Subgroups — C₂₃×#11233 #19158859

Order 5888.

694 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#341	C₂	1	2	yes	maximal
4	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#108	—	2	4	no	—
12	C₂×C₂₃×#108	—	2	2	no	—
13	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
15	C₂×C₂₃×C₄⋊C₈	C₄	1	4	yes	—
16	C₂₃×#341	—	2	4	no	—
17	C₂₃×#341	—	2	2	no	—
18	C₂₃×#336	—	2	4	no	—
19	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
33	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	—	2	2	no	—
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	—	2	1	no	—
39	C₂²×C₈×C₂₃	—	2	2	no	—
40	C₂²×C₈×C₂₃	—	2	2	no	—
41	C₂²×C₈×C₂₃	—	2	2	no	—
42	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₂₃	D₄	1	2	yes	—
45	C₂²×C₈×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	—	2	1	no	—
47	C₂×C₄²×C₂₃	—	2	1	no	—
48	C₂×C₄²×C₂₃	—	2	1	no	—
49	C₂×C₄²×C₂₃	—	2	1	no	—
50	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₂₃	D₄	1	2	yes	—
52	C₂₃×#108	—	2	8	no	—
53	C₂₃×C₄⋊C₈	—	2	8	no	—
54	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	2	1	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	#41	1	2	yes	—
72	C₂²×C₄×C₂₃	#41	1	2	yes	—
73	M₁₆×C₂₃	—	2	4	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	2	4	no	—
77	M₁₆×C₂₃	—	4	2	no	—
78	M₁₆×C₂₃	—	4	2	no	—
79	M₁₆×C₂₃	—	4	2	no	—
80	M₁₆×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	4	no	—
87	C₂×C₈×C₂₃	—	2	4	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	4	1	no	—
90	C₂×C₈×C₂₃	—	4	2	no	—
91	C₂×C₈×C₂₃	—	4	2	no	—
92	C₂×C₈×C₂₃	—	4	2	no	—
93	C₂×C₈×C₂₃	—	4	2	no	—
94	C₂×C₈×C₂₃	—	4	2	no	—
95	C₂×C₈×C₂₃	—	4	1	no	—
96	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	2	2	no	—
99	C₄²×C₂₃	—	2	2	no	—
100	C₄²×C₂₃	—	2	2	no	—
101	#11233	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	2	2	no	—
103	C₂³×C₂₃	—	2	2	no	—
104	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₂₃	#125	1	2	yes	—
108	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	#108	1	4	yes	complemented
118	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
123	C₂×C₄×C₂₃	#121	1	2	yes	—
124	C₂×C₄×C₂₃	#121	1	2	yes	—
125	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
131	C₂×C₄×C₂₃	#123	1	2	yes	—
132	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
133	C₂×C₄×C₂₃	#125	1	2	yes	—
134	C₂×C₄×C₂₃	#126	1	2	yes	—
135	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
136	C₈×C₂₃	—	2	4	no	—
137	C₈×C₂₃	—	4	2	no	—
138	C₈×C₂₃	—	4	4	no	—
139	C₈×C₂₃	—	4	4	no	—
140	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
141	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
142	C₂×#341	C₂×C₂₃	1	2	yes	—
143	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
144	#1728	C₂×C₂₃	1	2	yes	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	2	1	no	—
148	C₂²×C₂₃	—	2	1	no	—
149	C₂²×C₂₃	—	2	4	no	—
150	C₂²×C₂₃	—	2	4	no	—
151	C₂²×C₂₃	#380	1	1	yes	characteristic
152	C₂²×C₂₃	#389	1	2	yes	—
153	C₂²×C₂₃	#395	1	1	yes	characteristic
154	C₂²×C₂₃	#398	1	2	yes	—
155	C₂²×C₂₃	#400	1	1	yes	characteristic
156	C₂²×C₂₃	C₂×#108	1	2	yes	—
157	C₂²×C₂₃	#430	1	2	yes	—
158	C₄×C₂₃	—	2	4	no	—
159	C₄×C₂₃	—	4	2	no	—
160	C₄×C₂₃	#386	1	1	yes	characteristic
161	C₄×C₂₃	#387	1	2	yes	—
162	C₄×C₂₃	#391	1	1	yes	characteristic
163	C₄×C₂₃	#394	1	1	yes	characteristic
164	C₄×C₂₃	#396	1	2	yes	—
165	C₄×C₂₃	#398	1	1	yes	characteristic
166	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
