Subgroups — #20298721

Order 7232.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813644	C₂	1	1	yes	maximal, characteristic
2	#16813628	C₂	1	2	yes	maximal, complemented
3	#16813558	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813550	C₂	1	2	yes	maximal, complemented
5	#97×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268647	C₂²	1	2	yes	—
7	Dic₄₅₂	C₄	1	1	yes	characteristic, complemented
8	#14268636	C₄	1	2	yes	complemented
9	D₉₀₄	C₄	1	1	yes	characteristic, complemented
10	#14268631	C₂²	1	2	yes	—
11	#14268630	—	2	2	no	—
12	C₈×D₁₁₃	—	2	2	no	—
13	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₁₃	—	2	1	no	—
17	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
18	C₂²⋊D₁₁₃	—	2	2	no	—
19	D₄₅₂	C₂×C₄	1	2	yes	—
20	C₄×D₁₁₃	—	2	2	no	—
21	Dic₂₂₆	C₂×C₄	1	2	yes	—
22	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
23	C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₁₁₃	D₄	1	2	yes	—
25	#11744751	—	2	2	no	—
26	C₂²×C₁₁₃	#41	1	1	yes	characteristic
27	C₂×D₁₁₃	—	2	2	no	—
28	C₄×C₁₁₃	—	4	1	no	—
29	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
31	Dic₁₁₃	—	2	2	no	—
32	C₂×C₁₁₃	—	2	1	no	—
33	C₂×C₁₁₃	C₄×D₄	1	1	yes	characteristic
34	D₁₁₃	—	4	2	no	—
35	C₁₁₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	113	1	no	maximal, Sylow
37	#135	—	113	1	no	—
38	#131	—	113	2	no	—
39	#108	—	113	1	no	—
40	#104	—	113	2	no	—
41	#97	D₁₁₃	1	1	yes	characteristic, complemented
42	#41	—	113	2	no	—
43	Q₁₆	—	113	1	no	—
44	SD₁₆	—	113	2	no	—
45	D₈	—	113	1	no	—
46	M₁₆	—	113	2	no	—
47	M₁₆	—	226	2	no	—
48	C₂×C₈	—	226	2	no	—
49	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
50	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
51	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
52	Q₈	—	113	2	no	—
53	D₄	—	113	2	no	—
54	D₄	—	226	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	226	2	no	—
57	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
58	C₈	—	226	2	no	—
59	C₈	C₄×D₁₁₃	1	1	yes	characteristic
60	C₈	C₄×D₁₁₃	1	1	yes	characteristic
61	C₈	D₄₅₂	1	2	yes	—
62	C₂²	—	226	2	no	—
63	C₂²	#14268647	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	226	2	no	—
66	C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
67	C₄	C₂×D₄₅₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	452	2	no	—
70	C₂	C₄×D₄₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄₅₂
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#97×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₄×D₄
hypercenter	#97	D₁₁₃
hypocenter	C₁₁₃	#444
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #16813628	C₂	semidirect
3. #16813558	C₂	semidirect
4. #16813550	C₂	semidirect
5. #97×C₁₁₃	C₂	semidirect
7. Dic₄₅₂	C₄	semidirect
8. #14268636	C₄	semidirect
9. D₉₀₄	C₄	semidirect
35. C₁₁₃	#444	semidirect
41. #97	D₁₁₃	semidirect