Subgroups — #20294805

Order 6976.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₁₀₉	C₄	1	1	yes	characteristic
5	C₂×#11724784	C₄	1	1	yes	characteristic
6	C₈×D₁₀₉	—	2	1	no	—
7	C₈×D₁₀₉	—	2	1	no	—
8	#13182004	—	2	1	no	—
9	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₀₉	—	2	1	no	—
11	C₂²×D₁₀₉	C₈	1	1	yes	characteristic
12	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₁₀₉	C₈	1	1	yes	characteristic
14	C₄×D₁₀₉	—	2	1	no	—
15	C₄×D₁₀₉	—	2	1	no	—
16	C₈×C₁₀₉	D₄	1	1	yes	characteristic
17	C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	#11724784	—	2	1	no	—
19	C₂²×C₁₀₉	C₂×C₈	1	1	yes	characteristic
20	C₂×D₁₀₉	—	2	1	no	—
21	C₂×D₁₀₉	—	2	1	no	—
22	C₂×D₁₀₉	C₁₆	1	2	yes	complemented
23	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
25	Dic₁₀₉	—	2	1	no	—
26	C₂×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
27	C₂×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
28	C₂×C₁₀₉	M₃₂	1	1	yes	characteristic
29	D₁₀₉	—	2	2	no	—
30	C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₂²⋊C₁₆	—	109	1	no	maximal, Sylow
32	C₂²×C₈	—	109	1	no	—
33	C₂×C₁₆	—	109	1	no	—
34	C₂×C₁₆	D₁₀₉	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	109	1	no	—
36	C₂×C₈	—	109	1	no	—
37	C₂×C₈	—	218	1	no	—
38	C₂×C₈	—	218	1	no	—
39	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	—	218	1	no	—
42	C₂³	—	109	1	no	—
43	C₂×C₄	—	109	1	no	—
44	C₂×C₄	—	218	1	no	—
45	C₂×C₄	—	218	1	no	—
46	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
47	C₈	—	218	1	no	—
48	C₈	D₄₃₆	1	1	yes	characteristic
49	C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
50	C₂²	—	109	2	no	—
51	C₂²	—	218	1	no	—
52	C₂²	—	218	1	no	—
53	C₂²	C₈×D₁₀₉	1	1	yes	characteristic
54	C₄	—	218	1	no	—
55	C₄	#13182006	1	1	yes	characteristic
56	C₄	#13182012	1	1	yes	characteristic
57	C₂	—	218	2	no	—
58	C₂	#16812611	1	1	yes	characteristic
59	C₂	C₁₆×D₁₀₉	1	1	yes	characteristic
60	C₂	#16812638	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₁₆
Fitting	C₂×C₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₈
hypercenter	C₂×C₁₆	D₁₀₉
hypocenter	C₁₀₉	C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#13182004	C₂	semidirect
3. C₂×C₁₆×C₁₀₉	C₂	semidirect
22. C₂×D₁₀₉	C₁₆	semidirect
30. C₁₀₉	C₂²⋊C₁₆	semidirect
34. C₂×C₁₆	D₁₀₉	semidirect