Subgroups — C₂³×C₁₁×M₃₂ #14639814

Order 2816.

2334 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×M₃₂	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₁×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₄×C₁₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₁	C₄	1	14	yes	—
7	C₂³×C₈×C₁₁	C₂²	1	14	yes	—
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×M₃₂	C₂²	1	112	yes	direct product factor
10	C₂²×C₁₁×C₁₆	C₂²	1	28	yes	complemented
11	C₂⁵×C₁₁	C₈	1	1	yes	characteristic
12	C₂³×C₄×C₁₁	C₈	1	14	yes	—
13	C₂³×C₄×C₁₁	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₁	C₂×C₄	1	14	yes	—
15	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	C₂×C₄	1	14	yes	—
17	C₂²×C₈×C₁₁	C₂×C₄	1	84	yes	—
18	C₂²×C₈×C₁₁	C₂×C₄	1	7	yes	—
19	C₂²×C₈×C₁₁	C₂³	1	7	yes	—
20	C₂²×C₈×C₁₁	C₂³	1	28	yes	—
21	C₁₁×M₃₂	C₂³	1	64	yes	direct product factor
22	C₂×C₁₁×C₁₆	C₂³	1	56	yes	complemented
23	C₂³×M₃₂	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁴×C₁₁	—	2	8	no	—
25	C₂⁴×C₁₁	C₂×C₈	1	14	yes	—
26	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₁	C₂×C₈	1	84	yes	—
28	C₂²×C₄×C₁₁	C₂×C₈	1	14	yes	—
29	C₂²×C₄×C₁₁	C₂×C₈	1	7	yes	—
30	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
31	C₂²×C₄×C₁₁	C₂²×C₄	1	7	yes	—
32	C₂×C₈×C₁₁	C₂²×C₄	1	56	yes	—
33	C₂×C₈×C₁₁	C₂²×C₄	1	28	yes	—
34	C₂×C₈×C₁₁	C₂²×C₄	1	21	yes	—
35	C₂×C₈×C₁₁	C₂⁴	1	7	yes	—
36	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
37	C₁₁×C₁₆	C₂⁴	1	16	yes	complemented
38	C₂⁴×C₈	C₂×C₁₁	1	1	yes	characteristic
39	C₂²×M₃₂	C₂×C₁₁	1	28	yes	direct product factor
40	C₂³×C₁₆	C₂×C₁₁	1	2	yes	complemented
41	C₂³×C₁₁	—	2	56	no	—
42	C₂³×C₁₁	M₃₂	1	8	yes	direct product factor
43	C₂³×C₁₁	C₂²×C₈	1	28	yes	—
44	C₂³×C₁₁	C₂²×C₈	1	7	yes	—
45	C₂×C₄×C₁₁	C₂²×C₈	1	56	yes	—
46	C₂×C₄×C₁₁	C₂²×C₈	1	21	yes	—
47	C₂×C₄×C₁₁	C₂²×C₈	1	28	yes	—
48	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
49	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
50	C₈×C₁₁	C₂³×C₄	1	7	yes	—
51	C₈×C₁₁	C₂³×C₄	1	8	yes	—
52	C₈×C₁₁	C₂⁵	1	1	yes	characteristic
53	C₂⁴×C₄	C₄×C₁₁	1	1	yes	characteristic
54	C₂³×C₈	C₄×C₁₁	1	14	yes	—
55	C₂³×C₈	C₄×C₁₁	1	1	yes	characteristic
56	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
57	C₂³×C₈	C₂²×C₁₁	1	14	yes	—
58	C₂×M₃₂	C₂²×C₁₁	1	112	yes	direct product factor
59	C₂²×C₁₆	C₂²×C₁₁	1	28	yes	complemented
60	C₂²×C₁₁	—	2	56	no	—
61	C₂²×C₁₁	C₂×M₃₂	1	28	yes	direct product factor
62	C₂²×C₁₁	C₂³×C₈	1	8	yes	—
63	C₂²×C₁₁	C₂³×C₈	1	7	yes	—
64	C₄×C₁₁	C₂³×C₈	1	7	yes	—
65	C₄×C₁₁	C₂³×C₈	1	8	yes	—
66	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
67	C₂⁵	C₈×C₁₁	1	1	yes	characteristic
68	C₂³×C₄	C₈×C₁₁	1	14	yes	—
69	C₂³×C₄	C₈×C₁₁	1	1	yes	characteristic
70	C₂³×C₄	C₂×C₄×C₁₁	1	14	yes	—
