Subgroups — C₂₃×#19666 #19167292

Order 5888.

712 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×#3228	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3210	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1720	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1719	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	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	2	yes	—
13	C₂₃×#532	C₂²	1	2	yes	—
14	C₂₃×#500	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	1	yes	characteristic
16	C₂₃×#500	C₂²	1	2	yes	—
17	C₂₃×#498	C₂²	1	2	yes	—
18	C₂₃×#494	C₂²	1	1	yes	characteristic
19	C₂₃×#494	C₂²	1	2	yes	—
20	C₂₃×#493	C₂²	1	1	yes	characteristic
21	C₂₃×#492	C₂²	1	2	yes	—
22	C₂₃×#487	C₂²	1	2	yes	—
23	C₂₃×#487	C₂²	1	2	yes	—
24	C₂₃×#486	C₂²	1	2	yes	—
25	C₂₃×#486	C₂²	1	2	yes	—
26	C₂₃×#486	C₂²	1	2	yes	—
27	C₂₃×#486	C₂²	1	2	yes	—
28	C₂₃×#482	—	2	4	no	—
29	C₂₃×#478	—	2	4	no	—
30	C₂₃×#467	—	2	2	no	—
31	C₂₃×#466	—	2	2	no	—
32	C₂₃×#464	—	2	4	no	—
33	C₂₃×#452	—	2	4	no	—
34	C₄×Q₁₆×C₂₃	—	2	2	no	—
35	C₄×D₈×C₂₃	—	2	2	no	—
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#431	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	C₂×Q₁₆×C₂₃	—	2	2	no	—
45	C₂×Q₁₆×C₂₃	—	2	4	no	—
46	C₂×SD₁₆×C₂₃	—	2	4	no	—
47	C₂×D₈×C₂₃	—	2	2	no	—
48	C₂²×C₈×C₂₃	D₄	1	2	yes	—
49	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₂₃×#127	C₂³	1	2	yes	—
53	C₂₃×#125	—	2	2	no	—
54	C₂₃×#124	—	2	2	no	—
55	C₂₃×C₂²⋊Q₈	—	2	2	no	—
56	C₂₃×C₂²⋊Q₈	—	2	2	no	—
57	C₂₃×#121	—	2	2	no	—
58	C₂₃×#121	—	2	2	no	—
59	C₄×Q₈×C₂₃	—	2	2	no	—
60	C₄×Q₈×C₂₃	—	2	2	no	—
61	C₄×D₄×C₂₃	—	2	2	no	—
62	C₄×D₄×C₂₃	—	2	2	no	—
63	C₂₃×#117	—	2	2	no	—
64	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#107	—	2	2	no	—
66	C₂₃×#107	—	2	2	no	—
67	C₂₃×#107	—	2	4	no	—
68	C₂₃×#106	—	2	4	no	—
69	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
70	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
71	C₂₃×#103	—	2	4	no	—
72	C₂₃×#103	—	2	4	no	—
73	C₂₃×#102	—	2	4	no	—
74	C₂₃×#102	—	2	4	no	—
75	C₂₃×#102	—	2	4	no	—
76	C₂₃×#102	—	2	4	no	—
77	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
79	C₄×C₈×C₂₃	D₄	1	2	yes	—
80	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
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	#41×C₂₃	—	8	4	no	—
87	C₂×Q₈×C₂₃	—	2	2	no	—
88	C₂×Q₈×C₂₃	—	2	2	no	—
89	C₂×Q₈×C₂₃	—	2	2	no	—
90	C₂×D₄×C₂₃	—	2	2	no	—
91	C₂×D₄×C₂₃	—	2	2	no	—
92	C₂×D₄×C₂₃	—	2	2	no	—
93	C₂×D₄×C₂₃	—	4	2	no	—
94	C₂²×C₄×C₂₃	—	4	2	no	—
95	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₂₃	—	4	4	no	—
99	Q₁₆×C₂₃	—	8	4	no	—
100	SD₁₆×C₂₃	—	8	4	no	—
101	D₈×C₂₃	—	4	4	no	—
102	C₂×C₈×C₂₃	—	2	4	no	—
103	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
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	C₄⋊C₄×C₂₃	—	2	2	no	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
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	—	4	2	no	—
116	C₄⋊C₄×C₂₃	—	4	2	no	—
117	C₂²⋊C₄×C₂₃	—	4	2	no	—
118	C₂²⋊C₄×C₂₃	—	4	2	no	—
119	C₂²⋊C₄×C₂₃	—	4	2	no	—
120	C₄²×C₂₃	—	4	2	no	—
