Subgroups — #44548401

Order 32128.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632064	C₂	1	1	yes	maximal, characteristic
2	#30631508	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₀₀₈	—	2	1	no	—
5	C₂×D₂₀₀₈	—	2	1	no	—
6	#22548712	C₄	1	2	yes	—
7	#22548655	—	2	1	no	—
8	M₃₂×C₂₅₁	—	2	1	no	—
9	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₂×D₁₀₀₄	—	2	2	no	—
11	C₂×D₁₀₀₄	—	4	2	no	—
12	D₂₀₀₈	—	4	1	no	—
13	D₂₀₀₈	—	4	1	no	—
14	D₂₀₀₈	—	4	1	no	—
15	D₂₀₀₈	—	4	1	no	—
16	C₂₅₁⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
19	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₂₅₁	—	4	1	no	—
21	C₂²×D₂₅₁	—	4	2	no	—
22	D₁₀₀₄	—	4	2	no	—
23	D₁₀₀₄	—	4	2	no	—
24	D₁₀₀₄	—	8	4	no	—
25	C₂×C₄×C₂₅₁	—	2	1	no	—
26	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₂₅₁	D₈	1	1	yes	characteristic
28	C₈×C₂₅₁	D₈	1	1	yes	characteristic
29	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
30	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
31	C₂²×C₂₅₁	#100	1	1	yes	characteristic
32	C₂×D₂₅₁	—	8	4	no	—
33	C₂×D₂₅₁	—	8	2	no	—
34	C₄×C₂₅₁	—	2	2	no	—
35	C₄×C₂₅₁	#102	1	1	yes	characteristic
36	C₄×C₂₅₁	#102	1	1	yes	characteristic
37	C₂×C₂₅₁	—	2	1	no	—
38	C₂×C₂₅₁	#331	1	1	yes	characteristic
39	D₂₅₁	—	16	2	no	—
40	C₂₅₁	#1357	1	1	yes	characteristic, complemented, Sylow
41	#1357	—	251	1	no	maximal, Sylow
42	#493	—	251	1	no	—
43	#364	—	251	1	no	—
44	#364	D₂₅₁	1	1	yes	characteristic, complemented
45	C₂×D₈	—	502	1	no	—
46	C₂×D₈	—	502	1	no	—
47	#127	—	251	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	502	1	no	—
50	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
51	C₂×D₄	—	1004	2	no	—
52	C₂×D₄	—	502	2	no	—
53	D₈	—	1004	1	no	—
54	D₈	—	1004	1	no	—
55	D₈	—	1004	1	no	—
56	D₈	—	1004	1	no	—
57	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
58	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
59	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	1004	1	no	—
62	C₂³	—	1004	2	no	—
63	D₄	—	2008	4	no	—
64	D₄	—	1004	2	no	—
65	D₄	—	1004	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
68	C₈	#16860795	1	1	yes	characteristic
69	C₈	D₂₀₀₈	1	1	yes	characteristic
70	C₈	#16860797	1	1	yes	characteristic
71	C₈	#16860799	1	1	yes	characteristic
72	C₂²	—	2008	4	no	—
73	C₂²	—	2008	2	no	—
74	C₂²	#22548631	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#22548635	1	1	yes	characteristic
77	C₄	#22548637	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4016	2	no	—
80	C₂	#30631430	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631430
commutator	C₄²×C₂₅₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	#364×C₂₅₁	C₂
socle	C₂×C₂₅₁	#331
prime radical	#30632064	C₂
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	#364	D₂₅₁
hypocenter	C₂₅₁	#1357
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #30631508	C₂	semidirect
3. #364×C₂₅₁	C₂	semidirect
40. C₂₅₁	#1357	semidirect
44. #364	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632064	2
#30631508	2
#364×C₂₅₁	2
#1357	251