Subgroups — #44558976

Order 32128.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632454	C₂	1	2	yes	maximal
2	#30631736	C₂	1	6	yes	maximal, complemented
3	C₂×#22548651	C₂	1	6	yes	maximal, complemented
4	C₂×#131×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#22548722	—	2	8	no	—
6	C₂×#16860794	C₂²	1	12	yes	—
7	#22548651	C₂²	1	4	yes	—
8	#22548651	C₂²	1	12	yes	—
9	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
10	#131×C₂₅₁	C₄	1	8	yes	—
11	C₂×M₁₆×C₂₅₁	C₂²	1	3	yes	—
12	C₂²×C₈×C₂₅₁	C₂²	1	3	yes	—
13	#16860794	—	2	12	no	—
14	#16860794	C₂³	1	8	yes	—
15	C₂×C₂₅₁⋊C₈	—	2	12	no	—
16	#41×C₂₅₁	C₂×C₄	1	4	yes	—
17	#41×C₂₅₁	D₄	1	4	yes	—
18	C₂×Q₈×C₂₅₁	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₂₅₁	Q₈	1	3	yes	—
20	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
21	M₁₆×C₂₅₁	C₂×C₄	1	12	yes	—
22	C₂×C₈×C₂₅₁	C₂×C₄	1	12	yes	—
23	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₅₁	C₂³	1	3	yes	—
25	C₂₅₁⋊C₈	—	2	8	no	—
26	C₂³×C₂₅₁	C₂×Q₈	1	3	yes	—
27	Q₈×C₂₅₁	C₄⋊C₄	1	4	yes	—
28	D₄×C₂₅₁	C₄⋊C₄	1	12	yes	—
29	C₂×C₄×C₂₅₁	C₂²×C₄	1	6	yes	—
30	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
31	C₂×C₄×C₂₅₁	C₂×Q₈	1	3	yes	—
32	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
33	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
34	C₈×C₂₅₁	C₂²×C₄	1	6	yes	—
35	C₂²×C₂₅₁	—	4	3	no	—
36	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	6	yes	—
37	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
38	C₄×C₂₅₁	C₂×C₄⋊C₄	1	6	yes	—
39	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₂₅₁	—	2	1	no	—
42	C₂×C₂₅₁	—	2	6	no	—
43	C₂×C₂₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
44	C₂₅₁	#2916	1	1	yes	characteristic, complemented, Sylow
45	#2916	—	251	1	no	maximal, Sylow
46	#568	—	251	2	no	—
47	C₂×#131	D₂₅₁	1	1	yes	characteristic
48	#430	—	251	6	no	—
49	C₂×#108	—	251	6	no	—
50	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
51	#131	—	502	8	no	—
52	#131	Dic₂₅₁	1	8	yes	—
53	C₂×M₁₆	—	251	12	no	—
54	C₂×M₁₆	C₂×D₂₅₁	1	3	yes	—
55	C₂²×C₈	C₂×D₂₅₁	1	3	yes	—
56	#108	—	251	4	no	—
57	#108	—	251	12	no	—
58	#41	C₂×Dic₂₅₁	1	4	yes	—
59	#41	D₁₀₀₄	1	4	yes	—
60	C₂×Q₈	Dic₅₀₂	1	1	yes	characteristic
61	C₂×D₄	Dic₅₀₂	1	3	yes	—
62	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
63	M₁₆	—	502	12	no	—
64	M₁₆	—	251	8	no	—
65	M₁₆	C₂×Dic₂₅₁	1	12	yes	—
66	C₂×C₈	—	502	12	no	—
67	C₂×C₈	C₂×Dic₂₅₁	1	12	yes	—
68	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₂₅₁	1	3	yes	—
70	C₂³	C₂×Dic₅₀₂	1	3	yes	—
71	Q₈	C₄⋊Dic₂₅₁	1	4	yes	—
72	D₄	C₄⋊Dic₂₅₁	1	12	yes	—
73	C₂×C₄	C₂²×Dic₂₅₁	1	6	yes	—
74	C₂×C₄	C₂×D₁₀₀₄	1	6	yes	—
75	C₂×C₄	C₂×Dic₅₀₂	1	3	yes	—
76	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
77	C₈	—	502	8	no	—
78	C₈	C₂²×Dic₂₅₁	1	2	yes	—
79	C₈	C₂²×Dic₂₅₁	1	6	yes	—
80	C₂²	—	4	3	no	—
81	C₂²	C₂×C₄⋊Dic₂₅₁	1	6	yes	—
82	C₂²	C₂²×Dic₅₀₂	1	1	yes	characteristic
83	C₄	C₂×C₄⋊Dic₂₅₁	1	6	yes	—
84	C₄	C₂³×Dic₂₅₁	1	1	yes	characteristic
85	C₄	C₂²×D₁₀₀₄	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	6	no	—
88	C₂	C₂²×C₄⋊Dic₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₁₀₀₄
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#131×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₂₅₁	C₂²
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#131	D₂₅₁
hypocenter	C₂₅₁	#2916
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #30631736	C₂	semidirect
3. C₂×#22548651	C₂	semidirect
44. C₂₅₁	#2916	semidirect

Maximal subgroups

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

subgroup	index
#30632454	2
#30631736	2
C₂×#22548651	2
C₂×#131×C₂₅₁	2
#2916	251