Subgroups — C₈²×C₁₃×C₂₇ #72199864

Order 22464.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×C₁₃×C₂₇	C₂	1	3	yes	maximal
2	C₈²×C₉×C₁₃	C₃	1	1	yes	maximal, characteristic
3	C₄²×C₁₃×C₂₇	C₂²	1	1	yes	characteristic
4	C₂×C₈×C₁₃×C₂₇	C₄	1	6	yes	—
5	C₄×C₈×C₉×C₁₃	C₂×C₃	1	3	yes	—
6	C₂×C₄×C₁₃×C₂₇	C₂×C₄	1	3	yes	—
7	C₈×C₁₃×C₂₇	C₈	1	12	yes	direct product factor
8	C₃×C₈²×C₁₃	C₉	1	1	yes	characteristic
9	C₂×C₈×C₉×C₁₃	C₃×C₄	1	6	yes	—
10	C₄²×C₉×C₁₃	C₂²×C₃	1	1	yes	characteristic
11	C₈²×C₂₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₁₃×C₂₇	C₄²	1	1	yes	characteristic
13	C₄×C₁₃×C₂₇	C₂×C₈	1	6	yes	—
14	C₃×C₄×C₈×C₁₃	C₂×C₉	1	3	yes	—
15	C₂×C₄×C₉×C₁₃	C₂×C₃×C₄	1	3	yes	—
16	C₈×C₉×C₁₃	C₃×C₈	1	12	yes	—
17	C₄×C₈×C₂₇	C₂×C₁₃	1	3	yes	—
18	C₈²×C₁₃	C₂₇	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₁₃×C₂₇	C₄×C₈	1	3	yes	—
20	C₂×C₃×C₈×C₁₃	C₄×C₉	1	6	yes	—
21	C₃×C₄²×C₁₃	C₂²×C₉	1	1	yes	characteristic
22	C₈²×C₉	C₃×C₁₃	1	1	yes	characteristic
23	C₂²×C₉×C₁₃	C₃×C₄²	1	1	yes	characteristic
24	C₄×C₉×C₁₃	C₂×C₃×C₈	1	6	yes	—
25	C₂×C₈×C₂₇	C₄×C₁₃	1	6	yes	—
26	C₄²×C₂₇	C₂²×C₁₃	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂×C₂₇	1	3	yes	—
28	C₁₃×C₂₇	C₈²	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃×C₄×C₁₃	C₂×C₄×C₉	1	3	yes	—
30	C₃×C₈×C₁₃	C₈×C₉	1	12	yes	—
31	C₄×C₈×C₉	C₂×C₃×C₁₃	1	3	yes	—
32	C₂×C₉×C₁₃	C₃×C₄×C₈	1	3	yes	—
33	C₂×C₄×C₂₇	C₂×C₄×C₁₃	1	3	yes	—
34	C₈×C₂₇	C₈×C₁₃	1	12	yes	direct product factor
35	C₂×C₈×C₁₃	C₄×C₂₇	1	6	yes	—
36	C₄²×C₁₃	C₂²×C₂₇	1	1	yes	characteristic
37	C₃×C₈²	C₉×C₁₃	1	1	yes	characteristic
38	C₂²×C₃×C₁₃	C₄²×C₉	1	1	yes	characteristic
39	C₃×C₄×C₁₃	C₂×C₈×C₉	1	6	yes	—
40	C₂×C₈×C₉	C₃×C₄×C₁₃	1	6	yes	—
41	C₄²×C₉	C₂²×C₃×C₁₃	1	1	yes	characteristic
42	C₉×C₁₃	C₃×C₈²	1	1	yes	characteristic
43	C₂²×C₂₇	C₄²×C₁₃	1	1	yes	characteristic
44	C₄×C₂₇	C₂×C₈×C₁₃	1	6	yes	—
45	C₂×C₄×C₁₃	C₂×C₄×C₂₇	1	3	yes	—
46	C₈×C₁₃	C₈×C₂₇	1	12	yes	direct product factor
47	C₃×C₄×C₈	C₂×C₉×C₁₃	1	3	yes	—
48	C₂×C₃×C₁₃	C₄×C₈×C₉	1	3	yes	—
49	C₂×C₄×C₉	C₂×C₃×C₄×C₁₃	1	3	yes	—
50	C₈×C₉	C₃×C₈×C₁₃	1	12	yes	—
51	C₈²	C₁₃×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₂₇	C₄×C₈×C₁₃	1	3	yes	—
53	C₂²×C₁₃	C₄²×C₂₇	1	1	yes	characteristic
54	C₄×C₁₃	C₂×C₈×C₂₇	1	6	yes	—
55	C₂×C₃×C₈	C₄×C₉×C₁₃	1	6	yes	—
56	C₃×C₄²	C₂²×C₉×C₁₃	1	1	yes	characteristic
57	C₃×C₁₃	C₈²×C₉	1	1	yes	characteristic
58	C₂²×C₉	C₃×C₄²×C₁₃	1	1	yes	characteristic
59	C₄×C₉	C₂×C₃×C₈×C₁₃	1	6	yes	—
60	C₄×C₈	C₂×C₁₃×C₂₇	1	3	yes	—
61	C₂₇	C₈²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₁₃	C₄×C₈×C₂₇	1	3	yes	—
63	C₂×C₃×C₄	C₂×C₄×C₉×C₁₃	1	3	yes	—
64	C₃×C₈	C₈×C₉×C₁₃	1	12	yes	—
65	C₂×C₉	C₃×C₄×C₈×C₁₃	1	3	yes	—
66	C₂×C₈	C₄×C₁₃×C₂₇	1	6	yes	—
67	C₄²	C₂²×C₁₃×C₂₇	1	1	yes	characteristic
68	C₁₃	C₈²×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₃	C₄²×C₉×C₁₃	1	1	yes	characteristic
70	C₃×C₄	C₂×C₈×C₉×C₁₃	1	6	yes	—
71	C₉	C₃×C₈²×C₁₃	1	1	yes	characteristic
72	C₂×C₄	C₂×C₄×C₁₃×C₂₇	1	3	yes	—
73	C₈	C₈×C₁₃×C₂₇	1	12	yes	direct product factor
74	C₂×C₃	C₄×C₈×C₉×C₁₃	1	3	yes	—
75	C₂²	C₄²×C₁₃×C₂₇	1	1	yes	characteristic
76	C₄	C₂×C₈×C₁₃×C₂₇	1	6	yes	—
77	C₃	C₈²×C₉×C₁₃	1	1	yes	characteristic
78	C₂	C₄×C₈×C₁₃×C₂₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²×C₉	C₂²×C₃×C₁₃
socle, prime radical	C₂²×C₃×C₁₃	C₄²×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₈×C₁₃×C₂₇	C₈	direct (N × H)
11. C₈²×C₂₇	C₁₃	direct (N × H)
18. C₈²×C₁₃	C₂₇	direct (N × H)
28. C₁₃×C₂₇	C₈²	direct (N × H)
34. C₈×C₂₇	C₈×C₁₃	direct (N × H)
46. C₈×C₁₃	C₈×C₂₇	direct (N × H)
51. C₈²	C₁₃×C₂₇	direct (N × H)
61. C₂₇	C₈²×C₁₃	direct (N × H)
68. C₁₃	C₈²×C₂₇	direct (N × H)
73. C₈	C₈×C₁₃×C₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₈×C₁₃×C₂₇	2
C₈²×C₉×C₁₃	3
C₈²×C₂₇	13

Subgroups — C82×C13×C27 #72199864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈²×C₁₃×C₂₇ #72199864