Subgroups — C₁₀₉×#7846 #42125840

Order 27904.

582 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#97×C₁₀₉	C₂	1	3	yes	maximal
2	C₂×#322×C₁₀₉	C₂	1	3	yes	maximal
3	C₂×C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×C₁₀₉	C₄	1	12	yes	—
5	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
6	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
7	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
8	#322×C₁₀₉	C₄	1	12	yes	complemented
9	C₈²×C₁₀₉	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
11	C₂²×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
12	C₂×C₄²×C₁₀₉	C₂×C₄	1	6	yes	—
13	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
14	#97×C₁₀₉	—	2	24	no	—
15	C₄×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
16	C₄×C₈×C₁₀₉	C₂×C₄	1	12	yes	—
17	C₄×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
19	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
20	C₂×C₈×C₁₀₉	—	2	12	no	—
21	C₂×C₈×C₁₀₉	C₄²	1	12	yes	—
22	C₂×C₈×C₁₀₉	C₄²	1	12	yes	—
23	C₂×C₈×C₁₀₉	M₁₆	1	24	yes	—
24	C₄²×C₁₀₉	—	2	12	no	—
25	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₉	C₂²×C₄	1	3	yes	—
27	C₂³×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	12	no	—
29	C₂×C₄×C₁₀₉	C₂×C₄²	1	3	yes	—
30	C₂×C₄×C₁₀₉	C₂×C₄²	1	3	yes	—
31	C₂×C₄×C₁₀₉	C₂×M₁₆	1	12	yes	—
32	C₈×C₁₀₉	—	2	12	no	—
33	C₈×C₁₀₉	#97	1	24	yes	complemented
34	C₂²×C₁₀₉	C₄³	1	1	yes	characteristic
35	C₂²×C₁₀₉	C₄×M₁₆	1	6	yes	—
36	C₄×C₁₀₉	—	4	4	no	—
37	C₄×C₁₀₉	C₂×#97	1	6	yes	—
38	C₄×C₁₀₉	C₄×M₁₆	1	6	yes	—
39	#7846	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₁₀₉	#1455	1	4	yes	—
41	C₂×C₁₀₉	C₄×#97	1	3	yes	—
42	C₄×#97	C₂×C₁₀₉	1	3	yes	—
43	C₂×#322	C₂×C₁₀₉	1	3	yes	—
44	C₂×C₈²	C₂×C₁₀₉	1	1	yes	characteristic
45	C₁₀₉	#7846	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×#97	C₄×C₁₀₉	1	12	yes	—
47	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
48	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
49	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
50	#322	C₄×C₁₀₉	1	12	yes	complemented
51	C₈²	C₄×C₁₀₉	1	4	yes	complemented
52	C₂²×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
53	C₂²×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
54	C₂×C₄²	C₂×C₄×C₁₀₉	1	6	yes	—
55	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
56	#97	—	2	24	no	—
57	C₄×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
58	C₄×C₈	C₂×C₄×C₁₀₉	1	12	yes	—
59	C₄×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
60	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
61	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
62	C₂×C₈	—	2	12	no	—
63	C₂×C₈	C₄²×C₁₀₉	1	12	yes	—
64	C₂×C₈	C₄²×C₁₀₉	1	12	yes	—
65	C₂×C₈	M₁₆×C₁₀₉	1	24	yes	—
66	C₄²	—	2	12	no	—
67	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
68	C₄²	C₂²×C₄×C₁₀₉	1	3	yes	—
69	C₂³	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
70	C₂×C₄	—	2	12	no	—
71	C₂×C₄	C₂×C₄²×C₁₀₉	1	3	yes	—
72	C₂×C₄	C₂×C₄²×C₁₀₉	1	3	yes	—
73	C₂×C₄	C₂×M₁₆×C₁₀₉	1	12	yes	—
74	C₈	—	2	12	no	—
75	C₈	#97×C₁₀₉	1	24	yes	complemented
76	C₂²	C₄³×C₁₀₉	1	1	yes	characteristic
77	C₂²	C₄×M₁₆×C₁₀₉	1	6	yes	—
78	C₄	—	4	4	no	—
79	C₄	C₂×#97×C₁₀₉	1	6	yes	—
80	C₄	C₄×M₁₆×C₁₀₉	1	6	yes	—
81	C₂	C₁₀₉×#1455	1	4	yes	—
82	C₂	C₄×#97×C₁₀₉	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₀₉	C₂³
commutator	C₂²	C₄³×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. #322×C₁₀₉	C₄	semidirect
9. C₈²×C₁₀₉	C₄	semidirect
33. C₈×C₁₀₉	#97	semidirect
39. #7846	C₁₀₉	direct (N × H)
45. C₁₀₉	#7846	direct (N × H)
50. #322	C₄×C₁₀₉	semidirect
51. C₈²	C₄×C₁₀₉	semidirect
75. C₈	#97×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#97×C₁₀₉	2
C₂×#322×C₁₀₉	2
C₂×C₈²×C₁₀₉	2
#7846	109

Subgroups — C109×#7846 #42125840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7846 #42125840