Subgroups — #44548539

Order 32128.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632041	C₂	1	1	yes	maximal, characteristic
2	#30631484	C₂	1	1	yes	maximal, characteristic
3	#354×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#22548707	C₄	1	1	yes	characteristic, complemented
5	#22548639	—	2	1	no	—
6	#22548636	—	2	1	no	—
7	#22548634	—	2	1	no	—
8	C₄×C₂₅₁⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#124×C₂₅₁	C₂²	1	1	yes	characteristic
10	#99×C₂₅₁	—	2	1	no	—
11	#16860794	—	4	1	no	—
12	C₂×C₂₅₁⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₅₁	—	2	1	no	—
14	Dic₂₅₁⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
16	C₂×D₄×C₂₅₁	D₄	1	1	yes	characteristic
17	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
18	C₂²⋊C₄×C₂₅₁	—	4	1	no	—
19	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₅₁	—	4	1	no	—
21	C₂₅₁⋊C₈	—	4	1	no	—
22	C₂₅₁⋊C₈	—	8	1	no	—
23	C₂³×C₂₅₁	—	2	1	no	—
24	Q₈×C₂₅₁	—	4	1	no	—
25	D₄×C₂₅₁	—	4	1	no	—
26	C₂×C₄×C₂₅₁	—	2	1	no	—
27	C₂×C₄×C₂₅₁	—	2	1	no	—
28	C₂×C₄×C₂₅₁	—	8	1	no	—
29	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₅₁	—	4	1	no	—
31	C₂²×C₂₅₁	—	8	1	no	—
32	C₂²×C₂₅₁	#99	1	1	yes	characteristic
33	C₄×C₂₅₁	—	2	1	no	—
34	C₄×C₂₅₁	—	4	1	no	—
35	C₄×C₂₅₁	—	4	1	no	—
36	C₄×C₂₅₁	—	16	1	no	—
37	Dic₂₅₁	—	8	1	no	—
38	C₂×C₂₅₁	—	2	1	no	—
39	C₂×C₂₅₁	—	8	1	no	—
40	C₂×C₂₅₁	#354	1	1	yes	characteristic
41	C₂₅₁	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	251	1	no	maximal, Sylow
43	#488	—	251	1	no	—
44	#355	—	251	1	no	—
45	#354	D₂₅₁	1	1	yes	characteristic
46	#125	—	251	1	no	—
47	#124	C₂×D₂₅₁	1	1	yes	characteristic
48	#103	—	502	1	no	—
49	#102	—	502	1	no	—
50	#101	—	502	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	251	1	no	—
53	C₂×Q₈	D₁₀₀₄	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
55	M₁₆	—	1004	1	no	—
56	C₂×C₈	—	502	1	no	—
57	C₄⋊C₄	—	1004	1	no	—
58	C₄⋊C₄	—	502	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	1004	1	no	—
69	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
70	C₈	—	2008	1	no	—
71	C₈	—	1004	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#22548629	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	2008	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#30631481	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631481
commutator	C₄²×C₂₅₁	C₂×C₄
Frattini	#124	C₂×D₂₅₁
Fitting	#354×C₂₅₁	C₂
socle	C₂×C₂₅₁	#354
prime radical	C₂×D₄×C₂₅₁	D₄
cosocle	#124×C₂₅₁	C₂²
hypercenter	#354	D₂₅₁
hypocenter	C₂₅₁	#1414
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂×C₄×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #22548707	C₄	semidirect
8. C₄×C₂₅₁⋊C₈	C₄	semidirect
41. C₂₅₁	#1414	semidirect

Maximal subgroups

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

subgroup	index
#30632041	2
#30631484	2
#354×C₂₅₁	2
#1414	251