Subgroups — C₂×C₄×C₁₁×C₁₃×C₂₇ #72207656

Order 30888.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×C₁₃×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₁×C₁₃×C₂₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₄×C₉×C₁₁×C₁₃	C₃	1	1	yes	maximal, characteristic
4	C₂×C₁₁×C₁₃×C₂₇	C₄	1	2	yes	direct product factor
5	C₂×C₁₁×C₁₃×C₂₇	C₂²	1	1	yes	characteristic
6	C₁₁×C₁₃×C₂₇	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃×C₄×C₁₁×C₁₃	C₉	1	1	yes	characteristic
8	C₂×C₄×C₁₃×C₂₇	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₄×C₁₁×C₂₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₉×C₁₁×C₁₃	C₂×C₃	1	1	yes	characteristic
11	C₄×C₉×C₁₁×C₁₃	C₂×C₃	1	2	yes	—
12	C₂×C₉×C₁₁×C₁₃	C₃×C₄	1	2	yes	—
13	C₂×C₉×C₁₁×C₁₃	C₂²×C₃	1	1	yes	characteristic
14	C₂²×C₃×C₁₁×C₁₃	C₂×C₉	1	1	yes	characteristic
15	C₃×C₄×C₁₁×C₁₃	C₂×C₉	1	2	yes	—
16	C₂²×C₁₃×C₂₇	C₂×C₁₁	1	1	yes	characteristic
17	C₄×C₁₃×C₂₇	C₂×C₁₁	1	2	yes	direct product factor
18	C₉×C₁₁×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
19	C₂²×C₁₁×C₂₇	C₂×C₁₃	1	1	yes	characteristic
20	C₄×C₁₁×C₂₇	C₂×C₁₃	1	2	yes	direct product factor
21	C₂×C₄×C₁₁×C₁₃	C₂₇	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₄×C₉×C₁₃	C₃×C₁₁	1	1	yes	characteristic
23	C₂×C₃×C₁₁×C₁₃	C₄×C₉	1	2	yes	—
24	C₂×C₃×C₁₁×C₁₃	C₂²×C₉	1	1	yes	characteristic
25	C₂×C₄×C₉×C₁₁	C₃×C₁₃	1	1	yes	characteristic
26	C₂×C₁₃×C₂₇	C₄×C₁₁	1	2	yes	direct product factor
27	C₂×C₁₃×C₂₇	C₂²×C₁₁	1	1	yes	characteristic
28	C₂×C₁₁×C₂₇	C₄×C₁₃	1	2	yes	direct product factor
29	C₂×C₁₁×C₂₇	C₂²×C₁₃	1	1	yes	characteristic
30	C₂²×C₁₁×C₁₃	C₂×C₂₇	1	1	yes	characteristic
31	C₄×C₁₁×C₁₃	C₂×C₂₇	1	2	yes	direct product factor
32	C₂²×C₉×C₁₃	C₂×C₃×C₁₁	1	1	yes	characteristic
33	C₄×C₉×C₁₃	C₂×C₃×C₁₁	1	2	yes	—
34	C₃×C₁₁×C₁₃	C₂×C₄×C₉	1	1	yes	characteristic
35	C₂²×C₉×C₁₁	C₂×C₃×C₁₃	1	1	yes	characteristic
36	C₄×C₉×C₁₁	C₂×C₃×C₁₃	1	2	yes	—
37	C₁₃×C₂₇	C₂×C₄×C₁₁	1	1	yes	characteristic, direct product factor, Hall
38	C₂×C₃×C₄×C₁₃	C₉×C₁₁	1	1	yes	characteristic
39	C₁₁×C₂₇	C₂×C₄×C₁₃	1	1	yes	characteristic, direct product factor, Hall
