Subgroups — C₂₃×#21661 #19169287

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3135	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1963	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1938	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1764	C₂	1	4	yes	maximal, complemented
9	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#131	C₂²	1	2	yes	—
11	C₂₃×#533	C₂²	1	1	yes	characteristic
12	C₂₃×#532	C₂²	1	1	yes	characteristic
13	C₂₃×#491	C₄	1	2	yes	complemented
14	C₂₃×#490	C₄	1	2	yes	complemented
15	C₂₃×#489	C₄	1	4	yes	complemented
16	C₂₃×#446	—	2	2	no	—
17	C₈×Q₈×C₂₃	—	2	2	no	—
18	C₂₃×#436	—	2	2	no	—
19	C₈×D₄×C₂₃	—	2	2	no	—
20	C₂₃×#424	C₂²	1	2	yes	—
21	C₂₃×#423	C₂²	1	1	yes	characteristic
22	C₂₃×#423	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#104	C₂²	1	4	yes	—
24	C₂₃×#416	C₂²	1	4	yes	—
25	C₂₃×#408	C₂²	1	2	yes	—
26	C₂₃×#405	C₂²	1	2	yes	—
27	C₂×C₂₃×#97	C₂²	1	2	yes	—
28	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#376	C₂²	1	1	yes	characteristic
32	C₂₃×#341	C₂²	1	4	yes	—
33	C₂₃×#339	C₂²	1	4	yes	—
34	C₂×#41×C₂₃	C₂³	1	2	yes	—
35	C₂₃×#131	—	2	8	no	—
36	C₂×M₁₆×C₂₃	—	2	2	no	—
37	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
38	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	—	2	2	no	—
40	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
45	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
46	C₂₃×#125	C₂×C₄	1	2	yes	—
47	C₂₃×#124	C₂×C₄	1	2	yes	—
48	C₂₃×C₂²⋊Q₈	—	2	2	no	—
49	C₂₃×#121	—	2	2	no	—
50	C₄×Q₈×C₂₃	—	2	2	no	—
51	C₄×D₄×C₂₃	—	2	2	no	—
52	C₂₃×#117	C₂³	1	2	yes	—
53	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊C₈	—	2	4	no	—
56	C₂₃×C₄⋊C₈	—	2	2	no	—
57	C₂₃×C₄⋊C₈	—	2	2	no	—
58	C₂₃×#104	—	2	8	no	—
59	C₂₃×#103	C₂×C₄	1	8	yes	—
60	C₂₃×#102	C₂×C₄	1	8	yes	—
61	C₂₃×C₂²⋊C₈	—	2	4	no	—
62	C₂₃×#97	—	2	2	no	—
63	C₂₃×#97	—	2	4	no	—
64	C₂₃×#97	C₂×C₄	1	2	yes	—
65	C₂₃×#97	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₂₃	—	2	2	no	—
67	C₄×C₈×C₂₃	—	2	4	no	—
68	C₄×C₈×C₂₃	D₄	1	2	yes	—
69	C₄×C₈×C₂₃	D₄	1	2	yes	—
70	C₂₃×#95	—	2	2	no	—
71	#41×C₂₃	—	2	4	no	—
72	#41×C₂₃	—	2	4	no	—
73	C₂×Q₈×C₂₃	—	2	2	no	—
74	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×D₄×C₂₃	—	2	2	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₂₃	—	2	2	no	—
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	#41	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₂₃	—	2	4	no	—
83	M₁₆×C₂₃	—	2	4	no	—
84	M₁₆×C₂₃	—	4	4	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	4	4	no	—
91	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	#41	1	4	yes	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	4	no	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₂₃	—	2	2	no	—
101	C₂²⋊C₄×C₂₃	—	2	4	no	—
102	C₄²×C₂₃	—	2	2	no	—
103	C₄²×C₂₃	—	2	2	no	—
104	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₂₃	C₂×D₄	1	2	yes	—
107	C₄²×C₂₃	#41	1	4	yes	—
108	#21661	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₂₃	—	2	2	no	—
110	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
111	Q₈×C₂₃	—	2	4	no	—
112	Q₈×C₂₃	—	4	4	no	—
113	D₄×C₂₃	—	2	4	no	—
114	D₄×C₂₃	—	4	4	no	—
115	D₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
129	C₈×C₂₃	—	2	4	no	—
130	C₈×C₂₃	—	2	2	no	—
131	C₈×C₂₃	—	4	2	no	—
132	C₈×C₂₃	—	4	2	no	—
133	C₈×C₂₃	#124	1	2	yes	—
134	C₈×C₂₃	#124	1	2	yes	—
135	#3135	C₂×C₂₃	1	1	yes	characteristic
136	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#2931	C₂×C₂₃	1	1	yes	characteristic
138	#1995	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#1963	C₂×C₂₃	1	1	yes	characteristic
140	#1938	C₂×C₂₃	1	4	yes	complemented
141	#1838	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#1764	C₂×C₂₃	1	4	yes	complemented
143	#1726	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₂₃	—	2	2	no	—
145	C₂²×C₂₃	—	4	4	no	—
146	C₂²×C₂₃	—	4	2	no	—
147	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₂₃	#529	1	2	yes	—
