Subgroups — C₂₃×#8250 #19155876

Order 5888.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2966	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1524	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1500	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1302	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#529	C₂²	1	1	yes	characteristic
9	C₂₃×#446	C₄	1	1	yes	characteristic, complemented
10	C₈×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂₃×#436	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#435	—	2	1	no	—
13	C₈×D₄×C₂₃	—	2	1	no	—
14	C₈×D₄×C₂₃	C₄	1	1	yes	characteristic, complemented
15	C₂₃×#433	—	2	1	no	—
16	C₂₃×#432	C₂²	1	1	yes	characteristic
17	C₂₃×#431	C₂²	1	1	yes	characteristic
18	C₂₃×#405	—	2	1	no	—
19	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#376	C₂²	1	1	yes	characteristic
23	C₂₃×#336	—	2	1	no	—
24	C₂₃×#336	—	2	1	no	—
25	C₂₃×#330	C₄	1	2	yes	complemented
26	C₂₃×#329	C₄	1	2	yes	complemented
27	C₂₃×#328	C₄	1	2	yes	—
28	C₂₃×#327	C₄	1	2	yes	—
29	C₂₃×#322	—	2	1	no	—
30	C₈²×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	—	2	1	no	—
32	C₂×M₁₆×C₂₃	—	2	1	no	—
33	C₂²×C₈×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
37	C₂₃×#126	—	2	1	no	—
38	C₂₃×#125	C₂×C₄	1	1	yes	characteristic
39	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
40	C₂₃×#123	—	2	1	no	—
41	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂₃×#121	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₂₃	—	2	1	no	—
45	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂₃×#117	—	2	1	no	—
47	C₂×C₄²×C₂₃	—	2	1	no	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊C₈	—	2	1	no	—
50	C₂₃×C₄⋊C₈	—	2	1	no	—
51	C₂₃×C₄⋊C₈	—	2	1	no	—
52	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂₃×C₂²⋊C₈	—	2	1	no	—
55	C₂₃×C₂²⋊C₈	—	2	1	no	—
56	C₂₃×C₂²⋊C₈	—	2	1	no	—
57	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₂₃×#97	—	2	1	no	—
59	C₂₃×#97	—	2	1	no	—
60	C₂₃×#97	—	2	1	no	—
61	C₂₃×#97	C₂×C₄	1	2	yes	—
62	C₂₃×#97	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₂₃	—	2	1	no	—
64	C₄×C₈×C₂₃	—	2	1	no	—
65	C₄×C₈×C₂₃	—	2	1	no	—
66	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
70	C₂₃×#95	—	2	1	no	—
71	C₂₃×#95	C₂×C₄	1	2	yes	—
72	C₂×Q₈×C₂₃	C₄²	1	1	yes	characteristic
73	C₂×D₄×C₂₃	—	2	1	no	—
74	C₂×D₄×C₂₃	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	—	2	1	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
80	M₁₆×C₂₃	—	2	4	no	—
81	M₁₆×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	1	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	4	1	no	—
88	C₂×C₈×C₂₃	—	4	1	no	—
89	C₂×C₈×C₂₃	—	4	2	no	—
90	C₂×C₈×C₂₃	—	4	1	no	—
91	C₂×C₈×C₂₃	—	4	1	no	—
92	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₄⋊C₄×C₂₃	C₄²	1	1	yes	characteristic
101	C₄⋊C₄×C₂₃	C₄²	1	1	yes	characteristic
102	C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂²⋊C₄×C₂₃	—	2	1	no	—
107	C₄²×C₂₃	—	2	1	no	—
108	C₄²×C₂₃	—	4	1	no	—
109	C₄²×C₂₃	—	4	1	no	—
110	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₂₃	#41	1	1	yes	characteristic
114	#8250	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₂₃	—	2	1	no	—
116	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₂₃	—	4	1	no	—
118	D₄×C₂₃	—	4	1	no	—
119	D₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	2	1	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
138	C₈×C₂₃	—	2	4	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₈×C₂₃	—	4	2	no	—
144	C₈×C₂₃	—	4	2	no	—
145	C₈×C₂₃	—	4	1	no	—
146	C₈×C₂₃	—	4	1	no	—
147	#2966	C₂×C₂₃	1	1	yes	characteristic
148	#1746	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#1524	C₂×C₂₃	1	1	yes	characteristic
150	#1500	C₂×C₂₃	1	1	yes	characteristic
151	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#1302	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#1300	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	4	1	no	—
157	C₂²×C₂₃	—	4	1	no	—
158	C₂²×C₂₃	—	4	1	no	—
159	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₂₃	#443	1	2	yes	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	2	1	no	—
