Subgroups — C₂₃×#13283 #19160909

Order 5888.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1380	C₂	1	4	yes	maximal
5	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂₃×#500	—	2	2	no	—
8	C₂₃×#500	C₄	1	4	yes	complemented
9	C₂×C₂₃×#107	—	2	1	no	—
10	C₂×C₂₃×#107	C₂²	1	2	yes	—
11	C₂×C₂₃×#107	C₂²	1	2	yes	—
12	C₂×C₂₃×#106	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#382	—	2	1	no	—
15	C₂₃×#346	—	2	2	no	—
16	C₂²×Q₈×C₂₃	—	4	1	no	—
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	—	2	4	no	—
20	C₂₃×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
24	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₂₃	—	2	2	no	—
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂×C₁₆×C₂₃	Q₈	1	4	yes	complemented
28	C₂₃×#107	—	2	4	no	—
29	C₂₃×#107	—	2	2	no	—
30	C₂₃×#107	—	2	4	no	—
31	C₂₃×#107	C₂×C₄	1	4	yes	—
32	C₂₃×#106	—	4	2	no	—
33	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₂₃	D₄	1	2	yes	—
35	C₂₃×#95	—	4	1	no	—
36	C₂₃×#95	—	4	1	no	—
37	C₂×Q₈×C₂₃	—	4	4	no	—
38	C₂×Q₈×C₂₃	—	8	4	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₂₃	#41	1	2	yes	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₄⋊C₄×C₂₃	—	4	2	no	—
56	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₂₃	SD₁₆	1	2	yes	—
58	C₁₆×C₂₃	—	2	4	no	—
59	#13283	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
61	Q₈×C₂₃	—	8	8	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	4	4	no	—
65	C₂×C₄×C₂₃	—	8	4	no	—
66	C₂×C₄×C₂₃	—	8	2	no	—
67	C₂×C₄×C₂₃	—	8	2	no	—
68	C₂×C₄×C₂₃	—	8	2	no	—
69	C₂×C₄×C₂₃	#102	1	2	yes	—
70	C₂×C₄×C₂₃	#102	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₂₃	#103	1	4	yes	—
78	C₈×C₂₃	C₄×Q₈	1	2	yes	—
79	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
81	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
82	#1846	C₂×C₂₃	1	1	yes	characteristic
83	#1380	C₂×C₂₃	1	4	yes	—
84	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
85	C₂²×C₂₃	#486	1	1	yes	characteristic
86	C₂²×C₂₃	#500	1	1	yes	characteristic
87	C₂²×C₂₃	#509	1	2	yes	—
88	C₂²×C₂₃	#510	1	2	yes	—
89	C₄×C₂₃	—	2	2	no	—
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	8	4	no	—
92	C₄×C₂₃	—	16	2	no	—
93	C₄×C₂₃	#391	1	1	yes	characteristic
94	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
95	C₄×C₂₃	#487	1	1	yes	characteristic
96	C₄×C₂₃	#498	1	1	yes	characteristic
97	C₂×C₄⋊Q₈	—	2	1	no	—
98	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
99	#500	—	2	2	no	—
100	#500	C₄×C₂₃	1	4	yes	complemented
101	C₂×#107	—	2	1	no	—
102	C₂×#107	C₂²×C₂₃	1	2	yes	—
103	C₂×#107	C₂²×C₂₃	1	2	yes	—
104	C₂×#106	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
106	#382	—	2	1	no	—
107	#346	—	2	2	no	—
108	C₂×C₂₃	#1990	1	1	yes	characteristic
109	C₂×C₂₃	#2140	1	1	yes	characteristic
110	C₂×C₂₃	#2141	1	1	yes	characteristic
111	C₂×C₂₃	#2243	1	2	yes	—
112	C₂×C₂₃	#2255	1	2	yes	—
113	C₂²×Q₈	—	4	1	no	—
114	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	Q₈×C₂₃	1	4	yes	complemented
125	#107	—	2	4	no	—
126	#107	—	2	2	no	—
127	#107	—	2	4	no	—
128	#107	C₂×C₄×C₂₃	1	4	yes	—
129	#106	—	4	2	no	—
130	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
131	C₄×C₈	D₄×C₂₃	1	2	yes	—
132	#95	—	4	1	no	—
133	#95	—	4	1	no	—
134	C₂₃	#13283	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
143	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
144	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
148	C₂×C₈	#41×C₂₃	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
155	C₄²	SD₁₆×C₂₃	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₂₃×#102	1	2	yes	—
167	C₂×C₄	C₂₃×#102	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
170	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
174	C₈	C₂₃×#103	1	4	yes	—
175	C₈	C₄×Q₈×C₂₃	1	2	yes	—
176	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
177	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
178	C₂²	C₂₃×#486	1	1	yes	characteristic
179	C₂²	C₂₃×#500	1	1	yes	characteristic
180	C₂²	C₂₃×#509	1	2	yes	—
181	C₂²	C₂₃×#510	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	16	2	no	—
186	C₄	C₂₃×#391	1	1	yes	characteristic
187	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
188	C₄	C₂₃×#487	1	1	yes	characteristic
189	C₄	C₂₃×#498	1	1	yes	characteristic
190	C₂	C₂₃×#1990	1	1	yes	characteristic
191	C₂	C₂₃×#2140	1	1	yes	characteristic
192	C₂	C₂₃×#2141	1	1	yes	characteristic
193	C₂	C₂₃×#2243	1	2	yes	—
194	C₂	C₂₃×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂₃×#500	C₄	semidirect
27. C₂×C₁₆×C₂₃	Q₈	semidirect
59. #13283	C₂₃	direct (N × H)
100. #500	C₄×C₂₃	semidirect
124. C₂×C₁₆	Q₈×C₂₃	semidirect
134. C₂₃	#13283	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×#500	2
C₂×C₂₃×#346	2
C₂₃×#1846	2
C₂₃×#1380	2
#13283	23

Subgroups — C23×#13283 #19160909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13283 #19160909