Subgroups — C₂₃×#10288 #19157914

Order 5888.

554 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3226	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3206	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1962	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂₃×#500	—	2	1	no	—
12	C₂₃×#492	—	2	1	no	—
13	C₂₃×#487	—	2	1	no	—
14	C₂₃×#486	—	2	1	no	—
15	C₂₃×#480	—	2	2	no	—
16	C₂₃×#474	—	2	2	no	—
17	C₂₃×#461	—	2	2	no	—
18	C₂₃×#448	—	2	2	no	—
19	C₂₃×#431	C₂²	1	1	yes	characteristic
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂₃×#423	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#104	—	2	2	no	—
23	C₂₃×#391	—	2	2	no	—
24	C₂₃×#391	—	2	2	no	—
25	C₂₃×#382	—	2	2	no	—
26	C₄³×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#333	—	2	2	no	—
28	C₂₃×#332	—	2	2	no	—
29	C₂×#41×C₂₃	—	2	1	no	—
30	C₂²×Q₈×C₂₃	—	2	1	no	—
31	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
32	C₂×SD₁₆×C₂₃	—	2	4	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂₃×C₄⋊Q₈	—	2	1	no	—
35	C₂₃×C₄⋊Q₈	—	2	1	no	—
36	C₂₃×C₄⋊Q₈	—	2	1	no	—
37	C₂₃×C₄⋊Q₈	—	2	1	no	—
38	C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	—	2	1	no	—
41	C₂₃×C₄⋊Q₈	—	2	1	no	—
42	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₂₃×#127	D₄	1	1	yes	characteristic
44	C₂₃×#124	—	2	1	no	—
45	C₂₃×#121	—	2	1	no	—
46	C₂₃×#121	—	2	1	no	—
47	C₄×D₄×C₂₃	—	2	1	no	—
48	C₄×D₄×C₂₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
53	C₂×C₄²×C₂₃	—	2	1	no	—
54	C₂×C₄²×C₂₃	—	2	1	no	—
55	C₂×C₄²×C₂₃	D₄	1	2	yes	—
56	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₂₃×#107	—	2	4	no	—
58	C₂₃×C₄⋊C₈	—	2	1	no	—
59	C₂₃×C₄⋊C₈	—	2	1	no	—
60	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
61	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
62	C₂₃×#104	—	4	4	no	—
63	C₂₃×#103	—	2	4	no	—
64	C₂₃×#102	—	2	4	no	—
65	C₂₃×C₂²⋊C₈	—	2	1	no	—
66	C₂₃×C₂²⋊C₈	—	2	1	no	—
67	C₄×C₈×C₂₃	—	2	1	no	—
68	C₄×C₈×C₂₃	—	2	1	no	—
69	C₂₃×#95	—	2	8	no	—
70	#41×C₂₃	—	4	2	no	—
71	#41×C₂₃	—	4	2	no	—
72	C₂×Q₈×C₂₃	—	2	2	no	—
73	C₂×Q₈×C₂₃	—	2	1	no	—
74	C₂×Q₈×C₂₃	—	2	2	no	—
75	C₂×Q₈×C₂₃	—	4	4	no	—
76	C₂×Q₈×C₂₃	—	4	1	no	—
77	C₂×Q₈×C₂₃	—	4	1	no	—
78	C₂×Q₈×C₂₃	—	4	2	no	—
79	C₂×Q₈×C₂₃	—	4	2	no	—
80	C₂×D₄×C₂₃	—	2	1	no	—
81	C₂×D₄×C₂₃	—	2	1	no	—
82	C₂×D₄×C₂₃	—	2	1	no	—
83	C₂×D₄×C₂₃	—	4	1	no	—
84	C₂²×C₄×C₂₃	—	2	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	4	1	no	—
88	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	SD₁₆×C₂₃	—	8	4	no	—
92	M₁₆×C₂₃	—	4	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	4	1	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	4	1	no	—
101	C₄⋊C₄×C₂₃	—	4	1	no	—
102	C₄⋊C₄×C₂₃	—	4	1	no	—
103	C₄⋊C₄×C₂₃	—	4	1	no	—
104	C₄⋊C₄×C₂₃	—	4	1	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₂²⋊C₄×C₂₃	—	4	1	no	—
107	C₂²⋊C₄×C₂₃	—	4	1	no	—
108	C₄²×C₂₃	—	2	4	no	—
109	C₄²×C₂₃	—	2	2	no	—
110	C₄²×C₂₃	—	2	2	no	—
111	C₄²×C₂₃	—	4	2	no	—
112	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₂₃	#41	1	1	yes	characteristic
116	#10288	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₂₃	—	4	1	no	—
118	C₂³×C₂₃	#120	1	1	yes	characteristic
119	Q₈×C₂₃	—	4	4	no	—
120	Q₈×C₂₃	—	4	2	no	—
121	Q₈×C₂₃	—	8	2	no	—
122	Q₈×C₂₃	—	8	2	no	—
123	Q₈×C₂₃	—	8	1	no	—
124	Q₈×C₂₃	—	8	1	no	—
125	D₄×C₂₃	—	4	2	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	2	no	—
128	D₄×C₂₃	—	8	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	4	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	4	4	no	—
135	C₂×C₄×C₂₃	—	4	1	no	—
136	C₂×C₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	—	4	4	no	—
139	C₂×C₄×C₂₃	—	8	1	no	—
140	C₂×C₄×C₂₃	—	8	1	no	—
141	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
142	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#121	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	#127	1	1	yes	characteristic
147	C₂×C₄×C₂₃	#135	1	2	yes	—
148	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₈×C₂₃	—	4	2	no	—
150	C₈×C₂₃	—	4	2	no	—
151	C₈×C₂₃	—	8	1	no	—
