Subgroups — C₂₃×#23013 #19170639

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#464	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2367	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1959	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1936	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1880	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1763	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#464	C₄	1	8	yes	complemented
25	C₂₃×#441	—	2	4	no	—
26	C₄×SD₁₆×C₂₃	—	2	4	no	—
27	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#391	C₂²	1	1	yes	characteristic
42	C₂₃×#390	C₂²	1	1	yes	characteristic
43	C₂₃×#389	—	2	1	no	—
44	C₂₃×#389	—	2	1	no	—
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#389	C₂²	1	1	yes	characteristic
47	C₂₃×#388	—	2	1	no	—
48	C₂₃×#388	—	2	1	no	—
49	C₂₃×#385	—	2	1	no	—
50	C₂₃×#384	—	2	1	no	—
51	C₂₃×#384	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂×Q₁₆×C₂₃	—	2	2	no	—
67	C₂×Q₁₆×C₂₃	—	2	2	no	—
68	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×#124	—	2	2	no	—
77	C₂₃×#124	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₂₃	—	2	2	no	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	4	1	no	—
83	C₄×Q₈×C₂₃	—	4	1	no	—
84	C₄×Q₈×C₂₃	—	4	1	no	—
85	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	4	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	—	2	1	no	—
109	C₂×C₄²×C₂₃	—	2	1	no	—
110	C₂×C₄²×C₂₃	—	2	1	no	—
111	C₂×C₄²×C₂₃	—	2	1	no	—
112	C₂×C₄²×C₂₃	—	2	1	no	—
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
116	C₂₃×#106	—	2	2	no	—
117	C₂₃×#106	—	2	2	no	—
118	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#103	—	4	1	no	—
122	C₂₃×#103	C₂×C₄	1	4	yes	complemented
123	C₂₃×#103	D₄	1	4	yes	—
124	C₂₃×#102	—	2	2	no	—
125	C₂₃×#102	—	4	1	no	—
126	C₂₃×#102	C₂×C₄	1	4	yes	—
127	C₂₃×#97	—	2	2	no	—
128	C₂₃×#97	—	2	2	no	—
129	C₄×C₈×C₂₃	—	2	2	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂₃×#95	—	2	1	no	—
134	C₂₃×#95	—	2	1	no	—
135	C₂₃×#95	D₄	1	1	yes	characteristic
136	C₂₃×#95	D₄	1	1	yes	characteristic
137	C₂⁴×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
147	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
148	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₂₃	#41	1	2	yes	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	4	2	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	—	8	1	no	—
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	Q₁₆×C₂₃	—	2	8	no	—
174	SD₁₆×C₂₃	—	2	8	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
185	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
186	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
187	C₄⋊C₄×C₂₃	—	2	4	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	4	1	no	—
198	C₄⋊C₄×C₂₃	—	4	2	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	4	2	no	—
201	C₄⋊C₄×C₂₃	—	4	2	no	—
202	C₄⋊C₄×C₂₃	—	8	1	no	—
203	C₄⋊C₄×C₂₃	—	8	1	no	—
204	C₄⋊C₄×C₂₃	—	8	1	no	—
205	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
206	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	—	4	2	no	—
210	C₂²⋊C₄×C₂₃	—	4	2	no	—
211	C₂²⋊C₄×C₂₃	—	8	1	no	—
212	C₄²×C₂₃	—	2	4	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	2	1	no	—
217	C₄²×C₂₃	—	4	2	no	—
218	C₄²×C₂₃	—	4	2	no	—
219	C₄²×C₂₃	—	4	2	no	—
220	C₄²×C₂₃	—	8	1	no	—
221	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
222	C₄²×C₂₃	#41	1	2	yes	—
223	#23013	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	—	4	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	—	4	1	no	—
