Subgroups — C₁₀₉×#8135 #42126129

Order 27904.

382 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#1744	C₂	1	3	yes	maximal
2	C₁₀₉×#1273	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₁₀₉	C₂²	1	6	yes	—
4	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
5	#336×C₁₀₉	C₄	1	12	yes	—
6	C₂²×C₈×C₁₀₉	C₂×C₄	1	12	yes	—
7	C₂×C₄²×C₁₀₉	C₂×C₄	1	6	yes	—
8	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
9	C₄×C₈×C₁₀₉	—	2	12	no	—
10	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
11	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
12	C₂×C₈×C₁₀₉	—	2	12	no	—
13	C₂×C₈×C₁₀₉	—	2	24	no	—
14	C₄²×C₁₀₉	—	2	12	no	—
15	C₄²×C₁₀₉	#41	1	4	yes	—
16	C₂³×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	—	2	12	no	—
18	C₂×C₄×C₁₀₉	#117	1	6	yes	—
19	C₂×C₄×C₁₀₉	#131	1	12	yes	—
20	C₈×C₁₀₉	—	4	12	no	—
21	C₂²×C₁₀₉	#376	1	1	yes	characteristic
22	C₂²×C₁₀₉	#405	1	6	yes	—
23	C₄×C₁₀₉	—	4	4	no	—
24	C₄×C₁₀₉	#433	1	12	yes	—
25	#8135	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₁₀₉	#1454	1	4	yes	—
27	C₂×C₁₀₉	#1746	1	3	yes	—
28	#1744	C₂×C₁₀₉	1	3	yes	—
29	#1273	C₂×C₁₀₉	1	4	yes	—
30	C₁₀₉	#8135	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₄×C₈	C₂²×C₁₀₉	1	6	yes	—
32	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
33	#336	C₄×C₁₀₉	1	12	yes	—
34	C₂²×C₈	C₂×C₄×C₁₀₉	1	12	yes	—
35	C₂×C₄²	C₂×C₄×C₁₀₉	1	6	yes	—
36	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
37	C₄×C₈	—	2	12	no	—
38	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
39	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
40	C₂×C₈	—	2	12	no	—
41	C₂×C₈	—	2	24	no	—
42	C₄²	—	2	12	no	—
43	C₄²	#41×C₁₀₉	1	4	yes	—
44	C₂³	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
45	C₂×C₄	—	2	12	no	—
46	C₂×C₄	#117×C₁₀₉	1	6	yes	—
47	C₂×C₄	#131×C₁₀₉	1	12	yes	—
48	C₈	—	4	12	no	—
49	C₂²	#376×C₁₀₉	1	1	yes	characteristic
50	C₂²	#405×C₁₀₉	1	6	yes	—
51	C₄	—	4	4	no	—
52	C₄	#433×C₁₀₉	1	12	yes	—
53	C₂	C₁₀₉×#1454	1	4	yes	—
54	C₂	C₁₀₉×#1746	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₀₉	C₂³
commutator	C₂³	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
25. #8135	C₁₀₉	direct (N × H)
30. C₁₀₉	#8135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#1744	2
C₁₀₉×#1273	2
#8135	109

Subgroups — C109×#8135 #42126129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8135 #42126129