Subgroups — C₂₃×#19645 #19167271

Order 5888.

772 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3228	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3182	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1719	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1714	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	—	2	2	no	—
13	C₂₃×#533	C₂²	1	2	yes	—
14	C₂₃×#532	C₂²	1	2	yes	—
15	C₂₃×#500	C₂²	1	1	yes	characteristic
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#500	C₂²	1	2	yes	—
18	C₂₃×#498	C₂²	1	2	yes	—
19	C₂₃×#495	—	2	2	no	—
20	C₂₃×#494	C₂²	1	1	yes	characteristic
21	C₂₃×#494	C₂²	1	1	yes	characteristic
22	C₂₃×#493	C₂²	1	1	yes	characteristic
23	C₂₃×#492	C₂²	1	2	yes	—
24	C₂₃×#492	C₂²	1	2	yes	—
25	C₂₃×#487	C₂²	1	2	yes	—
26	C₂₃×#487	C₂²	1	2	yes	—
27	C₂₃×#486	C₂²	1	2	yes	—
28	C₂₃×#486	C₂²	1	2	yes	—
29	C₂₃×#486	C₂²	1	2	yes	—
30	C₂₃×#482	—	2	4	no	—
31	C₂₃×#478	—	2	4	no	—
32	C₂₃×#468	—	2	4	no	—
33	C₂₃×#467	—	2	2	no	—
34	C₂₃×#440	—	2	4	no	—
35	C₄×Q₁₆×C₂₃	—	2	2	no	—
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#322	C₂²	1	1	yes	characteristic
40	C₂₃×#322	C₂²	1	1	yes	characteristic
41	C₈²×C₂₃	C₂²	1	1	yes	characteristic
42	C₈²×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×#41×C₂₃	—	2	2	no	—
44	C₂₃×#135	—	4	8	no	—
45	C₂×Q₁₆×C₂₃	—	2	2	no	—
46	C₂×Q₁₆×C₂₃	—	2	2	no	—
47	C₂×SD₁₆×C₂₃	—	2	4	no	—
48	C₂×SD₁₆×C₂₃	—	2	4	no	—
49	C₂×D₈×C₂₃	—	2	2	no	—
50	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
51	C₂²×C₈×C₂₃	D₄	1	2	yes	—
52	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₂₃×#127	C₂³	1	2	yes	—
56	C₂₃×#125	—	2	2	no	—
57	C₂₃×#124	—	2	2	no	—
58	C₂₃×C₂²⋊Q₈	—	2	2	no	—
59	C₂₃×C₂²⋊Q₈	—	2	2	no	—
60	C₂₃×#121	—	2	2	no	—
61	C₂₃×#121	—	2	2	no	—
62	C₄×Q₈×C₂₃	—	2	2	no	—
63	C₄×Q₈×C₂₃	—	2	2	no	—
64	C₄×D₄×C₂₃	—	2	2	no	—
65	C₄×D₄×C₂₃	—	2	2	no	—
66	C₂₃×#117	—	2	2	no	—
67	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#107	—	2	2	no	—
69	C₂₃×#107	—	2	4	no	—
70	C₂₃×#107	—	2	4	no	—
71	C₂₃×#106	—	2	4	no	—
72	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₂₃×#103	—	2	4	no	—
74	C₂₃×#103	—	2	4	no	—
75	C₂₃×#102	—	2	4	no	—
76	C₂₃×#102	—	2	4	no	—
77	C₂₃×#102	—	2	4	no	—
78	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
79	C₂₃×#97	D₄	1	2	yes	—
80	C₄×C₈×C₂₃	D₄	1	2	yes	—
81	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
87	#41×C₂₃	—	4	8	no	—
88	C₂×Q₈×C₂₃	—	2	2	no	—
89	C₂×Q₈×C₂₃	—	2	2	no	—
90	C₂×Q₈×C₂₃	—	2	2	no	—
91	C₂×D₄×C₂₃	—	2	2	no	—
92	C₂×D₄×C₂₃	—	2	2	no	—
93	C₂×D₄×C₂₃	—	2	2	no	—
94	C₂×D₄×C₂₃	—	4	2	no	—
95	C₂²×C₄×C₂₃	—	4	2	no	—
96	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	Q₁₆×C₂₃	—	4	4	no	—
100	Q₁₆×C₂₃	—	4	4	no	—
101	SD₁₆×C₂₃	—	4	8	no	—
102	SD₁₆×C₂₃	—	4	8	no	—
103	D₈×C₂₃	—	4	4	no	—
104	M₁₆×C₂₃	—	2	4	no	—
105	C₂×C₈×C₂₃	—	2	4	no	—
106	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₄⋊C₄×C₂₃	—	2	2	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₄⋊C₄×C₂₃	—	4	2	no	—
119	C₄⋊C₄×C₂₃	—	4	2	no	—
120	C₂²⋊C₄×C₂₃	—	4	2	no	—
121	C₂²⋊C₄×C₂₃	—	4	2	no	—
122	C₂²⋊C₄×C₂₃	—	4	2	no	—
123	C₄²×C₂₃	—	4	2	no	—
124	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