71	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₄×C₁₁	1	7	yes	—
73	C₂²×C₈	C₂×C₄×C₁₁	1	84	yes	—
74	C₂²×C₈	C₂×C₄×C₁₁	1	14	yes	—
75	C₂²×C₈	C₂³×C₁₁	1	7	yes	—
76	C₂²×C₈	C₂³×C₁₁	1	28	yes	—
77	M₃₂	C₂³×C₁₁	1	64	yes	direct product factor
78	C₂×C₁₆	C₂³×C₁₁	1	56	yes	complemented
79	C₂×C₁₁	—	2	8	no	—
80	C₂×C₁₁	C₂²×M₃₂	1	14	yes	direct product factor
81	C₂×C₁₁	C₂⁴×C₈	1	1	yes	characteristic
82	C₂⁴	—	2	8	no	—
83	C₂⁴	C₂×C₈×C₁₁	1	14	yes	—
84	C₂⁴	C₂×C₈×C₁₁	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₈×C₁₁	1	84	yes	—
86	C₂²×C₄	C₂×C₈×C₁₁	1	14	yes	—
87	C₂²×C₄	C₂×C₈×C₁₁	1	7	yes	—
88	C₂²×C₄	C₂²×C₄×C₁₁	1	28	yes	—
89	C₂²×C₄	C₂²×C₄×C₁₁	1	7	yes	—
90	C₂×C₈	C₂²×C₄×C₁₁	1	21	yes	—
91	C₂×C₈	C₂²×C₄×C₁₁	1	56	yes	—
92	C₂×C₈	C₂²×C₄×C₁₁	1	28	yes	—
93	C₂×C₈	C₂⁴×C₁₁	1	7	yes	—
94	C₂×C₈	C₂⁴×C₁₁	1	8	yes	—
95	C₁₆	C₂⁴×C₁₁	1	16	yes	complemented
96	C₁₁	C₂³×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
97	C₂³	—	2	56	no	—
98	C₂³	C₁₁×M₃₂	1	8	yes	direct product factor
99	C₂³	C₂²×C₈×C₁₁	1	28	yes	—
100	C₂³	C₂²×C₈×C₁₁	1	7	yes	—
101	C₂×C₄	C₂²×C₈×C₁₁	1	56	yes	—
102	C₂×C₄	C₂²×C₈×C₁₁	1	21	yes	—
103	C₂×C₄	C₂²×C₈×C₁₁	1	28	yes	—
104	C₂×C₄	C₂³×C₄×C₁₁	1	8	yes	—
105	C₂×C₄	C₂³×C₄×C₁₁	1	7	yes	—
106	C₈	C₂³×C₄×C₁₁	1	7	yes	—
107	C₈	C₂³×C₄×C₁₁	1	8	yes	—
108	C₈	C₂⁵×C₁₁	1	1	yes	characteristic
109	C₂²	—	2	56	no	—
110	C₂²	C₂×C₁₁×M₃₂	1	28	yes	direct product factor
111	C₂²	C₂³×C₈×C₁₁	1	8	yes	—
112	C₂²	C₂³×C₈×C₁₁	1	7	yes	—
113	C₄	C₂³×C₈×C₁₁	1	7	yes	—
114	C₄	C₂³×C₈×C₁₁	1	8	yes	—
115	C₄	C₂⁴×C₄×C₁₁	1	1	yes	characteristic
116	C₂	—	2	8	no	—
117	C₂	C₂²×C₁₁×M₃₂	1	14	yes	direct product factor
118	C₂	C₂⁴×C₈×C₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈×C₁₁	C₂²
commutator	C₂	C₂⁴×C₈×C₁₁
Frattini, cosocle	C₈	C₂⁵×C₁₁
socle	C₂⁴×C₁₁	C₂×C₈
prime radical	C₂⁵×C₁₁	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₁×M₃₂	C₂	direct (N × H)
3. C₂³×C₁₁×C₁₆	C₂	semidirect
9. C₂×C₁₁×M₃₂	C₂²	direct (N × H)
10. C₂²×C₁₁×C₁₆	C₂²	semidirect
21. C₁₁×M₃₂	C₂³	direct (N × H)
22. C₂×C₁₁×C₁₆	C₂³	semidirect
23. C₂³×M₃₂	C₁₁	direct (N × H)
37. C₁₁×C₁₆	C₂⁴	semidirect
39. C₂²×M₃₂	C₂×C₁₁	direct (N × H)
40. C₂³×C₁₆	C₂×C₁₁	semidirect
42. C₂³×C₁₁	M₃₂	direct (N × H)
58. C₂×M₃₂	C₂²×C₁₁	direct (N × H)
59. C₂²×C₁₆	C₂²×C₁₁	semidirect
61. C₂²×C₁₁	C₂×M₃₂	direct (N × H)
77. M₃₂	C₂³×C₁₁	direct (N × H)
78. C₂×C₁₆	C₂³×C₁₁	semidirect
80. C₂×C₁₁	C₂²×M₃₂	direct (N × H)
95. C₁₆	C₂⁴×C₁₁	semidirect
96. C₁₁	C₂³×M₃₂	direct (N × H)
98. C₂³	C₁₁×M₃₂	direct (N × H)
110. C₂²	C₂×C₁₁×M₃₂	direct (N × H)
117. C₂	C₂²×C₁₁×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₁₁	2
C₂²×C₁₁×M₃₂	2
C₂³×C₁₁×C₁₆	2
C₂³×M₃₂	11

Subgroups — C23×C11×M32 #14639814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×M₃₂ #14639814