Subgroups — C₂×C₃²×C₅×C₂₄₃ #72199331

Order 21870.

464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₂₄₃	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃²×C₅×C₈₁	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₂₄₃	C₃	1	12	yes	maximal, direct product factor
4	C₂×C₃²×C₂₄₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃²×C₅×C₂₇	C₉	1	1	yes	characteristic
6	C₂×C₃×C₅×C₈₁	C₉	1	8	yes	—
7	C₂×C₃×C₅×C₈₁	C₃²	1	4	yes	—
8	C₂×C₅×C₂₄₃	C₃²	1	9	yes	direct product factor
9	C₃²×C₅×C₈₁	C₂×C₃	1	1	yes	characteristic
10	C₃×C₅×C₂₄₃	C₂×C₃	1	12	yes	direct product factor
11	C₃²×C₂₄₃	C₂×C₅	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃²×C₈₁	C₃×C₅	1	1	yes	characteristic
13	C₂×C₃×C₂₄₃	C₃×C₅	1	12	yes	direct product factor
14	C₃²×C₅×C₂₇	C₂×C₉	1	1	yes	characteristic
15	C₃×C₅×C₈₁	C₂×C₉	1	8	yes	—
16	C₃×C₅×C₈₁	C₂×C₃²	1	4	yes	—
17	C₅×C₂₄₃	C₂×C₃²	1	9	yes	direct product factor
18	C₂×C₃²×C₅×C₉	C₂₇	1	1	yes	characteristic
19	C₂×C₃×C₅×C₂₇	C₂₇	1	8	yes	—
20	C₂×C₃×C₅×C₂₇	C₃×C₉	1	4	yes	—
21	C₂×C₅×C₈₁	C₃×C₉	1	8	yes	—
22	C₂×C₅×C₈₁	C₃³	1	1	yes	characteristic
23	C₃²×C₈₁	C₂×C₃×C₅	1	1	yes	characteristic
24	C₃×C₂₄₃	C₂×C₃×C₅	1	12	yes	direct product factor
25	C₂×C₃²×C₂₇	C₅×C₉	1	1	yes	characteristic
26	C₂×C₃×C₈₁	C₅×C₉	1	8	yes	—
27	C₂×C₃×C₈₁	C₃²×C₅	1	4	yes	—
28	C₂×C₂₄₃	C₃²×C₅	1	9	yes	direct product factor
29	C₃²×C₅×C₉	C₂×C₂₇	1	1	yes	characteristic
30	C₃×C₅×C₂₇	C₂×C₂₇	1	8	yes	—
31	C₃×C₅×C₂₇	C₂×C₃×C₉	1	4	yes	—
32	C₅×C₈₁	C₂×C₃×C₉	1	8	yes	—
33	C₅×C₈₁	C₂×C₃³	1	1	yes	characteristic
34	C₂×C₃³×C₅	C₈₁	1	1	yes	characteristic
35	C₂×C₃×C₅×C₉	C₈₁	1	8	yes	—
36	C₂×C₃×C₅×C₉	C₃×C₂₇	1	4	yes	—
37	C₂×C₅×C₂₇	C₃×C₂₇	1	8	yes	—
38	C₂×C₅×C₂₇	C₃²×C₉	1	1	yes	characteristic
39	C₃²×C₂₇	C₂×C₅×C₉	1	1	yes	characteristic
40	C₃×C₈₁	C₂×C₅×C₉	1	8	yes	—
41	C₃×C₈₁	C₂×C₃²×C₅	1	4	yes	—
42	C₂₄₃	C₂×C₃²×C₅	1	9	yes	direct product factor
43	C₂×C₃²×C₉	C₅×C₂₇	1	1	yes	characteristic
44	C₂×C₃×C₂₇	C₅×C₂₇	1	8	yes	—
45	C₂×C₃×C₂₇	C₃×C₅×C₉	1	4	yes	—
46	C₂×C₈₁	C₃×C₅×C₉	1	8	yes	—
47	C₂×C₈₁	C₃³×C₅	1	1	yes	characteristic
48	C₃³×C₅	C₂×C₈₁	1	1	yes	characteristic
49	C₃×C₅×C₉	C₂×C₈₁	1	8	yes	—
50	C₃×C₅×C₉	C₂×C₃×C₂₇	1	4	yes	—
51	C₅×C₂₇	C₂×C₃×C₂₇	1	8	yes	—
52	C₅×C₂₇	C₂×C₃²×C₉	1	1	yes	characteristic
53	C₂×C₃²×C₅	C₂₄₃	1	9	yes	direct product factor
54	C₂×C₃²×C₅	C₃×C₈₁	1	4	yes	—
55	C₂×C₅×C₉	C₃×C₈₁	1	8	yes	—
56	C₂×C₅×C₉	C₃²×C₂₇	1	1	yes	characteristic
57	C₃²×C₉	C₂×C₅×C₂₇	1	1	yes	characteristic
58	C₃×C₂₇	C₂×C₅×C₂₇	1	8	yes	—
59	C₃×C₂₇	C₂×C₃×C₅×C₉	1	4	yes	—
60	C₈₁	C₂×C₃×C₅×C₉	1	8	yes	—
