Subgroups — #22553707

Order 8384.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523503	C₂	1	2	yes	maximal, complemented
2	C₂×#14523501	C₂	1	2	yes	maximal
3	#16862246	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862157	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₂	C₂²	1	2	yes	—
7	#14523503	—	2	4	no	—
8	#14523501	—	2	4	no	—
9	C₂×C₁₃₁⋊C₈	C₂²	1	2	yes	—
10	C₄⋊Dic₁₃₁	—	2	2	no	—
11	C₂×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
14	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	Dic₂₆₂	—	2	2	no	—
16	Dic₂₆₂	—	2	2	no	—
17	C₂×Dic₁₃₁	—	2	2	no	—
18	C₁₃₁⋊C₈	D₄	1	4	yes	—
19	C₂³×C₁₃₁	—	2	1	no	—
20	Q₈×C₁₃₁	—	2	2	no	—
21	D₄×C₁₃₁	—	2	2	no	—
22	C₂×C₄×C₁₃₁	—	2	1	no	—
23	C₂×C₄×C₁₃₁	D₄	1	2	yes	—
24	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₁₃₁	—	4	1	no	—
26	C₂²×C₁₃₁	—	4	2	no	—
27	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₃₁	—	2	2	no	—
29	C₄×C₁₃₁	—	4	1	no	—
30	C₄×C₁₃₁	C₂×D₄	1	2	yes	—
31	Dic₁₃₁	—	4	2	no	—
32	C₂×C₁₃₁	—	8	1	no	—
33	C₂×C₁₃₁	#127	1	1	yes	characteristic
34	C₂×C₁₃₁	#137	1	2	yes	—
35	C₁₃₁	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	131	1	no	maximal, Sylow
37	C₂×Q₁₆	—	131	2	no	—
38	C₂×SD₁₆	—	131	2	no	—
39	C₄⋊Q₈	—	131	1	no	—
40	#124	D₁₃₁	1	1	yes	characteristic
41	#97	—	131	1	no	—
42	C₂×Q₈	—	131	2	no	—
43	C₂×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
44	C₂×D₄	C₂×D₁₃₁	1	1	yes	characteristic
45	Q₁₆	—	262	4	no	—
46	SD₁₆	—	262	4	no	—
47	C₂×C₈	—	131	2	no	—
48	C₄⋊C₄	—	262	2	no	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	262	2	no	—
54	Q₈	—	262	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	262	2	no	—
58	C₂×C₄	C₂²⋊D₁₃₁	1	2	yes	—
59	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
60	C₈	—	131	4	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	524	2	no	—
67	C₄	D₄×D₁₃₁	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	#16862242	1	1	yes	characteristic
70	C₂	#16862276	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#124×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	#124	D₁₃₁
hypocenter	C₁₃₁	#497
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#14523503	C₂	semidirect
3. #16862246	C₂	semidirect
4. #16862157	C₂	semidirect
35. C₁₃₁	#497	semidirect