Subgroups — C₁₀₉×#2840 #28360094

Order 13952.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#393×C₁₀₉	C₂	1	12	yes	maximal, complemented
2	#379×C₁₀₉	C₂	1	3	yes	maximal
3	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
4	C₂²×Q₈×C₁₀₉	C₂²	1	4	yes	complemented
5	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	18	yes	—
6	#95×C₁₀₉	C₂²	1	12	yes	complemented
7	C₂⁴×C₁₀₉	—	4	6	no	—
8	C₂⁴×C₁₀₉	Q₈	1	4	yes	complemented
9	C₂⁴×C₁₀₉	C₂³	1	3	yes	—
10	C₂×Q₈×C₁₀₉	—	4	12	no	—
11	C₂²×C₄×C₁₀₉	C₂³	1	12	yes	—
12	C₂²⋊C₄×C₁₀₉	—	4	18	no	—
13	C₂³×C₁₀₉	—	2	18	no	—
14	C₂³×C₁₀₉	—	4	12	no	—
15	C₂³×C₁₀₉	—	4	16	no	—
16	C₂³×C₁₀₉	C₂×Q₈	1	6	yes	—
17	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
18	Q₈×C₁₀₉	—	4	16	no	—
19	C₂×C₄×C₁₀₉	—	2	36	no	—
20	C₂²×C₁₀₉	—	2	18	no	—
21	C₂²×C₁₀₉	—	4	16	no	—
22	C₂²×C₁₀₉	—	4	12	no	—
23	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
24	C₂²×C₁₀₉	H(5,𝔽₂)	1	6	yes	—
25	C₄×C₁₀₉	—	4	12	no	—
26	C₂×C₁₀₉	—	4	6	no	—
27	C₂×C₁₀₉	#543	1	3	yes	—
28	C₂×C₁₀₉	H(3,𝔽₄)	1	4	yes	—
29	#2840	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₁₀₉	#2840	1	1	yes	characteristic, direct product factor, Sylow
31	#393	C₂×C₁₀₉	1	12	yes	complemented
32	#379	C₂×C₁₀₉	1	3	yes	—
33	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
34	C₂²×Q₈	C₂²×C₁₀₉	1	4	yes	complemented
35	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	18	yes	—
36	#95	C₂²×C₁₀₉	1	12	yes	complemented
37	C₂⁴	—	4	6	no	—
38	C₂⁴	Q₈×C₁₀₉	1	4	yes	complemented
39	C₂⁴	C₂³×C₁₀₉	1	3	yes	—
40	C₂×Q₈	—	4	12	no	—
41	C₂²×C₄	C₂³×C₁₀₉	1	12	yes	—
42	C₂²⋊C₄	—	4	18	no	—
43	C₂³	—	2	18	no	—
44	C₂³	—	4	12	no	—
45	C₂³	—	4	16	no	—
46	C₂³	C₂×Q₈×C₁₀₉	1	6	yes	—
47	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
48	Q₈	—	4	16	no	—
49	C₂×C₄	—	2	36	no	—
50	C₂²	—	2	18	no	—
51	C₂²	—	4	16	no	—
52	C₂²	—	4	12	no	—
53	C₂²	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
54	C₂²	H(5,𝔽₂)×C₁₀₉	1	6	yes	—
55	C₄	—	4	12	no	—
56	C₂	—	4	6	no	—
57	C₂	#543×C₁₀₉	1	3	yes	—
58	C₂	H(3,𝔽₄)×C₁₀₉	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂⁵×C₁₀₉	C₂²
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. #393×C₁₀₉	C₂	semidirect
4. C₂²×Q₈×C₁₀₉	C₂²	semidirect
6. #95×C₁₀₉	C₂²	semidirect
8. C₂⁴×C₁₀₉	Q₈	semidirect
29. #2840	C₁₀₉	direct (N × H)
30. C₁₀₉	#2840	direct (N × H)
31. #393	C₂×C₁₀₉	semidirect
34. C₂²×Q₈	C₂²×C₁₀₉	semidirect
36. #95	C₂²×C₁₀₉	semidirect
38. C₂⁴	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#393×C₁₀₉	2
#379×C₁₀₉	2
#2840	109

Subgroups — C109×#2840 #28360094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2840 #28360094