Subgroups — C₁₀₉×#12902 #42130896

Order 27904.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2153	C₂	1	3	yes	maximal
2	C₁₀₉×#2151	C₂	1	3	yes	maximal
3	C₁₀₉×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	C₂²	1	3	yes	—
5	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	#405×C₁₀₉	C₂²	1	3	yes	—
7	#364×C₁₀₉	—	2	6	no	—
8	C₄⋊C₁₆×C₁₀₉	—	2	6	no	—
9	#338×C₁₀₉	C₄	1	1	yes	characteristic
10	H(3,ℤ/4)×C₁₀₉	C₄	1	3	yes	—
11	C₂×M₁₆×C₁₀₉	C₂×C₄	1	3	yes	—
12	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
13	#117×C₁₀₉	C₂×C₄	1	3	yes	—
14	M₃₂×C₁₀₉	—	2	6	no	—
15	C₂×C₁₆×C₁₀₉	—	2	3	no	—
16	C₂×C₁₆×C₁₀₉	—	2	3	no	—
17	#97×C₁₀₉	Q₈	1	6	yes	—
18	C₄×C₈×C₁₀₉	D₄	1	6	yes	—
19	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
20	M₁₆×C₁₀₉	—	2	6	no	—
21	C₂×C₈×C₁₀₉	C₄⋊C₄	1	6	yes	—
22	C₂×C₈×C₁₀₉	C₂×D₄	1	3	yes	—
23	C₂×C₈×C₁₀₉	C₂×Q₈	1	3	yes	—
24	C₄⋊C₄×C₁₀₉	—	2	3	no	—
25	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
26	C₄²×C₁₀₉	C₄⋊C₄	1	6	yes	—
27	C₁₆×C₁₀₉	—	4	1	no	—
28	C₁₆×C₁₀₉	—	4	3	no	—
29	C₂³×C₁₀₉	#131	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	6	no	—
31	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
32	C₂×C₄×C₁₀₉	#131	1	3	yes	—
33	C₈×C₁₀₉	—	2	6	no	—
34	C₈×C₁₀₉	#127	1	1	yes	characteristic
35	C₈×C₁₀₉	C₄⋊Q₈	1	3	yes	—
36	C₂²×C₁₀₉	—	4	1	no	—
37	C₂²×C₁₀₉	#424	1	3	yes	—
38	C₄×C₁₀₉	—	4	6	no	—
39	C₄×C₁₀₉	#384	1	1	yes	characteristic
40	C₄×C₁₀₉	#424	1	3	yes	—
41	#12902	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₁₀₉	—	2	3	no	—
43	C₂×C₁₀₉	#1843	1	1	yes	characteristic
44	#2153	C₂×C₁₀₉	1	3	yes	—
45	#2151	C₂×C₁₀₉	1	3	yes	—
46	#1747	C₂×C₁₀₉	1	1	yes	characteristic
47	C₁₀₉	#12902	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×M₃₂	C₂²×C₁₀₉	1	3	yes	—
49	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
50	#405	C₂²×C₁₀₉	1	3	yes	—
51	#364	—	2	6	no	—
52	C₄⋊C₁₆	—	2	6	no	—
53	#338	C₄×C₁₀₉	1	1	yes	characteristic
54	H(3,ℤ/4)	C₄×C₁₀₉	1	3	yes	—
55	C₂×M₁₆	C₂×C₄×C₁₀₉	1	3	yes	—
56	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
57	#117	C₂×C₄×C₁₀₉	1	3	yes	—
58	M₃₂	—	2	6	no	—
59	C₂×C₁₆	—	2	3	no	—
60	C₂×C₁₆	—	2	3	no	—
61	#97	Q₈×C₁₀₉	1	6	yes	—
62	C₄×C₈	D₄×C₁₀₉	1	6	yes	—
63	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	C₄⋊C₄×C₁₀₉	1	6	yes	—
66	C₂×C₈	C₂×D₄×C₁₀₉	1	3	yes	—
67	C₂×C₈	C₂×Q₈×C₁₀₉	1	3	yes	—
68	C₄⋊C₄	—	2	3	no	—
69	C₂²⋊C₄	—	2	3	no	—
70	C₄²	C₄⋊C₄×C₁₀₉	1	6	yes	—
71	C₁₆	—	4	1	no	—
72	C₁₆	—	4	3	no	—
73	C₂³	#131×C₁₀₉	1	1	yes	characteristic
74	C₂×C₄	—	2	6	no	—
75	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
76	C₂×C₄	#131×C₁₀₉	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	#127×C₁₀₉	1	1	yes	characteristic
79	C₈	C₄⋊Q₈×C₁₀₉	1	3	yes	—
80	C₂²	—	4	1	no	—
81	C₂²	#424×C₁₀₉	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	#384×C₁₀₉	1	1	yes	characteristic
84	C₄	#424×C₁₀₉	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	C₁₀₉×#1843	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₉	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1843
prime radical	C₂³×C₁₀₉	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
41. #12902	C₁₀₉	direct (N × H)
47. C₁₀₉	#12902	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×#2153	2
C₁₀₉×#2151	2
C₁₀₉×#1747	2
#12902	109

Subgroups — C109×#12902 #42130896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12902 #42130896