Subgroups — #20296066

Order 6976.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#89137	C₂	1	2	yes	maximal
2	#16812767	C₂	1	1	yes	maximal, characteristic
3	#16812762	C₂	1	2	yes	maximal
4	#16812753	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊#89137	—	2	2	no	—
8	C₄⋊#89137	C₂²	1	2	yes	—
9	C₄×#89137	—	2	2	no	—
10	C₄×#89137	C₂²	1	1	yes	characteristic
11	Q₈×D₁₀₉	C₂²	1	2	yes	—
12	Dic₄₃₆	C₄	1	1	yes	characteristic, complemented
13	#13182019	C₄	1	2	yes	complemented
14	C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#89137	—	2	2	no	—
17	C₂×#89137	D₄	1	2	yes	—
18	Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₄×D₁₀₉	—	2	2	no	—
20	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
21	Dic₂₁₈	—	2	2	no	—
22	Dic₂₁₈	C₂×C₄	1	2	yes	—
23	#11724786	—	2	1	no	—
24	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	#11724784	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
27	#89137	—	2	2	no	—
28	#89137	—	4	2	no	—
29	C₄×C₁₀₉	—	2	2	no	—
30	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₀₉	—	2	2	no	—
32	Dic₁₀₉	#41	1	1	yes	characteristic
33	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
34	D₁₀₉	#137	1	2	yes	—
35	C₁₀₉	#441	1	1	yes	characteristic, complemented, Sylow
36	#441	—	109	1	no	maximal, Sylow
37	C₂×Q₁₆	—	109	1	no	—
38	C₄×Q₈	—	109	2	no	—
39	#106	—	109	1	no	—
40	#103	—	109	2	no	—
41	#97	—	109	1	no	—
42	C₂×Q₈	—	109	2	no	—
43	Q₁₆	—	109	2	no	—
44	Q₁₆	—	109	1	no	—
45	Q₁₆	#89137	1	1	yes	characteristic, complemented
46	C₂×C₈	—	109	1	no	—
47	C₂×C₈	—	109	1	no	—
48	C₄⋊C₄	—	109	2	no	—
49	C₄⋊C₄	—	218	2	no	—
50	C₄²	—	109	1	no	—
51	C₄²	—	218	2	no	—
52	Q₈	—	109	2	no	—
53	Q₈	—	218	2	no	—
54	Q₈	C₂×#89137	1	2	yes	—
55	C₂×C₄	—	109	1	no	—
56	C₂×C₄	—	109	2	no	—
57	C₂×C₄	—	218	2	no	—
58	C₂×C₄	—	218	2	no	—
59	C₈	—	109	1	no	—
60	C₈	—	218	1	no	—
61	C₈	C₂×#89137	1	1	yes	characteristic
62	C₂²	—	109	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	109	1	no	—
65	C₄	—	436	2	no	—
66	C₄	—	218	2	no	—
67	C₄	—	218	2	no	—
68	C₄	C₂²×#89137	1	1	yes	characteristic
69	C₂	—	109	2	no	—
70	C₂	D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₄	C₂²×#89137
Fitting	Q₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₄
hypercenter	Q₁₆	#89137
hypocenter	C₁₀₉	#441
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Dic₄₃₆	C₄	semidirect
13. #13182019	C₄	semidirect
15. Q₁₆×C₁₀₉	C₄	semidirect
35. C₁₀₉	#441	semidirect
45. Q₁₆	#89137	semidirect