Subgroups — #20295876

Order 6976.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
2	C₄×#11724786	C₂	1	1	yes	maximal, characteristic
3	#16812664	C₂	1	1	yes	maximal, characteristic
4	C₂×#11724786	—	2	1	no	—
5	C₂×#11724786	—	2	1	no	—
6	C₂×Dic₂₁₈	C₄	1	1	yes	characteristic
7	#13182008	—	2	1	no	—
8	C₄×Dic₁₀₉	—	2	1	no	—
9	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₀₉	—	2	1	no	—
12	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₁₀₉	—	2	1	no	—
14	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
15	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
16	Dic₂₁₈	—	2	1	no	—
17	Dic₂₁₈	C₈	1	2	yes	complemented
18	#11724786	—	2	2	no	—
19	#11724786	—	4	1	no	—
20	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₁₀₉	—	4	1	no	—
22	C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
23	C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
24	Dic₁₀₉	—	2	2	no	—
25	Dic₁₀₉	—	2	1	no	—
26	Dic₁₀₉	SD₁₆	1	1	yes	characteristic
27	Dic₁₀₉	Q₁₆	1	1	yes	characteristic
28	C₂×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₁₀₉	#103	1	1	yes	characteristic
30	C₂×C₁₀₉	#104	1	1	yes	characteristic
31	C₁₀₉	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	109	1	no	maximal, Sylow
33	C₄×Q₈	—	109	1	no	—
34	C₄⋊C₈	—	109	1	no	—
35	C₄×C₈	—	109	1	no	—
36	C₂×Q₈	—	109	1	no	—
37	C₂×C₈	—	218	1	no	—
38	C₂×C₈	—	218	1	no	—
39	C₄⋊C₄	—	218	1	no	—
40	C₄⋊C₄	#89137	1	1	yes	characteristic
41	C₄²	—	109	1	no	—
42	C₄²	—	218	1	no	—
43	Q₈	—	109	2	no	—
44	Q₈	—	218	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	109	1	no	—
47	C₂×C₄	—	109	1	no	—
48	C₂×C₄	—	218	1	no	—
49	C₂×C₄	C₂×#89137	1	1	yes	characteristic
50	C₈	—	436	1	no	—
51	C₈	—	218	2	no	—
52	C₂²	C₂²⋊#89137	1	1	yes	characteristic
53	C₄	—	4	1	no	—
54	C₄	—	109	1	no	—
55	C₄	—	109	1	no	—
56	C₄	—	218	2	no	—
57	C₄	—	218	1	no	—
58	C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
59	C₄	#13182039	1	1	yes	characteristic
60	C₂	#16812756	1	1	yes	characteristic
61	C₂	#16812762	1	1	yes	characteristic
62	C₂	#16812763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₈
Frattini	C₂×C₄	C₂×#89137
Fitting	C₄⋊C₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²⋊C₄
hypercenter	C₄⋊C₄	#89137
hypocenter	C₁₀₉	Q₈⋊C₈
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. Dic₂₁₈	C₈	semidirect
31. C₁₀₉	Q₈⋊C₈	semidirect