Subgroups — C₂₃×#9550 #19157176

Order 5888.

378 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#100	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1478	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#431	C₂²	1	1	yes	characteristic
10	C₂₃×#431	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#100	C₄	1	1	yes	characteristic
13	C₂×C₂₃×#100	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#343	C₄	1	1	yes	characteristic
18	C₂₃×#343	C₄	1	1	yes	characteristic
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×D₄⋊C₈	—	2	1	no	—
22	C₂₃×D₄⋊C₈	—	2	1	no	—
23	C₂₃×D₄⋊C₈	—	2	1	no	—
24	C₂₃×D₄⋊C₈	—	2	1	no	—
25	C₂₃×#322	—	2	1	no	—
26	C₈²×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₄×D₄×C₂₃	—	2	1	no	—
37	C₄×D₄×C₂₃	—	2	1	no	—
38	C₄×D₄×C₂₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	1	no	—
43	C₂₃×C₄⋊C₈	—	2	1	no	—
44	C₂₃×C₄⋊C₈	—	2	1	no	—
45	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
47	C₂₃×#100	—	2	2	no	—
48	C₂₃×#100	C₈	1	4	yes	complemented
49	C₂₃×C₂²⋊C₈	—	2	1	no	—
50	C₂₃×C₂²⋊C₈	—	2	1	no	—
51	C₂₃×#97	—	2	1	no	—
52	C₂₃×#97	—	2	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	1	no	—
57	C₄×C₈×C₂₃	—	2	1	no	—
58	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂⁴×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	—	2	1	no	—
65	C₂×D₄×C₂₃	—	2	1	no	—
66	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	—	4	1	no	—
69	C₂²×C₄×C₂₃	—	4	1	no	—
70	C₂²×C₄×C₂₃	—	8	1	no	—
71	C₂²×C₄×C₂₃	—	8	1	no	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
75	M₁₆×C₂₃	—	2	1	no	—
76	M₁₆×C₂₃	—	2	1	no	—
77	M₁₆×C₂₃	—	2	1	no	—
78	M₁₆×C₂₃	—	2	1	no	—
79	M₁₆×C₂₃	—	2	1	no	—
80	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
81	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
82	M₁₆×C₂₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	1	no	—
85	C₂×C₈×C₂₃	—	2	1	no	—
86	C₂×C₈×C₂₃	—	2	1	no	—
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	2	1	no	—
89	C₂×C₈×C₂₃	—	2	1	no	—
90	C₂×C₈×C₂₃	—	2	1	no	—
91	C₂×C₈×C₂₃	—	4	2	no	—
92	C₂×C₈×C₂₃	—	4	1	no	—
93	C₂×C₈×C₂₃	—	4	1	no	—
94	C₂×C₈×C₂₃	—	8	1	no	—
95	C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂²⋊C₄×C₂₃	—	4	1	no	—
98	C₂²⋊C₄×C₂₃	—	4	1	no	—
99	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₂₃	#41	1	1	yes	characteristic
102	C₄²×C₂₃	#41	1	1	yes	characteristic
103	#9550	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	2	no	—
105	C₂³×C₂₃	—	4	1	no	—
106	C₂³×C₂₃	—	4	1	no	—
107	C₂³×C₂₃	—	8	1	no	—
108	C₂³×C₂₃	—	8	1	no	—
109	C₂³×C₂₃	#117	1	1	yes	characteristic
110	D₄×C₂₃	—	2	2	no	—
111	D₄×C₂₃	—	2	2	no	—
112	D₄×C₂₃	—	4	1	no	—
113	D₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	4	1	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	8	2	no	—
119	C₂×C₄×C₂₃	—	8	1	no	—
120	C₂×C₄×C₂₃	—	8	1	no	—
121	C₂×C₄×C₂₃	—	8	1	no	—
122	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#126	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	2	2	no	—
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	—	4	1	no	—
137	C₈×C₂₃	—	4	1	no	—
138	C₈×C₂₃	—	4	1	no	—
139	C₈×C₂₃	—	4	1	no	—
140	C₈×C₂₃	—	8	1	no	—
141	C₈×C₂₃	—	8	1	no	—
142	C₄×#100	C₂×C₂₃	1	1	yes	characteristic
143	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#1478	C₂×C₂₃	1	1	yes	characteristic
145	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
146	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#1283	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#1278	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂²×C₂₃	—	2	1	no	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	4	2	no	—
152	C₂²×C₂₃	—	8	2	no	—
153	C₂²×C₂₃	—	8	2	no	—
154	C₂²×C₂₃	—	8	1	no	—
155	C₂²×C₂₃	—	8	1	no	—
156	C₂²×C₂₃	#388	1	1	yes	characteristic
157	C₂²×C₂₃	#431	1	1	yes	characteristic
158	C₂²×C₂₃	#433	1	1	yes	characteristic
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	8	1	no	—