121	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
125	#19666	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₂₃	—	4	2	no	—
127	C₂³×C₂₃	#127	1	1	yes	characteristic
128	Q₈×C₂₃	—	4	4	no	—
129	Q₈×C₂₃	—	4	4	no	—
130	Q₈×C₂₃	—	4	4	no	—
131	D₄×C₂₃	—	4	4	no	—
132	D₄×C₂₃	—	4	4	no	—
133	D₄×C₂₃	—	4	4	no	—
134	D₄×C₂₃	—	8	4	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	4	2	no	—
140	C₂×C₄×C₂₃	—	4	2	no	—
141	C₂×C₄×C₂₃	—	8	4	no	—
142	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
144	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
145	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₂₃	—	2	2	no	—
149	C₈×C₂₃	—	2	2	no	—
150	C₈×C₂₃	—	2	2	no	—
151	C₈×C₂₃	—	2	2	no	—
152	C₈×C₂₃	—	2	2	no	—
153	C₈×C₂₃	#135	1	4	yes	—
154	#3228	C₂×C₂₃	1	2	yes	complemented
155	#3210	C₂×C₂₃	1	2	yes	complemented
156	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#3169	C₂×C₂₃	1	1	yes	characteristic
158	#1720	C₂×C₂₃	1	2	yes	complemented
159	#1719	C₂×C₂₃	1	2	yes	complemented
160	#1713	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#1709	C₂×C₂₃	1	1	yes	characteristic, complemented
162	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
163	#1706	C₂×C₂₃	1	1	yes	characteristic, complemented
164	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
165	C₂²×C₂₃	—	2	1	no	—
166	C₂²×C₂₃	—	2	1	no	—
167	C₂²×C₂₃	—	2	1	no	—
168	C₂²×C₂₃	—	8	2	no	—
169	C₂²×C₂₃	—	8	2	no	—
170	C₂²×C₂₃	—	8	2	no	—
171	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
172	C₄×C₂₃	—	2	1	no	—
173	C₄×C₂₃	—	2	1	no	—
174	C₄×C₂₃	—	2	1	no	—
175	C₄×C₂₃	—	8	2	no	—
176	C₄×C₂₃	—	8	2	no	—
177	C₄×C₂₃	—	8	2	no	—
178	C₄×C₂₃	C₂×#135	1	2	yes	—
179	C₄×C₂₃	#576	1	2	yes	—
180	C₄×C₂₃	#578	1	2	yes	—
181	#533	C₂²×C₂₃	1	2	yes	—
182	#532	C₂²×C₂₃	1	2	yes	—
183	#500	C₂²×C₂₃	1	1	yes	characteristic
184	#500	C₂²×C₂₃	1	1	yes	characteristic
185	#500	C₂²×C₂₃	1	2	yes	—
186	#498	C₂²×C₂₃	1	2	yes	—
187	#494	C₂²×C₂₃	1	1	yes	characteristic
188	#494	C₂²×C₂₃	1	2	yes	—
189	#493	C₂²×C₂₃	1	1	yes	characteristic
190	#492	C₂²×C₂₃	1	2	yes	—
191	#487	C₂²×C₂₃	1	2	yes	—
192	#487	C₂²×C₂₃	1	2	yes	—
193	#486	C₂²×C₂₃	1	2	yes	—
194	#486	C₂²×C₂₃	1	2	yes	—
195	#486	C₂²×C₂₃	1	2	yes	—
196	#486	C₂²×C₂₃	1	2	yes	—
197	#482	—	2	4	no	—
198	#478	—	2	4	no	—
199	#467	—	2	2	no	—
200	#466	—	2	2	no	—
201	#464	—	2	4	no	—
202	#452	—	2	4	no	—
203	C₄×Q₁₆	—	2	2	no	—
204	C₄×D₈	—	2	2	no	—
205	#431	C₂²×C₂₃	1	1	yes	characteristic
206	#431	C₂²×C₂₃	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
208	#322	C₂²×C₂₃	1	1	yes	characteristic
209	#322	C₂²×C₂₃	1	1	yes	characteristic
210	C₈²	C₂²×C₂₃	1	1	yes	characteristic
211	C₈²	C₂²×C₂₃	1	1	yes	characteristic
212	C₂×C₂₃	—	4	1	no	—
213	C₂×C₂₃	—	16	2	no	—
214	C₂×C₂₃	#3147	1	1	yes	characteristic
215	C₂×C₂₃	#3155	1	1	yes	characteristic
216	C₂×C₂₃	#3156	1	1	yes	characteristic
217	C₂×#41	—	2	2	no	—
218	C₂×Q₁₆	—	2	2	no	—
219	C₂×Q₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂²×C₈	D₄×C₂₃	1	2	yes	—
223	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
