Subgroups — C₂₃×#8605 #19156231

Order 5888.

462 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1477	C₂	1	2	yes	maximal
5	C₂₃×#1283	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
8	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#431	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂₃×#405	—	2	1	no	—
12	C₂₃×#405	—	2	1	no	—
13	C₂₃×#405	C₄	1	1	yes	characteristic
14	C₂₃×#405	C₄	1	1	yes	characteristic
15	C₄×M₁₆×C₂₃	—	2	1	no	—
16	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₃	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#335	—	2	1	no	—
21	C₂₃×#335	—	2	1	no	—
22	C₂₃×#334	—	2	1	no	—
23	C₂₃×Q₈⋊C₈	—	2	2	no	—
24	C₂₃×D₄⋊C₈	—	2	2	no	—
25	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂₃×#131	—	2	2	no	—
27	C₂₃×#131	—	2	2	no	—
28	C₂₃×#131	—	2	2	no	—
29	C₂₃×#131	C₈	1	4	yes	complemented
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	—	2	1	no	—
32	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	—	2	1	no	—
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₂₃	Q₈	1	1	yes	characteristic
41	C₄×D₄×C₂₃	—	2	1	no	—
42	C₄×D₄×C₂₃	—	2	1	no	—
43	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
44	C₄×D₄×C₂₃	Q₈	1	1	yes	characteristic
45	C₂₃×#117	—	2	1	no	—
46	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	—	2	1	no	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊C₈	—	2	2	no	—
50	C₂₃×C₄⋊C₈	—	2	2	no	—
51	C₂₃×C₄⋊C₈	—	2	2	no	—
52	C₂₃×C₂²⋊C₈	—	2	2	no	—
53	C₂₃×#97	—	2	1	no	—
54	C₂₃×#97	—	2	1	no	—
55	C₂₃×#97	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	1	no	—
57	C₄×C₈×C₂₃	—	2	1	no	—
58	C₄×C₈×C₂₃	—	2	1	no	—
59	C₄×C₈×C₂₃	—	2	2	no	—
60	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
64	#41×C₂₃	—	2	1	no	—
65	#41×C₂₃	—	2	1	no	—
66	#41×C₂₃	C₂×C₈	1	2	yes	—
67	#41×C₂₃	M₁₆	1	2	yes	—
68	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₂₃	—	2	1	no	—
70	C₂×D₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	—	2	1	no	—
72	C₂²×C₄×C₂₃	—	2	1	no	—
73	C₂²×C₄×C₂₃	—	2	1	no	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
77	M₁₆×C₂₃	—	2	4	no	—
78	M₁₆×C₂₃	—	2	1	no	—
79	M₁₆×C₂₃	—	2	1	no	—
80	M₁₆×C₂₃	—	2	4	no	—
81	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
82	M₁₆×C₂₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₂₃	—	2	4	no	—
84	C₂×C₈×C₂₃	—	2	1	no	—
85	C₂×C₈×C₂₃	—	2	1	no	—
86	C₂×C₈×C₂₃	—	2	1	no	—
87	C₂×C₈×C₂₃	—	2	1	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	4	2	no	—
90	C₂×C₈×C₂₃	—	4	2	no	—
91	C₂×C₈×C₂₃	—	8	2	no	—
92	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
97	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₄²×C₂₃	—	2	1	no	—
106	C₄²×C₂₃	—	2	1	no	—
107	C₄²×C₂₃	—	2	1	no	—
108	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
112	#8605	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	—	2	1	no	—
114	C₂³×C₂₃	#117	1	1	yes	characteristic
115	Q₈×C₂₃	—	2	1	no	—
116	Q₈×C₂₃	C₄⋊C₈	1	2	yes	—
117	D₄×C₂₃	—	2	2	no	—
118	D₄×C₂₃	—	2	2	no	—
119	D₄×C₂₃	—	2	1	no	—
120	D₄×C₂₃	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	2	1	no	—
126	C₂×C₄×C₂₃	—	2	1	no	—
127	C₂×C₄×C₂₃	—	2	1	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	1	no	—
131	C₂×C₄×C₂₃	—	4	1	no	—
132	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
142	C₈×C₂₃	—	2	2	no	—
143	C₈×C₂₃	—	2	1	no	—
144	C₈×C₂₃	—	2	2	no	—
145	C₈×C₂₃	—	2	1	no	—
146	C₈×C₂₃	—	8	2	no	—
147	C₈×C₂₃	—	8	2	no	—
148	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
149	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
150	C₄×#131	C₂×C₂₃	1	1	yes	characteristic
151	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
152	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#1477	C₂×C₂₃	1	2	yes	—
154	#1283	C₂×C₂₃	1	2	yes	complemented
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	2	1	no	—
157	C₂²×C₂₃	—	2	2	no	—
158	C₂²×C₂₃	—	4	1	no	—
159	C₂²×C₂₃	—	4	1	no	—
160	C₂²×C₂₃	#380	1	1	yes	characteristic
161	C₂²×C₂₃	#431	1	1	yes	characteristic
162	C₂²×C₂₃	#432	1	1	yes	characteristic
163	C₄×C₂₃	—	2	2	no	—
164	C₄×C₂₃	—	2	1	no	—
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	4	1	no	—
168	C₄×C₂₃	—	4	1	no	—
