Subgroups — C₂²×C₈×C₉×C₁₀₇ #72207608

Order 30816.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₉×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₉×C₁₀₇	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₈×C₁₀₇	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₉×C₁₀₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₉×C₁₀₇	C₄	1	3	yes	—
6	C₂×C₄×C₉×C₁₀₇	C₂²	1	3	yes	—
7	C₈×C₉×C₁₀₇	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₄×C₁₀₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×C₈×C₁₀₇	C₂×C₃	1	6	yes	—
10	C₂³×C₃×C₁₀₇	C₃×C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₁₀₇	C₃×C₄	1	3	yes	—
12	C₂×C₃×C₄×C₁₀₇	C₂²×C₃	1	3	yes	—
13	C₃×C₈×C₁₀₇	C₂²×C₃	1	4	yes	—
14	C₂²×C₉×C₁₀₇	C₈	1	4	yes	direct product factor
15	C₂²×C₉×C₁₀₇	C₂×C₄	1	3	yes	—
16	C₄×C₉×C₁₀₇	C₂×C₄	1	3	yes	—
17	C₄×C₉×C₁₀₇	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₇	C₉	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₉×C₁₀₇	C₂×C₈	1	6	yes	direct product factor
20	C₂×C₉×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₇	C₂×C₉	1	1	yes	characteristic
22	C₂×C₈×C₁₀₇	C₂×C₉	1	6	yes	direct product factor
23	C₂²×C₃×C₁₀₇	C₃×C₈	1	4	yes	—
24	C₂²×C₃×C₁₀₇	C₂×C₃×C₄	1	3	yes	—
25	C₃×C₄×C₁₀₇	C₂×C₃×C₄	1	3	yes	—
26	C₃×C₄×C₁₀₇	C₂³×C₃	1	1	yes	characteristic
27	C₉×C₁₀₇	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
28	C₂³×C₁₀₇	C₄×C₉	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	C₄×C₉	1	3	yes	—
30	C₂×C₄×C₁₀₇	C₂²×C₉	1	3	yes	—
31	C₈×C₁₀₇	C₂²×C₉	1	4	yes	direct product factor
32	C₂×C₃×C₁₀₇	C₂×C₃×C₈	1	6	yes	—
33	C₂×C₃×C₁₀₇	C₂²×C₃×C₄	1	1	yes	characteristic
34	C₂²×C₁₀₇	C₈×C₉	1	4	yes	direct product factor
35	C₂²×C₁₀₇	C₂×C₄×C₉	1	3	yes	—
36	C₄×C₁₀₇	C₂×C₄×C₉	1	3	yes	—
37	C₄×C₁₀₇	C₂³×C₉	1	1	yes	characteristic
38	C₃×C₁₀₇	C₂²×C₃×C₈	1	1	yes	characteristic
39	C₂²×C₈×C₉	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₁₀₇	C₂×C₈×C₉	1	6	yes	direct product factor
41	C₂×C₁₀₇	C₂²×C₄×C₉	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂×C₁₀₇	1	1	yes	characteristic
43	C₂×C₈×C₉	C₂×C₁₀₇	1	6	yes	direct product factor
44	C₁₀₇	C₂²×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₃×C₈	C₃×C₁₀₇	1	1	yes	characteristic
46	C₂³×C₉	C₄×C₁₀₇	1	1	yes	characteristic
47	C₂×C₄×C₉	C₄×C₁₀₇	1	3	yes	—
48	C₂×C₄×C₉	C₂²×C₁₀₇	1	3	yes	—
49	C₈×C₉	C₂²×C₁₀₇	1	4	yes	direct product factor
50	C₂²×C₃×C₄	C₂×C₃×C₁₀₇	1	1	yes	characteristic
51	C₂×C₃×C₈	C₂×C₃×C₁₀₇	1	6	yes	—
