Subgroups — C₂₃×#8570 #19156196

Order 5888.

472 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×#2295	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#391	C₄	1	1	yes	characteristic
11	C₂₃×#391	C₄	1	1	yes	characteristic
12	C₂₃×#389	C₄	1	1	yes	characteristic
13	C₂₃×#389	C₄	1	1	yes	characteristic
14	C₂₃×#387	—	2	1	no	—
15	C₂₃×#387	—	2	1	no	—
16	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
17	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#376	—	2	1	no	—
19	C₄³×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#336	—	2	1	no	—
21	C₂₃×#336	—	2	1	no	—
22	C₂₃×#336	—	2	1	no	—
23	C₂₃×Q₈⋊C₈	—	2	4	no	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₄×Q₈×C₂₃	—	2	1	no	—
28	C₄×Q₈×C₂₃	—	2	1	no	—
29	C₄×Q₈×C₂₃	D₄	1	2	yes	—
30	C₄×Q₈×C₂₃	Q₈	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	—	2	1	no	—
38	C₂×C₄²×C₂₃	—	2	1	no	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×C₄⋊C₈	—	2	4	no	—
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×C₄⋊C₈	—	2	2	no	—
49	C₄×C₈×C₂₃	—	2	2	no	—
50	C₄×C₈×C₂₃	—	2	2	no	—
51	C₂₃×#95	—	2	1	no	—
52	C₂₃×#95	—	2	1	no	—
53	C₂₃×#95	—	2	1	no	—
54	C₂₃×#95	—	2	1	no	—
55	C₂₃×#95	—	2	1	no	—
56	C₂₃×#95	—	2	1	no	—
57	C₂₃×#95	—	2	1	no	—
58	C₂×Q₈×C₂₃	—	2	1	no	—
59	C₂×Q₈×C₂₃	—	2	2	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₂₃	—	2	2	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	—	2	2	no	—
90	C₄²×C₂₃	—	2	2	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	4	1	no	—
95	C₄²×C₂₃	—	4	2	no	—
96	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₂₃	#41	1	1	yes	characteristic
99	C₄²×C₂₃	#41	1	1	yes	characteristic
100	#8570	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₂₃	—	2	4	no	—
103	Q₈×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	4	1	no	—
120	C₂×C₄×C₂₃	—	4	1	no	—
121	C₂×C₄×C₂₃	—	4	1	no	—
122	C₂×C₄×C₂₃	—	4	1	no	—
123	C₂×C₄×C₂₃	—	4	1	no	—
124	C₂×C₄×C₂₃	—	4	1	no	—
125	C₂×C₄×C₂₃	—	4	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	#104	1	2	yes	—
130	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₂₃	#131	1	2	yes	—
137	C₂×C₄×C₂₃	#131	1	2	yes	—
138	C₂×C₄×C₂₃	#135	1	2	yes	—
139	C₂×C₄×C₂₃	#137	1	2	yes	—
140	C₈×C₂₃	—	4	4	no	—
141	C₈×C₂₃	—	8	2	no	—
142	#2295	C₂×C₂₃	1	1	yes	characteristic
143	#1831	C₂×C₂₃	1	1	yes	characteristic
144	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
145	C₂×Q₈⋊C₈	C₂×C₂₃	1	2	yes	—
146	#1279	C₂×C₂₃	1	2	yes	—
147	C₂²×C₂₃	#380	1	1	yes	characteristic
148	C₂²×C₂₃	#408	1	1	yes	characteristic
149	C₂²×C₂₃	#408	1	1	yes	characteristic
150	C₂²×C₂₃	#416	1	1	yes	characteristic
151	C₂²×C₂₃	#419	1	1	yes	characteristic
152	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
153	C₂²×C₂₃	#421	1	1	yes	characteristic
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	4	2	no	—
158	C₄×C₂₃	—	4	2	no	—
159	C₄×C₂₃	—	4	1	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	4	1	no	—
162	C₄×C₂₃	—	8	1	no	—
163	C₄×C₂₃	—	8	1	no	—
164	C₄×C₂₃	#424	1	2	yes	—
