Subgroups — #20294811

Order 6976.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812705	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182004	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#13182029	C₄	1	1	yes	characteristic
5	#13182013	C₄	1	1	yes	characteristic
6	#13182012	—	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	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂²⋊D₁₀₉	—	2	1	no	—
13	D₄₃₆	C₈	1	1	yes	characteristic
14	C₄×D₁₀₉	—	2	1	no	—
15	Dic₂₁₈	C₈	1	1	yes	characteristic
16	C₈×C₁₀₉	D₄	1	1	yes	characteristic
17	C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	#11724784	—	2	1	no	—
19	C₂²×C₁₀₉	M₁₆	1	1	yes	characteristic
20	C₂×D₁₀₉	—	2	1	no	—
21	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
23	Dic₁₀₉	—	2	1	no	—
24	C₂×C₁₀₉	—	2	1	no	—
25	C₂×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
26	D₁₀₉	—	4	1	no	—
27	C₁₀₉	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	109	1	no	maximal, Sylow
29	#131	—	109	1	no	—
30	M₃₂	D₁₀₉	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	109	1	no	—
32	#41	—	109	1	no	—
33	M₁₆	—	109	1	no	—
34	M₁₆	—	218	1	no	—
35	C₂×C₈	—	218	1	no	—
36	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	218	1	no	—
39	Q₈	—	109	1	no	—
40	D₄	—	109	1	no	—
41	D₄	—	218	1	no	—
42	C₂×C₄	—	218	1	no	—
43	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
44	C₈	—	218	1	no	—
45	C₈	D₄₃₆	1	1	yes	characteristic
46	C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
47	C₂²	—	218	1	no	—
48	C₂²	#13182012	1	1	yes	characteristic
49	C₄	—	218	1	no	—
50	C₄	#13182006	1	1	yes	characteristic
51	C₄	C₈×D₁₀₉	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	436	1	no	—
54	C₂	#16812611	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₀₉
Fitting	M₃₂×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²⋊C₈
hypercenter	M₃₂	D₁₀₉
hypocenter	C₁₀₉	#350
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#13182004	C₂	semidirect
3. M₃₂×C₁₀₉	C₂	semidirect
27. C₁₀₉	#350	semidirect
30. M₃₂	D₁₀₉	semidirect