149	C₄×C₂₃	—	2	2	no	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	4	4	no	—
152	C₄×C₂₃	—	4	4	no	—
153	C₄×C₂₃	—	4	2	no	—
154	C₄×C₂₃	C₄×#41	1	2	yes	—
155	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
156	C₄×C₂₃	#532	1	1	yes	characteristic
157	C₂×#131	C₂²×C₂₃	1	2	yes	—
158	#533	C₂²×C₂₃	1	1	yes	characteristic
159	#532	C₂²×C₂₃	1	1	yes	characteristic
160	#491	C₄×C₂₃	1	2	yes	complemented
161	#490	C₄×C₂₃	1	2	yes	complemented
162	#489	C₄×C₂₃	1	4	yes	complemented
163	#446	—	2	2	no	—
164	C₈×Q₈	—	2	2	no	—
165	#436	—	2	2	no	—
166	C₈×D₄	—	2	2	no	—
167	#424	C₂²×C₂₃	1	2	yes	—
168	#423	C₂²×C₂₃	1	1	yes	characteristic
169	#423	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×#104	C₂²×C₂₃	1	4	yes	—
171	#416	C₂²×C₂₃	1	4	yes	—
172	#408	C₂²×C₂₃	1	2	yes	—
173	#405	C₂²×C₂₃	1	2	yes	—
174	C₂×#97	C₂²×C₂₃	1	2	yes	—
175	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
178	#376	C₂²×C₂₃	1	1	yes	characteristic
179	#341	C₂²×C₂₃	1	4	yes	—
180	#339	C₂²×C₂₃	1	4	yes	—
181	C₂×C₂₃	—	2	2	no	—
182	C₂×C₂₃	—	8	2	no	—
183	C₂×C₂₃	C₄×#124	1	1	yes	characteristic
184	C₂×C₂₃	#2955	1	2	yes	—
185	C₂×#41	C₂³×C₂₃	1	2	yes	—
186	#131	—	2	8	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
189	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
194	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
195	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
196	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
197	#125	C₂×C₄×C₂₃	1	2	yes	—
198	#124	C₂×C₄×C₂₃	1	2	yes	—
199	C₂²⋊Q₈	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	#117	C₂³×C₂₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	#104	—	2	8	no	—
210	#103	C₂×C₄×C₂₃	1	8	yes	—
211	#102	C₂×C₄×C₂₃	1	8	yes	—
212	C₂²⋊C₈	—	2	4	no	—
213	#97	—	2	2	no	—
214	#97	—	2	4	no	—
215	#97	C₂×C₄×C₂₃	1	2	yes	—
216	#97	C₂×C₄×C₂₃	1	2	yes	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	D₄×C₂₃	1	2	yes	—
220	C₄×C₈	D₄×C₂₃	1	2	yes	—
221	#95	—	2	2	no	—
222	C₂₃	#21661	1	1	yes	characteristic, direct product factor, Sylow
223	#41	—	2	4	no	—
224	#41	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	#41×C₂₃	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	4	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
244	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
245	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₈	#41×C₂₃	1	4	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
257	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₂₃	1	2	yes	—
259	C₄²	#41×C₂₃	1	4	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
276	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
279	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₂₃×#124	1	2	yes	—
285	C₈	C₂₃×#124	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
290	C₂²	C₂₃×#529	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	2	no	—
296	C₄	C₄×#41×C₂₃	1	2	yes	—
297	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
298	C₄	C₂₃×#532	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	8	2	no	—
301	C₂	C₄×C₂₃×#124	1	1	yes	characteristic
302	C₂	C₂₃×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2960	C₂	semidirect
4. C₂₃×#1995	C₂	semidirect
6. C₂₃×#1938	C₂	semidirect
7. C₂₃×#1838	C₂	semidirect
8. C₂₃×#1764	C₂	semidirect
9. C₂₃×#1726	C₂	semidirect
13. C₂₃×#491	C₄	semidirect
14. C₂₃×#490	C₄	semidirect
15. C₂₃×#489	C₄	semidirect
108. #21661	C₂₃	direct (N × H)
136. #2960	C₂×C₂₃	semidirect
138. #1995	C₂×C₂₃	semidirect
140. #1938	C₂×C₂₃	semidirect
141. #1838	C₂×C₂₃	semidirect
142. #1764	C₂×C₂₃	semidirect
143. #1726	C₂×C₂₃	semidirect
160. #491	C₄×C₂₃	semidirect
161. #490	C₄×C₂₃	semidirect
162. #489	C₄×C₂₃	semidirect
222. C₂₃	#21661	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×#3135	2
C₂₃×#2960	2
C₂₃×#2931	2
C₂₃×#1995	2
C₂₃×#1963	2
C₂₃×#1938	2
C₂₃×#1838	2
C₂₃×#1764	2
C₂₃×#1726	2
#21661	23

Subgroups — C23×#21661 #19169287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21661 #19169287