Subgroups — C₂₃×#7668 #19155294

Order 5888.

568 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1271	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₂₃	—	2	1	no	—
6	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
11	C₂₃×#391	—	2	1	no	—
12	C₂₃×#389	—	2	1	no	—
13	C₂₃×#389	C₂²	1	2	yes	—
14	C₂₃×#384	C₄	1	1	yes	characteristic
15	C₂₃×#380	—	2	1	no	—
16	C₂₃×#380	C₄	1	2	yes	—
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#336	C₂²	1	2	yes	—
19	C₂₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	—	2	4	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	4	2	no	—
28	C₂²×C₈×C₂₃	—	4	2	no	—
29	C₂₃×C₄⋊Q₈	—	4	2	no	—
30	C₂₃×C₂²⋊Q₈	—	4	2	no	—
31	C₂₃×C₂²⋊Q₈	—	4	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄²×C₂₃	—	4	2	no	—
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	C₂³	1	1	yes	characteristic
44	C₂₃×C₄⋊C₈	—	4	2	no	—
45	C₂₃×C₂²⋊C₈	—	4	2	no	—
46	C₂₃×#95	—	2	2	no	—
47	C₂₃×#95	—	2	2	no	—
48	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂×Q₈×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	—	4	1	no	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	D₈	1	2	yes	—
65	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	4	8	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	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	4	2	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₂²⋊C₄×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	—	2	2	no	—
87	C₄²×C₂₃	—	2	2	no	—
88	C₄²×C₂₃	—	4	1	no	—
89	C₄²×C₂₃	—	4	1	no	—
90	#7668	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	1	no	—
92	C₂³×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	2	1	no	—
95	C₂³×C₂₃	#101	1	2	yes	—
96	C₂³×C₂₃	#102	1	2	yes	—
97	C₂³×C₂₃	#103	1	2	yes	—
98	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₂₃	—	8	4	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	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	4	no	—
109	C₂×C₄×C₂₃	—	2	4	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	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	#99	1	2	yes	—
123	C₂×C₄×C₂₃	#102	1	2	yes	—
124	C₂×C₄×C₂₃	#103	1	2	yes	—
125	C₂×C₄×C₂₃	#104	1	2	yes	—
126	C₂×C₄×C₂₃	#131	1	2	yes	—
127	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
131	C₈×C₂₃	—	4	4	no	—
132	C₈×C₂₃	—	8	2	no	—
133	#2857	C₂×C₂₃	1	1	yes	characteristic
134	#1797	C₂×C₂₃	1	1	yes	characteristic
135	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
136	#1271	C₂×C₂₃	1	4	yes	complemented
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	—	2	2	no	—
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	2	2	no	—
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	#332	1	4	yes	—
144	C₂²×C₂₃	#408	1	1	yes	characteristic
145	C₂²×C₂₃	C₂×#99	1	1	yes	characteristic
146	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
147	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
148	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
149	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
150	C₂²×C₂₃	#421	1	1	yes	characteristic
151	C₄×C₂₃	—	2	2	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	2	1	no	—
155	C₄×C₂₃	—	2	2	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	4	1	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	—	8	2	no	—
160	C₄×C₂₃	#327	1	4	yes	—
161	C₂³×C₈	—	2	1	no	—
162	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₂²⋊Q₈	—	2	1	no	—
164	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	2	1	no	—
166	C₂×C₂²⋊C₈	—	2	1	no	—
167	#391	—	2	1	no	—
168	#389	—	2	1	no	—
169	#389	C₂²×C₂₃	1	2	yes	—
170	#384	C₄×C₂₃	1	1	yes	characteristic
171	#380	—	2	1	no	—
172	#380	C₄×C₂₃	1	2	yes	—
173	#336	—	2	1	no	—
174	#336	C₂²×C₂₃	1	2	yes	—
175	#336	C₂²×C₂₃	1	2	yes	—
176	C₂×C₂₃	—	2	4	no	—
177	C₂×C₂₃	#1472	1	1	yes	characteristic
178	C₂×C₂₃	#1480	1	1	yes	characteristic
179	C₂×C₂₃	#1493	1	1	yes	characteristic
180	C₂×C₂₃	C₂×#327	1	1	yes	characteristic
181	C₂×C₂₃	#1520	1	1	yes	characteristic
182	C₂×C₂₃	#1531	1	1	yes	characteristic
183	C₂×C₂₃	C₂×#332	1	1	yes	characteristic
184	C₂²×Q₈	—	2	1	no	—
185	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
187	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	4	2	no	—
192	C₂²×C₈	—	4	2	no	—
193	C₄⋊Q₈	—	4	2	no	—
194	C₂²⋊Q₈	—	4	2	no	—
195	C₂²⋊Q₈	—	4	2	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₄²	—	4	2	no	—
204	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	C₄⋊C₈	—	4	2	no	—
209	C₂²⋊C₈	—	4	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	C₂₃	#7668	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	D₈×C₂₃	1	2	yes	—
230	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
231	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
232	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	8	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	C₂₃×#101	1	2	yes	—
260	C₂³	C₂₃×#102	1	2	yes	—
261	C₂³	C₂₃×#103	1	2	yes	—
262	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
263	Q₈	—	8	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂₃×#99	1	2	yes	—
287	C₂×C₄	C₂₃×#102	1	2	yes	—
288	C₂×C₄	C₂₃×#103	1	2	yes	—
289	C₂×C₄	C₂₃×#104	1	2	yes	—
290	C₂×C₄	C₂₃×#131	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	8	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂₃×#332	1	4	yes	—
304	C₂²	C₂₃×#408	1	1	yes	characteristic
305	C₂²	C₂×C₂₃×#99	1	1	yes	characteristic
306	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
307	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
308	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
309	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
310	C₂²	C₂₃×#421	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₂₃×#327	1	4	yes	—
321	C₂	—	2	4	no	—
322	C₂	C₂₃×#1472	1	1	yes	characteristic
323	C₂	C₂₃×#1480	1	1	yes	characteristic
324	C₂	C₂₃×#1493	1	1	yes	characteristic
325	C₂	C₂×C₂₃×#327	1	1	yes	characteristic
326	C₂	C₂₃×#1520	1	1	yes	characteristic
327	C₂	C₂₃×#1531	1	1	yes	characteristic
328	C₂	C₂×C₂₃×#332	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1271	C₂	semidirect
90. #7668	C₂₃	direct (N × H)
136. #1271	C₂×C₂₃	semidirect
212. C₂₃	#7668	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×#2857	2
C₂₃×#1797	2
C₂×C₂₃×#336	2
C₂₃×#1271	2
#7668	23

Subgroups — C23×#7668 #19155294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7668 #19155294