Subgroups — C₂²×C₈×C₂₄₃ #72186708

Order 7776.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₂₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₂₄₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₈×C₈₁	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₂₄₃	C₄	1	1	yes	characteristic
5	C₂×C₄×C₂₄₃	C₄	1	3	yes	—
6	C₂×C₄×C₂₄₃	C₂²	1	3	yes	—
7	C₈×C₂₄₃	C₂²	1	4	yes	direct product factor
8	C₂²×C₄×C₈₁	C₂×C₃	1	1	yes	characteristic
9	C₂×C₈×C₈₁	C₂×C₃	1	6	yes	—
10	C₂²×C₂₄₃	C₈	1	4	yes	direct product factor
11	C₂²×C₂₄₃	C₂×C₄	1	3	yes	—
12	C₄×C₂₄₃	C₂×C₄	1	3	yes	—
13	C₄×C₂₄₃	C₂³	1	1	yes	characteristic
14	C₂²×C₈×C₂₇	C₉	1	1	yes	characteristic
15	C₂³×C₈₁	C₃×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈₁	C₃×C₄	1	3	yes	—
17	C₂×C₄×C₈₁	C₂²×C₃	1	3	yes	—
18	C₈×C₈₁	C₂²×C₃	1	4	yes	—
19	C₂×C₂₄₃	C₂×C₈	1	6	yes	direct product factor
20	C₂×C₂₄₃	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₇	C₂×C₉	1	1	yes	characteristic
22	C₂×C₈×C₂₇	C₂×C₉	1	6	yes	—
23	C₂²×C₈₁	C₃×C₈	1	4	yes	—
24	C₂²×C₈₁	C₂×C₃×C₄	1	3	yes	—
25	C₄×C₈₁	C₂×C₃×C₄	1	3	yes	—
26	C₄×C₈₁	C₂³×C₃	1	1	yes	characteristic
27	C₂²×C₈×C₉	C₂₇	1	1	yes	characteristic
28	C₂₄₃	C₂²×C₈	1	1	yes	characteristic, direct product factor, Sylow
29	C₂³×C₂₇	C₄×C₉	1	1	yes	characteristic
30	C₂×C₄×C₂₇	C₄×C₉	1	3	yes	—
31	C₂×C₄×C₂₇	C₂²×C₉	1	3	yes	—
32	C₈×C₂₇	C₂²×C₉	1	4	yes	—
33	C₂×C₈₁	C₂×C₃×C₈	1	6	yes	—
34	C₂×C₈₁	C₂²×C₃×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₂₇	1	1	yes	characteristic
36	C₂×C₈×C₉	C₂×C₂₇	1	6	yes	—
37	C₂²×C₂₇	C₈×C₉	1	4	yes	—
38	C₂²×C₂₇	C₂×C₄×C₉	1	3	yes	—
39	C₄×C₂₇	C₂×C₄×C₉	1	3	yes	—
40	C₄×C₂₇	C₂³×C₉	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₈₁	1	1	yes	characteristic
42	C₈₁	C₂²×C₃×C₈	1	1	yes	characteristic
43	C₂³×C₉	C₄×C₂₇	1	1	yes	characteristic
44	C₂×C₄×C₉	C₄×C₂₇	1	3	yes	—
45	C₂×C₄×C₉	C₂²×C₂₇	1	3	yes	—
46	C₈×C₉	C₂²×C₂₇	1	4	yes	—
47	C₂×C₂₇	C₂×C₈×C₉	1	6	yes	—
48	C₂×C₂₇	C₂²×C₄×C₉	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂×C₈₁	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂×C₈₁	1	6	yes	—
51	C₂²×C₉	C₈×C₂₇	1	4	yes	—
52	C₂²×C₉	C₂×C₄×C₂₇	1	3	yes	—
53	C₄×C₉	C₂×C₄×C₂₇	1	3	yes	—
54	C₄×C₉	C₂³×C₂₇	1	1	yes	characteristic
55	C₂²×C₈	C₂₄₃	1	1	yes	characteristic, direct product factor, Sylow
56	C₂₇	C₂²×C₈×C₉	1	1	yes	characteristic
57	C₂³×C₃	C₄×C₈₁	1	1	yes	characteristic
58	C₂×C₃×C₄	C₄×C₈₁	1	3	yes	—
59	C₂×C₃×C₄	C₂²×C₈₁	1	3	yes	—
60	C₃×C₈	C₂²×C₈₁	1	4	yes	—
61	C₂×C₉	C₂×C₈×C₂₇	1	6	yes	—
62	C₂×C₉	C₂²×C₄×C₂₇	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₂₄₃	1	1	yes	characteristic
64	C₂×C₈	C₂×C₂₄₃	1	6	yes	direct product factor
65	C₂²×C₃	C₈×C₈₁	1	4	yes	—
66	C₂²×C₃	C₂×C₄×C₈₁	1	3	yes	—
67	C₃×C₄	C₂×C₄×C₈₁	1	3	yes	—
68	C₃×C₄	C₂³×C₈₁	1	1	yes	characteristic
69	C₉	C₂²×C₈×C₂₇	1	1	yes	characteristic
70	C₂³	C₄×C₂₄₃	1	1	yes	characteristic
71	C₂×C₄	C₄×C₂₄₃	1	3	yes	—
72	C₂×C₄	C₂²×C₂₄₃	1	3	yes	—
73	C₈	C₂²×C₂₄₃	1	4	yes	direct product factor
74	C₂×C₃	C₂×C₈×C₈₁	1	6	yes	—
75	C₂×C₃	C₂²×C₄×C₈₁	1	1	yes	characteristic
76	C₂²	C₈×C₂₄₃	1	4	yes	direct product factor
77	C₂²	C₂×C₄×C₂₄₃	1	3	yes	—
78	C₄	C₂×C₄×C₂₄₃	1	3	yes	—
79	C₄	C₂³×C₂₄₃	1	1	yes	characteristic
80	C₃	C₂²×C₈×C₈₁	1	1	yes	characteristic
81	C₂	C₂×C₈×C₂₄₃	1	6	yes	direct product factor
82	C₂	C₂²×C₄×C₂₄₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈₁	C₂³×C₃
socle, prime radical	C₂³×C₃	C₄×C₈₁
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×C₂₄₃	C₂	direct (N × H)
7. C₈×C₂₄₃	C₂²	direct (N × H)
10. C₂²×C₂₄₃	C₈	direct (N × H)
19. C₂×C₂₄₃	C₂×C₈	direct (N × H)
28. C₂₄₃	C₂²×C₈	direct (N × H)
55. C₂²×C₈	C₂₄₃	direct (N × H)
64. C₂×C₈	C₂×C₂₄₃	direct (N × H)
73. C₈	C₂²×C₂₄₃	direct (N × H)
76. C₂²	C₈×C₂₄₃	direct (N × H)
81. C₂	C₂×C₈×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
C₂²×C₄×C₂₄₃	2
C₂×C₈×C₂₄₃	2
C₂²×C₈×C₈₁	3

Subgroups — C22×C8×C243 #72186708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₂₄₃ #72186708