61	C₈₁	C₂×C₃³×C₅	1	1	yes	characteristic
62	C₂×C₃³	C₅×C₈₁	1	1	yes	characteristic
63	C₂×C₃×C₉	C₅×C₈₁	1	8	yes	—
64	C₂×C₃×C₉	C₃×C₅×C₂₇	1	4	yes	—
65	C₂×C₂₇	C₃×C₅×C₂₇	1	8	yes	—
66	C₂×C₂₇	C₃²×C₅×C₉	1	1	yes	characteristic
67	C₃²×C₅	C₂×C₂₄₃	1	9	yes	direct product factor
68	C₃²×C₅	C₂×C₃×C₈₁	1	4	yes	—
69	C₅×C₉	C₂×C₃×C₈₁	1	8	yes	—
70	C₅×C₉	C₂×C₃²×C₂₇	1	1	yes	characteristic
71	C₂×C₃×C₅	C₃×C₂₄₃	1	12	yes	direct product factor
72	C₂×C₃×C₅	C₃²×C₈₁	1	1	yes	characteristic
73	C₃³	C₂×C₅×C₈₁	1	1	yes	characteristic
74	C₃×C₉	C₂×C₅×C₈₁	1	8	yes	—
75	C₃×C₉	C₂×C₃×C₅×C₂₇	1	4	yes	—
76	C₂₇	C₂×C₃×C₅×C₂₇	1	8	yes	—
77	C₂₇	C₂×C₃²×C₅×C₉	1	1	yes	characteristic
78	C₂×C₃²	C₅×C₂₄₃	1	9	yes	direct product factor
79	C₂×C₃²	C₃×C₅×C₈₁	1	4	yes	—
80	C₂×C₉	C₃×C₅×C₈₁	1	8	yes	—
81	C₂×C₉	C₃²×C₅×C₂₇	1	1	yes	characteristic
82	C₃×C₅	C₂×C₃×C₂₄₃	1	12	yes	direct product factor
83	C₃×C₅	C₂×C₃²×C₈₁	1	1	yes	characteristic
84	C₂×C₅	C₃²×C₂₄₃	1	1	yes	characteristic, direct product factor, Hall
85	C₃²	C₂×C₅×C₂₄₃	1	9	yes	direct product factor
86	C₃²	C₂×C₃×C₅×C₈₁	1	4	yes	—
87	C₉	C₂×C₃×C₅×C₈₁	1	8	yes	—
88	C₉	C₂×C₃²×C₅×C₂₇	1	1	yes	characteristic
89	C₂×C₃	C₃×C₅×C₂₄₃	1	12	yes	direct product factor
90	C₂×C₃	C₃²×C₅×C₈₁	1	1	yes	characteristic
91	C₅	C₂×C₃²×C₂₄₃	1	1	yes	characteristic, direct product factor, Sylow
92	C₃	C₂×C₃×C₅×C₂₄₃	1	12	yes	direct product factor
93	C₃	C₂×C₃²×C₅×C₈₁	1	1	yes	characteristic
94	C₂	C₃²×C₅×C₂₄₃	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈₁	C₂×C₃³×C₅
socle, prime radical	C₂×C₃³×C₅	C₈₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₂₄₃	C₂	direct (N × H)
3. C₂×C₃×C₅×C₂₄₃	C₃	direct (N × H)
4. C₂×C₃²×C₂₄₃	C₅	direct (N × H)
8. C₂×C₅×C₂₄₃	C₃²	direct (N × H)
10. C₃×C₅×C₂₄₃	C₂×C₃	direct (N × H)
11. C₃²×C₂₄₃	C₂×C₅	direct (N × H)
13. C₂×C₃×C₂₄₃	C₃×C₅	direct (N × H)
17. C₅×C₂₄₃	C₂×C₃²	direct (N × H)
24. C₃×C₂₄₃	C₂×C₃×C₅	direct (N × H)
28. C₂×C₂₄₃	C₃²×C₅	direct (N × H)
42. C₂₄₃	C₂×C₃²×C₅	direct (N × H)
53. C₂×C₃²×C₅	C₂₄₃	direct (N × H)
67. C₃²×C₅	C₂×C₂₄₃	direct (N × H)
71. C₂×C₃×C₅	C₃×C₂₄₃	direct (N × H)
78. C₂×C₃²	C₅×C₂₄₃	direct (N × H)
82. C₃×C₅	C₂×C₃×C₂₄₃	direct (N × H)
84. C₂×C₅	C₃²×C₂₄₃	direct (N × H)
85. C₃²	C₂×C₅×C₂₄₃	direct (N × H)
89. C₂×C₃	C₃×C₅×C₂₄₃	direct (N × H)
91. C₅	C₂×C₃²×C₂₄₃	direct (N × H)
92. C₃	C₂×C₃×C₅×C₂₄₃	direct (N × H)
94. C₂	C₃²×C₅×C₂₄₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₂₄₃	2
C₂×C₃²×C₅×C₈₁	3
C₂×C₃×C₅×C₂₄₃	3
C₂×C₃²×C₂₄₃	5

Subgroups — C2×C32×C5×C243 #72199331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅×C₂₄₃ #72199331