Subgroups — C₂×C₂₃×#1311 #19159023

Order 5888.

366 conjugacy classes of subgroups.

The subgroup lattice has 106 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1311	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₂₃	C₄	1	4	yes	—
5	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₄×C₁₆×C₂₃	C₄	1	4	yes	—
8	C₄×C₁₆×C₂₃	C₂²	1	4	yes	—
9	C₈²×C₂₃	C₄	1	2	yes	—
10	C₈²×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₃	C₈	1	4	yes	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₂₃	C₈	1	16	yes	complemented
15	C₂×C₁₆×C₂₃	C₂×C₄	1	8	yes	—
16	C₄×C₈×C₂₃	C₈	1	4	yes	—
17	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
23	C₂×C₈×C₂₃	—	2	8	no	—
24	C₂×C₈×C₂₃	C₄²	1	2	yes	—
25	C₂×C₈×C₂₃	C₂×C₈	1	8	yes	—
26	C₂×C₈×C₂₃	C₂×C₈	1	8	yes	—
27	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
28	C₄²×C₂₃	C₄²	1	1	yes	characteristic
29	C₄²×C₂₃	C₂×C₈	1	2	yes	—
30	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₁₆×C₂₃	C₂×C₈	1	16	yes	complemented
32	C₂×#1311	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₂₃	C₄×C₈	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
36	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
37	C₂×C₄×C₂₃	M₃₂	1	8	yes	—
38	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
40	C₈×C₂₃	—	2	8	no	—
41	C₈×C₂₃	C₄×C₈	1	4	yes	—
42	C₈×C₂₃	C₂²×C₈	1	4	yes	—
43	C₂×C₄×C₁₆	C₂×C₂₃	1	2	yes	—
44	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
45	#1311	C₂×C₂₃	1	4	yes	direct product factor
46	C₂²×C₂₃	C₈²	1	2	yes	—
47	C₂²×C₂₃	#346	1	4	yes	—
48	C₂²×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
49	C₄×C₂₃	C₈²	1	2	yes	—
50	C₄×C₂₃	#346	1	4	yes	—
51	C₄×C₂₃	C₂×C₄×C₈	1	2	yes	—
52	C₄×C₂₃	C₂×M₃₂	1	4	yes	—
53	C₂²×C₁₆	C₄×C₂₃	1	4	yes	—
54	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
55	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
56	C₄×C₁₆	C₄×C₂₃	1	4	yes	—
57	C₄×C₁₆	C₂²×C₂₃	1	4	yes	—
58	C₈²	C₄×C₂₃	1	2	yes	—
59	C₈²	C₂²×C₂₃	1	2	yes	—
60	C₂×C₂₃	#1311	1	4	yes	direct product factor
61	C₂×C₂₃	C₂×C₈²	1	1	yes	characteristic
62	C₂×C₂₃	C₂×#346	1	2	yes	—
63	C₂²×C₈	C₈×C₂₃	1	4	yes	—
64	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
65	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
66	C₂×C₁₆	C₈×C₂₃	1	16	yes	complemented
67	C₂×C₁₆	C₂×C₄×C₂₃	1	8	yes	—
68	C₄×C₈	C₈×C₂₃	1	4	yes	—
69	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
70	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
71	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
72	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
73	C₂₃	C₂×#1311	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	C₄²×C₂₃	1	2	yes	—
78	C₂×C₈	C₂×C₈×C₂₃	1	8	yes	—
79	C₂×C₈	C₂×C₈×C₂₃	1	8	yes	—
80	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
81	C₄²	C₄²×C₂₃	1	1	yes	characteristic
82	C₄²	C₂×C₈×C₂₃	1	2	yes	—
83	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
84	C₁₆	C₂×C₈×C₂₃	1	16	yes	complemented
85	C₂³	C₄×C₈×C₂₃	1	1	yes	characteristic
86	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
87	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
88	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
89	C₂×C₄	C₂₃×M₃₂	1	8	yes	—
90	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
91	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
92	C₈	—	2	8	no	—
93	C₈	C₄×C₈×C₂₃	1	4	yes	—
94	C₈	C₂²×C₈×C₂₃	1	4	yes	—
95	C₂²	C₈²×C₂₃	1	2	yes	—
96	C₂²	C₂₃×#346	1	4	yes	—
97	C₂²	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
98	C₄	C₈²×C₂₃	1	2	yes	—
99	C₄	C₂₃×#346	1	4	yes	—
100	C₄	C₂×C₄×C₈×C₂₃	1	2	yes	—
101	C₄	C₂×C₂₃×M₃₂	1	4	yes	—
102	C₂	C₂₃×#1311	1	4	yes	direct product factor
103	C₂	C₂×C₈²×C₂₃	1	1	yes	characteristic
104	C₂	C₂×C₂₃×#346	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₂₃	C₂²
commutator	C₂	C₂×C₈²×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1311	C₂	direct (N × H)
14. C₂×C₁₆×C₂₃	C₈	semidirect
31. C₁₆×C₂₃	C₂×C₈	semidirect
32. C₂×#1311	C₂₃	direct (N × H)
45. #1311	C₂×C₂₃	direct (N × H)
60. C₂×C₂₃	#1311	direct (N × H)
66. C₂×C₁₆	C₈×C₂₃	semidirect
73. C₂₃	C₂×#1311	direct (N × H)
84. C₁₆	C₂×C₈×C₂₃	semidirect
102. C₂	C₂₃×#1311	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₂₃	2
C₂×C₈²×C₂₃	2
C₂₃×#1311	2
C₂×#1311	23

Subgroups — C2×C23×#1311 #19159023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1311 #19159023