Subgroups — #44548504

Order 32128.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548654	C₂	1	1	yes	maximal, characteristic
2	#30631521	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₂₅₁	C₄	1	1	yes	characteristic
5	#22548654	—	2	1	no	—
6	#22548654	—	2	1	no	—
7	C₂×C₂₅₁⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂₅₁⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₅₁	—	2	1	no	—
11	C₂×C₄×D₂₅₁	C₈	1	1	yes	characteristic
12	C₂×C₂₅₁⋊C₈	C₈	1	1	yes	characteristic
13	C₈×D₂₅₁	—	2	1	no	—
14	C₈×D₂₅₁	—	2	1	no	—
15	C₂₅₁⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₂₅₁	D₄	1	1	yes	characteristic
18	C₁₆×C₂₅₁	D₄	1	1	yes	characteristic
19	C₂²×D₂₅₁	C₁₆	1	1	yes	characteristic
20	C₂×Dic₂₅₁	C₁₆	1	1	yes	characteristic
21	C₄×D₂₅₁	—	2	1	no	—
22	C₄×D₂₅₁	—	2	1	no	—
23	C₂₅₁⋊C₈	—	2	1	no	—
24	C₂×C₄×C₂₅₁	C₂×C₈	1	1	yes	characteristic
25	C₈×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₅₁	M₁₆	1	1	yes	characteristic
27	C₂²×C₂₅₁	C₂×C₁₆	1	1	yes	characteristic
28	C₂×D₂₅₁	—	2	1	no	—
29	C₂×D₂₅₁	—	4	1	no	—
30	C₄×C₂₅₁	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₂₅₁	M₃₂	1	1	yes	characteristic
32	Dic₂₅₁	—	2	1	no	—
33	C₂×C₂₅₁	—	2	1	no	—
34	C₂×C₂₅₁	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₂₅₁	—	4	1	no	—
36	C₂₅₁	#1400	1	1	yes	characteristic, complemented, Sylow
37	#1400	—	251	1	no	maximal, Sylow
38	C₂×M₃₂	—	251	1	no	—
39	M₆₄	—	251	1	no	—
40	M₆₄	D₂₅₁	1	1	yes	characteristic, complemented
41	C₂²×C₈	—	251	1	no	—
42	M₃₂	—	502	1	no	—
43	M₃₂	—	502	1	no	—
44	C₂×C₁₆	—	251	1	no	—
45	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	502	1	no	—
48	C₂²×C₄	—	251	1	no	—
49	C₂×C₈	—	502	1	no	—
50	C₂×C₈	—	502	1	no	—
51	C₂×C₈	—	251	1	no	—
52	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
53	C₁₆	—	502	1	no	—
54	C₁₆	C₂²⋊D₂₅₁	1	1	yes	characteristic
55	C₁₆	D₁₀₀₄	1	1	yes	characteristic
56	C₂³	—	251	1	no	—
57	C₂×C₄	—	502	1	no	—
58	C₂×C₄	—	502	1	no	—
59	C₂×C₄	—	251	1	no	—
60	C₂×C₄	C₈×D₂₅₁	1	1	yes	characteristic
61	C₈	—	502	1	no	—
62	C₈	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
63	C₈	#16860793	1	1	yes	characteristic
64	C₂²	—	1004	1	no	—
65	C₂²	—	502	1	no	—
66	C₂²	C₁₆×D₂₅₁	1	1	yes	characteristic
67	C₄	—	502	1	no	—
68	C₄	(C₂×D₂₅₁)⋊C₈	1	1	yes	characteristic
69	C₄	#22548654	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1004	1	no	—
72	C₂	(C₂×D₂₅₁)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₈	(C₂×D₂₅₁)⋊C₄
commutator	C₂²×C₂₅₁	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₅₁
Fitting	M₆₄×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂²⋊C₁₆
prime radical	C₂²×D₂₅₁	C₁₆
cosocle	C₂×C₁₆×C₂₅₁	C₂²
hypercenter	M₆₄	D₂₅₁
hypocenter	C₂₅₁	#1400
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #30631521	C₂	semidirect
3. M₆₄×C₂₅₁	C₂	semidirect
36. C₂₅₁	#1400	semidirect
40. M₆₄	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548654	2
#30631521	2
M₆₄×C₂₅₁	2
#1400	251