Subgroups — C₂³×C₂₃×M₃₂ #19210247

Order 5888.

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

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×M₃₂	C₂	direct (N × H)
3. C₂³×C₁₆×C₂₃	C₂	semidirect
9. C₂×C₂₃×M₃₂	C₂²	direct (N × H)
10. C₂²×C₁₆×C₂₃	C₂²	semidirect
21. C₂₃×M₃₂	C₂³	direct (N × H)
22. C₂×C₁₆×C₂₃	C₂³	semidirect
36. C₁₆×C₂₃	C₂⁴	semidirect
37. C₂³×M₃₂	C₂₃	direct (N × H)
39. C₂³×C₂₃	M₃₂	direct (N × H)
51. C₂²×M₃₂	C₂×C₂₃	direct (N × H)
52. C₂³×C₁₆	C₂×C₂₃	semidirect
54. C₂²×C₂₃	C₂×M₃₂	direct (N × H)
65. C₂×M₃₂	C₂²×C₂₃	direct (N × H)
66. C₂²×C₁₆	C₂²×C₂₃	semidirect
68. C₂×C₂₃	C₂²×M₃₂	direct (N × H)
80. M₃₂	C₂³×C₂₃	direct (N × H)
81. C₂×C₁₆	C₂³×C₂₃	semidirect
82. C₂₃	C₂³×M₃₂	direct (N × H)
96. C₁₆	C₂⁴×C₂₃	semidirect
98. C₂³	C₂₃×M₃₂	direct (N × H)
110. C₂²	C₂×C₂₃×M₃₂	direct (N × H)
117. C₂	C₂²×C₂₃×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×C₂₃	2
C₂²×C₂₃×M₃₂	2
C₂³×C₁₆×C₂₃	2
C₂³×M₃₂	23

Subgroups — C23×C23×M32 #19210247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×M₃₂ #19210247