Subgroups — C₂₃×#9267 #19156893

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3206	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1547	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	—	2	1	no	—
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂₃×#492	—	2	1	no	—
12	C₂₃×#487	—	2	1	no	—
13	C₂₃×#461	—	2	2	no	—
14	C₂₃×#448	C₄	1	4	yes	—
15	C₂₃×#436	—	2	1	no	—
16	C₂₃×#435	—	2	2	no	—
17	C₈×D₄×C₂₃	—	2	1	no	—
18	C₂₃×#431	C₂²	1	1	yes	characteristic
19	C₂₃×#423	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂₃×#408	—	2	1	no	—
23	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#343	—	2	1	no	—
25	C₂₃×#343	—	2	1	no	—
26	C₂₃×#333	—	2	1	no	—
27	C₂₃×#333	—	2	1	no	—
28	C₂₃×#332	—	2	1	no	—
29	C₂₃×#331	—	2	1	no	—
30	C₂×#41×C₂₃	—	2	1	no	—
31	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×SD₁₆×C₂₃	—	2	4	no	—
33	C₂₃×#131	—	4	2	no	—
34	C₂₃×#131	—	4	2	no	—
35	C₂×M₁₆×C₂₃	—	2	2	no	—
36	C₂×M₁₆×C₂₃	—	2	1	no	—
37	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	—	2	2	no	—
40	C₂²×C₈×C₂₃	—	2	1	no	—
41	C₂²×C₈×C₂₃	—	2	1	no	—
42	C₂₃×C₄⋊Q₈	—	2	1	no	—
43	C₂₃×C₄⋊Q₈	—	2	1	no	—
44	C₂₃×C₄⋊Q₈	—	2	1	no	—
45	C₂₃×C₄⋊Q₈	—	2	1	no	—
46	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₂₃×#127	D₄	1	1	yes	characteristic
48	C₂₃×#124	—	2	1	no	—
49	C₂₃×#121	—	2	1	no	—
50	C₂₃×#121	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₂₃	—	2	1	no	—
52	C₄×D₄×C₂₃	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×C₄⋊C₈	—	2	1	no	—
57	C₂₃×C₄⋊C₈	—	2	1	no	—
58	C₂₃×C₄⋊C₈	—	2	1	no	—
59	C₂₃×C₄⋊C₈	—	2	1	no	—
60	C₂₃×C₄⋊C₈	—	2	1	no	—
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₂₃×#103	—	2	4	no	—
65	C₂₃×C₂²⋊C₈	—	2	2	no	—
66	C₂₃×C₂²⋊C₈	—	2	2	no	—
67	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₂₃×#97	—	2	1	no	—
69	C₄×C₈×C₂₃	—	2	1	no	—
70	C₄×C₈×C₂₃	—	2	1	no	—
71	#41×C₂₃	—	4	1	no	—
72	#41×C₂₃	—	4	1	no	—
73	#41×C₂₃	—	4	1	no	—
74	#41×C₂₃	—	4	1	no	—
75	C₂×Q₈×C₂₃	—	2	1	no	—
76	C₂×Q₈×C₂₃	—	2	2	no	—
77	C₂×Q₈×C₂₃	—	2	2	no	—
78	C₂×Q₈×C₂₃	—	8	2	no	—
79	C₂×D₄×C₂₃	—	2	2	no	—
80	C₂×D₄×C₂₃	—	2	1	no	—
81	C₂×D₄×C₂₃	—	2	1	no	—
82	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₂₃	—	2	2	no	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
88	SD₁₆×C₂₃	—	8	4	no	—
89	M₁₆×C₂₃	—	2	1	no	—
90	M₁₆×C₂₃	—	2	1	no	—
91	M₁₆×C₂₃	—	2	1	no	—
92	M₁₆×C₂₃	—	4	4	no	—
93	M₁₆×C₂₃	—	4	1	no	—
94	M₁₆×C₂₃	—	4	1	no	—
95	M₁₆×C₂₃	SD₁₆	1	2	yes	complemented
96	C₂×C₈×C₂₃	—	2	1	no	—
97	C₂×C₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	2	1	no	—
99	C₂×C₈×C₂₃	—	2	1	no	—
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	4	4	no	—
102	C₂×C₈×C₂₃	—	4	1	no	—
103	C₂×C₈×C₂₃	—	4	1	no	—
104	C₂×C₈×C₂₃	—	4	1	no	—
105	C₂×C₈×C₂₃	—	4	1	no	—
106	C₂×C₈×C₂₃	—	8	1	no	—
107	C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₄⋊C₄×C₂₃	—	4	1	no	—
109	C₄⋊C₄×C₂₃	—	4	1	no	—
110	C₄⋊C₄×C₂₃	—	4	1	no	—
111	C₄⋊C₄×C₂₃	—	4	1	no	—
112	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
113	C₂²⋊C₄×C₂₃	—	2	2	no	—
114	C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₂₃	#41	1	1	yes	characteristic
119	#9267	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	2	2	no	—
121	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
122	Q₈×C₂₃	—	4	4	no	—
123	Q₈×C₂₃	—	4	1	no	—
124	Q₈×C₂₃	—	4	1	no	—
125	Q₈×C₂₃	—	8	2	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	2	no	—
128	D₄×C₂₃	—	4	2	no	—
129	D₄×C₂₃	—	4	1	no	—
130	D₄×C₂₃	—	4	1	no	—
131	D₄×C₂₃	—	4	1	no	—
132	D₄×C₂₃	—	4	1	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	1	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	4	1	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	4	2	no	—
140	C₂×C₄×C₂₃	—	8	2	no	—
141	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
144	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
149	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
151	C₈×C₂₃	—	2	2	no	—
152	C₈×C₂₃	—	4	2	no	—
153	C₈×C₂₃	—	4	1	no	—
154	C₈×C₂₃	—	4	1	no	—
155	C₈×C₂₃	—	4	1	no	—
156	C₈×C₂₃	—	8	1	no	—
157	C₈×C₂₃	—	8	1	no	—
158	#3206	C₂×C₂₃	1	1	yes	characteristic
159	#2935	C₂×C₂₃	1	1	yes	characteristic
160	#1997	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#1547	C₂×C₂₃	1	1	yes	characteristic
162	C₂×#333	C₂×C₂₃	1	1	yes	characteristic, complemented
163	#1301	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#1283	C₂×C₂₃	1	1	yes	characteristic, complemented
165	C₂²×C₂₃	—	2	1	no	—
