Subgroups — #20295464

Order 6976.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812721	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812644	C₂	1	2	yes	maximal, complemented
4	D₁₇₄₄	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#13182029	—	2	1	no	—
7	C₂×D₄₃₆	—	2	1	no	—
8	Dic₄₃₆	C₂²	1	1	yes	characteristic, complemented
9	#13182018	—	2	1	no	—
10	D₈₇₂	—	2	1	no	—
11	D₈₇₂	C₂²	1	2	yes	—
12	D₈₇₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂²×D₁₀₉	—	4	1	no	—
16	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂²⋊D₁₀₉	—	4	1	no	—
18	D₄₃₆	—	2	2	no	—
19	D₄₃₆	—	2	1	no	—
20	D₄₃₆	—	4	1	no	—
21	C₄×D₁₀₉	—	4	1	no	—
22	Dic₂₁₈	—	2	1	no	—
23	C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₈×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₀₉	D₈	1	1	yes	characteristic
26	C₂×D₁₀₉	—	4	1	no	—
27	C₂×D₁₀₉	—	4	2	no	—
28	C₂×D₁₀₉	—	8	2	no	—
29	C₄×C₁₀₉	D₈	1	1	yes	characteristic
30	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	Dic₁₀₉	—	4	1	no	—
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
34	D₁₀₉	—	8	1	no	—
35	D₁₀₉	—	8	2	no	—
36	C₁₀₉	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	109	1	no	maximal, Sylow
38	#135	—	109	1	no	—
39	C₂×D₈	—	109	1	no	—
40	SD₃₂	—	109	2	no	—
41	D₁₆	—	109	2	no	—
42	M₃₂	D₁₀₉	1	1	yes	characteristic, complemented
43	#41	—	218	1	no	—
44	C₂×D₄	—	218	1	no	—
45	Q₁₆	—	109	1	no	—
46	SD₁₆	—	218	1	no	—
47	D₈	—	109	2	no	—
48	D₈	—	109	1	no	—
49	D₈	—	218	1	no	—
50	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
51	C₁₆	C₂×D₁₀₉	1	2	yes	complemented
52	C₂³	—	436	1	no	—
53	Q₈	—	218	1	no	—
54	D₄	—	436	1	no	—
55	D₄	—	436	1	no	—
56	D₄	—	218	2	no	—
57	D₄	—	218	1	no	—
58	C₂×C₄	—	436	1	no	—
59	C₂×C₄	D₄₃₆	1	1	yes	characteristic
60	C₈	D₄₃₆	1	1	yes	characteristic
61	C₈	C₂²×D₁₀₉	1	1	yes	characteristic
62	C₂²	—	872	2	no	—
63	C₂²	—	436	1	no	—
64	C₂²	—	436	2	no	—
65	C₂²	D₈₇₂	1	1	yes	characteristic
66	C₄	—	436	1	no	—
67	C₄	D₈₇₂	1	1	yes	characteristic
68	C₄	C₂×D₄₃₆	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	872	2	no	—
71	C₂	—	872	1	no	—
72	C₂	C₂×D₈₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈₇₂
commutator	C₈×C₁₀₉	C₂³
Frattini	C₈	C₂²×D₁₀₉
Fitting	M₃₂×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×D₈
hypercenter	M₃₂	D₁₀₉
hypocenter	C₁₀₉	#509
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #16812721	C₂	semidirect
2. C₂×D₈₇₂	C₂	semidirect
3. #16812644	C₂	semidirect
4. D₁₇₄₄	C₂	semidirect
5. M₃₂×C₁₀₉	C₂	semidirect
8. Dic₄₃₆	C₂²	semidirect
12. D₈₇₂	C₂²	semidirect
14. C₁₆×C₁₀₉	C₂²	semidirect
36. C₁₀₉	#509	semidirect
42. M₃₂	D₁₀₉	semidirect
51. C₁₆	C₂×D₁₀₉	semidirect