Subgroups — C₁₀₉×#12523 #42130517

Order 27904.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2254	C₂	1	4	yes	maximal
2	C₁₀₉×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₀₉	—	2	2	no	—
5	#500×C₁₀₉	C₂²	1	4	yes	—
6	#494×C₁₀₉	—	2	2	no	—
7	#367×C₁₀₉	—	2	8	no	—
8	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₀₉	—	4	4	no	—
11	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
12	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
13	#107×C₁₀₉	—	2	8	no	—
14	#107×C₁₀₉	—	4	4	no	—
15	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
16	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₁₀₉	—	4	4	no	—
18	Q₁₆×C₁₀₉	—	8	8	no	—
19	C₂×C₈×C₁₀₉	D₈	1	4	yes	—
20	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
21	C₄⋊C₄×C₁₀₉	—	4	4	no	—
22	C₄⋊C₄×C₁₀₉	—	4	4	no	—
23	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
24	C₁₆×C₁₀₉	Q₁₆	1	8	yes	complemented
25	Q₈×C₁₀₉	—	8	8	no	—
26	C₂×C₄×C₁₀₉	—	8	4	no	—
27	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
29	C₈×C₁₀₉	SD₃₂	1	8	yes	—
30	C₈×C₁₀₉	C₂×Q₁₆	1	4	yes	—
31	C₂²×C₁₀₉	#493	1	1	yes	characteristic
32	C₄×C₁₀₉	—	16	4	no	—
33	C₄×C₁₀₉	#494	1	2	yes	—
34	C₄×C₁₀₉	C₂×SD₃₂	1	4	yes	—
35	#12523	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₁₀₉	#1713	1	1	yes	characteristic
37	C₂×C₁₀₉	#2248	1	2	yes	—
38	#2254	C₂×C₁₀₉	1	4	yes	—
39	#1713	C₂×C₁₀₉	1	2	yes	—
40	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
41	C₁₀₉	#12523	1	1	yes	characteristic, direct product factor, Sylow
42	#500	—	2	2	no	—
43	#500	C₂²×C₁₀₉	1	4	yes	—
44	#494	—	2	2	no	—
45	#367	—	2	8	no	—
46	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
47	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
48	C₂×Q₁₆	—	4	4	no	—
49	C₄⋊Q₈	—	2	4	no	—
50	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
51	#107	—	2	8	no	—
52	#107	—	4	4	no	—
53	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
54	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
55	C₂×Q₈	—	4	4	no	—
56	Q₁₆	—	8	8	no	—
57	C₂×C₈	D₈×C₁₀₉	1	4	yes	—
58	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
59	C₄⋊C₄	—	4	4	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
62	C₁₆	Q₁₆×C₁₀₉	1	8	yes	complemented
63	Q₈	—	8	8	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
66	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
67	C₈	SD₃₂×C₁₀₉	1	8	yes	—
68	C₈	C₂×Q₁₆×C₁₀₉	1	4	yes	—
69	C₂²	#493×C₁₀₉	1	1	yes	characteristic
70	C₄	—	16	4	no	—
71	C₄	#494×C₁₀₉	1	2	yes	—
72	C₄	C₂×SD₃₂×C₁₀₉	1	4	yes	—
73	C₂	C₁₀₉×#1713	1	1	yes	characteristic
74	C₂	C₁₀₉×#2248	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
24. C₁₆×C₁₀₉	Q₁₆	semidirect
35. #12523	C₁₀₉	direct (N × H)
41. C₁₀₉	#12523	direct (N × H)
62. C₁₆	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2254	2
C₁₀₉×#1713	2
C₈×C₁₆×C₁₀₉	2
#12523	109

Subgroups — C109×#12523 #42130517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12523 #42130517