Subgroups — #20298403

Order 7232.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₅₂	C₂	1	1	yes	maximal, characteristic
2	#16813556	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₂₆	—	2	1	no	—
5	Dic₄₅₂	—	2	1	no	—
6	Dic₄₅₂	C₄	1	2	yes	complemented
7	C₄⋊Dic₁₁₃	—	2	1	no	—
8	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₁₃	—	2	1	no	—
10	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
11	C₂×Dic₁₁₃	—	4	1	no	—
12	C₂×Dic₁₁₃	—	4	1	no	—
13	Dic₂₂₆	—	2	2	no	—
14	Dic₂₂₆	—	4	1	no	—
15	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₁₁₃	D₈	1	1	yes	characteristic
18	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₁₁₃	SD₁₆	1	1	yes	characteristic
20	Dic₁₁₃	—	4	2	no	—
21	Dic₁₁₃	—	8	1	no	—
22	C₂×C₁₁₃	#102	1	1	yes	characteristic
23	C₂×C₁₁₃	SD₃₂	1	1	yes	characteristic
24	C₂×C₁₁₃	Q₃₂	1	1	yes	characteristic
25	C₁₁₃	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	113	1	no	maximal, Sylow
27	C₂×Q₁₆	—	113	1	no	—
28	C₂×C₁₆	D₁₁₃	1	1	yes	characteristic
29	#107	—	113	1	no	—
30	C₂×Q₈	—	226	1	no	—
31	Q₁₆	—	113	2	no	—
32	Q₁₆	—	226	1	no	—
33	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
34	C₄⋊C₄	—	226	1	no	—
35	C₁₆	—	2	1	no	—
36	Q₈	—	452	1	no	—
37	Q₈	—	226	2	no	—
38	C₂×C₄	—	452	1	no	—
39	C₂×C₄	—	452	1	no	—
40	C₂×C₄	D₄₅₂	1	1	yes	characteristic
41	C₈	C₄×D₁₁₃	1	1	yes	characteristic
42	C₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
43	C₂²	D₉₀₄	1	1	yes	characteristic
44	C₄	—	904	1	no	—
45	C₄	—	452	2	no	—
46	C₄	#14268624	1	1	yes	characteristic
47	C₄	#14268636	1	1	yes	characteristic
48	C₂	#16813544	1	1	yes	characteristic
49	C₂	#16813567	1	1	yes	characteristic
50	C₂	Dic₉₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₉₀₄
commutator	C₈×C₁₁₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₁₃
Fitting	C₂×C₁₆×C₁₁₃	C₂
socle	C₂²×C₁₁₃	D₈
hypercenter	C₂×C₁₆	D₁₁₃
hypocenter	C₁₁₃	#358
LCS	G ▷ C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
6. Dic₄₅₂	C₄	semidirect
25. C₁₁₃	#358	semidirect