Subgroups — #22553751

Order 8384.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16862265	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₀₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862192	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862190	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862185	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523512	—	2	1	no	—
9	Q₈×D₁₃₁	—	2	1	no	—
10	Dic₅₂₄	C₂²	1	1	yes	characteristic
11	Dic₅₂₄	C₂²	1	1	yes	characteristic
12	#14523503	—	2	1	no	—
13	#14523501	—	2	1	no	—
14	C₈×D₁₃₁	C₂²	1	1	yes	characteristic
15	C₁₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
17	D₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
19	Dic₂₆₂	—	2	1	no	—
20	Dic₂₆₂	—	2	1	no	—
21	Dic₂₆₂	—	4	1	no	—
22	C₂²⋊D₁₃₁	—	4	1	no	—
23	C₂×Dic₁₃₁	—	4	1	no	—
24	C₄×D₁₃₁	—	4	1	no	—
25	C₄×D₁₃₁	D₄	1	1	yes	characteristic
26	C₁₃₁⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₁₃₁	—	2	1	no	—
28	D₄×C₁₃₁	—	2	1	no	—
29	C₈×C₁₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₁₃₁	—	4	1	no	—
31	C₂×D₁₃₁	D₈	1	1	yes	characteristic
32	C₄×C₁₃₁	—	4	1	no	—
33	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
34	Dic₁₃₁	—	4	1	no	—
35	Dic₁₃₁	—	4	1	no	—
36	Dic₁₃₁	D₈	1	1	yes	characteristic
37	C₂×C₁₃₁	—	8	1	no	—
38	C₂×C₁₃₁	C₂×D₈	1	1	yes	characteristic
39	D₁₃₁	—	2	1	no	—
40	C₁₃₁	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	131	1	no	maximal, Sylow
42	#135	—	131	1	no	—
43	C₂×Q₁₆	—	131	1	no	—
44	Q₃₂	—	131	1	no	—
45	Q₃₂	—	131	1	no	—
46	SD₃₂	—	131	1	no	—
47	SD₃₂	D₁₃₁	1	1	yes	characteristic, complemented
48	M₃₂	—	131	1	no	—
49	#41	—	262	1	no	—
50	C₂×Q₈	—	262	1	no	—
51	Q₁₆	—	262	1	no	—
52	Q₁₆	—	131	1	no	—
53	Q₁₆	—	131	1	no	—
54	Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
55	SD₁₆	—	262	1	no	—
56	D₈	C₂×D₁₃₁	1	1	yes	characteristic
57	C₂×C₈	—	131	1	no	—
58	C₁₆	—	131	1	no	—
59	C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	262	1	no	—
62	Q₈	—	262	1	no	—
63	Q₈	—	524	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	524	1	no	—
66	C₂×C₄	—	524	1	no	—
67	C₂×C₄	—	524	1	no	—
68	C₂×C₄	—	131	1	no	—
69	C₈	—	131	1	no	—
70	C₈	C₂²×D₁₃₁	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	—	131	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	524	1	no	—
75	C₄	—	524	1	no	—
76	C₄	—	131	1	no	—
77	C₄	D₄×D₁₃₁	1	1	yes	characteristic
78	C₂	—	262	1	no	—
79	C₂	—	8	1	no	—
80	C₂	D₈×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₃₁
commutator	C₈×C₁₃₁	C₂³
Frattini	C₈	C₂²×D₁₃₁
Fitting	SD₃₂×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂×D₈
hypercenter	SD₃₂	D₁₃₁
hypocenter	C₁₃₁	#510
LCS	G ▷ C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₁₃₁	C₂	semidirect
3. Dic₁₀₄₈	C₂	semidirect
4. #16862192	C₂	semidirect
5. #16862190	C₂	semidirect
6. #16862185	C₂	semidirect
7. SD₃₂×C₁₃₁	C₂	semidirect
40. C₁₃₁	#510	semidirect
47. SD₃₂	D₁₃₁	semidirect