224	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
225	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
226	#127	C₂³×C₂₃	1	2	yes	—
227	#125	—	2	2	no	—
228	#124	—	2	2	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	#121	—	2	2	no	—
232	#121	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	—	2	2	no	—
237	#117	—	2	2	no	—
238	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
239	#107	—	2	2	no	—
240	#107	—	2	2	no	—
241	#107	—	2	4	no	—
242	#106	—	2	4	no	—
243	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
244	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
245	#103	—	2	4	no	—
246	#103	—	2	4	no	—
247	#102	—	2	4	no	—
248	#102	—	2	4	no	—
249	#102	—	2	4	no	—
250	#102	—	2	4	no	—
251	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
252	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
253	C₄×C₈	D₄×C₂₃	1	2	yes	—
254	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
260	C₂₃	#19666	1	1	yes	characteristic, direct product factor, Sylow
261	#41	—	8	4	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
273	Q₁₆	—	4	4	no	—
274	Q₁₆	—	8	4	no	—
275	SD₁₆	—	8	4	no	—
276	D₈	—	4	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
283	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₄²	—	4	2	no	—
296	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
297	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
299	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
300	C₂³	—	4	2	no	—
301	C₂³	C₂₃×#127	1	1	yes	characteristic
302	Q₈	—	4	4	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	8	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
317	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
318	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	C₂₃×#135	1	4	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	C₂×C₂₃×#135	1	2	yes	—
342	C₄	C₂₃×#576	1	2	yes	—
343	C₄	C₂₃×#578	1	2	yes	—
344	C₂	—	4	1	no	—
345	C₂	—	16	2	no	—
346	C₂	C₂₃×#3147	1	1	yes	characteristic
347	C₂	C₂₃×#3155	1	1	yes	characteristic
348	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×#3169	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×#3210	C₂	semidirect
3. C₂₃×#3170	C₂	semidirect
5. C₂₃×#1720	C₂	semidirect
6. C₂₃×#1719	C₂	semidirect
7. C₂₃×#1713	C₂	semidirect
8. C₂₃×#1709	C₂	semidirect
9. C₂₃×#1707	C₂	semidirect
10. C₂₃×#1706	C₂	semidirect
11. C₂₃×#1450	C₂	semidirect
125. #19666	C₂₃	direct (N × H)
154. #3228	C₂×C₂₃	semidirect
155. #3210	C₂×C₂₃	semidirect
156. #3170	C₂×C₂₃	semidirect
158. #1720	C₂×C₂₃	semidirect
159. #1719	C₂×C₂₃	semidirect
160. #1713	C₂×C₂₃	semidirect
161. #1709	C₂×C₂₃	semidirect
162. #1707	C₂×C₂₃	semidirect
163. #1706	C₂×C₂₃	semidirect
164. #1450	C₂×C₂₃	semidirect
260. C₂₃	#19666	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×#3228	2
C₂₃×#3210	2
C₂₃×#3170	2
C₂₃×#3169	2
C₂₃×#1720	2
C₂₃×#1719	2
C₂₃×#1713	2
C₂₃×#1709	2
C₂₃×#1707	2
C₂₃×#1706	2
C₂₃×#1450	2
#19666	23

Subgroups — C23×#19666 #19167292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19666 #19167292