152	#3226	C₂×C₂₃	1	1	yes	characteristic
153	#3206	C₂×C₂₃	1	1	yes	characteristic
154	#2862	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#1962	C₂×C₂₃	1	1	yes	characteristic
156	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
158	#1276	C₂×C₂₃	1	1	yes	characteristic, complemented
159	C₂²×C₂₃	—	2	1	no	—
160	C₂²×C₂₃	—	2	1	no	—
161	C₂²×C₂₃	—	8	1	no	—
162	C₂²×C₂₃	—	8	1	no	—
163	C₂²×C₂₃	—	8	1	no	—
164	C₂²×C₂₃	#392	1	1	yes	characteristic
165	C₂²×C₂₃	#449	1	2	yes	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	4	4	no	—
169	C₄×C₂₃	—	8	2	no	—
170	C₄×C₂₃	—	8	2	no	—
171	C₄×C₂₃	—	8	1	no	—
172	C₄×C₂₃	#455	1	2	yes	—
173	C₄×C₂₃	#456	1	2	yes	—
174	C₄×C₂₃	#463	1	2	yes	—
175	C₄×C₂₃	#495	1	1	yes	characteristic
176	C₄×C₂₃	#496	1	1	yes	characteristic
177	#532	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
180	#500	—	2	1	no	—
181	#492	—	2	1	no	—
182	#487	—	2	1	no	—
183	#486	—	2	1	no	—
184	#480	—	2	2	no	—
185	#474	—	2	2	no	—
186	#461	—	2	2	no	—
187	#448	—	2	2	no	—
188	#431	C₂²×C₂₃	1	1	yes	characteristic
189	#431	C₂²×C₂₃	1	1	yes	characteristic
190	#423	C₂²×C₂₃	1	1	yes	characteristic
191	C₂×#104	—	2	2	no	—
192	#391	—	2	2	no	—
193	#391	—	2	2	no	—
194	#382	—	2	2	no	—
195	C₄³	C₂²×C₂₃	1	1	yes	characteristic
196	#333	—	2	2	no	—
197	#332	—	2	2	no	—
198	C₂×C₂₃	—	2	2	no	—
199	C₂×C₂₃	—	16	1	no	—
200	C₂×C₂₃	#2004	1	1	yes	characteristic
201	C₂×C₂₃	#2005	1	1	yes	characteristic
202	C₂×C₂₃	#2012	1	1	yes	characteristic
203	C₂×#41	—	2	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
217	#127	D₄×C₂₃	1	1	yes	characteristic
218	#124	—	2	1	no	—
219	#121	—	2	1	no	—
220	#121	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	D₄×C₂₃	1	2	yes	—
230	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
231	#107	—	2	4	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
235	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
236	#104	—	4	4	no	—
237	#103	—	2	4	no	—
238	#102	—	2	4	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	#95	—	2	8	no	—
244	C₂₃	#10288	1	1	yes	characteristic, direct product factor, Sylow
245	#41	—	4	2	no	—
246	#41	—	4	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
266	SD₁₆	—	8	4	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	4	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
290	C₄²	#41×C₂₃	1	1	yes	characteristic
291	C₂³	—	4	1	no	—
292	C₂³	C₂₃×#120	1	1	yes	characteristic
293	Q₈	—	4	4	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	8	1	no	—
298	Q₈	—	8	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
316	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
317	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
319	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
320	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
321	C₂×C₄	C₂₃×#135	1	2	yes	—
322	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₂₃×#392	1	1	yes	characteristic
332	C₂²	C₂₃×#449	1	2	yes	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	4	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	C₂₃×#455	1	2	yes	—
340	C₄	C₂₃×#456	1	2	yes	—
341	C₄	C₂₃×#463	1	2	yes	—
342	C₄	C₂₃×#495	1	1	yes	characteristic
343	C₄	C₂₃×#496	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	1	no	—
346	C₂	C₂₃×#2004	1	1	yes	characteristic
347	C₂	C₂₃×#2005	1	1	yes	characteristic
348	C₂	C₂₃×#2012	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ 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×#2862	C₂	semidirect
5. C₂₃×#1548	C₂	semidirect
6. C₂₃×#1546	C₂	semidirect
7. C₂₃×#1276	C₂	semidirect
116. #10288	C₂₃	direct (N × H)
154. #2862	C₂×C₂₃	semidirect
156. #1548	C₂×C₂₃	semidirect
157. #1546	C₂×C₂₃	semidirect
158. #1276	C₂×C₂₃	semidirect
244. C₂₃	#10288	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×#3226	2
C₂₃×#3206	2
C₂₃×#2862	2
C₂₃×#1962	2
C₂₃×#1548	2
C₂₃×#1546	2
C₂₃×#1276	2
#10288	23

Subgroups — C23×#10288 #19157914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10288 #19157914