228	C₂³×C₂₃	—	4	1	no	—
229	C₂³×C₂₃	—	4	1	no	—
230	C₂³×C₂₃	—	4	1	no	—
231	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₂₃	—	2	2	no	—
233	Q₈×C₂₃	—	2	2	no	—
234	Q₈×C₂₃	—	2	2	no	—
235	Q₈×C₂₃	—	2	2	no	—
236	Q₈×C₂₃	—	2	2	no	—
237	Q₈×C₂₃	—	4	1	no	—
238	Q₈×C₂₃	—	4	1	no	—
239	Q₈×C₂₃	C₄×D₄	1	4	yes	—
240	D₄×C₂₃	—	2	2	no	—
241	D₄×C₂₃	—	2	2	no	—
242	D₄×C₂₃	—	4	1	no	—
243	D₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	8	4	no	—
281	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
282	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
283	C₂×C₄×C₂₃	#137	1	2	yes	—
284	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
285	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
289	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
290	C₈×C₂₃	—	2	2	no	—
291	C₈×C₂₃	—	2	2	no	—
292	C₈×C₂₃	—	2	2	no	—
293	C₈×C₂₃	—	4	1	no	—
294	C₂×#464	C₂×C₂₃	1	1	yes	characteristic
295	C₂×#441	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
297	#2367	C₂×C₂₃	1	1	yes	characteristic
298	Q₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1959	C₂×C₂₃	1	1	yes	characteristic
300	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1936	C₂×C₂₃	1	1	yes	characteristic
302	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1880	C₂×C₂₃	1	1	yes	characteristic
304	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1864	C₂×C₂₃	1	1	yes	characteristic
307	#1771	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#1763	C₂×C₂₃	1	1	yes	characteristic, complemented
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	4	2	no	—
311	C₂²×C₂₃	—	4	2	no	—
312	C₂²×C₂₃	—	8	1	no	—
313	C₂²×C₂₃	—	8	1	no	—
314	C₂²×C₂₃	—	8	1	no	—
315	C₂²×C₂₃	—	8	1	no	—
316	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
317	C₂²×C₂₃	D₄²	1	1	yes	characteristic
318	C₂²×C₂₃	#548	1	1	yes	characteristic
319	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
320	C₂²×C₂₃	#575	1	1	yes	characteristic
321	C₂²×C₂₃	#577	1	1	yes	characteristic
322	C₂²×C₂₃	#577	1	1	yes	characteristic
323	C₄×C₂₃	—	2	4	no	—
324	C₄×C₂₃	—	2	2	no	—
325	C₄×C₂₃	—	2	2	no	—
326	C₄×C₂₃	—	2	2	no	—
327	C₄×C₂₃	—	4	2	no	—
328	C₄×C₂₃	—	4	2	no	—
329	C₄×C₂₃	—	4	1	no	—
330	C₄×C₂₃	—	4	2	no	—
331	C₄×C₂₃	—	4	1	no	—
332	C₄×C₂₃	—	4	1	no	—
333	C₄×C₂₃	—	8	1	no	—
334	C₄×C₂₃	—	8	1	no	—
335	C₄×C₂₃	—	8	1	no	—
336	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
337	C₄×C₂₃	#520	1	1	yes	characteristic
338	C₄×C₂₃	#547	1	1	yes	characteristic
339	C₄×C₂₃	#553	1	1	yes	characteristic
340	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
341	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
348	#464	C₄×C₂₃	1	8	yes	complemented
349	#441	—	2	4	no	—
350	C₄×SD₁₆	—	2	4	no	—
351	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
359	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
360	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
365	#391	C₂²×C₂₃	1	1	yes	characteristic
366	#390	C₂²×C₂₃	1	1	yes	characteristic
367	#389	—	2	1	no	—
368	#389	—	2	1	no	—
369	#389	C₂²×C₂₃	1	1	yes	characteristic
370	#389	C₂²×C₂₃	1	1	yes	characteristic
371	#388	—	2	1	no	—
372	#388	—	2	1	no	—
373	#385	—	2	1	no	—
374	#384	—	2	1	no	—
375	#384	C₂²×C₂₃	1	1	yes	characteristic
376	C₄×C₄⋊C₄	—	2	1	no	—
377	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
378	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
379	C₄×C₂²⋊C₄	—	2	1	no	—
380	#340	C₂²×C₂₃	1	1	yes	characteristic
