Subgroups — #44550485

Order 32128.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	1	2	yes	maximal
2	#30631442	C₂	1	4	yes	maximal
3	#376×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₅₁⋊C₈	C₂²	1	4	yes	—
5	C₄⋊(C₂₅₁⋊C₈)	—	2	4	no	—
6	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
8	#95×C₂₅₁	C₄	1	4	yes	—
9	C₂×C₂₅₁⋊C₈	—	2	4	no	—
10	C₂×C₂₅₁⋊C₈	—	2	8	no	—
11	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
12	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
14	C₄²×C₂₅₁	—	2	4	no	—
15	C₄²×C₂₅₁	D₄	1	2	yes	—
16	C₄²×C₂₅₁	Q₈	1	2	yes	—
17	C₂₅₁⋊C₈	—	4	4	no	—
18	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₅₁	—	2	4	no	—
20	C₂×C₄×C₂₅₁	—	2	4	no	—
21	C₂×C₄×C₂₅₁	—	2	8	no	—
22	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
23	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₂₅₁	#41	1	4	yes	—
26	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₅₁	#117	1	2	yes	—
28	C₂²×C₂₅₁	#131	1	4	yes	—
29	C₄×C₂₅₁	—	2	4	no	—
30	C₄×C₂₅₁	—	4	4	no	—
31	C₄×C₂₅₁	#124	1	2	yes	—
32	C₄×C₂₅₁	#125	1	2	yes	—
33	C₂×C₂₅₁	#382	1	1	yes	characteristic
34	C₂×C₂₅₁	#424	1	2	yes	—
35	C₂×C₂₅₁	#433	1	4	yes	—
36	C₂₅₁	#1832	1	1	yes	characteristic, complemented, Sylow
37	#1832	—	251	1	no	maximal, Sylow
38	C₂×C₄⋊C₈	—	251	2	no	—
39	#376	D₂₅₁	1	1	yes	characteristic
40	#336	—	251	4	no	—
41	C₂²×C₈	—	251	4	no	—
42	C₂×C₄²	C₂×D₂₅₁	1	2	yes	—
43	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
44	C₄⋊C₈	—	502	4	no	—
45	#95	Dic₂₅₁	1	4	yes	—
46	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	—
47	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
48	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
49	C₂×C₈	—	502	4	no	—
50	C₂×C₈	—	502	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	D₁₀₀₄	1	2	yes	—
53	C₄²	Dic₅₀₂	1	2	yes	—
54	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	8	no	—
58	C₂×C₄	C₄⋊Dic₂₅₁	1	4	yes	—
59	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₅₀₂	1	1	yes	characteristic
61	C₂×C₄	#16860810	1	4	yes	—
62	C₈	—	1004	4	no	—
63	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
64	C₂²	#22548674	1	2	yes	—
65	C₂²	#22548722	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	4	4	no	—
68	C₄	#22548703	1	2	yes	—
69	C₄	#22548707	1	2	yes	—
70	C₂	#30631553	1	1	yes	characteristic
71	C₂	#30631714	1	2	yes	—
72	C₂	#30631748	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₁
commutator, socle, prime radical	C₂³×C₂₅₁	C₂²×C₄
Fitting	#376×C₂₅₁	C₂
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#376	D₂₅₁
hypocenter	C₂₅₁	#1832
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
36. C₂₅₁	#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₂₅₁⋊C₈)	2
#30631442	2
#376×C₂₅₁	2
#1832	251