Subgroups — #44549962

Order 32128.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632092	C₂	1	1	yes	maximal, characteristic
2	#30632085	C₂	1	2	yes	maximal
3	#30632067	C₂	1	1	yes	maximal, characteristic
4	#30632038	C₂	1	2	yes	maximal
5	C₈²×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₀₄	—	2	2	no	—
7	#22548715	C₂²	1	2	yes	—
8	#22548715	C₂²	1	2	yes	—
9	#22548649	—	2	2	no	—
10	#22548647	—	2	4	no	—
11	#22548640	—	2	4	no	—
12	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₀₂	—	2	2	no	—
16	C₂×Dic₅₀₂	—	2	2	no	—
17	Dic₁₀₀₄	—	4	4	no	—
18	C₄⋊Dic₂₅₁	—	2	2	no	—
19	C₄⋊Dic₂₅₁	—	2	2	no	—
20	C₄⋊Dic₂₅₁	—	2	2	no	—
21	C₄⋊Dic₂₅₁	—	2	2	no	—
22	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
23	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
24	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
25	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
26	Dic₅₀₂	—	4	4	no	—
27	Dic₅₀₂	—	4	4	no	—
28	C₂×Dic₂₅₁	—	4	2	no	—
29	C₂×Dic₂₅₁	—	4	2	no	—
30	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₅₁	—	2	2	no	—
34	C₈×C₂₅₁	SD₁₆	1	4	yes	—
35	C₈×C₂₅₁	Q₁₆	1	4	yes	complemented
36	C₂²×C₂₅₁	#127	1	1	yes	characteristic
37	C₄×C₂₅₁	C₂×SD₁₆	1	2	yes	—
38	C₄×C₂₅₁	C₂×Q₁₆	1	2	yes	—
39	C₄×C₂₅₁	#135	1	2	yes	—
40	Dic₂₅₁	—	8	2	no	—
41	Dic₂₅₁	—	8	2	no	—
42	C₂×C₂₅₁	#492	1	1	yes	characteristic
43	C₂×C₂₅₁	#494	1	1	yes	characteristic
44	C₂×C₂₅₁	#495	1	1	yes	characteristic
45	C₂₅₁	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	251	1	no	maximal, Sylow
47	#500	—	251	1	no	—
48	#498	—	251	2	no	—
49	#494	—	251	1	no	—
50	#487	—	251	2	no	—
51	C₈²	D₂₅₁	1	1	yes	characteristic
52	C₂×Q₁₆	—	502	2	no	—
53	C₄⋊Q₈	—	251	2	no	—
54	C₄⋊Q₈	—	251	2	no	—
55	#107	—	502	2	no	—
56	#106	—	502	4	no	—
57	#103	—	502	4	no	—
58	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
59	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
60	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
61	C₂×Q₈	—	502	2	no	—
62	C₂×Q₈	—	502	2	no	—
63	Q₁₆	—	1004	4	no	—
64	C₂×C₈	D₁₀₀₄	1	2	yes	—
65	C₂×C₈	D₁₀₀₄	1	2	yes	—
66	C₂×C₈	D₁₀₀₄	1	2	yes	—
67	C₄⋊C₄	—	502	2	no	—
68	C₄⋊C₄	—	502	2	no	—
69	C₄⋊C₄	—	502	2	no	—
70	C₄⋊C₄	—	502	2	no	—
71	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
72	Q₈	—	1004	4	no	—
73	Q₈	—	1004	4	no	—
74	C₂×C₄	—	1004	2	no	—
75	C₂×C₄	—	1004	2	no	—
76	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
77	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
78	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	#16860799	1	4	yes	—
81	C₈	Dic₁₀₀₄	1	4	yes	complemented
82	C₂²	#22548712	1	1	yes	characteristic
83	C₄	—	2008	2	no	—
84	C₄	—	2008	2	no	—
85	C₄	C₂×#16860799	1	2	yes	—
86	C₄	C₂×Dic₁₀₀₄	1	2	yes	—
87	C₄	#22548737	1	2	yes	—
88	C₂	#30632060	1	1	yes	characteristic
89	C₂	#30632067	1	1	yes	characteristic
90	C₂	#30632071	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22548712
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	C₈²×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#127
hypercenter	C₈²	D₂₅₁
hypocenter	C₂₅₁	#1707
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
35. C₈×C₂₅₁	Q₁₆	semidirect
45. C₂₅₁	#1707	semidirect
81. C₈	Dic₁₀₀₄	semidirect

Maximal subgroups

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

subgroup	index
#30632092	2
#30632085	2
#30632067	2
#30632038	2
C₈²×C₂₅₁	2
#1707	251