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	4	1	no	—
167	C₄×C₂₃	—	4	1	no	—
168	C₄×C₂₃	—	4	1	no	—
169	C₄×C₂₃	—	8	2	no	—
170	C₄×C₂₃	#421	1	1	yes	characteristic
171	C₄×C₂₃	#421	1	1	yes	characteristic
172	#529	C₂²×C₂₃	1	1	yes	characteristic
173	#446	C₄×C₂₃	1	1	yes	characteristic, complemented
174	C₈×Q₈	C₄×C₂₃	1	1	yes	characteristic, complemented
175	#436	C₄×C₂₃	1	1	yes	characteristic, complemented
176	#435	—	2	1	no	—
177	C₈×D₄	—	2	1	no	—
178	C₈×D₄	C₄×C₂₃	1	1	yes	characteristic, complemented
179	#433	—	2	1	no	—
180	#432	C₂²×C₂₃	1	1	yes	characteristic
181	#431	C₂²×C₂₃	1	1	yes	characteristic
182	#405	—	2	1	no	—
183	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
186	#376	C₂²×C₂₃	1	1	yes	characteristic
187	#336	—	2	1	no	—
188	#336	—	2	1	no	—
189	#330	C₄×C₂₃	1	2	yes	complemented
190	#329	C₄×C₂₃	1	2	yes	complemented
191	#328	C₄×C₂₃	1	2	yes	—
192	#327	C₄×C₂₃	1	2	yes	—
193	#322	—	2	1	no	—
194	C₈²	—	2	1	no	—
195	C₂×C₂₃	—	2	2	no	—
196	C₂×C₂₃	—	8	1	no	—
197	C₂×C₂₃	#1757	1	1	yes	characteristic
198	C₂×C₂₃	#1759	1	1	yes	characteristic
199	C₂×C₂₃	#1764	1	1	yes	characteristic
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
206	#126	—	2	1	no	—
207	#125	C₂×C₄×C₂₃	1	1	yes	characteristic
208	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
209	#123	—	2	1	no	—
210	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
211	#121	C₂×C₄×C₂₃	1	1	yes	characteristic
212	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
215	#117	—	2	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
222	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
227	#97	—	2	1	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	C₂×C₄×C₂₃	1	2	yes	—
231	#97	C₂×C₄×C₂₃	1	2	yes	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
236	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
237	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
238	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
239	#95	—	2	1	no	—
240	#95	C₂×C₄×C₂₃	1	2	yes	—
241	C₂₃	#8250	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×Q₈	C₄²×C₂₃	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	C₄²×C₂₃	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	4	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₄²×C₂₃	1	1	yes	characteristic
271	C₄⋊C₄	C₄²×C₂₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
281	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
283	C₄²	#41×C₂₃	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
286	Q₈	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
306	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
307	C₈	—	2	4	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
322	C₂²	C₂₃×#443	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	8	2	no	—
332	C₄	C₂₃×#421	1	1	yes	characteristic
333	C₄	C₂₃×#421	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	8	1	no	—
336	C₂	C₂₃×#1757	1	1	yes	characteristic
337	C₂	C₂₃×#1759	1	1	yes	characteristic
338	C₂	C₂₃×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×C₂²⋊C₄
prime radical	C₂₃×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1746	C₂	semidirect
5. C₈×M₁₆×C₂₃	C₂	semidirect
6. C₂₃×#1302	C₂	semidirect
7. C₂₃×#1300	C₂	semidirect
9. C₂₃×#446	C₄	semidirect
10. C₈×Q₈×C₂₃	C₄	semidirect
11. C₂₃×#436	C₄	semidirect
14. C₈×D₄×C₂₃	C₄	semidirect
25. C₂₃×#330	C₄	semidirect
26. C₂₃×#329	C₄	semidirect
114. #8250	C₂₃	direct (N × H)
148. #1746	C₂×C₂₃	semidirect
151. C₈×M₁₆	C₂×C₂₃	semidirect
152. #1302	C₂×C₂₃	semidirect
153. #1300	C₂×C₂₃	semidirect
173. #446	C₄×C₂₃	semidirect
174. C₈×Q₈	C₄×C₂₃	semidirect
175. #436	C₄×C₂₃	semidirect
178. C₈×D₄	C₄×C₂₃	semidirect
189. #330	C₄×C₂₃	semidirect
190. #329	C₄×C₂₃	semidirect
241. C₂₃	#8250	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2966	2
C₂₃×#1746	2
C₂₃×#1524	2
C₂₃×#1500	2
C₈×M₁₆×C₂₃	2
C₂₃×#1302	2
C₂₃×#1300	2
#8250	23

Subgroups — C23×#8250 #19155876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8250 #19155876