Subgroups — C₂₃×#19754 #19167380

Order 5888.

728 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3184	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3182	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3141	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1719	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1718	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1717	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1711	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#533	C₂²	1	1	yes	characteristic
11	C₂₃×#532	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×#500	C₂²	1	2	yes	—
14	C₂₃×#498	C₂²	1	1	yes	characteristic
15	C₂₃×#493	C₂²	1	1	yes	characteristic
16	C₂₃×#492	C₂²	1	2	yes	—
17	C₂₃×#491	—	2	2	no	—
18	C₂₃×#489	—	2	4	no	—
19	C₂₃×#487	C₂²	1	2	yes	—
20	C₂₃×#487	C₂²	1	2	yes	—
21	C₂₃×#487	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×#469	—	2	2	no	—
29	C₂₃×#468	—	2	4	no	—
30	C₂₃×#442	—	2	2	no	—
31	C₂₃×#440	—	2	4	no	—
32	C₂₃×#419	—	2	2	no	—
33	C₂₃×#417	—	2	4	no	—
34	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
35	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#322	C₂²	1	2	yes	—
37	C₂₃×#322	C₂²	1	1	yes	characteristic
38	C₈²×C₂₃	C₂²	1	1	yes	characteristic
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂²×Q₈×C₂₃	—	2	1	no	—
41	C₂×SD₁₆×C₂₃	—	2	4	no	—
42	C₂×D₈×C₂₃	—	2	2	no	—
43	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
44	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
45	C₂₃×C₄⋊Q₈	—	2	2	no	—
46	C₂₃×C₄⋊Q₈	—	2	1	no	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₂₃×#127	C₂³	1	2	yes	—
51	C₂₃×#125	—	2	1	no	—
52	C₂₃×#124	—	2	2	no	—
53	C₂₃×C₂²⋊Q₈	—	2	2	no	—
54	C₂₃×#121	—	2	2	no	—
55	C₂₃×#121	—	2	2	no	—
56	C₄×Q₈×C₂₃	—	2	2	no	—
57	C₄×D₄×C₂₃	—	2	2	no	—
58	C₄×D₄×C₂₃	—	2	2	no	—
59	C₂₃×#117	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×#107	—	2	4	no	—
63	C₂₃×#106	—	2	2	no	—
64	C₂₃×#103	—	2	4	no	—
65	C₂₃×#103	—	2	4	no	—
66	C₂₃×#103	—	2	4	no	—
67	C₂₃×#103	—	2	4	no	—
68	C₂₃×#103	—	2	4	no	—
69	C₂₃×#102	—	2	4	no	—
70	C₂₃×#102	—	2	4	no	—
71	C₂₃×#102	—	2	4	no	—
72	C₂₃×#102	—	2	4	no	—
73	C₂₃×#97	D₄	1	4	yes	—
74	C₂₃×#97	D₄	1	2	yes	—
75	C₄×C₈×C₂₃	C₂³	1	2	yes	—
76	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₂₃	C₂³	1	2	yes	—
78	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	#41×C₂₃	—	4	8	no	—
80	C₂×Q₈×C₂₃	—	2	2	no	—
81	C₂×Q₈×C₂₃	—	2	2	no	—
82	C₂×Q₈×C₂₃	—	2	2	no	—
83	C₂×Q₈×C₂₃	—	4	4	no	—
84	C₂×Q₈×C₂₃	—	4	1	no	—
85	C₂×D₄×C₂₃	—	2	2	no	—
86	C₂×D₄×C₂₃	—	2	2	no	—
87	C₂×D₄×C₂₃	—	2	2	no	—
88	C₂×D₄×C₂₃	—	4	2	no	—
89	C₂²×C₄×C₂₃	—	4	1	no	—
90	C₂²×C₄×C₂₃	—	4	2	no	—
91	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	SD₁₆×C₂₃	—	4	8	no	—
94	D₈×C₂₃	—	4	4	no	—
95	M₁₆×C₂₃	—	4	2	no	—
96	M₁₆×C₂₃	—	4	4	no	—
97	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₄⋊C₄×C₂₃	—	4	2	no	—
109	C₂²⋊C₄×C₂₃	—	4	1	no	—
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₂²⋊C₄×C₂₃	—	4	2	no	—
112	C₄²×C₂₃	—	4	1	no	—
113	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₂₃	C₂×D₄	1	2	yes	—
115	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
116	#19754	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₂₃	—	4	2	no	—
118	C₂³×C₂₃	#127	1	1	yes	characteristic
119	Q₈×C₂₃	—	4	4	no	—
120	Q₈×C₂₃	—	4	4	no	—
121	Q₈×C₂₃	—	4	4	no	—
122	Q₈×C₂₃	—	8	2	no	—
123	D₄×C₂₃	—	4	4	no	—
124	D₄×C₂₃	—	4	4	no	—
125	D₄×C₂₃	—	4	4	no	—
126	D₄×C₂₃	—	8	4	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	2	4	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	8	4	no	—
133	C₂×C₄×C₂₃	—	8	2	no	—
134	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#127	1	2	yes	—
136	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
138	C₈×C₂₃	—	2	2	no	—
139	C₈×C₂₃	—	2	4	no	—
140	C₈×C₂₃	—	2	2	no	—
141	C₈×C₂₃	—	2	4	no	—
142	#3184	C₂×C₂₃	1	1	yes	characteristic
143	#3182	C₂×C₂₃	1	2	yes	complemented
144	#3141	C₂×C₂₃	1	2	yes	complemented
145	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#1719	C₂×C₂₃	1	2	yes	complemented
147	#1718	C₂×C₂₃	1	2	yes	complemented
