Subgroups — #20294795

Order 6976.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182013	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#13182013	—	2	2	no	—
4	#13182013	C₄	1	4	yes	complemented
5	C₂×#11724784	—	2	2	no	—
6	C₂²×C₄×C₁₀₉	C₄	1	2	yes	—
7	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₁₀₉	—	2	2	no	—
9	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₀₉	—	2	2	no	—
11	C₂×C₄×C₁₀₉	—	4	2	no	—
12	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
15	C₂×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
16	#11724784	—	2	4	no	—
17	C₂²×C₁₀₉	—	2	1	no	—
18	C₂²×C₁₀₉	—	2	1	no	—
19	C₂²×C₁₀₉	C₄²	1	1	yes	characteristic
20	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₁₀₉	—	4	2	no	—
23	C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
24	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₁₀₉	—	2	1	no	—
26	C₂×C₁₀₉	—	2	1	no	—
27	C₂×C₁₀₉	#95	1	1	yes	characteristic
28	C₂×C₁₀₉	#100	1	1	yes	characteristic
29	C₂×C₁₀₉	#101	1	1	yes	characteristic
30	C₁₀₉	#343	1	1	yes	characteristic, complemented, Sylow
31	#343	—	109	1	no	maximal, Sylow
32	C₂×M₁₆	—	109	2	no	—
33	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
34	C₂²×C₄	Dic₁₀₉	1	2	yes	—
35	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
36	M₁₆	—	109	4	no	—
37	M₁₆	—	218	2	no	—
38	C₂×C₈	—	218	2	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₂³	C₂×Dic₁₀₉	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	Dic₂₁₈	1	1	yes	characteristic
44	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
45	C₂×C₄	D₄₃₆	1	1	yes	characteristic
46	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
47	C₈	—	218	4	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₄×Dic₁₀₉	1	1	yes	characteristic
51	C₂²	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₁₀₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#13182006	1	2	yes	—
55	C₄	#13182008	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#16812606	1	1	yes	characteristic
59	C₂	#16812616	1	1	yes	characteristic
60	C₂	#16812618	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₁₀₉
commutator	C₂²×C₁₀₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₀₉
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#343
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #13182013	C₄	semidirect
30. C₁₀₉	#343	semidirect