128	#19645	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₂₃	—	4	2	no	—
130	C₂³×C₂₃	#127	1	1	yes	characteristic
131	Q₈×C₂₃	—	4	4	no	—
132	Q₈×C₂₃	—	4	4	no	—
133	Q₈×C₂₃	—	4	4	no	—
134	D₄×C₂₃	—	4	4	no	—
135	D₄×C₂₃	—	4	4	no	—
136	D₄×C₂₃	—	4	4	no	—
137	D₄×C₂₃	—	8	4	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	4	2	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	—	4	2	no	—
144	C₂×C₄×C₂₃	—	8	4	no	—
145	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
147	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
148	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₂₃	—	2	2	no	—
152	C₈×C₂₃	—	2	2	no	—
153	C₈×C₂₃	—	2	2	no	—
154	C₈×C₂₃	—	2	2	no	—
155	C₈×C₂₃	#135	1	4	yes	complemented
156	C₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
157	#3228	C₂×C₂₃	1	2	yes	complemented
158	#3182	C₂×C₂₃	1	2	yes	complemented
159	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
160	#3147	C₂×C₂₃	1	1	yes	characteristic
161	#1719	C₂×C₂₃	1	2	yes	complemented
162	#1714	C₂×C₂₃	1	2	yes	complemented
163	#1713	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#1709	C₂×C₂₃	1	1	yes	characteristic, complemented
165	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
166	#1706	C₂×C₂₃	1	1	yes	characteristic, complemented
167	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
168	C₂²×C₂₃	—	2	1	no	—
169	C₂²×C₂₃	—	2	1	no	—
170	C₂²×C₂₃	—	2	1	no	—
171	C₂²×C₂₃	—	8	2	no	—
172	C₂²×C₂₃	—	8	2	no	—
173	C₂²×C₂₃	—	8	2	no	—
174	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
175	C₄×C₂₃	—	2	1	no	—
176	C₄×C₂₃	—	2	1	no	—
177	C₄×C₂₃	—	2	1	no	—
178	C₄×C₂₃	—	8	2	no	—
179	C₄×C₂₃	—	8	2	no	—
180	C₄×C₂₃	—	8	2	no	—
181	C₄×C₂₃	C₂×#135	1	2	yes	—
182	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
183	C₄×C₂₃	#578	1	2	yes	—
184	C₂×#135	—	2	2	no	—
185	#533	C₂²×C₂₃	1	2	yes	—
186	#532	C₂²×C₂₃	1	2	yes	—
187	#500	C₂²×C₂₃	1	1	yes	characteristic
188	#500	C₂²×C₂₃	1	2	yes	—
189	#500	C₂²×C₂₃	1	2	yes	—
190	#498	C₂²×C₂₃	1	2	yes	—
191	#495	—	2	2	no	—
192	#494	C₂²×C₂₃	1	1	yes	characteristic
193	#494	C₂²×C₂₃	1	1	yes	characteristic
194	#493	C₂²×C₂₃	1	1	yes	characteristic
195	#492	C₂²×C₂₃	1	2	yes	—
196	#492	C₂²×C₂₃	1	2	yes	—
197	#487	C₂²×C₂₃	1	2	yes	—
198	#487	C₂²×C₂₃	1	2	yes	—
199	#486	C₂²×C₂₃	1	2	yes	—
200	#486	C₂²×C₂₃	1	2	yes	—
201	#486	C₂²×C₂₃	1	2	yes	—
202	#482	—	2	4	no	—
203	#478	—	2	4	no	—
204	#468	—	2	4	no	—
205	#467	—	2	2	no	—
206	#440	—	2	4	no	—
207	C₄×Q₁₆	—	2	2	no	—
208	#431	C₂²×C₂₃	1	1	yes	characteristic
209	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
211	#322	C₂²×C₂₃	1	1	yes	characteristic
212	#322	C₂²×C₂₃	1	1	yes	characteristic
213	C₈²	C₂²×C₂₃	1	1	yes	characteristic
214	C₈²	C₂²×C₂₃	1	1	yes	characteristic
215	C₂×C₂₃	—	4	1	no	—
216	C₂×C₂₃	—	16	2	no	—
217	C₂×C₂₃	#3147	1	1	yes	characteristic
218	C₂×C₂₃	#3151	1	1	yes	characteristic
219	C₂×C₂₃	#3156	1	1	yes	characteristic
220	C₂×#41	—	2	2	no	—
221	#135	—	4	8	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×SD₁₆	—	2	4	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×D₈	—	2	2	no	—
227	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
228	C₂²×C₈	D₄×C₂₃	1	2	yes	—
