Subgroups — #44548345

Order 32128.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₅₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₅₀₂	C₄	1	1	yes	characteristic
5	C₂×C₂₅₁⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₂₅₁⋊C₈)	C₄	1	1	yes	characteristic
7	Dic₂₅₁⋊C₈	—	2	1	no	—
8	C₈×Dic₂₅₁	—	2	1	no	—
9	C₂×C₁₆×C₂₅₁	—	2	1	no	—
10	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₀₂	C₈	1	1	yes	characteristic
12	C₂×C₂₅₁⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₅₁	C₈	1	1	yes	characteristic
14	Dic₂₅₁⋊C₄	—	2	1	no	—
15	C₄×Dic₂₅₁	—	2	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	—	2	2	no	—
21	Dic₅₀₂	—	2	1	no	—
22	Dic₅₀₂	C₁₆	1	2	yes	complemented
23	C₂×Dic₂₅₁	—	2	1	no	—
24	C₂×Dic₂₅₁	—	2	1	no	—
25	C₂₅₁⋊C₈	—	4	1	no	—
26	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₅₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₅₁	M₁₆	1	1	yes	characteristic
29	C₈×C₂₅₁	—	2	1	no	—
30	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
31	C₈×C₂₅₁	Q₁₆	1	1	yes	characteristic
32	C₂²×C₂₅₁	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₂₅₁	—	2	1	no	—
34	C₄×C₂₅₁	#103	1	1	yes	characteristic
35	C₄×C₂₅₁	#104	1	1	yes	characteristic
36	C₄×C₂₅₁	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₂₅₁	M₃₂	1	1	yes	characteristic
38	Dic₂₅₁	—	2	2	no	—
39	Dic₂₅₁	—	4	1	no	—
40	C₂×C₂₅₁	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₂₅₁	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₂₅₁	#350	1	1	yes	characteristic
43	C₂₅₁	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	251	1	no	maximal, Sylow
45	C₈×Q₈	—	251	1	no	—
46	C₄⋊C₁₆	—	251	1	no	—
47	C₄×C₁₆	D₂₅₁	1	1	yes	characteristic
48	C₄×Q₈	—	251	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	502	1	no	—
51	C₄⋊C₈	—	502	1	no	—
52	C₄⋊C₈	—	251	1	no	—
53	C₄×C₈	—	502	1	no	—
54	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
55	C₂×Q₈	—	251	1	no	—
56	C₂×C₈	—	502	1	no	—
57	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
58	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
59	C₄⋊C₄	—	502	1	no	—
60	C₄⋊C₄	—	251	1	no	—
61	C₄²	—	502	1	no	—
62	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	1004	1	no	—
65	Q₈	—	502	1	no	—
66	Q₈	—	251	2	no	—
67	C₂×C₄	—	502	1	no	—
68	C₂×C₄	—	502	1	no	—
69	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
70	C₂×C₄	C₈×D₂₅₁	1	1	yes	characteristic
71	C₂×C₄	#16860793	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	1004	1	no	—
74	C₈	#16860799	1	1	yes	characteristic
75	C₈	Dic₁₀₀₄	1	1	yes	characteristic
76	C₂²	(C₂×D₂₅₁)⋊C₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	1004	1	no	—
79	C₄	—	502	2	no	—
80	C₄	#22548640	1	1	yes	characteristic
81	C₄	#22548643	1	1	yes	characteristic
82	C₄	C₁₆×D₂₅₁	1	1	yes	characteristic
83	C₄	#22548654	1	1	yes	characteristic
84	C₂	#30631419	1	1	yes	characteristic
85	C₂	(C₂×D₂₅₁)⋊C₁₆	1	1	yes	characteristic
86	C₂	#30631470	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₁₀₀₄
commutator	C₄×C₂₅₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	C₄×C₁₆×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂²⋊C₈
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	C₄×C₁₆	D₂₅₁
hypocenter	C₂₅₁	Q₈⋊C₁₆
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
22. Dic₅₀₂	C₁₆	semidirect
43. C₂₅₁	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₅₀₂	2
C₄⋊(C₂₅₁⋊C₁₆)	2
C₄×C₁₆×C₂₅₁	2
Q₈⋊C₁₆	251