Subgroups — #22553513

Order 8384.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523502	C₂	1	2	yes	maximal, complemented
2	#16862243	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862168	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
7	C₂×D₅₂₄	—	2	1	no	—
8	C₂×D₅₂₄	—	2	1	no	—
9	C₂×D₅₂₄	C₂²	1	2	yes	—
10	#14523502	—	2	4	no	—
11	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
12	Dic₁₃₁⋊C₄	—	2	1	no	—
13	C₄×Dic₁₃₁	—	2	1	no	—
14	C₂×C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂²×D₁₃₁	—	2	2	no	—
17	Dic₂₆₂	—	2	1	no	—
18	Dic₂₆₂	D₄	1	2	yes	complemented
19	D₅₂₄	—	2	2	no	—
20	D₅₂₄	—	2	2	no	—
21	D₅₂₄	—	4	2	no	—
22	D₅₂₄	—	4	2	no	—
23	C₂×Dic₁₃₁	—	2	1	no	—
24	C₂×Dic₁₃₁	—	2	1	no	—
25	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₈×C₁₃₁	—	2	2	no	—
29	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₁₃₁	—	4	2	no	—
31	C₂×D₁₃₁	—	4	2	no	—
32	C₂×D₁₃₁	—	4	2	no	—
33	C₄×C₁₃₁	—	2	1	no	—
34	C₄×C₁₃₁	SD₁₆	1	2	yes	complemented
35	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₃₁	#41	1	1	yes	characteristic
37	Dic₁₃₁	—	2	2	no	—
38	Dic₁₃₁	—	4	1	no	—
39	C₂×C₁₃₁	#121	1	1	yes	characteristic
40	C₂×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₁₃₁	—	8	2	no	—
43	C₁₃₁	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	131	1	no	maximal, Sylow
45	C₂×SD₁₆	—	131	2	no	—
46	#127	—	131	1	no	—
47	C₄×Q₈	—	131	1	no	—
48	C₄⋊C₈	D₁₃₁	1	1	yes	characteristic, complemented
49	#102	—	131	2	no	—
50	C₂×Q₈	—	131	1	no	—
51	C₂×D₄	—	262	1	no	—
52	C₂×D₄	—	262	1	no	—
53	C₂×D₄	—	131	2	no	—
54	SD₁₆	—	262	4	no	—
55	C₂×C₈	C₂×D₁₃₁	1	2	yes	—
56	C₄⋊C₄	—	262	1	no	—
57	C₄⋊C₄	—	131	1	no	—
58	C₄²	—	262	1	no	—
59	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
60	C₂³	—	262	2	no	—
61	Q₈	—	262	1	no	—
62	Q₈	—	131	2	no	—
63	D₄	—	262	2	no	—
64	D₄	—	262	2	no	—
65	D₄	—	524	2	no	—
66	D₄	—	524	2	no	—
67	C₂×C₄	—	262	1	no	—
68	C₂×C₄	—	262	1	no	—
69	C₂×C₄	D₅₂₄	1	1	yes	characteristic
70	C₂×C₄	D₅₂₄	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	524	2	no	—
74	C₂²	—	524	2	no	—
75	C₂²	—	524	2	no	—
76	C₂²	C₂×D₅₂₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	262	2	no	—
79	C₄	—	524	1	no	—
80	C₄	#14523502	1	2	yes	complemented
81	C₄	D₄×D₁₃₁	1	1	yes	characteristic
82	C₄	#14523514	1	1	yes	characteristic
83	C₂	—	1048	2	no	—
84	C₂	#16862221	1	1	yes	characteristic
85	C₂	C₂×#14523502	1	1	yes	characteristic
86	C₂	#16862274	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₂₄
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	C₄⋊C₈×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	C₄⋊C₈	D₁₃₁
hypocenter	C₁₃₁	#460
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14523502	C₂	semidirect
3. C₄×Dic₂₆₂	C₂	semidirect
4. #16862168	C₂	semidirect
5. C₄⋊C₈×C₁₃₁	C₂	semidirect
18. Dic₂₆₂	D₄	semidirect
34. C₄×C₁₃₁	SD₁₆	semidirect
43. C₁₃₁	#460	semidirect
48. C₄⋊C₈	D₁₃₁	semidirect
80. C₄	#14523502	semidirect