40	C₂×C₁₁×C₁₃	C₄×C₂₇	1	2	yes	direct product factor
41	C₂×C₁₁×C₁₃	C₂²×C₂₇	1	1	yes	characteristic
42	C₂×C₃×C₄×C₁₁	C₉×C₁₃	1	1	yes	characteristic
43	C₂×C₉×C₁₃	C₃×C₄×C₁₁	1	2	yes	—
44	C₂×C₉×C₁₃	C₂²×C₃×C₁₁	1	1	yes	characteristic
45	C₂×C₄×C₂₇	C₁₁×C₁₃	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₉×C₁₁	C₃×C₄×C₁₃	1	2	yes	—
47	C₂×C₉×C₁₁	C₂²×C₃×C₁₃	1	1	yes	characteristic
48	C₂²×C₃×C₁₃	C₂×C₉×C₁₁	1	1	yes	characteristic
49	C₃×C₄×C₁₃	C₂×C₉×C₁₁	1	2	yes	—
50	C₁₁×C₁₃	C₂×C₄×C₂₇	1	1	yes	characteristic, direct product factor, Hall
51	C₂²×C₃×C₁₁	C₂×C₉×C₁₃	1	1	yes	characteristic
52	C₃×C₄×C₁₁	C₂×C₉×C₁₃	1	2	yes	—
53	C₉×C₁₃	C₂×C₃×C₄×C₁₁	1	1	yes	characteristic
54	C₂²×C₂₇	C₂×C₁₁×C₁₃	1	1	yes	characteristic
55	C₄×C₂₇	C₂×C₁₁×C₁₃	1	2	yes	direct product factor
56	C₂×C₄×C₁₃	C₁₁×C₂₇	1	1	yes	characteristic, direct product factor, Hall
57	C₉×C₁₁	C₂×C₃×C₄×C₁₃	1	1	yes	characteristic
58	C₂×C₄×C₁₁	C₁₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
59	C₂×C₃×C₁₃	C₄×C₉×C₁₁	1	2	yes	—
60	C₂×C₃×C₁₃	C₂²×C₉×C₁₁	1	1	yes	characteristic
61	C₂×C₄×C₉	C₃×C₁₁×C₁₃	1	1	yes	characteristic
62	C₂×C₃×C₁₁	C₄×C₉×C₁₃	1	2	yes	—
63	C₂×C₃×C₁₁	C₂²×C₉×C₁₃	1	1	yes	characteristic
64	C₂×C₂₇	C₄×C₁₁×C₁₃	1	2	yes	direct product factor
65	C₂×C₂₇	C₂²×C₁₁×C₁₃	1	1	yes	characteristic
66	C₂²×C₁₃	C₂×C₁₁×C₂₇	1	1	yes	characteristic
67	C₄×C₁₃	C₂×C₁₁×C₂₇	1	2	yes	direct product factor
68	C₂²×C₁₁	C₂×C₁₃×C₂₇	1	1	yes	characteristic
69	C₄×C₁₁	C₂×C₁₃×C₂₇	1	2	yes	direct product factor
70	C₃×C₁₃	C₂×C₄×C₉×C₁₁	1	1	yes	characteristic
71	C₂²×C₉	C₂×C₃×C₁₁×C₁₃	1	1	yes	characteristic
72	C₄×C₉	C₂×C₃×C₁₁×C₁₃	1	2	yes	—
73	C₃×C₁₁	C₂×C₄×C₉×C₁₃	1	1	yes	characteristic
74	C₂₇	C₂×C₄×C₁₁×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₁₃	C₄×C₁₁×C₂₇	1	2	yes	direct product factor
76	C₂×C₁₃	C₂²×C₁₁×C₂₇	1	1	yes	characteristic
77	C₂×C₃×C₄	C₉×C₁₁×C₁₃	1	1	yes	characteristic
78	C₂×C₁₁	C₄×C₁₃×C₂₇	1	2	yes	direct product factor
79	C₂×C₁₁	C₂²×C₁₃×C₂₇	1	1	yes	characteristic
80	C₂×C₉	C₃×C₄×C₁₁×C₁₃	1	2	yes	—
81	C₂×C₉	C₂²×C₃×C₁₁×C₁₃	1	1	yes	characteristic
82	C₁₃	C₂×C₄×C₁₁×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₃	C₂×C₉×C₁₁×C₁₃	1	1	yes	characteristic
