Subgroups — C₁₀₉×#12522 #42130516

Order 27904.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2254	C₂	1	2	yes	maximal
2	C₁₀₉×#2252	C₂	1	2	yes	maximal
3	C₁₀₉×#1710	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₀₉	C₂²	1	4	yes	—
6	#487×C₁₀₉	—	2	4	no	—
7	#367×C₁₀₉	—	2	4	no	—
8	#366×C₁₀₉	—	2	4	no	—
9	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
13	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
14	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
15	#107×C₁₀₉	—	2	4	no	—
16	#107×C₁₀₉	—	2	4	no	—
17	#103×C₁₀₉	—	4	8	no	—
18	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
19	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₁₀₉	—	4	4	no	—
21	C₂×C₈×C₁₀₉	D₈	1	4	yes	—
22	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₉	—	4	4	no	—
25	C₄⋊C₄×C₁₀₉	—	4	4	no	—
26	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₁₀₉	Q₁₆	1	4	yes	—
28	C₁₆×C₁₀₉	Q₁₆	1	4	yes	—
29	Q₈×C₁₀₉	—	8	8	no	—
30	C₂×C₄×C₁₀₉	—	8	4	no	—
31	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
33	C₈×C₁₀₉	—	2	4	no	—
34	C₈×C₁₀₉	C₂×Q₁₆	1	2	yes	—
35	C₈×C₁₀₉	C₂×Q₁₆	1	2	yes	—
36	C₂²×C₁₀₉	#493	1	1	yes	characteristic
37	C₄×C₁₀₉	—	16	4	no	—
38	C₄×C₁₀₉	#494	1	1	yes	characteristic
39	C₄×C₁₀₉	#494	1	1	yes	characteristic
40	C₄×C₁₀₉	#508	1	4	yes	—
41	#12522	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₁₀₉	#1713	1	1	yes	characteristic
43	C₂×C₁₀₉	#2249	1	2	yes	—
44	#2254	C₂×C₁₀₉	1	2	yes	—
45	#2252	C₂×C₁₀₉	1	2	yes	—
46	#1710	C₂×C₁₀₉	1	2	yes	—
47	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
48	C₁₀₉	#12522	1	1	yes	characteristic, direct product factor, Sylow
49	#500	C₂²×C₁₀₉	1	4	yes	—
50	#487	—	2	4	no	—
51	#367	—	2	4	no	—
52	#366	—	2	4	no	—
53	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
54	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
55	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	4	no	—
57	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
58	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
59	#107	—	2	4	no	—
60	#107	—	2	4	no	—
61	#103	—	4	8	no	—
62	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
63	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
64	C₂×Q₈	—	4	4	no	—
65	C₂×C₈	D₈×C₁₀₉	1	4	yes	—
66	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
68	C₄⋊C₄	—	4	4	no	—
69	C₄⋊C₄	—	4	4	no	—
70	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
71	C₁₆	Q₁₆×C₁₀₉	1	4	yes	—
72	C₁₆	Q₁₆×C₁₀₉	1	4	yes	—
73	Q₈	—	8	8	no	—
74	C₂×C₄	—	8	4	no	—
75	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
76	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Q₁₆×C₁₀₉	1	2	yes	—
79	C₈	C₂×Q₁₆×C₁₀₉	1	2	yes	—
80	C₂²	#493×C₁₀₉	1	1	yes	characteristic
81	C₄	—	16	4	no	—
82	C₄	#494×C₁₀₉	1	1	yes	characteristic
83	C₄	#494×C₁₀₉	1	1	yes	characteristic
84	C₄	#508×C₁₀₉	1	4	yes	—
85	C₂	C₁₀₉×#1713	1	1	yes	characteristic
86	C₂	C₁₀₉×#2249	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
41. #12522	C₁₀₉	direct (N × H)
48. C₁₀₉	#12522	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×#2254	2
C₁₀₉×#2252	2
C₁₀₉×#1710	2
C₈×C₁₆×C₁₀₉	2
#12522	109

Subgroups — C109×#12522 #42130516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12522 #42130516