Subgroups — #20295468

Order 6976.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812722	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812645	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812644	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812642	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812638	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	—	2	1	no	—
9	Q₈×D₁₀₉	—	2	1	no	—
10	Dic₄₃₆	C₂²	1	1	yes	characteristic
11	#13182019	—	2	1	no	—
12	#13182016	—	2	1	no	—
13	D₈₇₂	C₂²	1	1	yes	characteristic
14	C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	#13182004	C₂²	1	1	yes	characteristic
16	Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
17	Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	Q₈×C₁₀₉	—	2	1	no	—
20	Q₈×C₁₀₉	—	2	1	no	—
21	D₄₃₆	—	2	1	no	—
22	D₄₃₆	—	4	1	no	—
23	C₄×D₁₀₉	—	4	1	no	—
24	C₄×D₁₀₉	—	4	1	no	—
25	C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	Dic₂₁₈	—	2	1	no	—
27	Dic₂₁₈	—	4	1	no	—
28	C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	#11724784	D₄	1	1	yes	characteristic
30	C₂×D₁₀₉	—	4	1	no	—
31	C₂×D₁₀₉	D₈	1	1	yes	characteristic
32	C₄×C₁₀₉	—	4	1	no	—
33	C₄×C₁₀₉	—	4	1	no	—
34	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	Dic₁₀₉	—	4	1	no	—
36	Dic₁₀₉	D₈	1	1	yes	characteristic
37	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
38	D₁₀₉	—	2	1	no	—
39	D₁₀₉	—	8	1	no	—
40	C₁₀₉	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	109	1	no	maximal, Sylow
42	#135	—	109	1	no	—
43	C₂×Q₁₆	—	109	1	no	—
44	Q₃₂	—	109	1	no	—
45	Q₃₂	D₁₀₉	1	1	yes	characteristic, complemented
46	SD₃₂	—	109	1	no	—
47	SD₃₂	—	109	1	no	—
48	M₃₂	—	109	1	no	—
49	#41	—	218	1	no	—
50	C₂×Q₈	—	218	1	no	—
51	Q₁₆	—	109	1	no	—
52	Q₁₆	—	218	1	no	—
53	Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
54	Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
55	SD₁₆	—	218	1	no	—
56	D₈	—	109	1	no	—
57	C₂×C₈	—	109	1	no	—
58	C₁₆	—	109	1	no	—
59	C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	436	1	no	—
63	Q₈	—	218	1	no	—
64	D₄	—	436	1	no	—
65	D₄	—	218	1	no	—
66	C₂×C₄	—	109	1	no	—
67	C₂×C₄	—	436	1	no	—
68	C₂×C₄	—	436	1	no	—
69	C₈	—	109	1	no	—
70	C₈	C₂²×D₁₀₉	1	1	yes	characteristic
71	C₂²	—	109	1	no	—
72	C₂²	—	436	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	109	1	no	—
76	C₄	—	436	1	no	—
77	C₄	D₄×D₁₀₉	1	1	yes	characteristic
78	C₂	—	872	1	no	—
79	C₂	—	218	1	no	—
80	C₂	D₈×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₀₉
commutator	C₈×C₁₀₉	C₂³
Frattini	C₈	C₂²×D₁₀₉
Fitting	Q₃₂×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×D₈
hypercenter	Q₃₂	D₁₀₉
hypocenter	C₁₀₉	#510
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₁₀₉	C₂	semidirect
3. #16812645	C₂	semidirect
4. #16812644	C₂	semidirect
5. #16812642	C₂	semidirect
6. #16812638	C₂	semidirect
7. Q₃₂×C₁₀₉	C₂	semidirect
40. C₁₀₉	#510	semidirect
45. Q₃₂	D₁₀₉	semidirect