Subgroups — #20295831

Order 6976.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812745	C₂	1	2	yes	maximal, complemented
2	#16812727	C₂	1	6	yes	maximal, complemented
3	#16812721	C₂	1	3	yes	maximal, complemented
4	C₂×D₈₇₂	C₂	1	3	yes	maximal, complemented
5	#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#13182030	—	2	2	no	—
7	#13182029	C₂²	1	6	yes	complemented
8	C₂×D₄₃₆	C₂²	1	6	yes	complemented
9	D₄×D₁₀₉	—	2	6	no	—
10	Dic₄₃₆	C₂²	1	1	yes	characteristic, complemented
11	#13182018	C₂²	1	6	yes	complemented
12	D₈₇₂	C₂²	1	3	yes	complemented
13	D₈₇₂	C₂²	1	6	yes	complemented
14	#41×C₁₀₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₀₉	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₀₉	C₂²	1	3	yes	complemented
17	C₂²×D₁₀₉	—	2	6	no	—
18	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
19	D₄×C₁₀₉	D₄	1	3	yes	—
20	C₂×C₄×C₁₀₉	C₂³	1	3	yes	—
21	C₂²⋊D₁₀₉	—	2	6	no	—
22	D₄₃₆	—	2	6	no	—
23	D₄₃₆	C₂³	1	6	yes	—
24	C₄×D₁₀₉	—	2	6	no	—
25	Dic₂₁₈	C₂³	1	2	yes	—
26	C₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₈×C₁₀₉	C₂³	1	3	yes	—
28	C₂²×C₁₀₉	C₂×D₄	1	3	yes	—
29	C₂×D₁₀₉	—	2	6	no	—
30	C₂×D₁₀₉	—	8	6	no	—
31	C₄×C₁₀₉	C₂×D₄	1	3	yes	—
32	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
33	Dic₁₀₉	—	2	2	no	—
34	C₂×C₁₀₉	—	2	3	no	—
35	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
36	D₁₀₉	—	4	6	no	—
37	C₁₀₉	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	109	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	109	2	no	—
40	AGL(1,ℤ/8)	—	109	6	no	—
41	#135	—	109	3	no	—
42	C₂×D₈	—	109	3	no	—
43	#131	D₁₀₉	1	1	yes	characteristic, complemented
44	#41	—	109	6	no	—
45	#41	—	218	2	no	—
46	#41	C₂×D₁₀₉	1	1	yes	characteristic
47	C₂×D₄	—	109	6	no	—
48	C₂×D₄	—	218	6	no	—
49	Q₁₆	—	109	1	no	—
50	SD₁₆	—	109	6	no	—
51	D₈	—	109	3	no	—
52	D₈	—	109	6	no	—
53	M₁₆	C₂×D₁₀₉	1	3	yes	complemented
54	C₂×C₈	C₂×D₁₀₉	1	3	yes	complemented
55	C₂³	—	218	6	no	—
56	Q₈	—	109	2	no	—
57	Q₈	D₄₃₆	1	1	yes	characteristic
58	D₄	—	109	6	no	—
59	D₄	—	218	6	no	—
60	D₄	—	218	6	no	—
61	D₄	D₄₃₆	1	3	yes	—
62	C₂×C₄	—	218	6	no	—
63	C₂×C₄	C₂²×D₁₀₉	1	3	yes	—
64	C₈	C₂²×D₁₀₉	1	1	yes	characteristic
65	C₈	C₂²×D₁₀₉	1	3	yes	—
66	C₂²	—	872	6	no	—
67	C₂²	—	218	6	no	—
68	C₂²	C₂×D₄₃₆	1	3	yes	—
69	C₄	—	218	2	no	—
70	C₄	C₂×D₄₃₆	1	3	yes	—
71	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	436	6	no	—
74	C₂	C₂²×D₄₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₃₆
commutator	C₄×C₁₀₉	C₂⁴
Frattini	C₄	C₂³×D₁₀₉
Fitting	#131×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×D₄
hypercenter	#131	D₁₀₉
hypocenter	C₁₀₉	#576
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #16812745	C₂	semidirect
2. #16812727	C₂	semidirect
3. #16812721	C₂	semidirect
4. C₂×D₈₇₂	C₂	semidirect
5. #131×C₁₀₉	C₂	semidirect
7. #13182029	C₂²	semidirect
8. C₂×D₄₃₆	C₂²	semidirect
10. Dic₄₃₆	C₂²	semidirect
11. #13182018	C₂²	semidirect
12. D₈₇₂	C₂²	semidirect
13. D₈₇₂	C₂²	semidirect
15. M₁₆×C₁₀₉	C₂²	semidirect
16. C₂×C₈×C₁₀₉	C₂²	semidirect
37. C₁₀₉	#576	semidirect
43. #131	D₁₀₉	semidirect
53. M₁₆	C₂×D₁₀₉	semidirect
54. C₂×C₈	C₂×D₁₀₉	semidirect