Subgroups — #22553349

Order 8384.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#16862204	C₂	1	2	yes	maximal
3	#16862180	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862177	C₂	1	2	yes	maximal, complemented
5	#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₈×D₁₃₁	C₄	1	4	yes	complemented
9	C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
10	(C₂×D₁₃₁)⋊C₄	—	2	2	no	—
11	C₄×Dic₁₃₁	—	2	2	no	—
12	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
14	C₂²×D₁₃₁	D₄	1	1	yes	characteristic
15	C₂×Dic₁₃₁	—	2	2	no	—
16	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
17	C₄×D₁₃₁	C₂×C₄	1	2	yes	—
18	C₄×D₁₃₁	Q₈	1	2	yes	—
19	C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₃₁	—	2	2	no	—
21	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₈×C₁₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₃₁	—	2	1	no	—
25	C₂×D₁₃₁	—	2	1	no	—
26	C₂×D₁₃₁	C₄⋊C₄	1	2	yes	—
27	C₄×C₁₃₁	—	4	2	no	—
28	C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
30	Dic₁₃₁	—	4	2	no	—
31	Dic₁₃₁	C₄⋊C₄	1	2	yes	—
32	C₂×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₁₃₁	#135	1	1	yes	characteristic
34	C₂×C₁₃₁	#135	1	1	yes	characteristic
35	D₁₃₁	—	2	2	no	—
36	C₁₃₁	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	131	1	no	maximal, Sylow
38	C₂²×C₈	—	131	1	no	—
39	#117	—	131	2	no	—
40	#107	—	131	1	no	—
41	#107	D₁₃₁	1	1	yes	characteristic, complemented
42	#106	—	131	2	no	—
43	C₂²×C₄	—	131	1	no	—
44	C₂×C₈	—	131	1	no	—
45	C₂×C₈	—	131	4	no	—
46	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
47	C₄⋊C₄	—	131	2	no	—
48	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
49	C₂²⋊C₄	—	262	2	no	—
50	C₄²	—	262	2	no	—
51	C₂³	—	131	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	262	2	no	—
54	C₂×C₄	—	131	1	no	—
55	C₂×C₄	—	131	2	no	—
56	C₂×C₄	—	131	2	no	—
57	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
58	C₈	—	131	2	no	—
59	C₈	C₄×D₁₃₁	1	2	yes	—
60	C₂²	—	262	1	no	—
61	C₂²	—	262	1	no	—
62	C₂²	—	131	2	no	—
63	C₂²	D₄×D₁₃₁	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	524	2	no	—
66	C₄	—	131	2	no	—
67	C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
68	C₄	Q₈×D₁₃₁	1	1	yes	characteristic
69	C₂	—	262	2	no	—
70	C₂	C₄⋊C₄×D₁₃₁	1	1	yes	characteristic
71	C₂	#16862265	1	1	yes	characteristic
72	C₂	#16862269	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₁
commutator	C₄×C₁₃₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#107×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	#107	D₁₃₁
hypocenter	C₁₃₁	#427
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
3. #16862180	C₂	semidirect
4. #16862177	C₂	semidirect
5. #107×C₁₃₁	C₂	semidirect
8. C₈×D₁₃₁	C₄	semidirect
36. C₁₃₁	#427	semidirect
41. #107	D₁₃₁	semidirect