163	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
164	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
165	C₄×C₂₃	#435	1	1	yes	characteristic
166	C₄×C₂₃	#436	1	1	yes	characteristic
167	C₄×C₂₃	#471	1	1	yes	characteristic
168	C₄×C₂₃	#472	1	1	yes	characteristic
169	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
170	#431	C₂²×C₂₃	1	1	yes	characteristic
171	#431	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×#100	C₄×C₂₃	1	1	yes	characteristic
174	C₂×#100	C₄×C₂₃	1	1	yes	characteristic
175	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
176	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
178	#343	C₄×C₂₃	1	1	yes	characteristic
179	#343	C₄×C₂₃	1	1	yes	characteristic
180	#335	—	2	1	no	—
181	#334	—	2	1	no	—
182	D₄⋊C₈	—	2	1	no	—
183	D₄⋊C₈	—	2	1	no	—
184	D₄⋊C₈	—	2	1	no	—
185	D₄⋊C₈	—	2	1	no	—
186	#322	—	2	1	no	—
187	C₈²	—	2	1	no	—
188	C₂×C₂₃	—	2	1	no	—
189	C₂×C₂₃	—	2	1	no	—
190	C₂×C₂₃	—	8	2	no	—
191	C₂×C₂₃	#1923	1	1	yes	characteristic
192	C₂×C₂₃	#1931	1	1	yes	characteristic
193	C₂×C₂₃	#1932	1	1	yes	characteristic
194	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
195	C₂³×C₄	—	2	1	no	—
196	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
198	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
213	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
214	#100	—	2	2	no	—
215	#100	C₈×C₂₃	1	4	yes	complemented
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	#97	—	2	1	no	—
219	#97	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
226	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
227	C₂₃	#9550	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	8	1	no	—
239	C₂²×C₄	—	8	1	no	—
240	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	1	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	2	1	no	—
247	M₁₆	—	2	1	no	—
248	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
249	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
250	M₁₆	M₁₆×C₂₃	1	2	yes	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
269	C₄²	#41×C₂₃	1	1	yes	characteristic
270	C₄²	#41×C₂₃	1	1	yes	characteristic
271	C₂³	—	2	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	C₂₃×#117	1	1	yes	characteristic
277	D₄	—	2	2	no	—
278	D₄	—	2	2	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
292	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
293	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
294	C₂×C₄	C₂₃×#126	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
298	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	—	2	2	no	—
302	C₈	—	2	2	no	—
303	C₈	—	4	1	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	8	1	no	—
308	C₈	—	8	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	C₂₃×#388	1	1	yes	characteristic
317	C₂²	C₂₃×#431	1	1	yes	characteristic
318	C₂²	C₂₃×#433	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
324	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
325	C₄	C₂₃×#435	1	1	yes	characteristic
326	C₄	C₂₃×#436	1	1	yes	characteristic
327	C₄	C₂₃×#471	1	1	yes	characteristic
328	C₄	C₂₃×#472	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂₃×#1923	1	1	yes	characteristic
333	C₂	C₂₃×#1931	1	1	yes	characteristic
334	C₂	C₂₃×#1932	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#388
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1565	C₂	semidirect
5. C₈×M₁₆×C₂₃	C₂	semidirect
6. C₂₃×#1283	C₂	semidirect
7. C₂₃×#1278	C₂	semidirect
48. C₂₃×#100	C₈	semidirect
103. #9550	C₂₃	direct (N × H)
143. #1565	C₂×C₂₃	semidirect
146. C₈×M₁₆	C₂×C₂₃	semidirect
147. #1283	C₂×C₂₃	semidirect
148. #1278	C₂×C₂₃	semidirect
215. #100	C₈×C₂₃	semidirect
227. C₂₃	#9550	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#100	2
C₂₃×#1565	2
C₂₃×#1478	2
C₂×C₂₃×D₄⋊C₈	2
C₈×M₁₆×C₂₃	2
C₂₃×#1283	2
C₂₃×#1278	2
#9550	23

Subgroups — C23×#9550 #19157176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9550 #19157176