229	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
230	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
231	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
232	#127	C₂³×C₂₃	1	2	yes	—
233	#125	—	2	2	no	—
234	#124	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	#121	—	2	2	no	—
238	#121	—	2	2	no	—
239	C₄×Q₈	—	2	2	no	—
240	C₄×Q₈	—	2	2	no	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	#117	—	2	2	no	—
244	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
245	#107	—	2	2	no	—
246	#107	—	2	4	no	—
247	#107	—	2	4	no	—
248	#106	—	2	4	no	—
249	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
250	#103	—	2	4	no	—
251	#103	—	2	4	no	—
252	#102	—	2	4	no	—
253	#102	—	2	4	no	—
254	#102	—	2	4	no	—
255	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
256	#97	D₄×C₂₃	1	2	yes	—
257	C₄×C₈	D₄×C₂₃	1	2	yes	—
258	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
264	C₂₃	#19645	1	1	yes	characteristic, direct product factor, Sylow
265	#41	—	4	8	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
277	Q₁₆	—	4	4	no	—
278	Q₁₆	—	4	4	no	—
279	SD₁₆	—	4	8	no	—
280	SD₁₆	—	4	8	no	—
281	D₈	—	4	4	no	—
282	M₁₆	—	2	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
287	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
288	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
305	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
306	C₂³	—	4	2	no	—
307	C₂³	C₂₃×#127	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	8	4	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
323	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
324	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	C₂₃×#135	1	4	yes	complemented
333	C₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₂×C₂₃×#135	1	2	yes	—
348	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
349	C₄	C₂₃×#578	1	2	yes	—
350	C₂	—	4	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₂₃×#3147	1	1	yes	characteristic
353	C₂	C₂₃×#3151	1	1	yes	characteristic
354	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3147	C₂
LCS	G ▷ 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
1. C₂₃×#3228	C₂	semidirect
2. C₂₃×#3182	C₂	semidirect
3. C₂₃×#3170	C₂	semidirect
5. C₂₃×#1719	C₂	semidirect
6. C₂₃×#1714	C₂	semidirect
7. C₂₃×#1713	C₂	semidirect
8. C₂₃×#1709	C₂	semidirect
9. C₂₃×#1707	C₂	semidirect
10. C₂₃×#1706	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
128. #19645	C₂₃	direct (N × H)
155. C₈×C₂₃	#135	semidirect
157. #3228	C₂×C₂₃	semidirect
158. #3182	C₂×C₂₃	semidirect
159. #3170	C₂×C₂₃	semidirect
161. #1719	C₂×C₂₃	semidirect
162. #1714	C₂×C₂₃	semidirect
163. #1713	C₂×C₂₃	semidirect
164. #1709	C₂×C₂₃	semidirect
165. #1707	C₂×C₂₃	semidirect
166. #1706	C₂×C₂₃	semidirect
167. C₈×M₁₆	C₂×C₂₃	semidirect
264. C₂₃	#19645	direct (N × H)
332. C₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3228	2
C₂₃×#3182	2
C₂₃×#3170	2
C₂₃×#3147	2
C₂₃×#1719	2
C₂₃×#1714	2
C₂₃×#1713	2
C₂₃×#1709	2
C₂₃×#1707	2
C₂₃×#1706	2
C₈×M₁₆×C₂₃	2
#19645	23

Subgroups — C23×#19645 #19167271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19645 #19167271