84	C₃×C₄	C₂×C₉×C₁₁×C₁₃	1	2	yes	—
85	C₁₁	C₂×C₄×C₁₃×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
86	C₉	C₂×C₃×C₄×C₁₁×C₁₃	1	1	yes	characteristic
87	C₂×C₄	C₁₁×C₁₃×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃	C₄×C₉×C₁₁×C₁₃	1	2	yes	—
89	C₂×C₃	C₂²×C₉×C₁₁×C₁₃	1	1	yes	characteristic
90	C₂²	C₂×C₁₁×C₁₃×C₂₇	1	1	yes	characteristic
91	C₄	C₂×C₁₁×C₁₃×C₂₇	1	2	yes	direct product factor
92	C₃	C₂×C₄×C₉×C₁₁×C₁₃	1	1	yes	characteristic
93	C₂	C₄×C₁₁×C₁₃×C₂₇	1	2	yes	direct product factor
94	C₂	C₂²×C₁₁×C₁₃×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉	C₂²×C₃×C₁₁×C₁₃
socle, prime radical	C₂²×C₃×C₁₁×C₁₃	C₂×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₁×C₁₃×C₂₇	C₂	direct (N × H)
4. C₂×C₁₁×C₁₃×C₂₇	C₄	direct (N × H)
6. C₁₁×C₁₃×C₂₇	C₂×C₄	direct (N × H)
8. C₂×C₄×C₁₃×C₂₇	C₁₁	direct (N × H)
9. C₂×C₄×C₁₁×C₂₇	C₁₃	direct (N × H)
17. C₄×C₁₃×C₂₇	C₂×C₁₁	direct (N × H)
20. C₄×C₁₁×C₂₇	C₂×C₁₃	direct (N × H)
21. C₂×C₄×C₁₁×C₁₃	C₂₇	direct (N × H)
26. C₂×C₁₃×C₂₇	C₄×C₁₁	direct (N × H)
28. C₂×C₁₁×C₂₇	C₄×C₁₃	direct (N × H)
31. C₄×C₁₁×C₁₃	C₂×C₂₇	direct (N × H)
37. C₁₃×C₂₇	C₂×C₄×C₁₁	direct (N × H)
39. C₁₁×C₂₇	C₂×C₄×C₁₃	direct (N × H)
40. C₂×C₁₁×C₁₃	C₄×C₂₇	direct (N × H)
45. C₂×C₄×C₂₇	C₁₁×C₁₃	direct (N × H)
50. C₁₁×C₁₃	C₂×C₄×C₂₇	direct (N × H)
55. C₄×C₂₇	C₂×C₁₁×C₁₃	direct (N × H)
56. C₂×C₄×C₁₃	C₁₁×C₂₇	direct (N × H)
58. C₂×C₄×C₁₁	C₁₃×C₂₇	direct (N × H)
64. C₂×C₂₇	C₄×C₁₁×C₁₃	direct (N × H)
67. C₄×C₁₃	C₂×C₁₁×C₂₇	direct (N × H)
69. C₄×C₁₁	C₂×C₁₃×C₂₇	direct (N × H)
74. C₂₇	C₂×C₄×C₁₁×C₁₃	direct (N × H)
75. C₂×C₁₃	C₄×C₁₁×C₂₇	direct (N × H)
78. C₂×C₁₁	C₄×C₁₃×C₂₇	direct (N × H)
82. C₁₃	C₂×C₄×C₁₁×C₂₇	direct (N × H)
85. C₁₁	C₂×C₄×C₁₃×C₂₇	direct (N × H)
87. C₂×C₄	C₁₁×C₁₃×C₂₇	direct (N × H)
91. C₄	C₂×C₁₁×C₁₃×C₂₇	direct (N × H)
93. C₂	C₄×C₁₁×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
C₂²×C₁₁×C₁₃×C₂₇	2
C₄×C₁₁×C₁₃×C₂₇	2
C₂×C₄×C₉×C₁₁×C₁₃	3
C₂×C₄×C₁₃×C₂₇	11
C₂×C₄×C₁₁×C₂₇	13

Subgroups — C2×C4×C11×C13×C27 #72207656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₁×C₁₃×C₂₇ #72207656