Subgroups — #20295138

Order 6976.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
2	#16812631	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812620	C₂	1	2	yes	maximal, complemented
4	#16812609	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₉	—	2	2	no	—
7	C₂×#11724784	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	C₂²⋊Dic₁₀₉	—	2	2	no	—
10	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₁₀₉	C₂²	1	2	yes	—
12	D₈×C₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₁₀₉	—	2	2	no	—
15	D₄×C₁₀₉	—	2	2	no	—
16	D₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
17	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₀₉	—	2	2	no	—
19	C₂×Dic₁₀₉	—	4	4	no	—
20	C₂×Dic₁₀₉	D₄	1	2	yes	—
21	C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
22	#11724784	—	2	1	no	—
23	C₂²×C₁₀₉	—	2	4	no	—
24	C₂²×C₁₀₉	—	4	2	no	—
25	C₂²×C₁₀₉	—	4	2	no	—
26	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₁₀₉	#41	1	1	yes	characteristic
29	Dic₁₀₉	—	2	2	no	—
30	Dic₁₀₉	—	4	2	no	—
31	C₂×C₁₀₉	—	4	4	no	—
32	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
33	C₂×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
34	C₁₀₉	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	109	1	no	maximal, Sylow
36	C₂×D₈	D₁₀₉	1	1	yes	characteristic
37	C₄×D₄	—	109	2	no	—
38	#106	—	109	1	no	—
39	#102	—	109	2	no	—
40	#97	—	109	1	no	—
41	C₂×D₄	C₂×D₁₀₉	1	2	yes	—
42	C₂²×C₄	—	218	2	no	—
43	D₈	Dic₁₀₉	1	4	yes	complemented
44	C₂×C₈	—	109	1	no	—
45	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
46	C₄⋊C₄	—	109	2	no	—
47	C₂²⋊C₄	—	218	2	no	—
48	C₄²	—	109	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₁₀₉	1	4	yes	complemented
52	C₂×C₄	—	109	2	no	—
53	C₂×C₄	—	436	4	no	—
54	C₂×C₄	—	218	2	no	—
55	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
56	C₈	—	218	1	no	—
57	C₈	C₂×Dic₁₀₉	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
62	C₄	—	436	2	no	—
63	C₄	—	218	2	no	—
64	C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
65	C₄	#13182028	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	D₄×Dic₁₀₉	1	1	yes	characteristic
68	C₂	#16812723	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	C₂×D₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	C₂×D₈	D₁₀₉
hypocenter	C₁₀₉	#442
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₁₀₉	C₂	semidirect
2. #16812631	C₂	semidirect
3. #16812620	C₂	semidirect
4. #16812609	C₂	semidirect
12. D₈×C₁₀₉	C₄	semidirect
16. D₄×C₁₀₉	C₂×C₄	semidirect
21. C₈×C₁₀₉	C₂×C₄	semidirect
34. C₁₀₉	#442	semidirect
43. D₈	Dic₁₀₉	semidirect
51. D₄	C₂×Dic₁₀₉	semidirect
57. C₈	C₂×Dic₁₀₉	semidirect