Subgroups — #44550506

Order 32128.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631711	C₂	1	2	yes	maximal
2	#30631452	C₂	1	4	yes	maximal
3	C₂×#97×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16860794	C₂²	1	4	yes	—
5	C₄⋊(C₂₅₁⋊C₈)	—	2	4	no	—
6	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
8	#97×C₂₅₁	C₄	1	4	yes	—
9	#16860794	—	2	8	no	—
10	C₂×C₂₅₁⋊C₈	—	2	4	no	—
11	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
12	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₂₅₁	—	2	4	no	—
14	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
15	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
16	C₄²×C₂₅₁	D₄	1	2	yes	—
17	C₂₅₁⋊C₈	—	4	4	no	—
18	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
19	C₂×C₄×C₂₅₁	—	4	2	no	—
20	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
21	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₁	#41	1	4	yes	—
24	C₈×C₂₅₁	—	2	4	no	—
25	C₂²×C₂₅₁	—	2	2	no	—
26	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₅₁	#125	1	2	yes	—
28	C₄×C₂₅₁	—	2	4	no	—
29	C₄×C₂₅₁	#117	1	2	yes	—
30	C₄×C₂₅₁	#124	1	2	yes	—
31	C₂×C₂₅₁	—	2	2	no	—
32	C₂×C₂₅₁	#382	1	1	yes	characteristic
33	C₂×C₂₅₁	#430	1	2	yes	—
34	C₂₅₁	#1838	1	1	yes	characteristic, complemented, Sylow
35	#1838	—	251	1	no	maximal, Sylow
36	#423	—	251	2	no	—
37	C₂×#97	D₂₅₁	1	1	yes	characteristic
38	#341	—	251	4	no	—
39	C₂×M₁₆	—	251	4	no	—
40	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
41	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
42	C₄⋊C₈	—	502	4	no	—
43	#97	Dic₂₅₁	1	4	yes	—
44	C₂²×C₄	Dic₅₀₂	1	2	yes	—
45	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
46	M₁₆	—	502	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	502	4	no	—
49	C₂×C₈	C₂×Dic₂₅₁	1	4	yes	—
50	C₄²	C₂×Dic₂₅₁	1	2	yes	—
51	C₄²	D₁₀₀₄	1	2	yes	—
52	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊Dic₂₅₁	1	4	yes	—
55	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
56	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
57	C₂×C₄	#16860810	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	1004	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
62	C₂²	#22548707	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#22548674	1	2	yes	—
65	C₄	#22548703	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#30631553	1	1	yes	characteristic
68	C₂	#30631736	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₀₄
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×#97×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#97	D₂₅₁
hypocenter	C₂₅₁	#1838
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
34. C₂₅₁	#1838	semidirect

Maximal subgroups

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

subgroup	index
#30631711	2
#30631452	2
C₂×#97×C₂₅₁	2
#1838	251