381	#340	C₂²×C₂₃	1	1	yes	characteristic
382	#340	C₂²×C₂₃	1	1	yes	characteristic
383	#340	C₂²×C₂₃	1	1	yes	characteristic
384	#336	C₂²×C₂₃	1	1	yes	characteristic
385	#336	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×C₂₃	—	8	2	no	—
387	C₂×C₂₃	#2361	1	1	yes	characteristic
388	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
389	C₂×C₂₃	#2952	1	1	yes	characteristic
390	C₂×C₂₃	#3266	1	1	yes	characteristic
391	C₂×C₂₃	#3271	1	1	yes	characteristic
392	C₂×C₂₃	#3369	1	1	yes	characteristic
393	C₂×C₂₃	#3371	1	1	yes	characteristic
394	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
395	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
396	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
397	C₂³×C₄	—	2	1	no	—
398	C₂×Q₁₆	—	2	2	no	—
399	C₂×Q₁₆	—	2	2	no	—
400	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
401	C₂×SD₁₆	—	2	2	no	—
402	C₂×SD₁₆	—	2	2	no	—
403	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
404	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
408	#124	—	2	2	no	—
409	#124	C₂×C₄×C₂₃	1	4	yes	—
410	C₄×Q₈	—	2	2	no	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	—	2	2	no	—
413	C₄×Q₈	—	2	2	no	—
414	C₄×Q₈	—	4	1	no	—
415	C₄×Q₈	—	4	1	no	—
416	C₄×Q₈	—	4	1	no	—
417	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	4	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	—	2	1	no	—
443	C₂×C₄²	—	2	1	no	—
444	C₂×C₄²	—	2	1	no	—
445	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	#106	—	2	2	no	—
449	#106	—	2	2	no	—
450	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
451	#103	—	2	2	no	—
452	#103	—	2	2	no	—
453	#103	—	4	1	no	—
454	#103	C₂×C₄×C₂₃	1	4	yes	complemented
455	#103	D₄×C₂₃	1	4	yes	—
456	#102	—	2	2	no	—
457	#102	—	4	1	no	—
458	#102	C₂×C₄×C₂₃	1	4	yes	—
459	#97	—	2	2	no	—
460	#97	—	2	2	no	—
461	C₄×C₈	—	2	2	no	—
462	#95	—	2	1	no	—
463	#95	—	2	1	no	—
464	#95	—	2	1	no	—
465	#95	—	2	1	no	—
466	#95	—	2	1	no	—
467	#95	D₄×C₂₃	1	1	yes	characteristic
468	#95	D₄×C₂₃	1	1	yes	characteristic
469	C₂₃	#23013	1	1	yes	characteristic, direct product factor, Sylow
470	C₂⁴	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	2	1	no	—
479	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
480	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
481	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
482	C₂×Q₈	#41×C₂₃	1	2	yes	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	4	2	no	—
497	C₂²×C₄	—	4	2	no	—
498	C₂²×C₄	—	8	1	no	—
499	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
506	Q₁₆	—	2	8	no	—
507	SD₁₆	—	2	8	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
518	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
519	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	4	1	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	2	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	—	4	2	no	—
535	C₄⋊C₄	—	8	1	no	—
536	C₄⋊C₄	—	8	1	no	—
537	C₄⋊C₄	—	8	1	no	—
538	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
539	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	4	2	no	—
544	C₂²⋊C₄	—	8	1	no	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	1	no	—
549	C₄²	—	2	1	no	—
550	C₄²	—	4	2	no	—
551	C₄²	—	4	2	no	—
552	C₄²	—	4	2	no	—
553	C₄²	—	8	1	no	—
554	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
555	C₄²	#41×C₂₃	1	2	yes	—
556	C₂³	—	2	2	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	4	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	—	4	1	no	—
