Subgroups — #44548337

Order 32128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631805	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22548737	C₄	1	1	yes	characteristic
5	#22548721	—	2	1	no	—
6	#22548651	C₄	1	1	yes	characteristic
7	#22548643	—	2	1	no	—
8	C₄×C₂₅₁⋊C₈	—	2	1	no	—
9	M₃₂×C₂₅₁	—	2	1	no	—
10	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	#16860810	—	2	1	no	—
12	Dic₁₀₀₄	C₈	1	1	yes	characteristic, complemented
13	#16860799	—	2	1	no	—
14	D₂₀₀₈	C₈	1	1	yes	characteristic, complemented
15	#16860794	—	2	1	no	—
16	#16860793	—	4	1	no	—
17	C₂×C₂₅₁⋊C₈	—	2	2	no	—
18	C₈×D₂₅₁	—	4	1	no	—
19	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
21	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
22	C₁₆×C₂₅₁	—	4	1	no	—
23	Dic₅₀₂	—	2	1	no	—
24	D₁₀₀₄	—	2	1	no	—
25	C₂²⋊D₂₅₁	—	4	1	no	—
26	C₄×D₂₅₁	—	4	1	no	—
27	C₂₅₁⋊C₈	—	2	4	no	—
28	C₂₅₁⋊C₈	—	4	1	no	—
29	C₂×C₄×C₂₅₁	—	2	1	no	—
30	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₂₅₁	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₅₁	M₁₆	1	1	yes	characteristic
33	C₈×C₂₅₁	D₈	1	1	yes	characteristic
34	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₅₁	#104	1	1	yes	characteristic
36	C₂×D₂₅₁	—	4	1	no	—
37	C₄×C₂₅₁	—	2	2	no	—
38	C₄×C₂₅₁	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₅₁	#102	1	1	yes	characteristic
40	Dic₂₅₁	—	4	1	no	—
41	C₂×C₂₅₁	—	2	1	no	—
42	C₂×C₂₅₁	D₄⋊C₈	1	1	yes	characteristic
43	D₂₅₁	—	8	1	no	—
44	C₂₅₁	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	251	1	no	maximal, Sylow
46	#443	—	251	1	no	—
47	#364	D₂₅₁	1	1	yes	characteristic, complemented
48	C₈²	—	251	1	no	—
49	#135	—	251	1	no	—
50	#131	—	502	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	251	1	no	—
53	#104	—	502	1	no	—
54	C₄×C₈	—	502	1	no	—
55	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
56	#41	—	502	1	no	—
57	Q₁₆	—	251	1	no	—
58	SD₁₆	—	502	1	no	—
59	D₈	—	251	1	no	—
60	M₁₆	—	1004	1	no	—
61	M₁₆	—	502	1	no	—
62	C₂×C₈	—	1004	1	no	—
63	C₂×C₈	—	502	2	no	—
64	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
66	C₄²	D₁₀₀₄	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	502	1	no	—
69	D₄	—	1004	1	no	—
70	D₄	—	502	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	1004	1	no	—
73	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
74	C₈	—	1004	1	no	—
75	C₈	—	502	4	no	—
76	C₈	C₈×D₂₅₁	1	1	yes	characteristic
77	C₈	#16860793	1	1	yes	characteristic
78	C₈	#16860795	1	1	yes	characteristic
79	C₈	#16860797	1	1	yes	characteristic
80	C₂²	—	1004	1	no	—
81	C₂²	#22548641	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	1004	1	no	—
84	C₄	(C₂×D₂₅₁)⋊C₈	1	1	yes	characteristic
85	C₄	#22548635	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2008	1	no	—
88	C₂	#30631414	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#16860795
commutator	C₈×C₂₅₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	#364×C₂₅₁	C₂
socle	C₂×C₂₅₁	D₄⋊C₈
prime radical	#22548737	C₄
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	#364	D₂₅₁
hypocenter	C₂₅₁	#1335
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×C₂₅₁⋊C₈	C₂	semidirect
3. #364×C₂₅₁	C₂	semidirect
12. Dic₁₀₀₄	C₈	semidirect
14. D₂₀₀₈	C₈	semidirect
44. C₂₅₁	#1335	semidirect
47. #364	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30631805	2
C₈×C₂₅₁⋊C₈	2
#364×C₂₅₁	2
#1335	251