Subgroups — C₂²×C₂₃×#504 #19210248

Order 5888.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#504	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₂₃×M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂³×C₁₆×C₂₃	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₂₃×#131	C₄	1	12	yes	—
7	C₂×C₂₃×#131	C₂²	1	12	yes	—
8	C₂²×M₁₆×C₂₃	C₄	1	3	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	3	yes	—
10	C₂₃×#504	C₂²	1	64	yes	direct product factor
11	C₂×C₂₃×M₃₂	C₂²	1	36	yes	complemented
12	C₂²×C₁₆×C₂₃	C₂²	1	3	yes	—
13	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₂₃	C₂²	1	36	yes	complemented
15	C₂×#41×C₂₃	C₈	1	12	yes	—
16	C₂×#41×C₂₃	C₂×C₄	1	12	yes	—
17	C₂²×Q₈×C₂₃	C₈	1	1	yes	characteristic
18	C₂²×D₄×C₂₃	C₈	1	3	yes	—
19	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
20	C₂₃×#131	C₂×C₄	1	48	yes	—
21	C₂₃×#131	C₂³	1	16	yes	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	36	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	18	yes	—
25	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂³	1	18	yes	—
27	C₂₃×M₃₂	C₂³	1	48	yes	complemented
28	C₂×C₁₆×C₂₃	C₂³	1	18	yes	—
29	C₂×C₁₆×C₂₃	C₂³	1	6	yes	—
30	C₂×C₁₆×C₂₃	C₂³	1	48	yes	complemented
31	C₂⁴×C₂₃	C₂×C₈	1	3	yes	—
32	#41×C₂₃	C₂×C₈	1	48	yes	—
33	#41×C₂₃	C₂²×C₄	1	16	yes	—
34	C₂×Q₈×C₂₃	C₂×C₈	1	12	yes	—
35	C₂×D₄×C₂₃	C₂×C₈	1	36	yes	—
36	C₂²×C₄×C₂₃	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₂₃	C₂×C₈	1	18	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂²×C₄	1	18	yes	—
40	M₁₆×C₂₃	C₂²×C₄	1	48	yes	—
41	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₂₃	C₂²×C₄	1	18	yes	—
43	C₂×C₈×C₂₃	C₂²×C₄	1	36	yes	—
44	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
45	C₂×C₈×C₂₃	C₂⁴	1	12	yes	—
46	C₁₆×C₂₃	C₂⁴	1	4	yes	—
47	C₁₆×C₂₃	C₂⁴	1	12	yes	—
48	C₂²×#504	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₃	—	2	12	no	—
50	C₂³×C₂₃	C₂²×C₈	1	18	yes	—
51	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
52	Q₈×C₂₃	C₂²×C₈	1	16	yes	—
53	D₄×C₂₃	C₂²×C₈	1	48	yes	—
54	C₂×C₄×C₂₃	C₂²×C₈	1	3	yes	—
55	C₂×C₄×C₂₃	C₂²×C₈	1	18	yes	—
56	C₂×C₄×C₂₃	C₂²×C₈	1	36	yes	—
57	C₂×C₄×C₂₃	C₂³×C₄	1	3	yes	—
58	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
59	C₈×C₂₃	C₂³×C₄	1	3	yes	—
60	C₈×C₂₃	C₂³×C₄	1	12	yes	—
61	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
62	C₂²×#131	C₂×C₂₃	1	1	yes	characteristic
63	C₂×#504	C₂×C₂₃	1	24	yes	direct product factor
64	C₂²×M₃₂	C₂×C₂₃	1	3	yes	complemented
65	C₂³×C₁₆	C₂×C₂₃	1	3	yes	complemented
66	C₂²×C₂₃	—	2	36	no	—
67	C₂²×C₂₃	#504	1	4	yes	direct product factor
68	C₂²×C₂₃	C₂³×C₈	1	12	yes	—
69	C₂²×C₂₃	C₂³×C₈	1	3	yes	—
70	C₄×C₂₃	C₂³×C₈	1	3	yes	—
71	C₄×C₂₃	C₂³×C₈	1	12	yes	—
72	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
73	C₂²×#41	C₄×C₂₃	1	1	yes	characteristic
74	C₂×#131	C₄×C₂₃	1	12	yes	—
75	C₂×#131	C₂²×C₂₃	1	12	yes	—
76	C₂²×M₁₆	C₄×C₂₃	1	3	yes	—
77	C₂³×C₈	C₂²×C₂₃	1	3	yes	—
78	#504	C₂²×C₂₃	1	64	yes	direct product factor
79	C₂×M₃₂	C₂²×C₂₃	1	36	yes	complemented
80	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
81	C₂²×C₁₆	C₂²×C₂₃	1	3	yes	—
82	C₂²×C₁₆	C₂²×C₂₃	1	36	yes	complemented
83	C₂×C₂₃	—	2	12	no	—
84	C₂×C₂₃	C₂×#504	1	6	yes	direct product factor
85	C₂×C₂₃	C₂⁴×C₈	1	1	yes	characteristic