52	C₂²×C₉	C₈×C₁₀₇	1	4	yes	direct product factor
53	C₂²×C₉	C₂×C₄×C₁₀₇	1	3	yes	—
54	C₄×C₉	C₂×C₄×C₁₀₇	1	3	yes	—
55	C₄×C₉	C₂³×C₁₀₇	1	1	yes	characteristic
56	C₂²×C₈	C₉×C₁₀₇	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃	C₃×C₄×C₁₀₇	1	1	yes	characteristic
58	C₂×C₃×C₄	C₃×C₄×C₁₀₇	1	3	yes	—
59	C₂×C₃×C₄	C₂²×C₃×C₁₀₇	1	3	yes	—
60	C₃×C₈	C₂²×C₃×C₁₀₇	1	4	yes	—
61	C₂×C₉	C₂×C₈×C₁₀₇	1	6	yes	direct product factor
62	C₂×C₉	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₉×C₁₀₇	1	1	yes	characteristic
64	C₂×C₈	C₂×C₉×C₁₀₇	1	6	yes	direct product factor
65	C₂²×C₃	C₃×C₈×C₁₀₇	1	4	yes	—
66	C₂²×C₃	C₂×C₃×C₄×C₁₀₇	1	3	yes	—
67	C₃×C₄	C₂×C₃×C₄×C₁₀₇	1	3	yes	—
68	C₃×C₄	C₂³×C₃×C₁₀₇	1	1	yes	characteristic
69	C₉	C₂²×C₈×C₁₀₇	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³	C₄×C₉×C₁₀₇	1	1	yes	characteristic
71	C₂×C₄	C₄×C₉×C₁₀₇	1	3	yes	—
72	C₂×C₄	C₂²×C₉×C₁₀₇	1	3	yes	—
73	C₈	C₂²×C₉×C₁₀₇	1	4	yes	direct product factor
74	C₂×C₃	C₂×C₃×C₈×C₁₀₇	1	6	yes	—
75	C₂×C₃	C₂²×C₃×C₄×C₁₀₇	1	1	yes	characteristic
76	C₂²	C₈×C₉×C₁₀₇	1	4	yes	direct product factor
77	C₂²	C₂×C₄×C₉×C₁₀₇	1	3	yes	—
78	C₄	C₂×C₄×C₉×C₁₀₇	1	3	yes	—
79	C₄	C₂³×C₉×C₁₀₇	1	1	yes	characteristic
80	C₃	C₂²×C₃×C₈×C₁₀₇	1	1	yes	characteristic
81	C₂	C₂×C₈×C₉×C₁₀₇	1	6	yes	direct product factor
82	C₂	C₂²×C₄×C₉×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂³×C₃×C₁₀₇
socle, prime radical	C₂³×C₃×C₁₀₇	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×C₉×C₁₀₇	C₂	direct (N × H)
7. C₈×C₉×C₁₀₇	C₂²	direct (N × H)
14. C₂²×C₉×C₁₀₇	C₈	direct (N × H)
18. C₂²×C₈×C₁₀₇	C₉	direct (N × H)
19. C₂×C₉×C₁₀₇	C₂×C₈	direct (N × H)
22. C₂×C₈×C₁₀₇	C₂×C₉	direct (N × H)
27. C₉×C₁₀₇	C₂²×C₈	direct (N × H)
31. C₈×C₁₀₇	C₂²×C₉	direct (N × H)
34. C₂²×C₁₀₇	C₈×C₉	direct (N × H)
39. C₂²×C₈×C₉	C₁₀₇	direct (N × H)
40. C₂×C₁₀₇	C₂×C₈×C₉	direct (N × H)
43. C₂×C₈×C₉	C₂×C₁₀₇	direct (N × H)
44. C₁₀₇	C₂²×C₈×C₉	direct (N × H)
49. C₈×C₉	C₂²×C₁₀₇	direct (N × H)
52. C₂²×C₉	C₈×C₁₀₇	direct (N × H)
56. C₂²×C₈	C₉×C₁₀₇	direct (N × H)
61. C₂×C₉	C₂×C₈×C₁₀₇	direct (N × H)
64. C₂×C₈	C₂×C₉×C₁₀₇	direct (N × H)
69. C₉	C₂²×C₈×C₁₀₇	direct (N × H)
73. C₈	C₂²×C₉×C₁₀₇	direct (N × H)
76. C₂²	C₈×C₉×C₁₀₇	direct (N × H)
81. C₂	C₂×C₈×C₉×C₁₀₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₉×C₁₀₇	2
C₂×C₈×C₉×C₁₀₇	2
C₂²×C₃×C₈×C₁₀₇	3
C₂²×C₈×C₉	107

Subgroups — C22×C8×C9×C107 #72207608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₉×C₁₀₇ #72207608