562	C₂³	—	4	1	no	—
563	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
564	Q₈	—	2	2	no	—
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	2	2	no	—
569	Q₈	—	4	1	no	—
570	Q₈	—	4	1	no	—
571	Q₈	C₄×D₄×C₂₃	1	4	yes	—
572	D₄	—	2	2	no	—
573	D₄	—	2	2	no	—
574	D₄	—	4	1	no	—
575	D₄	—	4	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	8	4	no	—
613	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
614	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
615	C₂×C₄	C₂₃×#137	1	2	yes	—
616	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
617	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
618	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
619	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
620	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
621	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
622	C₈	—	2	2	no	—
623	C₈	—	2	2	no	—
624	C₈	—	2	2	no	—
625	C₈	—	4	1	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	8	1	no	—
630	C₂²	—	8	1	no	—
631	C₂²	—	8	1	no	—
632	C₂²	—	8	1	no	—
633	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
634	C₂²	D₄²×C₂₃	1	1	yes	characteristic
635	C₂²	C₂₃×#548	1	1	yes	characteristic
636	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
637	C₂²	C₂₃×#575	1	1	yes	characteristic
638	C₂²	C₂₃×#577	1	1	yes	characteristic
639	C₂²	C₂₃×#577	1	1	yes	characteristic
640	C₄	—	2	2	no	—
641	C₄	—	2	2	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	4	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	1	no	—
649	C₄	—	4	2	no	—
650	C₄	—	8	1	no	—
651	C₄	—	8	1	no	—
652	C₄	—	8	1	no	—
653	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
654	C₄	C₂₃×#520	1	1	yes	characteristic
655	C₄	C₂₃×#547	1	1	yes	characteristic
656	C₄	C₂₃×#553	1	1	yes	characteristic
657	C₂	—	8	2	no	—
658	C₂	C₂₃×#2361	1	1	yes	characteristic
659	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
660	C₂	C₂₃×#2952	1	1	yes	characteristic
661	C₂	C₂₃×#3266	1	1	yes	characteristic
662	C₂	C₂₃×#3271	1	1	yes	characteristic
663	C₂	C₂₃×#3369	1	1	yes	characteristic
664	C₂	C₂₃×#3371	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#441	C₂	semidirect
5. Q₈×C₄⋊C₄×C₂₃	C₂	semidirect
7. C₂₃×#1948	C₂	semidirect
9. C₂₃×#1933	C₂	semidirect
11. C₂₃×#1878	C₂	semidirect
12. C₂₃×#1865	C₂	semidirect
14. C₂₃×#1771	C₂	semidirect
15. C₂₃×#1763	C₂	semidirect
24. C₂₃×#464	C₄	semidirect
68. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
85. C₄×Q₈×C₂₃	C₂×C₄	semidirect
118. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
122. C₂₃×#103	C₂×C₄	semidirect
223. #23013	C₂₃	direct (N × H)
295. C₂×#441	C₂×C₂₃	semidirect
298. Q₈×C₄⋊C₄	C₂×C₂₃	semidirect
300. #1948	C₂×C₂₃	semidirect
302. #1933	C₂×C₂₃	semidirect
304. #1878	C₂×C₂₃	semidirect
305. #1865	C₂×C₂₃	semidirect
307. #1771	C₂×C₂₃	semidirect
308. #1763	C₂×C₂₃	semidirect
348. #464	C₄×C₂₃	semidirect
400. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
417. C₄×Q₈	C₂×C₄×C₂₃	semidirect
450. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
454. #103	C₂×C₄×C₂₃	semidirect
469. C₂₃	#23013	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×#464	2
C₂×C₂₃×#441	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#2367	2
Q₈×C₄⋊C₄×C₂₃	2
C₂₃×#1959	2
C₂₃×#1948	2
C₂₃×#1936	2
C₂₃×#1933	2
C₂₃×#1880	2
C₂₃×#1878	2
C₂₃×#1865	2
C₂₃×#1864	2
C₂₃×#1771	2
C₂₃×#1763	2
#23013	23

Subgroups — C23×#23013 #19170639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23013 #19170639