165	C₄×C₂₃	#433	1	2	yes	—
166	C₄×C₂₃	#464	1	2	yes	—
167	C₄×C₂₃	#479	1	2	yes	—
168	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	2	1	no	—
170	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
172	#391	C₄×C₂₃	1	1	yes	characteristic
173	#391	C₄×C₂₃	1	1	yes	characteristic
174	#389	C₄×C₂₃	1	1	yes	characteristic
175	#389	C₄×C₂₃	1	1	yes	characteristic
176	#387	—	2	1	no	—
177	#387	—	2	1	no	—
178	C₄×C₄⋊C₄	—	2	1	no	—
179	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
180	#376	—	2	1	no	—
181	C₄³	C₂²×C₂₃	1	1	yes	characteristic
182	#336	—	2	1	no	—
183	#336	—	2	1	no	—
184	#336	—	2	1	no	—
185	Q₈⋊C₈	—	2	4	no	—
186	C₂×C₂₃	#1482	1	1	yes	characteristic
187	C₂×C₂₃	#1485	1	1	yes	characteristic
188	C₂×C₂₃	#1494	1	1	yes	characteristic
189	C₂×C₂₃	#1497	1	1	yes	characteristic
190	C₂×C₂₃	#1798	1	1	yes	characteristic
191	C₂×C₂₃	#1805	1	1	yes	characteristic
192	C₂×C₂₃	#1806	1	1	yes	characteristic
193	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	D₄×C₂₃	1	2	yes	—
199	C₄×Q₈	Q₈×C₂₃	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	—	2	2	no	—
220	#95	—	2	1	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	C₂₃	#8570	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
267	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
268	C₄²	#41×C₂₃	1	1	yes	characteristic
269	C₄²	#41×C₂₃	1	1	yes	characteristic
270	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	4	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	C₂₃×#104	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
300	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
301	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
302	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
303	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂×C₄	C₂₃×#131	1	2	yes	—
306	C₂×C₄	C₂₃×#131	1	2	yes	—
307	C₂×C₄	C₂₃×#135	1	2	yes	—
308	C₂×C₄	C₂₃×#137	1	2	yes	—
309	C₈	—	4	4	no	—
310	C₈	—	8	2	no	—
311	C₂²	C₂₃×#380	1	1	yes	characteristic
312	C₂²	C₂₃×#408	1	1	yes	characteristic
313	C₂²	C₂₃×#408	1	1	yes	characteristic
314	C₂²	C₂₃×#416	1	1	yes	characteristic
315	C₂²	C₂₃×#419	1	1	yes	characteristic
316	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
317	C₂²	C₂₃×#421	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	C₂₃×#424	1	2	yes	—
329	C₄	C₂₃×#433	1	2	yes	—
330	C₄	C₂₃×#464	1	2	yes	—
331	C₄	C₂₃×#479	1	2	yes	—
332	C₂	C₂₃×#1482	1	1	yes	characteristic
333	C₂	C₂₃×#1485	1	1	yes	characteristic
334	C₂	C₂₃×#1494	1	1	yes	characteristic
335	C₂	C₂₃×#1497	1	1	yes	characteristic
336	C₂	C₂₃×#1798	1	1	yes	characteristic
337	C₂	C₂₃×#1805	1	1	yes	characteristic
338	C₂	C₂₃×#1806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
100. #8570	C₂₃	direct (N × H)
227. C₂₃	#8570	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×#2295	2
C₂₃×#1831	2
C₄×C₂₃×C₄⋊C₈	2
C₂×C₂₃×Q₈⋊C₈	2
C₂₃×#1279	2
#8570	23

Subgroups — C23×#8570 #19156196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8570 #19156196