167	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
168	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
170	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×#108	—	2	4	no	—
172	C₂×#108	—	2	2	no	—
173	C₂×C₄⋊C₈	—	2	2	no	—
174	C₂×C₄⋊C₈	—	2	2	no	—
175	C₂×C₄⋊C₈	C₄×C₂₃	1	4	yes	—
176	#341	—	2	4	no	—
177	#341	—	2	2	no	—
178	#336	—	2	4	no	—
179	C₂×C₂₃	—	2	1	no	—
180	C₂×C₂₃	—	2	2	no	—
181	C₂×C₂₃	—	2	1	no	—
182	C₂×C₂₃	#1441	1	1	yes	characteristic
183	C₂×C₂₃	#1810	1	1	yes	characteristic
184	C₂×C₂₃	#1811	1	1	yes	characteristic
185	C₂×C₂₃	#1950	1	4	yes	—
186	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
187	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
200	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
210	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
211	C₂²×C₈	D₄×C₂₃	1	2	yes	—
212	C₂²×C₈	D₄×C₂₃	1	2	yes	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
218	C₂×C₄²	D₄×C₂₃	1	2	yes	—
219	#108	—	2	8	no	—
220	C₄⋊C₈	—	2	8	no	—
221	C₂₃	#11233	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
230	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
237	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	#41×C₂₃	1	2	yes	—
240	C₂²×C₄	#41×C₂₃	1	2	yes	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
272	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
273	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
274	C₂³	C₂₃×#125	1	2	yes	—
275	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	C₂₃×#108	1	4	yes	complemented
285	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
288	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
289	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
290	C₂×C₄	C₂₃×#121	1	2	yes	—
291	C₂×C₄	C₂₃×#121	1	2	yes	—
292	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
293	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
294	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
295	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
296	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
297	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
298	C₂×C₄	C₂₃×#123	1	2	yes	—
299	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
300	C₂×C₄	C₂₃×#125	1	2	yes	—
301	C₂×C₄	C₂₃×#126	1	2	yes	—
302	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	C₂₃×#380	1	1	yes	characteristic
314	C₂²	C₂₃×#389	1	2	yes	—
315	C₂²	C₂₃×#395	1	1	yes	characteristic
316	C₂²	C₂₃×#398	1	2	yes	—
317	C₂²	C₂₃×#400	1	1	yes	characteristic
318	C₂²	C₂×C₂₃×#108	1	2	yes	—
319	C₂²	C₂₃×#430	1	2	yes	—
320	C₄	—	2	4	no	—
321	C₄	—	4	2	no	—
322	C₄	C₂₃×#386	1	1	yes	characteristic
323	C₄	C₂₃×#387	1	2	yes	—
324	C₄	C₂₃×#391	1	1	yes	characteristic
325	C₄	C₂₃×#394	1	1	yes	characteristic
326	C₄	C₂₃×#396	1	2	yes	—
327	C₄	C₂₃×#398	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	2	no	—
330	C₂	—	2	1	no	—
331	C₂	C₂₃×#1441	1	1	yes	characteristic
332	C₂	C₂₃×#1810	1	1	yes	characteristic
333	C₂	C₂₃×#1811	1	1	yes	characteristic
334	C₂	C₂₃×#1950	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
101. #11233	C₂₃	direct (N × H)
117. C₂×C₄×C₂₃	#108	semidirect
221. C₂₃	#11233	direct (N × H)
284. C₂×C₄	C₂₃×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#108	2
C₂²×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#341	2
C₂₃×#1728	2
#11233	23

Subgroups — C23×#11233 #19158859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11233 #19158859