Subgroups — C₁₀₉×#12516 #42130510

Order 27904.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2247	C₂	1	4	yes	maximal
2	C₁₀₉×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₁₀₉	—	2	8	no	—
5	#500×C₁₀₉	—	2	4	no	—
6	#494×C₁₀₉	C₂²	1	4	yes	—
7	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×Q₁₆×C₁₀₉	—	2	8	no	—
10	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
11	Q₃₂×C₁₀₉	—	4	16	no	—
12	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
13	#107×C₁₀₉	—	4	8	no	—
14	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
15	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₉	—	4	4	no	—
17	Q₁₆×C₁₀₉	—	4	16	no	—
18	C₂×C₈×C₁₀₉	D₈	1	4	yes	—
19	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
20	C₄⋊C₄×C₁₀₉	—	4	8	no	—
21	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
22	C₁₆×C₁₀₉	D₈	1	8	yes	—
23	Q₈×C₁₀₉	—	8	8	no	—
24	C₂×C₄×C₁₀₉	—	8	4	no	—
25	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
27	C₈×C₁₀₉	Q₃₂	1	8	yes	complemented
28	C₈×C₁₀₉	C₂×D₈	1	4	yes	—
29	C₂²×C₁₀₉	#493	1	1	yes	characteristic
30	C₄×C₁₀₉	—	16	4	no	—
31	C₄×C₁₀₉	#493	1	2	yes	—
32	C₄×C₁₀₉	C₂×Q₃₂	1	4	yes	—
33	#12516	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₁₀₉	#1712	1	1	yes	characteristic
35	C₂×C₁₀₉	#2247	1	2	yes	—
36	#2247	C₂×C₁₀₉	1	4	yes	—
37	#1713	C₂×C₁₀₉	1	2	yes	—
38	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
39	C₁₀₉	#12516	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×Q₃₂	—	2	8	no	—
41	#500	—	2	4	no	—
42	#494	C₂²×C₁₀₉	1	4	yes	—
43	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
44	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	8	no	—
46	C₄⋊Q₈	—	2	4	no	—
47	Q₃₂	—	4	16	no	—
48	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
49	#107	—	4	8	no	—
50	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
51	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
52	C₂×Q₈	—	4	4	no	—
53	Q₁₆	—	4	16	no	—
54	C₂×C₈	D₈×C₁₀₉	1	4	yes	—
55	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
56	C₄⋊C₄	—	4	8	no	—
57	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
58	C₁₆	D₈×C₁₀₉	1	8	yes	—
59	Q₈	—	8	8	no	—
60	C₂×C₄	—	8	4	no	—
61	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
62	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
63	C₈	Q₃₂×C₁₀₉	1	8	yes	complemented
64	C₈	C₂×D₈×C₁₀₉	1	4	yes	—
65	C₂²	#493×C₁₀₉	1	1	yes	characteristic
66	C₄	—	16	4	no	—
67	C₄	#493×C₁₀₉	1	2	yes	—
68	C₄	C₂×Q₃₂×C₁₀₉	1	4	yes	—
69	C₂	C₁₀₉×#1712	1	1	yes	characteristic
70	C₂	C₁₀₉×#2247	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₈×C₁₀₉	Q₃₂	semidirect
33. #12516	C₁₀₉	direct (N × H)
39. C₁₀₉	#12516	direct (N × H)
63. C₈	Q₃₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2247	2
C₁₀₉×#1713	2
C₈×C₁₆×C₁₀₉	2
#12516	109

Subgroups — C109×#12516 #42130510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12516 #42130510