Subgroups — C₂₃×#11470 #19159096

Order 5888.

392 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1323	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1315	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#525	C₄	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	—	2	1	no	—
10	C₂²×C₁₆×C₂₃	—	2	1	no	—
11	C₂₃×#435	—	2	1	no	—
12	C₂₃×#435	—	2	1	no	—
13	C₈×D₄×C₂₃	—	2	1	no	—
14	C₈×D₄×C₂₃	—	2	1	no	—
15	C₂₃×#424	C₂²	1	1	yes	characteristic
16	C₂₃×#408	C₂²	1	1	yes	characteristic
17	C₂₃×#407	C₄	1	1	yes	characteristic
18	C₂₃×#405	C₄	1	1	yes	characteristic
19	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
20	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
21	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×#131	—	4	2	no	—
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	4	2	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×#123	—	2	1	no	—
38	C₂₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
39	C₂₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
40	C₂₃×#121	C₈	1	1	yes	characteristic
41	C₂₃×#121	C₈	1	1	yes	characteristic
42	C₂₃×#120	—	2	1	no	—
43	C₄×D₄×C₂₃	—	2	1	no	—
44	C₄×D₄×C₂₃	—	2	1	no	—
45	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
46	C₂₃×M₃₂	—	4	2	no	—
47	C₂×C₁₆×C₂₃	—	2	2	no	—
48	C₂×C₁₆×C₂₃	—	2	2	no	—
49	C₂×C₁₆×C₂₃	—	4	2	no	—
50	C₂₃×C₄⋊C₈	—	2	1	no	—
51	C₂₃×C₄⋊C₈	—	2	1	no	—
52	C₂₃×C₂²⋊C₈	—	2	1	no	—
53	C₂₃×C₂²⋊C₈	—	2	1	no	—
54	C₂₃×C₂²⋊C₈	—	2	1	no	—
55	C₂₃×C₂²⋊C₈	—	2	1	no	—
56	C₂₃×#97	—	2	1	no	—
57	C₄×C₈×C₂₃	—	2	1	no	—
58	C₂⁴×C₂₃	M₁₆	1	1	yes	characteristic
59	#41×C₂₃	—	4	1	no	—
60	#41×C₂₃	—	4	1	no	—
61	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	—	2	1	no	—
65	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	4	1	no	—
68	C₂²×C₄×C₂₃	—	4	1	no	—
69	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	M₁₆×C₂₃	—	4	2	no	—
76	M₁₆×C₂₃	—	4	1	no	—
77	M₁₆×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	1	no	—
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	4	1	no	—
84	C₂×C₈×C₂₃	—	4	1	no	—
85	C₂×C₈×C₂₃	—	4	1	no	—
86	C₂×C₈×C₂₃	—	4	2	no	—
87	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
94	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
95	C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂²⋊C₄×C₂₃	—	2	1	no	—
98	C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₄²×C₂₃	—	2	1	no	—
101	C₁₆×C₂₃	—	4	2	no	—
102	C₁₆×C₂₃	—	4	2	no	—
103	#11470	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	2	no	—
105	C₂³×C₂₃	—	2	1	no	—
106	C₂³×C₂₃	—	4	1	no	—
107	C₂³×C₂₃	—	4	1	no	—
108	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
111	Q₈×C₂₃	—	4	1	no	—
112	D₄×C₂₃	—	4	1	no	—
113	D₄×C₂₃	—	4	2	no	—
114	D₄×C₂₃	—	4	2	no	—
115	D₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	4	1	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	1	no	—
130	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
134	C₈×C₂₃	—	2	1	no	—
135	C₈×C₂₃	—	2	1	no	—
136	C₈×C₂₃	—	2	1	no	—
137	C₈×C₂₃	—	4	1	no	—
138	C₈×C₂₃	—	4	1	no	—
139	C₈×C₂₃	#99	1	2	yes	—
140	#2929	C₂×C₂₃	1	1	yes	characteristic
141	#2112	C₂×C₂₃	1	1	yes	characteristic, complemented
142	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#1323	C₂×C₂₃	1	2	yes	complemented
144	#1315	C₂×C₂₃	1	2	yes	complemented
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	1	no	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	2	1	no	—
153	C₂²×C₂₃	—	4	4	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	—	4	1	no	—
156	C₂²×C₂₃	—	4	1	no	—
157	C₂²×C₂₃	#350	1	2	yes	—
158	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	4	1	no	—
163	C₄×C₂₃	—	4	1	no	—
164	C₄×C₂₃	—	4	1	no	—
165	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
166	C₄×C₂₃	#413	1	1	yes	characteristic
167	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	#525	C₄×C₂₃	1	1	yes	characteristic
170	C₂×M₃₂	—	2	1	no	—
171	C₂²×C₁₆	—	2	1	no	—
172	#435	—	2	1	no	—
173	#435	—	2	1	no	—
174	C₈×D₄	—	2	1	no	—
175	C₈×D₄	—	2	1	no	—
176	#424	C₂²×C₂₃	1	1	yes	characteristic
177	#408	C₂²×C₂₃	1	1	yes	characteristic
178	#407	C₄×C₂₃	1	1	yes	characteristic
179	#405	C₄×C₂₃	1	1	yes	characteristic
180	C₂²⋊C₁₆	—	2	1	no	—
181	C₂²⋊C₁₆	—	2	1	no	—
182	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
183	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
184	C₂×C₂₃	—	2	2	no	—
185	C₂×C₂₃	—	4	1	no	—
186	C₂×C₂₃	—	4	1	no	—
187	C₂×C₂₃	—	8	1	no	—
188	C₂×C₂₃	#1459	1	1	yes	characteristic
189	C₂×C₂₃	C₂×#350	1	1	yes	characteristic
190	C₂×C₂₃	#2117	1	1	yes	characteristic
191	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
193	#131	—	4	2	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
196	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	4	2	no	—
199	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
205	#123	—	2	1	no	—
206	C₂²⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
207	C₂²⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
208	#121	C₈×C₂₃	1	1	yes	characteristic
209	#121	C₈×C₂₃	1	1	yes	characteristic
210	#120	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
214	M₃₂	—	4	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	—	4	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	#97	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₂₃	#11470	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	M₁₆×C₂₃	1	1	yes	characteristic
228	#41	—	4	1	no	—
229	#41	—	4	1	no	—
230	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
257	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₁₆	—	4	2	no	—
271	C₁₆	—	4	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
277	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
278	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
279	Q₈	—	4	1	no	—
280	D₄	—	4	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
299	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	C₂₃×#99	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	C₂₃×#350	1	2	yes	—
321	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
329	C₄	C₂₃×#413	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	4	1	no	—
332	C₂	—	4	1	no	—
333	C₂	—	8	1	no	—
334	C₂	C₂₃×#1459	1	1	yes	characteristic
335	C₂	C₂×C₂₃×#350	1	1	yes	characteristic
336	C₂	C₂₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂₃×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2112	C₂	semidirect
3. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
4. C₂₃×#1323	C₂	semidirect
5. C₂₃×#1315	C₂	semidirect
103. #11470	C₂₃	direct (N × H)
141. #2112	C₂×C₂₃	semidirect
142. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
143. #1323	C₂×C₂₃	semidirect
144. #1315	C₂×C₂₃	semidirect
226. C₂₃	#11470	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×#2929	2
C₂₃×#2112	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂₃×#1323	2
C₂₃×#1315	2
#11470	23

Subgroups — C23×#11470 #19159096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11470 #19159096