Subgroups — C₁₀₉×#2846 #28360100

Order 13952.

3422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#120×C₁₀₉	C₂	1	7	yes	maximal, complemented
3	#379×C₁₀₉	C₂	1	7	yes	maximal, complemented
4	C₂⁵×C₁₀₉	—	2	28	no	—
5	C₂⁵×C₁₀₉	C₂²	1	7	yes	—
6	C₂²×D₄×C₁₀₉	C₂²	1	7	yes	complemented
7	#120×C₁₀₉	—	2	28	no	—
8	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	21	yes	complemented
9	C₂⁴×C₁₀₉	—	2	224	no	—
10	C₂⁴×C₁₀₉	—	2	84	no	—
11	C₂⁴×C₁₀₉	D₄	1	28	yes	complemented
12	C₂⁴×C₁₀₉	C₂³	1	7	yes	—
13	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
14	C₂×D₄×C₁₀₉	—	2	42	no	—
15	C₂²×C₄×C₁₀₉	C₂³	1	7	yes	complemented
16	C₂²⋊C₄×C₁₀₉	—	2	42	no	—
17	C₂³×C₁₀₉	—	2	256	no	—
18	C₂³×C₁₀₉	—	2	224	no	—
19	C₂³×C₁₀₉	—	2	168	no	—
20	C₂³×C₁₀₉	—	2	28	no	—
21	C₂³×C₁₀₉	—	2	7	no	—
22	C₂³×C₁₀₉	C₂×D₄	1	42	yes	—
23	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
24	D₄×C₁₀₉	—	4	28	no	—
25	C₂×C₄×C₁₀₉	—	2	21	no	—
26	C₂²×C₁₀₉	—	2	224	no	—
27	C₂²×C₁₀₉	—	2	84	no	—
28	C₂²×C₁₀₉	—	4	7	no	—
29	C₂²×C₁₀₉	#120	1	28	yes	complemented
30	C₂²×C₁₀₉	C₂²×D₄	1	7	yes	—
31	C₄×C₁₀₉	—	4	7	no	—
32	C₂×C₁₀₉	—	2	28	no	—
33	C₂×C₁₀₉	—	8	1	no	—
34	C₂×C₁₀₉	C₂×#120	1	7	yes	—
35	#2846	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₁₀₉	#2846	1	1	yes	characteristic, direct product factor, Sylow
37	C₂⁶	C₂×C₁₀₉	1	1	yes	characteristic, complemented
38	C₂×#120	C₂×C₁₀₉	1	7	yes	complemented
39	#379	C₂×C₁₀₉	1	7	yes	complemented
40	C₂⁵	—	2	28	no	—
41	C₂⁵	C₂²×C₁₀₉	1	7	yes	—
42	C₂²×D₄	C₂²×C₁₀₉	1	7	yes	complemented
43	#120	—	2	28	no	—
44	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	21	yes	complemented
45	C₂⁴	—	2	224	no	—
46	C₂⁴	—	2	84	no	—
47	C₂⁴	D₄×C₁₀₉	1	28	yes	complemented
48	C₂⁴	C₂³×C₁₀₉	1	7	yes	—
49	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic, complemented
50	C₂×D₄	—	2	42	no	—
51	C₂²×C₄	C₂³×C₁₀₉	1	7	yes	complemented
52	C₂²⋊C₄	—	2	42	no	—
53	C₂³	—	2	256	no	—
54	C₂³	—	2	224	no	—
55	C₂³	—	2	168	no	—
56	C₂³	—	2	28	no	—
57	C₂³	—	2	7	no	—
58	C₂³	C₂×D₄×C₁₀₉	1	42	yes	—
59	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
60	D₄	—	4	28	no	—
61	C₂×C₄	—	2	21	no	—
62	C₂²	—	2	224	no	—
63	C₂²	—	2	84	no	—
64	C₂²	—	4	7	no	—
65	C₂²	#120×C₁₀₉	1	28	yes	complemented
66	C₂²	C₂²×D₄×C₁₀₉	1	7	yes	—
67	C₄	—	4	7	no	—
68	C₂	—	2	28	no	—
69	C₂	—	8	1	no	—
70	C₂	C₂×#120×C₁₀₉	1	7	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₁₀₉	C₂	semidirect
2. C₂×#120×C₁₀₉	C₂	semidirect
3. #379×C₁₀₉	C₂	semidirect
6. C₂²×D₄×C₁₀₉	C₂²	semidirect
8. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
11. C₂⁴×C₁₀₉	D₄	semidirect
13. C₂⁴×C₁₀₉	C₂³	semidirect
15. C₂²×C₄×C₁₀₉	C₂³	semidirect
29. C₂²×C₁₀₉	#120	semidirect
35. #2846	C₁₀₉	direct (N × H)
36. C₁₀₉	#2846	direct (N × H)
37. C₂⁶	C₂×C₁₀₉	semidirect
38. C₂×#120	C₂×C₁₀₉	semidirect
39. #379	C₂×C₁₀₉	semidirect
42. C₂²×D₄	C₂²×C₁₀₉	semidirect
44. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
47. C₂⁴	D₄×C₁₀₉	semidirect
49. C₂⁴	C₂³×C₁₀₉	semidirect
51. C₂²×C₄	C₂³×C₁₀₉	semidirect
65. C₂²	#120×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂⁶×C₁₀₉	2
C₂×#120×C₁₀₉	2
#379×C₁₀₉	2
#2846	109

Subgroups — C109×#2846 #28360100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2846 #28360100