148	#1717	C₂×C₂₃	1	2	yes	complemented
149	#1711	C₂×C₂₃	1	2	yes	complemented
150	#1455	C₂×C₂₃	1	1	yes	characteristic, complemented
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	2	2	no	—
153	C₂²×C₂₃	—	8	2	no	—
154	C₂²×C₂₃	—	8	2	no	—
155	C₂²×C₂₃	—	8	2	no	—
156	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
157	C₄×C₂₃	—	2	2	no	—
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	8	2	no	—
160	C₄×C₂₃	—	8	2	no	—
161	C₄×C₂₃	—	8	2	no	—
162	C₄×C₂₃	#575	1	2	yes	—
163	C₄×C₂₃	#575	1	4	yes	—
164	#533	C₂²×C₂₃	1	1	yes	characteristic
165	#532	C₂²×C₂₃	1	2	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
167	#500	C₂²×C₂₃	1	2	yes	—
168	#498	C₂²×C₂₃	1	1	yes	characteristic
169	#493	C₂²×C₂₃	1	1	yes	characteristic
170	#492	C₂²×C₂₃	1	2	yes	—
171	#491	—	2	2	no	—
172	#489	—	2	4	no	—
173	#487	C₂²×C₂₃	1	2	yes	—
174	#487	C₂²×C₂₃	1	2	yes	—
175	#487	C₂²×C₂₃	1	2	yes	—
176	#487	C₂²×C₂₃	1	2	yes	—
177	#487	C₂²×C₂₃	1	2	yes	—
178	#486	C₂²×C₂₃	1	2	yes	—
179	#486	C₂²×C₂₃	1	2	yes	—
180	#486	C₂²×C₂₃	1	2	yes	—
181	#486	C₂²×C₂₃	1	2	yes	—
182	#469	—	2	2	no	—
183	#468	—	2	4	no	—
184	#442	—	2	2	no	—
185	#440	—	2	4	no	—
186	#419	—	2	2	no	—
187	#417	—	2	4	no	—
188	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
189	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
190	#322	C₂²×C₂₃	1	2	yes	—
191	#322	C₂²×C₂₃	1	1	yes	characteristic
192	C₈²	C₂²×C₂₃	1	1	yes	characteristic
193	C₂×C₂₃	—	4	1	no	—
194	C₂×C₂₃	—	16	2	no	—
195	C₂×C₂₃	#3153	1	2	yes	—
196	C₂×C₂₃	#3153	1	1	yes	characteristic
197	C₂×#41	—	2	2	no	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×D₈	—	2	2	no	—
201	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
202	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
208	#127	C₂³×C₂₃	1	2	yes	—
209	#125	—	2	1	no	—
210	#124	—	2	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	#121	—	2	2	no	—
213	#121	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	#117	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	#106	—	2	2	no	—
222	#103	—	2	4	no	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#102	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#97	D₄×C₂₃	1	4	yes	—
232	#97	D₄×C₂₃	1	2	yes	—
233	C₄×C₈	C₂³×C₂₃	1	2	yes	—
234	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₂₃	1	2	yes	—
236	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
237	C₂₃	#19754	1	1	yes	characteristic, direct product factor, Sylow
238	#41	—	4	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	SD₁₆	—	4	8	no	—
253	D₈	—	4	4	no	—
254	M₁₆	—	4	2	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
257	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
259	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₂₃	1	2	yes	—
274	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
275	C₂³	—	4	2	no	—
276	C₂³	C₂₃×#127	1	1	yes	characteristic
277	Q₈	—	4	4	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	4	no	—
280	Q₈	—	8	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
293	C₂×C₄	C₂₃×#127	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₂₃×#575	1	2	yes	—
312	C₄	C₂₃×#575	1	4	yes	—
313	C₂	—	4	1	no	—
314	C₂	—	16	2	no	—
315	C₂	C₂₃×#3153	1	2	yes	—
316	C₂	C₂₃×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3184	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
2. C₂₃×#3182	C₂	semidirect
3. C₂₃×#3141	C₂	semidirect
4. C₂₃×#3139	C₂	semidirect
5. C₂₃×#1719	C₂	semidirect
6. C₂₃×#1718	C₂	semidirect
7. C₂₃×#1717	C₂	semidirect
8. C₂₃×#1711	C₂	semidirect
9. C₂₃×#1455	C₂	semidirect
116. #19754	C₂₃	direct (N × H)
143. #3182	C₂×C₂₃	semidirect
144. #3141	C₂×C₂₃	semidirect
145. #3139	C₂×C₂₃	semidirect
146. #1719	C₂×C₂₃	semidirect
147. #1718	C₂×C₂₃	semidirect
148. #1717	C₂×C₂₃	semidirect
149. #1711	C₂×C₂₃	semidirect
150. #1455	C₂×C₂₃	semidirect
237. C₂₃	#19754	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×#3184	2
C₂₃×#3182	2
C₂₃×#3141	2
C₂₃×#3139	2
C₂₃×#1719	2
C₂₃×#1718	2
C₂₃×#1717	2
C₂₃×#1711	2
C₂₃×#1455	2
#19754	23

Subgroups — C23×#19754 #19167380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19754 #19167380