169	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₂₃	#407	1	1	yes	characteristic
171	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₄×C₂₃	#423	1	1	yes	characteristic
173	C₄×C₂₃	#472	1	1	yes	characteristic
174	C₄×C₂₃	#473	1	1	yes	characteristic
175	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
176	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
177	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
178	#431	C₂²×C₂₃	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
180	#405	—	2	1	no	—
181	#405	—	2	1	no	—
182	#405	C₄×C₂₃	1	1	yes	characteristic
183	#405	C₄×C₂₃	1	1	yes	characteristic
184	C₄×M₁₆	—	2	1	no	—
185	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
186	C₂×C₄×C₈	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
188	#335	—	2	1	no	—
189	#335	—	2	1	no	—
190	#335	—	2	1	no	—
191	#334	—	2	1	no	—
192	Q₈⋊C₈	—	2	2	no	—
193	D₄⋊C₈	—	2	2	no	—
194	C₂×C₂₃	—	2	1	no	—
195	C₂×C₂₃	—	2	1	no	—
196	C₂×C₂₃	—	4	2	no	—
197	C₂×C₂₃	#1797	1	1	yes	characteristic
198	C₂×C₂₃	#1814	1	1	yes	characteristic
199	C₂×C₂₃	#1816	1	1	yes	characteristic
200	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	C₈×C₂₃	1	4	yes	complemented
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
208	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
214	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
215	C₄×Q₈	Q₈×C₂₃	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
219	C₄×D₄	Q₈×C₂₃	1	1	yes	characteristic
220	#117	—	2	1	no	—
221	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
236	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
237	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
238	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
239	C₂₃	#8605	1	1	yes	characteristic, direct product factor, Sylow
240	#41	—	2	1	no	—
241	#41	—	2	1	no	—
242	#41	C₂×C₈×C₂₃	1	2	yes	—
243	#41	M₁₆×C₂₃	1	2	yes	—
244	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
253	M₁₆	—	2	4	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
258	M₁₆	M₁₆×C₂₃	1	2	yes	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	8	2	no	—
268	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
273	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
277	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
287	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	C₂₃×#117	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	C₂₃×C₄⋊C₈	1	2	yes	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	1	no	—
295	D₄	C₂₃×C₄⋊C₈	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
309	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
310	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
312	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
314	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
315	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
316	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	—	8	2	no	—
322	C₈	—	8	2	no	—
323	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
324	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂₃×#380	1	1	yes	characteristic
331	C₂²	C₂₃×#431	1	1	yes	characteristic
332	C₂²	C₂₃×#432	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
340	C₄	C₂₃×#407	1	1	yes	characteristic
341	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
342	C₄	C₂₃×#423	1	1	yes	characteristic
343	C₄	C₂₃×#472	1	1	yes	characteristic
344	C₄	C₂₃×#473	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	2	no	—
348	C₂	C₂₃×#1797	1	1	yes	characteristic
349	C₂	C₂₃×#1814	1	1	yes	characteristic
350	C₂	C₂₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1564	C₂	semidirect
3. C₂×C₂₃×#335	C₂	semidirect
5. C₂₃×#1283	C₂	semidirect
29. C₂₃×#131	C₈	semidirect
112. #8605	C₂₃	direct (N × H)
151. #1564	C₂×C₂₃	semidirect
152. C₂×#335	C₂×C₂₃	semidirect
154. #1283	C₂×C₂₃	semidirect
204. #131	C₈×C₂₃	semidirect
239. C₂₃	#8605	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#131	2
C₂₃×#1564	2
C₂×C₂₃×#335	2
C₂₃×#1477	2
C₂₃×#1283	2
#8605	23

Subgroups — C23×#8605 #19156231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8605 #19156231