166	C₂²×C₂₃	—	2	1	no	—
167	C₂²×C₂₃	—	4	2	no	—
168	C₂²×C₂₃	—	4	2	no	—
169	C₂²×C₂₃	—	4	2	no	—
170	C₂²×C₂₃	#386	1	1	yes	characteristic
171	C₂²×C₂₃	C₄×SD₁₆	1	1	yes	characteristic
172	C₂²×C₂₃	#440	1	1	yes	characteristic
173	C₄×C₂₃	—	2	1	no	—
174	C₄×C₂₃	—	2	1	no	—
175	C₄×C₂₃	—	2	1	no	—
176	C₄×C₂₃	—	4	2	no	—
177	C₄×C₂₃	—	8	2	no	—
178	C₄×C₂₃	#450	1	1	yes	characteristic
179	C₄×C₂₃	#452	1	1	yes	characteristic
180	C₄×C₂₃	#461	1	1	yes	characteristic
181	C₄×C₂₃	#463	1	1	yes	characteristic
182	C₄×C₂₃	#471	1	1	yes	characteristic
183	C₄×C₂₃	#473	1	1	yes	characteristic
184	C₂×#131	—	2	1	no	—
185	#532	C₂²×C₂₃	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
187	#492	—	2	1	no	—
188	#487	—	2	1	no	—
189	#461	—	2	2	no	—
190	#448	C₄×C₂₃	1	4	yes	—
191	#436	—	2	1	no	—
192	#435	—	2	2	no	—
193	C₈×D₄	—	2	1	no	—
194	#431	C₂²×C₂₃	1	1	yes	characteristic
195	#423	C₂²×C₂₃	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
197	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
198	#408	—	2	1	no	—
199	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
200	#343	—	2	1	no	—
201	#343	—	2	1	no	—
202	#333	—	2	1	no	—
203	#333	—	2	1	no	—
204	#332	—	2	1	no	—
205	#331	—	2	1	no	—
206	C₂×C₂₃	—	2	1	no	—
207	C₂×C₂₃	—	2	1	no	—
208	C₂×C₂₃	—	8	2	no	—
209	C₂×C₂₃	#1877	1	1	yes	characteristic
210	C₂×C₂₃	#1908	1	1	yes	characteristic
211	C₂×C₂₃	#1914	1	1	yes	characteristic
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
214	C₂×SD₁₆	—	2	4	no	—
215	#131	—	4	2	no	—
216	#131	—	4	2	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
220	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
229	#127	D₄×C₂₃	1	1	yes	characteristic
230	#124	—	2	1	no	—
231	#121	—	2	1	no	—
232	#121	C₂×C₄×C₂₃	1	2	yes	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	D₄×C₂₃	1	2	yes	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
245	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
246	#103	—	2	4	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₂₃	#9267	1	1	yes	characteristic, direct product factor, Sylow
254	#41	—	4	1	no	—
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	#41	—	4	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	8	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
270	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
271	SD₁₆	—	8	4	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	4	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	SD₁₆×C₂₃	1	2	yes	complemented
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	8	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
301	C₄²	#41×C₂₃	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	8	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
324	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
325	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
326	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
327	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
328	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
329	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
330	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
331	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	8	1	no	—
339	C₈	—	8	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	C₂₃×#386	1	1	yes	characteristic
346	C₂²	C₄×SD₁₆×C₂₃	1	1	yes	characteristic
347	C₂²	C₂₃×#440	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₂₃×#450	1	1	yes	characteristic
354	C₄	C₂₃×#452	1	1	yes	characteristic
355	C₄	C₂₃×#461	1	1	yes	characteristic
356	C₄	C₂₃×#463	1	1	yes	characteristic
357	C₄	C₂₃×#471	1	1	yes	characteristic
358	C₄	C₂₃×#473	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂₃×#1877	1	1	yes	characteristic
363	C₂	C₂₃×#1908	1	1	yes	characteristic
364	C₂	C₂₃×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1997	C₂	semidirect
5. C₂×C₂₃×#333	C₂	semidirect
6. C₂₃×#1301	C₂	semidirect
7. C₂₃×#1283	C₂	semidirect
95. M₁₆×C₂₃	SD₁₆	semidirect
119. #9267	C₂₃	direct (N × H)
160. #1997	C₂×C₂₃	semidirect
162. C₂×#333	C₂×C₂₃	semidirect
163. #1301	C₂×C₂₃	semidirect
164. #1283	C₂×C₂₃	semidirect
253. C₂₃	#9267	direct (N × H)
278. M₁₆	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3206	2
C₂₃×#2935	2
C₂₃×#1997	2
C₂₃×#1547	2
C₂×C₂₃×#333	2
C₂₃×#1301	2
C₂₃×#1283	2
#9267	23

Subgroups — C23×#9267 #19156893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9267 #19156893