Subgroups — #22553613

Order 8384.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16862238	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₅₂₄	C₂	1	1	yes	maximal, characteristic
3	#16862180	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862178	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862168	C₂	1	1	yes	maximal, characteristic
6	#16862168	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₃₁	—	2	1	no	—
10	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
13	Dic₁₃₁⋊C₄	—	2	1	no	—
14	Dic₁₃₁⋊C₄	—	2	1	no	—
15	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
16	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂²×D₁₃₁	—	2	1	no	—
20	D₅₂₄	—	2	1	no	—
21	D₅₂₄	Q₈	1	2	yes	—
22	C₂×Dic₁₃₁	—	2	1	no	—
23	C₂×Dic₁₃₁	—	2	1	no	—
24	C₂×Dic₁₃₁	—	2	1	no	—
25	C₄×D₁₃₁	—	4	2	no	—
26	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₈×C₁₃₁	—	2	1	no	—
30	C₈×C₁₃₁	—	2	1	no	—
31	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₁₃₁	—	2	2	no	—
33	C₂×D₁₃₁	—	4	1	no	—
34	C₂×D₁₃₁	—	4	1	no	—
35	C₄×C₁₃₁	—	2	1	no	—
36	C₄×C₁₃₁	—	2	1	no	—
37	C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₁₃₁	#41	1	1	yes	characteristic
39	Dic₁₃₁	—	4	1	no	—
40	Dic₁₃₁	—	4	1	no	—
41	Dic₁₃₁	—	4	1	no	—
42	C₂×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₁₃₁	#135	1	1	yes	characteristic
44	C₂×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₁₃₁	—	4	2	no	—
46	C₁₃₁	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	131	1	no	maximal, Sylow
48	#125	—	131	1	no	—
49	C₄×D₄	—	131	1	no	—
50	#107	—	131	1	no	—
51	#106	—	131	1	no	—
52	C₄⋊C₈	D₁₃₁	1	1	yes	characteristic, complemented
53	#102	—	131	1	no	—
54	#102	—	131	1	no	—
55	C₂×D₄	—	131	1	no	—
56	C₂²×C₄	—	262	1	no	—
57	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
58	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
59	C₄⋊C₄	—	262	1	no	—
60	C₄⋊C₄	—	262	1	no	—
61	C₄⋊C₄	—	131	1	no	—
62	C₄⋊C₄	—	131	1	no	—
63	C₄⋊C₄	—	131	1	no	—
64	C₂²⋊C₄	—	262	1	no	—
65	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
66	C₂³	—	262	1	no	—
67	D₄	—	262	1	no	—
68	D₄	—	131	2	no	—
69	C₂×C₄	—	262	1	no	—
70	C₂×C₄	—	262	1	no	—
71	C₂×C₄	—	262	1	no	—
72	C₂×C₄	—	524	2	no	—
73	C₂×C₄	D₅₂₄	1	1	yes	characteristic
74	C₂×C₄	D₅₂₄	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₂²	—	262	2	no	—
79	C₂²	—	524	1	no	—
80	C₂²	—	524	1	no	—
81	C₂²	C₂×D₅₂₄	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	524	1	no	—
85	C₄	—	524	1	no	—
86	C₄	—	524	1	no	—
87	C₄	Q₈×D₁₃₁	1	1	yes	characteristic
88	C₄	#14523512	1	1	yes	characteristic
89	C₂	—	524	2	no	—
90	C₂	#16862226	1	1	yes	characteristic
91	C₂	#16862268	1	1	yes	characteristic
92	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	#478
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. #16862238	C₂	semidirect
3. #16862180	C₂	semidirect
4. #16862178	C₂	semidirect
7. C₄⋊C₈×C₁₃₁	C₂	semidirect
46. C₁₃₁	#478	semidirect
52. C₄⋊C₈	D₁₃₁	semidirect