Subgroups — C₁₀₉×#8804 #42126798

Order 27904.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#1455	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1291	C₂	1	3	yes	maximal
3	C₁₀₉×#1288	C₂	1	3	yes	maximal
4	#432×C₁₀₉	C₂²	1	3	yes	—
5	C₄×M₁₆×C₁₀₉	C₂²	1	3	yes	—
6	#383×C₁₀₉	C₂²	1	1	yes	characteristic
7	#322×C₁₀₉	C₄	1	3	yes	complemented
8	C₈²×C₁₀₉	C₄	1	1	yes	characteristic, complemented
9	C₂×M₁₆×C₁₀₉	—	2	3	no	—
10	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
11	C₄⋊C₈×C₁₀₉	—	2	3	no	—
12	C₂²⋊C₈×C₁₀₉	—	2	3	no	—
13	#97×C₁₀₉	—	2	3	no	—
14	#97×C₁₀₉	—	2	3	no	—
15	C₄×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
16	C₄×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
17	#95×C₁₀₉	—	2	3	no	—
18	C₂²×C₄×C₁₀₉	—	4	1	no	—
19	C₂²×C₄×C₁₀₉	#41	1	3	yes	—
20	M₁₆×C₁₀₉	—	4	6	no	—
21	C₂×C₈×C₁₀₉	—	2	3	no	—
22	C₂×C₈×C₁₀₉	—	2	3	no	—
23	C₂×C₈×C₁₀₉	—	4	3	no	—
24	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	#41	1	3	yes	—
26	C₂³×C₁₀₉	#126	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	4	3	no	—
28	C₂×C₄×C₁₀₉	—	8	3	no	—
29	C₂×C₄×C₁₀₉	#117	1	3	yes	—
30	C₂×C₄×C₁₀₉	#126	1	3	yes	—
31	C₈×C₁₀₉	—	2	6	no	—
32	C₈×C₁₀₉	—	4	3	no	—
33	C₈×C₁₀₉	—	8	3	no	—
34	C₂²×C₁₀₉	—	2	3	no	—
35	C₂²×C₁₀₉	#383	1	1	yes	characteristic
36	C₄×C₁₀₉	—	2	3	no	—
37	C₄×C₁₀₉	—	2	3	no	—
38	C₄×C₁₀₉	—	16	1	no	—
39	#8804	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₁₀₉	—	4	1	no	—
41	C₂×C₁₀₉	#1841	1	3	yes	—
42	#1455	C₂×C₁₀₉	1	1	yes	characteristic
43	#1291	C₂×C₁₀₉	1	3	yes	—
44	#1288	C₂×C₁₀₉	1	3	yes	—
45	C₁₀₉	#8804	1	1	yes	characteristic, direct product factor, Sylow
46	#432	C₂²×C₁₀₉	1	3	yes	—
47	C₄×M₁₆	C₂²×C₁₀₉	1	3	yes	—
48	#383	C₂²×C₁₀₉	1	1	yes	characteristic
49	#322	C₄×C₁₀₉	1	3	yes	complemented
50	C₈²	C₄×C₁₀₉	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
53	C₄⋊C₈	—	2	3	no	—
54	C₂²⋊C₈	—	2	3	no	—
55	#97	—	2	3	no	—
56	#97	—	2	3	no	—
57	C₄×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
58	C₄×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
59	#95	—	2	3	no	—
60	C₂²×C₄	—	4	1	no	—
61	C₂²×C₄	#41×C₁₀₉	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	4	3	no	—
66	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
67	C₄²	#41×C₁₀₉	1	3	yes	—
68	C₂³	#126×C₁₀₉	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	8	3	no	—
71	C₂×C₄	#117×C₁₀₉	1	3	yes	—
72	C₂×C₄	#126×C₁₀₉	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	8	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#383×C₁₀₉	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	16	1	no	—
81	C₂	—	4	1	no	—
82	C₂	C₁₀₉×#1841	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#383
commutator	C₄²	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
7. #322×C₁₀₉	C₄	semidirect
8. C₈²×C₁₀₉	C₄	semidirect
39. #8804	C₁₀₉	direct (N × H)
45. C₁₀₉	#8804	direct (N × H)
49. #322	C₄×C₁₀₉	semidirect
50. C₈²	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#1455	2
C₁₀₉×#1291	2
C₁₀₉×#1288	2
#8804	109

Subgroups — C109×#8804 #42126798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8804 #42126798