86	C₂×#41	C₈×C₂₃	1	12	yes	—
87	C₂×#41	C₂×C₄×C₂₃	1	12	yes	—
88	C₂²×Q₈	C₈×C₂₃	1	1	yes	characteristic
89	C₂²×D₄	C₈×C₂₃	1	3	yes	—
90	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
91	#131	C₂×C₄×C₂₃	1	48	yes	—
92	#131	C₂³×C₂₃	1	16	yes	—
93	C₂×M₁₆	C₂×C₄×C₂₃	1	36	yes	—
94	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
95	C₂²×C₈	C₂×C₄×C₂₃	1	18	yes	—
96	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₂₃	1	18	yes	—
98	M₃₂	C₂³×C₂₃	1	48	yes	complemented
99	C₂×C₁₆	C₂³×C₂₃	1	6	yes	—
100	C₂×C₁₆	C₂³×C₂₃	1	18	yes	—
101	C₂×C₁₆	C₂³×C₂₃	1	48	yes	complemented
102	C₂₃	C₂²×#504	1	1	yes	characteristic, direct product factor, Sylow
103	C₂⁴	C₂×C₈×C₂₃	1	3	yes	—
104	#41	C₂×C₈×C₂₃	1	48	yes	—
105	#41	C₂²×C₄×C₂₃	1	16	yes	—
106	C₂×Q₈	C₂×C₈×C₂₃	1	12	yes	—
107	C₂×D₄	C₂×C₈×C₂₃	1	36	yes	—
108	C₂²×C₄	C₂×C₈×C₂₃	1	3	yes	—
109	C₂²×C₄	C₂×C₈×C₂₃	1	18	yes	—
110	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×C₄×C₂₃	1	18	yes	—
112	M₁₆	C₂²×C₄×C₂₃	1	48	yes	—
113	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
114	C₂×C₈	C₂²×C₄×C₂₃	1	18	yes	—
115	C₂×C₈	C₂²×C₄×C₂₃	1	36	yes	—
116	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
117	C₂×C₈	C₂⁴×C₂₃	1	12	yes	—
118	C₁₆	C₂⁴×C₂₃	1	4	yes	—
119	C₁₆	C₂⁴×C₂₃	1	12	yes	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×C₈×C₂₃	1	18	yes	—
122	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
123	Q₈	C₂²×C₈×C₂₃	1	16	yes	—
124	D₄	C₂²×C₈×C₂₃	1	48	yes	—
125	C₂×C₄	C₂²×C₈×C₂₃	1	3	yes	—
126	C₂×C₄	C₂²×C₈×C₂₃	1	18	yes	—
127	C₂×C₄	C₂²×C₈×C₂₃	1	36	yes	—
128	C₂×C₄	C₂³×C₄×C₂₃	1	3	yes	—
129	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
130	C₈	C₂³×C₄×C₂₃	1	3	yes	—
131	C₈	C₂³×C₄×C₂₃	1	12	yes	—
132	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
133	C₂²	—	2	36	no	—
134	C₂²	C₂₃×#504	1	4	yes	direct product factor
135	C₂²	C₂³×C₈×C₂₃	1	12	yes	—
136	C₂²	C₂³×C₈×C₂₃	1	3	yes	—
137	C₄	C₂³×C₈×C₂₃	1	3	yes	—
138	C₄	C₂³×C₈×C₂₃	1	12	yes	—
139	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×C₂₃×#504	1	6	yes	direct product factor
142	C₂	C₂⁴×C₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆×C₂₃	C₂²
commutator	C₂	C₂⁴×C₈×C₂₃
Frattini, cosocle	C₈	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂²×C₈
prime radical	C₂²×#41×C₂₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#504	C₂	direct (N × H)
3. C₂²×C₂₃×M₃₂	C₂	semidirect
4. C₂³×C₁₆×C₂₃	C₂	semidirect
10. C₂₃×#504	C₂²	direct (N × H)
11. C₂×C₂₃×M₃₂	C₂²	semidirect
14. C₂²×C₁₆×C₂₃	C₂²	semidirect
27. C₂₃×M₃₂	C₂³	semidirect
30. C₂×C₁₆×C₂₃	C₂³	semidirect
48. C₂²×#504	C₂₃	direct (N × H)
63. C₂×#504	C₂×C₂₃	direct (N × H)
64. C₂²×M₃₂	C₂×C₂₃	semidirect
65. C₂³×C₁₆	C₂×C₂₃	semidirect
67. C₂²×C₂₃	#504	direct (N × H)
78. #504	C₂²×C₂₃	direct (N × H)
79. C₂×M₃₂	C₂²×C₂₃	semidirect
82. C₂²×C₁₆	C₂²×C₂₃	semidirect
84. C₂×C₂₃	C₂×#504	direct (N × H)
98. M₃₂	C₂³×C₂₃	semidirect
101. C₂×C₁₆	C₂³×C₂₃	semidirect
102. C₂₃	C₂²×#504	direct (N × H)
134. C₂²	C₂₃×#504	direct (N × H)
141. C₂	C₂×C₂₃×#504	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×C₂₃×#131	2
C₂×C₂₃×#504	2
C₂²×C₂₃×M₃₂	2
C₂³×C₁₆×C₂₃	2
C₂²×#504	23

Subgroups — C22×C23×#504 #19210248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#504 #19210248