Subgroups — #20295064

Order 6976.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812658	C₂	1	2	yes	maximal
2	#16812633	C₂	1	2	yes	maximal, complemented
3	#16812631	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
6	C₂²⋊Dic₁₀₉	—	2	2	no	—
7	C₄×Dic₁₀₉	—	2	2	no	—
8	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₁₀₉	C₄	1	4	yes	complemented
10	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
13	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	Q₈	1	2	yes	—
16	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₀₉	—	2	4	no	—
18	C₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₁₀₉	—	2	2	no	—
21	C₂²×C₁₀₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
24	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
26	Dic₁₀₉	—	4	4	no	—
27	C₂×C₁₀₉	—	2	2	no	—
28	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₁₀₉	#135	1	2	yes	—
30	C₁₀₉	#427	1	1	yes	characteristic, complemented, Sylow
31	#427	—	109	1	no	maximal, Sylow
32	C₂²×C₈	D₁₀₉	1	1	yes	characteristic
33	#117	—	109	2	no	—
34	#107	—	109	2	no	—
35	#106	—	109	2	no	—
36	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
37	C₂×C₈	Dic₁₀₉	1	4	yes	complemented
38	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
39	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
40	C₄⋊C₄	—	109	4	no	—
41	C₂²⋊C₄	—	218	2	no	—
42	C₄²	—	218	2	no	—
43	C₂³	D₄₃₆	1	1	yes	characteristic
44	C₂×C₄	—	218	4	no	—
45	C₂×C₄	Dic₂₁₈	1	2	yes	—
46	C₂×C₄	D₄₃₆	1	1	yes	characteristic
47	C₂×C₄	C₂×Dic₁₀₉	1	2	yes	—
48	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
49	C₈	C₂×Dic₁₀₉	1	2	yes	—
50	C₈	C₂×Dic₁₀₉	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊Dic₁₀₉	1	2	yes	—
53	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
54	C₄	—	436	4	no	—
55	C₄	C₄⋊Dic₁₀₉	1	2	yes	—
56	C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
57	C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
60	C₂	#16812721	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄₃₆
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂²×C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂²×C₈	D₁₀₉
hypocenter	C₁₀₉	#427
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #16812633	C₂	semidirect
3. #16812631	C₂	semidirect
9. C₂×C₈×C₁₀₉	C₄	semidirect
30. C₁₀₉	#427	semidirect
37. C₂×C₈	Dic₁₀₉	semidirect