Subgroups — C₂₃×#11326 #19158952

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	2	yes	maximal
2	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#341	C₂	1	2	yes	maximal
4	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#108	—	2	4	no	—
12	C₂×C₂₃×#108	—	2	4	no	—
13	C₂×C₂₃×#108	—	2	4	no	—
14	C₂₃×#423	—	2	4	no	—
15	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
16	C₂₃×#341	—	2	4	no	—
17	C₂₃×#336	—	2	2	no	—
18	C₂₃×#336	C₄	1	2	yes	—
19	C₂₃×#336	C₄	1	2	yes	—
20	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	—	2	1	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
34	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
35	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	—	2	2	no	—
39	C₂²×C₈×C₂₃	—	2	2	no	—
40	C₂²×C₈×C₂₃	—	2	1	no	—
41	C₂²×C₈×C₂₃	—	2	1	no	—
42	C₂²×C₈×C₂₃	—	2	2	no	—
43	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₂₃	D₄	1	4	yes	—
45	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₂₃	—	2	2	no	—
47	C₂×C₄²×C₂₃	—	2	2	no	—
48	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₂₃	D₄	1	2	yes	—
51	C₂₃×#108	—	2	16	no	—
52	C₂₃×C₄⋊C₈	—	2	8	no	—
53	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	#41	1	2	yes	—
68	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	4	2	no	—
76	M₁₆×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	4	1	no	—
87	C₂×C₈×C₂₃	—	4	2	no	—
88	C₂×C₈×C₂₃	—	4	1	no	—
89	C₂×C₈×C₂₃	—	4	1	no	—
90	C₂×C₈×C₂₃	—	4	2	no	—
91	C₂×C₈×C₂₃	—	4	1	no	—
92	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
93	C₄²×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	#11326	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₃	—	2	2	no	—
99	C₂³×C₂₃	—	2	2	no	—
100	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
102	C₂³×C₂₃	#125	1	1	yes	characteristic
103	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
104	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	4	8	no	—
111	C₂×C₄×C₂₃	#108	1	4	yes	complemented
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#120	1	2	yes	—
117	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#121	1	2	yes	—
119	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#123	1	2	yes	—
125	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#125	1	2	yes	—
127	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
130	C₈×C₂₃	—	2	8	no	—
131	C₈×C₂₃	—	4	4	no	—
132	C₈×C₂₃	—	4	2	no	—
133	C₈×C₂₃	—	4	2	no	—
134	C₂²×#108	C₂×C₂₃	1	2	yes	—
135	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
136	C₂×#341	C₂×C₂₃	1	2	yes	—
137	#1728	C₂×C₂₃	1	1	yes	characteristic
138	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	2	2	no	—
141	C₂²×C₂₃	—	2	4	no	—
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	2	2	no	—
144	C₂²×C₂₃	#384	1	1	yes	characteristic
145	C₂²×C₂₃	#385	1	1	yes	characteristic
146	C₂²×C₂₃	#389	1	1	yes	characteristic
147	C₂²×C₂₃	#395	1	2	yes	—
148	C₂²×C₂₃	#398	1	2	yes	—
149	C₂²×C₂₃	C₂×#108	1	2	yes	—
150	C₂²×C₂₃	#430	1	2	yes	—
151	C₄×C₂₃	—	2	4	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	#385	1	2	yes	—
154	C₄×C₂₃	#389	1	2	yes	—
155	C₄×C₂₃	#392	1	1	yes	characteristic
156	C₄×C₂₃	#395	1	1	yes	characteristic
157	C₄×C₂₃	#398	1	1	yes	characteristic
158	C₄×C₂₃	#399	1	1	yes	characteristic
159	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
163	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×#108	—	2	4	no	—
165	C₂×#108	—	2	4	no	—
166	C₂×#108	—	2	4	no	—
167	#423	—	2	4	no	—
168	C₂×C₄⋊C₈	—	2	2	no	—
169	#341	—	2	4	no	—
170	#336	—	2	2	no	—
171	#336	C₄×C₂₃	1	2	yes	—
172	#336	C₄×C₂₃	1	2	yes	—
173	C₂×C₂₃	—	2	2	no	—
174	C₂×C₂₃	—	2	2	no	—
175	C₂×C₂₃	#1444	1	1	yes	characteristic
176	C₂×C₂₃	#1848	1	1	yes	characteristic
177	C₂×C₂₃	#1849	1	1	yes	characteristic
178	C₂×C₂₃	#1855	1	2	yes	—
179	C₂×C₂₃	#1950	1	2	yes	—
180	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
181	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
194	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
195	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
204	C₂²×C₈	D₄×C₂₃	1	4	yes	—
205	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	2	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	2	yes	—
211	#108	—	2	16	no	—
212	C₄⋊C₈	—	2	8	no	—
213	C₂₃	#11326	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
219	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
226	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	#41×C₂₃	1	2	yes	—
229	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
261	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
262	C₂³	C₂₃×#125	1	1	yes	characteristic
263	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
264	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	C₂₃×#108	1	4	yes	complemented
272	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
275	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂₃×#120	1	2	yes	—
277	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
278	C₂×C₄	C₂₃×#121	1	2	yes	—
279	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
280	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
282	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
283	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂×C₄	C₂₃×#123	1	2	yes	—
285	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
286	C₂×C₄	C₂₃×#125	1	2	yes	—
287	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
288	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
289	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
290	C₈	—	2	8	no	—
291	C₈	—	4	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	C₂₃×#384	1	1	yes	characteristic
300	C₂²	C₂₃×#385	1	1	yes	characteristic
301	C₂²	C₂₃×#389	1	1	yes	characteristic
302	C₂²	C₂₃×#395	1	2	yes	—
303	C₂²	C₂₃×#398	1	2	yes	—
304	C₂²	C₂×C₂₃×#108	1	2	yes	—
305	C₂²	C₂₃×#430	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	C₂₃×#385	1	2	yes	—
309	C₄	C₂₃×#389	1	2	yes	—
310	C₄	C₂₃×#392	1	1	yes	characteristic
311	C₄	C₂₃×#395	1	1	yes	characteristic
312	C₄	C₂₃×#398	1	1	yes	characteristic
313	C₄	C₂₃×#399	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	C₂₃×#1444	1	1	yes	characteristic
317	C₂	C₂₃×#1848	1	1	yes	characteristic
318	C₂	C₂₃×#1849	1	1	yes	characteristic
319	C₂	C₂₃×#1855	1	2	yes	—
320	C₂	C₂₃×#1950	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
97. #11326	C₂₃	direct (N × H)
111. C₂×C₄×C₂₃	#108	semidirect
213. C₂₃	#11326	direct (N × H)
271. C₂×C₄	C₂₃×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#108	2
C₂×C₂₃×#423	2
C₂×C₂₃×#341	2
C₂₃×#1728	2
C₂×C₂₃×#336	2
#11326	23

Subgroups — C23×#11326 #19158952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11326 #19158952