Subgroups — C₁₀₉×#12899 #42130893

Order 27904.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆×C₁₀₉	C₂	1	6	yes	maximal
2	C₁₀₉×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₁₀₉	C₂²	1	4	yes	—
4	C₂×#97×C₁₀₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
6	#376×C₁₀₉	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₁₀₉	—	2	12	no	—
8	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
11	C₂×C₁₆×C₁₀₉	—	2	12	no	—
12	#97×C₁₀₉	—	2	6	no	—
13	C₄×C₈×C₁₀₉	D₄	1	6	yes	—
14	C₄×C₈×C₁₀₉	Q₈	1	6	yes	—
15	#95×C₁₀₉	C₈	1	4	yes	—
16	C₂²×C₄×C₁₀₉	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	—	2	6	no	—
19	C₂×C₈×C₁₀₉	C₄⋊C₄	1	6	yes	—
20	C₂×C₈×C₁₀₉	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₁₀₉	C₂×Q₈	1	3	yes	—
22	C₄²×C₁₀₉	C₄⋊C₄	1	6	yes	—
23	C₄²×C₁₀₉	M₁₆	1	6	yes	—
24	C₁₆×C₁₀₉	—	4	4	no	—
25	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	—	2	12	no	—
27	C₂×C₄×C₁₀₉	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₁₀₉	C₂×M₁₆	1	3	yes	—
30	C₈×C₁₀₉	—	2	6	no	—
31	C₈×C₁₀₉	#127	1	1	yes	characteristic
32	C₈×C₁₀₉	C₄⋊Q₈	1	3	yes	—
33	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	3	yes	—
34	C₂²×C₁₀₉	#504	1	4	yes	—
35	C₄×C₁₀₉	—	2	12	no	—
36	C₄×C₁₀₉	#384	1	1	yes	characteristic
37	C₄×C₁₀₉	#423	1	3	yes	—
38	#12899	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₁₀₉	#1842	1	1	yes	characteristic
40	C₂×C₁₀₉	#2151	1	6	yes	—
41	C₂×C₄⋊C₁₆	C₂×C₁₀₉	1	6	yes	—
42	#1726	C₂×C₁₀₉	1	1	yes	characteristic
43	C₁₀₉	#12899	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²×C₁₆	C₂²×C₁₀₉	1	4	yes	—
45	C₂×#97	C₄×C₁₀₉	1	3	yes	—
46	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
47	#376	C₄×C₁₀₉	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	12	no	—
49	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
50	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄×C₁₀₉	1	3	yes	—
52	C₂×C₁₆	—	2	12	no	—
53	#97	—	2	6	no	—
54	C₄×C₈	D₄×C₁₀₉	1	6	yes	—
55	C₄×C₈	Q₈×C₁₀₉	1	6	yes	—
56	#95	C₈×C₁₀₉	1	4	yes	—
57	C₂²×C₄	C₂×C₈×C₁₀₉	1	6	yes	—
58	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊C₄×C₁₀₉	1	6	yes	—
61	C₂×C₈	C₂×D₄×C₁₀₉	1	3	yes	—
62	C₂×C₈	C₂×Q₈×C₁₀₉	1	3	yes	—
63	C₄²	C₄⋊C₄×C₁₀₉	1	6	yes	—
64	C₄²	M₁₆×C₁₀₉	1	6	yes	—
65	C₁₆	—	4	4	no	—
66	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊C₈×C₁₀₉	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
70	C₂×C₄	C₂×M₁₆×C₁₀₉	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	#127×C₁₀₉	1	1	yes	characteristic
73	C₈	C₄⋊Q₈×C₁₀₉	1	3	yes	—
74	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	3	yes	—
75	C₂²	#504×C₁₀₉	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#384×C₁₀₉	1	1	yes	characteristic
78	C₄	#423×C₁₀₉	1	3	yes	—
79	C₂	C₁₀₉×#1842	1	1	yes	characteristic
80	C₂	C₁₀₉×#2151	1	6	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂³
commutator	C₂³	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
38. #12899	C₁₀₉	direct (N × H)
43. C₁₀₉	#12899	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆×C₁₀₉	2
C₁₀₉×#1726	2
#12899	109

Subgroups — C109×#12899 #42130893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12899 #42130893