Subgroups — C₂₃×#13551 #19161177

Order 5888.

214 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×M₆₄	C₂	1	2	yes	maximal
2	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1398	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#405	C₄	1	1	yes	characteristic
7	C₂₃×M₆₄	C₄	1	4	yes	complemented
8	C₂₃×M₆₄	C₂²	1	4	yes	—
9	C₂×C₂₃×C₃₂	C₄	1	4	yes	complemented
10	C₂₃×#346	C₂²	1	2	yes	—
11	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
12	C₂×M₁₆×C₂₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂₃×#117	C₈	1	1	yes	characteristic
15	C₂₃×M₃₂	C₈	1	4	yes	—
16	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×#97	C₈	1	2	yes	—
22	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂₃×C₃₂	C₂×C₄	1	8	yes	—
24	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	C₄²	1	2	yes	—
27	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂²⋊C₄×C₂₃	—	2	1	no	—
33	C₄²×C₂₃	C₂×C₈	1	2	yes	—
34	C₁₆×C₂₃	C₄²	1	1	yes	characteristic
35	C₁₆×C₂₃	C₄²	1	1	yes	characteristic
36	C₁₆×C₂₃	C₂×C₈	1	4	yes	—
37	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
38	#13551	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂³×C₂₃	M₃₂	1	1	yes	characteristic
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
42	C₂×C₄×C₂₃	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
44	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
45	C₈×C₂₃	—	2	2	no	—
46	C₈×C₂₃	C₄×C₈	1	2	yes	—
47	C₈×C₂₃	C₂×C₄²	1	1	yes	characteristic
48	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×M₆₄	C₂×C₂₃	1	2	yes	—
50	#2108	C₂×C₂₃	1	1	yes	characteristic
51	#1398	C₂×C₂₃	1	4	yes	complemented
52	C₂²×C₂₃	—	4	1	no	—
53	C₂²×C₂₃	#346	1	2	yes	—
54	C₂²×C₂₃	C₂×M₃₂	1	1	yes	characteristic
55	C₄×C₂₃	—	4	2	no	—
56	C₄×C₂₃	#346	1	2	yes	—
57	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
58	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
60	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
61	#405	C₄×C₂₃	1	1	yes	characteristic
62	M₆₄	C₄×C₂₃	1	4	yes	complemented
63	M₆₄	C₂²×C₂₃	1	4	yes	—
64	C₂×C₃₂	C₄×C₂₃	1	4	yes	complemented
65	#346	C₂²×C₂₃	1	2	yes	—
66	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	—	2	2	no	—
69	C₂×C₂₃	C₂×#346	1	1	yes	characteristic
70	C₂×M₁₆	C₈×C₂₃	1	1	yes	characteristic
71	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
72	#117	C₈×C₂₃	1	1	yes	characteristic
73	M₃₂	C₈×C₂₃	1	4	yes	—
74	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
76	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
77	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
78	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
79	#97	C₈×C₂₃	1	2	yes	—
80	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
81	C₃₂	C₂×C₄×C₂₃	1	8	yes	—
82	C₂₃	#13551	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	C₂×C₈	C₄²×C₂₃	1	2	yes	—
86	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
87	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
88	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
89	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×C₈×C₂₃	1	2	yes	—
93	C₁₆	C₄²×C₂₃	1	1	yes	characteristic
94	C₁₆	C₄²×C₂₃	1	1	yes	characteristic
95	C₁₆	C₂×C₈×C₂₃	1	4	yes	—
96	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
97	C₂³	C₂₃×M₃₂	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
100	C₂×C₄	C₂₃×M₃₂	1	1	yes	characteristic
101	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
102	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	C₄×C₈×C₂₃	1	2	yes	—
105	C₈	C₂×C₄²×C₂₃	1	1	yes	characteristic
106	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₂₃×#346	1	2	yes	—
109	C₂²	C₂×C₂₃×M₃₂	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₂₃×#346	1	2	yes	—
112	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
113	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₂₃×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₂₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂×C₂₃	C₂×#346
prime radical	C₂³×C₂₃	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1398	C₂	semidirect
7. C₂₃×M₆₄	C₄	semidirect
9. C₂×C₂₃×C₃₂	C₄	semidirect
38. #13551	C₂₃	direct (N × H)
51. #1398	C₂×C₂₃	semidirect
62. M₆₄	C₄×C₂₃	semidirect
64. C₂×C₃₂	C₄×C₂₃	semidirect
82. C₂₃	#13551	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×C₂₃×M₆₄	2
C₂₃×#2108	2
C₂₃×#1398	2
#13551	23

Subgroups — C23×#13551 #19161177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13551 #19161177