Subgroups — C₂²×M₃₂×C₁₀₉ #28360659

Order 13952.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂×C₁₀₉	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₂³×C₄×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₁₀₉	C₄	1	6	yes	—
7	C₂²×C₈×C₁₀₉	C₂²	1	6	yes	—
8	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	M₃₂×C₁₀₉	C₂²	1	16	yes	direct product factor
10	C₂×C₁₆×C₁₀₉	C₂²	1	12	yes	complemented
11	C₂⁴×C₁₀₉	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₁₀₉	C₈	1	6	yes	—
13	C₂²×C₄×C₁₀₉	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₉	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
17	C₂×C₈×C₁₀₉	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₁₀₉	C₂³	1	3	yes	—
20	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
21	C₁₆×C₁₀₉	C₂³	1	8	yes	complemented
22	C₂³×C₁₀₉	—	2	4	no	—
23	C₂³×C₁₀₉	C₂×C₈	1	6	yes	—
24	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₀₉	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
30	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
31	C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
32	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₀₉	—	2	12	no	—
34	C₂²×C₁₀₉	M₃₂	1	4	yes	direct product factor
35	C₂²×C₁₀₉	C₂²×C₈	1	4	yes	—
36	C₂²×C₁₀₉	C₂²×C₈	1	3	yes	—
37	C₄×C₁₀₉	C₂²×C₈	1	3	yes	—
38	C₄×C₁₀₉	C₂²×C₈	1	4	yes	—
39	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₀₉	—	2	4	no	—
41	C₂×C₁₀₉	C₂×M₃₂	1	6	yes	direct product factor
42	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
43	C₂²×M₃₂	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₁₀₉	C₂²×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₈	C₂×C₁₀₉	1	1	yes	characteristic
46	C₂×M₃₂	C₂×C₁₀₉	1	12	yes	direct product factor
47	C₂²×C₁₆	C₂×C₁₀₉	1	2	yes	complemented
48	C₂³×C₄	C₄×C₁₀₉	1	1	yes	characteristic
49	C₂²×C₈	C₄×C₁₀₉	1	1	yes	characteristic
50	C₂²×C₈	C₄×C₁₀₉	1	6	yes	—
51	C₂²×C₈	C₂²×C₁₀₉	1	6	yes	—
52	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
53	M₃₂	C₂²×C₁₀₉	1	16	yes	direct product factor
54	C₂×C₁₆	C₂²×C₁₀₉	1	12	yes	complemented
55	C₂⁴	C₈×C₁₀₉	1	1	yes	characteristic
56	C₂²×C₄	C₈×C₁₀₉	1	6	yes	—
57	C₂²×C₄	C₈×C₁₀₉	1	1	yes	characteristic
58	C₂²×C₄	C₂×C₄×C₁₀₉	1	6	yes	—
59	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
60	C₂×C₈	C₂×C₄×C₁₀₉	1	12	yes	—
61	C₂×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
62	C₂×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
63	C₂×C₈	C₂³×C₁₀₉	1	3	yes	—
64	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
65	C₁₆	C₂³×C₁₀₉	1	8	yes	complemented
66	C₂³	—	2	4	no	—
67	C₂³	C₂×C₈×C₁₀₉	1	6	yes	—
68	C₂³	C₂×C₈×C₁₀₉	1	1	yes	characteristic
69	C₂×C₄	C₂×C₈×C₁₀₉	1	12	yes	—
70	C₂×C₄	C₂×C₈×C₁₀₉	1	3	yes	—
71	C₂×C₄	C₂×C₈×C₁₀₉	1	6	yes	—
72	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
73	C₂×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
74	C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
75	C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
76	C₈	C₂⁴×C₁₀₉	1	1	yes	characteristic
77	C₂²	—	2	12	no	—
78	C₂²	M₃₂×C₁₀₉	1	4	yes	direct product factor
79	C₂²	C₂²×C₈×C₁₀₉	1	4	yes	—
80	C₂²	C₂²×C₈×C₁₀₉	1	3	yes	—
81	C₄	C₂²×C₈×C₁₀₉	1	3	yes	—
82	C₄	C₂²×C₈×C₁₀₉	1	4	yes	—
83	C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×M₃₂×C₁₀₉	1	6	yes	direct product factor
86	C₂	C₂³×C₈×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂²
commutator	C₂	C₂³×C₈×C₁₀₉
Frattini, cosocle	C₈	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₈
prime radical	C₂⁴×C₁₀₉	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₃₂×C₁₀₉	C₂	direct (N × H)
3. C₂²×C₁₆×C₁₀₉	C₂	semidirect
9. M₃₂×C₁₀₉	C₂²	direct (N × H)
10. C₂×C₁₆×C₁₀₉	C₂²	semidirect
21. C₁₆×C₁₀₉	C₂³	semidirect
34. C₂²×C₁₀₉	M₃₂	direct (N × H)
41. C₂×C₁₀₉	C₂×M₃₂	direct (N × H)
43. C₂²×M₃₂	C₁₀₉	direct (N × H)
44. C₁₀₉	C₂²×M₃₂	direct (N × H)
46. C₂×M₃₂	C₂×C₁₀₉	direct (N × H)
47. C₂²×C₁₆	C₂×C₁₀₉	semidirect
53. M₃₂	C₂²×C₁₀₉	direct (N × H)
54. C₂×C₁₆	C₂²×C₁₀₉	semidirect
65. C₁₆	C₂³×C₁₀₉	semidirect
78. C₂²	M₃₂×C₁₀₉	direct (N × H)
85. C₂	C₂×M₃₂×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₁₀₉	2
C₂×M₃₂×C₁₀₉	2
C₂²×C₁₆×C₁₀₉	2
C₂²×M₃₂	109

Subgroups — C22×M32×C109 #28360659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₃₂×C₁₀₉ #28360659