
35	C₂×C₂₅₁	#543	1	3	yes	—
36	C₂×C₂₅₁	H(3,𝔽₄)	1	4	yes	—
37	C₂₅₁	#2840	1	1	yes	characteristic, complemented, Sylow
38	#2840	—	251	1	no	maximal, Sylow
39	#393	—	251	12	no	—
40	#379	—	251	2	no	—
41	#379	D₂₅₁	1	1	yes	characteristic
42	C₂⁵	C₂×D₂₅₁	1	1	yes	characteristic
43	C₂²×Q₈	—	251	4	no	—
44	C₂×C₂²⋊C₄	—	251	12	no	—
45	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	6	yes	—
46	#95	—	251	12	no	—
47	C₂⁴	—	4	6	no	—
48	C₂⁴	Dic₅₀₂	1	4	yes	complemented
49	C₂⁴	C₂²×D₂₅₁	1	3	yes	—
50	C₂×Q₈	—	1004	12	no	—
51	C₂²×C₄	—	251	8	no	—
52	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
53	C₂²⋊C₄	—	4	6	no	—
54	C₂²⋊C₄	—	1004	12	no	—
55	C₂³	—	2	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	4	12	no	—
58	C₂³	—	4	16	no	—
59	C₂³	C₂×Dic₅₀₂	1	6	yes	—
60	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
61	Q₈	—	1004	16	no	—
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	502	24	no	—
64	C₂²	—	2	6	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	4	16	no	—
67	C₂²	—	4	12	no	—
68	C₂²	C₂²×Dic₅₀₂	1	1	yes	characteristic
69	C₂²	#22548760	1	6	yes	—
70	C₄	—	4	4	no	—
71	C₄	—	1004	8	no	—
72	C₂	—	4	6	no	—
73	C₂	#30632313	1	3	yes	—
74	C₂	#30632430	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#379×C₂₅₁	C₂
prime radical	C₂⁵×C₂₅₁	C₂²
hypercenter	#379	D₂₅₁
hypocenter	C₂₅₁	#2840
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #379

Complements

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

N	H ≅ G/N	type
1. #30631599	C₂	semidirect
4. C₂²×Dic₅₀₂	C₂²	semidirect
6. C₂₅₁⋊#95	C₂²	semidirect
13. C₂⁴×C₂₅₁	Q₈	semidirect
37. C₂₅₁	#2840	semidirect
48. C₂⁴	Dic₅₀₂	semidirect

Maximal subgroups

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

subgroup	index
#30631599	2
#30631543	2
#379×C₂₅₁	2
#2840	251