Subgroups — C₂×#126×C₁₀₉ #20294686

Order 6976.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#126×C₁₀₉	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊C₄×C₁₀₉	C₂	1	3	yes	maximal, complemented
3	C₂×C₂²⋊C₄×C₁₀₉	C₂	1	3	yes	maximal
4	C₂×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₁₀₉	C₂²	1	3	yes	—
7	C₂²×C₄×C₁₀₉	C₂²	1	3	yes	—
8	C₄⋊C₄×C₁₀₉	C₂²	1	12	yes	complemented
9	C₂²⋊C₄×C₁₀₉	C₂²	1	12	yes	—
10	C₄²×C₁₀₉	C₂²	1	4	yes	complemented
11	C₂³×C₁₀₉	—	2	6	no	—
12	C₂³×C₁₀₉	C₂³	1	2	yes	—
13	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	—	2	6	no	—
15	C₂×C₄×C₁₀₉	—	2	6	no	—
16	C₂×C₄×C₁₀₉	C₂³	1	6	yes	—
17	C₂×C₄×C₁₀₉	C₂³	1	6	yes	—
18	C₂²×C₁₀₉	—	2	6	no	—
19	C₂²×C₁₀₉	—	4	4	no	—
20	C₂²×C₁₀₉	#41	1	6	yes	—
21	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
22	C₄×C₁₀₉	—	2	6	no	—
23	C₄×C₁₀₉	—	2	6	no	—
24	C₂×C₁₀₉	—	4	2	no	—
25	C₂×C₁₀₉	#126	1	4	yes	direct product factor
26	C₂×C₁₀₉	C₂×#41	1	3	yes	—
27	C₁₀₉	C₂×#126	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×#126	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#126	C₂×C₁₀₉	1	8	yes	direct product factor
30	C₂×C₄⋊C₄	C₂×C₁₀₉	1	3	yes	complemented
31	C₂×C₂²⋊C₄	C₂×C₁₀₉	1	3	yes	—
32	C₂×C₄²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
33	C₂⁴	C₂²×C₁₀₉	1	1	yes	characteristic
34	C₂²×C₄	C₂²×C₁₀₉	1	3	yes	—
35	C₂²×C₄	C₂²×C₁₀₉	1	3	yes	—
36	C₄⋊C₄	C₂²×C₁₀₉	1	12	yes	complemented
37	C₂²⋊C₄	C₂²×C₁₀₉	1	12	yes	—
38	C₄²	C₂²×C₁₀₉	1	4	yes	complemented
39	C₂³	—	2	6	no	—
40	C₂³	C₂³×C₁₀₉	1	2	yes	—
41	C₂³	C₂³×C₁₀₉	1	1	yes	characteristic
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	—	2	6	no	—
44	C₂×C₄	C₂³×C₁₀₉	1	6	yes	—
45	C₂×C₄	C₂³×C₁₀₉	1	6	yes	—
46	C₂²	—	2	6	no	—
47	C₂²	—	4	4	no	—
48	C₂²	#41×C₁₀₉	1	6	yes	—
49	C₂²	C₂⁴×C₁₀₉	1	1	yes	characteristic
50	C₄	—	2	6	no	—
51	C₄	—	2	6	no	—
52	C₂	—	4	2	no	—
53	C₂	#126×C₁₀₉	1	4	yes	direct product factor
54	C₂	C₂×#41×C₁₀₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #126×C₁₀₉	C₂	direct (N × H)
2. C₂×C₄⋊C₄×C₁₀₉	C₂	semidirect
4. C₂×C₄²×C₁₀₉	C₂	semidirect
8. C₄⋊C₄×C₁₀₉	C₂²	semidirect
10. C₄²×C₁₀₉	C₂²	semidirect
25. C₂×C₁₀₉	#126	direct (N × H)
27. C₁₀₉	C₂×#126	direct (N × H)
28. C₂×#126	C₁₀₉	direct (N × H)
29. #126	C₂×C₁₀₉	direct (N × H)
30. C₂×C₄⋊C₄	C₂×C₁₀₉	semidirect
32. C₂×C₄²	C₂×C₁₀₉	semidirect
36. C₄⋊C₄	C₂²×C₁₀₉	semidirect
38. C₄²	C₂²×C₁₀₉	semidirect
53. C₂	#126×C₁₀₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#126×C₁₀₉	2
C₂×C₄⋊C₄×C₁₀₉	2
C₂×C₂²⋊C₄×C₁₀₉	2
C₂×C₄²×C₁₀₉	2
C₂×#126	109

Subgroups — C2×#126×C109 #20294686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#126×C₁₀₉ #20294686