Subgroups — #30631978

Order 16064.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22548715	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₀₀₄	C₂	1	1	yes	maximal, characteristic
3	#22548647	C₂	1	2	yes	maximal, complemented
4	#22548637	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₀₂	—	2	1	no	—
7	C₂×D₁₀₀₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₅₁	—	2	1	no	—
9	C₄⋊Dic₂₅₁	—	2	1	no	—
10	C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
11	C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
12	(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
13	C₂×C₈×C₂₅₁	C₂²	1	2	yes	—
14	C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₅₁	—	2	1	no	—
16	Dic₅₀₂	—	4	2	no	—
17	D₁₀₀₄	—	2	1	no	—
18	D₁₀₀₄	Q₈	1	2	yes	complemented
19	C₂×Dic₂₅₁	—	2	1	no	—
20	C₂×Dic₂₅₁	—	2	2	no	—
21	C₄×D₂₅₁	—	4	2	no	—
22	C₂×C₄×C₂₅₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
25	C₈×C₂₅₁	—	2	2	no	—
26	C₂²×C₂₅₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₅₁	—	2	2	no	—
28	C₂×D₂₅₁	—	4	1	no	—
29	C₂×D₂₅₁	—	4	1	no	—
30	C₄×C₂₅₁	—	2	1	no	—
31	C₄×C₂₅₁	SD₁₆	1	2	yes	—
32	C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₅₁	#41	1	1	yes	characteristic
34	Dic₂₅₁	—	4	1	no	—
35	Dic₂₅₁	—	4	2	no	—
36	C₂×C₂₅₁	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₂₅₁	—	4	2	no	—
40	C₂₅₁	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	251	1	no	maximal, Sylow
42	C₄⋊Q₈	—	251	1	no	—
43	C₄×D₄	—	251	1	no	—
44	#106	—	251	2	no	—
45	C₄⋊C₈	D₂₅₁	1	1	yes	characteristic, complemented
46	#102	—	251	2	no	—
47	C₂×Q₈	—	502	1	no	—
48	C₂×D₄	—	251	1	no	—
49	C₂²×C₄	—	502	1	no	—
50	C₂×C₈	C₂×D₂₅₁	1	2	yes	—
51	C₄⋊C₄	—	502	1	no	—
52	C₄⋊C₄	—	251	2	no	—
53	C₄⋊C₄	—	251	1	no	—
54	C₂²⋊C₄	—	502	1	no	—
55	C₄²	C₂×D₂₅₁	1	1	yes	characteristic
56	C₂³	—	502	1	no	—
57	Q₈	—	1004	2	no	—
58	D₄	—	502	1	no	—
59	D₄	—	251	2	no	—
60	C₂×C₄	—	1004	2	no	—
61	C₂×C₄	—	502	1	no	—
62	C₂×C₄	—	502	2	no	—
63	C₂×C₄	D₁₀₀₄	1	1	yes	characteristic
64	C₂×C₄	D₁₀₀₄	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	1004	1	no	—
68	C₂²	—	1004	1	no	—
69	C₂²	—	502	2	no	—
70	C₂²	C₂×D₁₀₀₄	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	1004	2	no	—
73	C₄	—	1004	1	no	—
74	C₄	#16860799	1	2	yes	—
75	C₄	Q₈×D₂₅₁	1	1	yes	characteristic
76	C₄	#16860809	1	1	yes	characteristic
77	C₂	—	1004	2	no	—
78	C₂	#22548695	1	1	yes	characteristic
79	C₂	C₂×#16860799	1	1	yes	characteristic
80	C₂	#22548743	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₀₄
commutator, cosocle	C₂×C₄×C₂₅₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₁
Fitting	C₄⋊C₈×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×D₄
prime radical	C₂×D₁₀₀₄	C₂²
hypercenter	C₄⋊C₈	D₂₅₁
hypocenter	C₂₅₁	#476
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #22548715	C₂	semidirect
3. #22548647	C₂	semidirect
5. C₄⋊C₈×C₂₅₁	C₂	semidirect
18. D₁₀₀₄	Q₈	semidirect
40. C₂₅₁	#476	semidirect
45. C₄⋊C₈	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#22548715	2
C₄×D₁₀₀₄	2
#22548647	2
#22548637	2
C₄⋊C₈×C₂₅₁	2
#476	251