Subgroups — #20298517

Order 7232.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268626	C₂	1	3	yes	maximal
2	#16813529	C₂	1	3	yes	maximal
3	C₂²×Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₁₃	C₂²	1	1	yes	characteristic
5	C₂²×Dic₁₁₃	C₂²	1	3	yes	—
6	#14268626	—	2	6	no	—
7	C₂×Q₈×C₁₁₃	—	2	6	no	—
8	C₂²×C₄×C₁₁₃	C₂²	1	3	yes	—
9	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
10	Q₈×C₁₁₃	—	2	8	no	—
11	C₂×C₄×C₁₁₃	—	2	6	no	—
12	C₂×C₄×C₁₁₃	D₄	1	6	yes	—
13	C₂×Dic₁₁₃	—	2	3	no	—
14	C₂×Dic₁₁₃	—	2	3	no	—
15	C₂×Dic₁₁₃	—	2	3	no	—
16	C₂×Dic₁₁₃	Q₈	1	6	yes	complemented
17	C₂²×C₁₁₃	C₂×D₄	1	3	yes	—
18	C₂²×C₁₁₃	C₂×Q₈	1	3	yes	—
19	C₂²×C₁₁₃	#41	1	1	yes	characteristic
20	C₄×C₁₁₃	—	2	6	no	—
21	Dic₁₁₃	—	2	6	no	—
22	Dic₁₁₃	—	4	1	no	—
23	C₂×C₁₁₃	#120	1	1	yes	characteristic
24	C₂×C₁₁₃	C₂²⋊Q₈	1	3	yes	—
25	C₂×C₁₁₃	C₄⋊Q₈	1	3	yes	—
26	C₁₁₃	#395	1	1	yes	characteristic, complemented, Sylow
27	#395	—	113	1	no	maximal, Sylow
28	C₂²×Q₈	D₁₁₃	1	1	yes	characteristic
29	C₂×C₄⋊C₄	—	113	3	no	—
30	#95	—	113	3	no	—
31	C₂×Q₈	—	2	6	no	—
32	C₂²×C₄	—	113	3	no	—
33	C₂²×C₄	—	113	1	no	—
34	C₂²×C₄	C₂×D₁₁₃	1	3	yes	—
35	C₄⋊C₄	—	226	6	no	—
36	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
37	Q₈	—	2	8	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	—	113	6	no	—
40	C₂×C₄	—	226	3	no	—
41	C₂×C₄	—	226	3	no	—
42	C₂×C₄	—	226	3	no	—
43	C₂×C₄	C₂²⋊D₁₁₃	1	6	yes	—
44	C₂²	C₂×C₂²⋊D₁₁₃	1	3	yes	—
45	C₂²	Q₈×D₁₁₃	1	3	yes	—
46	C₂²	#14268648	1	1	yes	characteristic
47	C₄	—	2	6	no	—
48	C₄	—	452	1	no	—
49	C₄	—	226	6	no	—
50	C₂	#16813591	1	1	yes	characteristic
51	C₂	#16813598	1	3	yes	—
52	C₂	#16813620	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₁₃
commutator, socle	C₂³×C₁₁₃	C₂³
Fitting	C₂²×Q₈×C₁₁₃	C₂
hypercenter	C₂²×Q₈	D₁₁₃
hypocenter	C₁₁₃	#395
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
16. C₂×Dic₁₁₃	Q₈	semidirect
26. C₁₁₃	#395	semidirect