Subgroups — C₂₃×#11467 #19159093

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#435	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×#1322	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1314	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₂₃	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	C₂₃×#435	—	2	1	no	—
14	C₂₃×#435	—	2	1	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#97	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	C₂²×D₄×C₂₃	C₈	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	4	2	no	—
29	C₂²×C₈×C₂₃	—	4	2	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂×C₄	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	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₂₃	—	2	1	no	—
39	C₄×D₄×C₂₃	—	2	1	no	—
40	C₄×D₄×C₂₃	—	2	1	no	—
41	C₄×D₄×C₂₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₂₃	C₈	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₂₃	C₈	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	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	C₂₃×#97	—	2	1	no	—
58	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
59	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	4	1	no	—
66	C₂²×C₄×C₂₃	—	4	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	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	M₁₆×C₂₃	—	4	2	no	—
79	M₁₆×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	1	no	—
85	C₂×C₈×C₂₃	—	4	1	no	—
86	C₂×C₈×C₂₃	—	4	1	no	—
87	C₂×C₈×C₂₃	—	4	1	no	—
88	C₂×C₈×C₂₃	—	4	1	no	—
89	C₂×C₈×C₂₃	—	4	1	no	—
90	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₄⋊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	C₄²×C₂₃	—	2	1	no	—
102	C₁₆×C₂₃	—	4	2	no	—
103	C₁₆×C₂₃	—	4	2	no	—
104	#11467	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	2	2	no	—
106	C₂³×C₂₃	—	2	2	no	—
107	C₂³×C₂₃	—	2	1	no	—
108	C₂³×C₂₃	—	2	2	no	—
109	C₂³×C₂₃	—	4	1	no	—
110	C₂³×C₂₃	—	4	1	no	—
111	C₂³×C₂₃	—	4	1	no	—
112	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
113	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
114	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
115	D₄×C₂₃	—	4	2	no	—
116	D₄×C₂₃	—	4	2	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	—	2	1	no	—
126	C₂×C₄×C₂₃	—	2	2	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	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	1	no	—
132	C₂×C₄×C₂₃	—	4	1	no	—
133	C₂×C₄×C₂₃	—	4	1	no	—
134	C₂×C₄×C₂₃	—	4	1	no	—
135	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	4	1	no	—
143	C₈×C₂₃	—	4	1	no	—
144	C₈×C₂₃	#99	1	2	yes	—
145	C₂×#435	C₂×C₂₃	1	1	yes	characteristic
146	#2112	C₂×C₂₃	1	1	yes	characteristic, complemented
147	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#1322	C₂×C₂₃	1	2	yes	complemented
149	#1314	C₂×C₂₃	1	2	yes	complemented
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	2	1	no	—
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	2	1	no	—
157	C₂²×C₂₃	—	4	4	no	—
158	C₂²×C₂₃	—	4	1	no	—
159	C₂²×C₂₃	—	4	1	no	—
160	C₂²×C₂₃	—	4	1	no	—
161	C₂²×C₂₃	—	8	2	no	—
162	C₂²×C₂₃	#349	1	2	yes	—
163	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₂₃	#504	1	2	yes	—
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	2	1	no	—
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	4	1	no	—
169	C₄×C₂₃	—	4	1	no	—
170	C₄×C₂₃	—	4	1	no	—
171	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
172	C₄×C₂₃	#413	1	1	yes	characteristic
173	C₂²×M₁₆	C₄×C₂₃	1	1	yes	characteristic
174	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
176	C₂×M₃₂	—	2	1	no	—
177	C₂²×C₁₆	—	2	1	no	—
178	#435	—	2	1	no	—
179	#435	—	2	1	no	—
180	#435	—	2	1	no	—
181	#435	—	2	1	no	—
182	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
183	C₂×C₂²⋊C₈	C₄×C₂₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
186	C₂²⋊C₁₆	—	2	1	no	—
187	C₂²⋊C₁₆	—	2	1	no	—
188	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
189	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
190	C₂×C₂₃	—	2	2	no	—
191	C₂×C₂₃	—	2	2	no	—
192	C₂×C₂₃	—	4	1	no	—
193	C₂×C₂₃	—	8	1	no	—
194	C₂×C₂₃	#1459	1	1	yes	characteristic
195	C₂×C₂₃	#2113	1	1	yes	characteristic
196	C₂×C₂₃	C₂×#349	1	1	yes	characteristic
197	C₂²×D₄	C₈×C₂₃	1	1	yes	characteristic
198	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
199	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	4	2	no	—
203	C₂²×C₈	—	4	2	no	—
204	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	C₈×C₂₃	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	C₈×C₂₃	1	1	yes	characteristic
219	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
220	M₃₂	—	4	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	—	4	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	2	1	no	—
230	#97	—	2	1	no	—
231	#97	—	2	1	no	—
232	C₂₃	#11467	1	1	yes	characteristic, direct product factor, Sylow
233	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
234	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
250	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₁₆	—	4	2	no	—
278	C₁₆	—	4	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
287	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
288	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	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	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
310	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	C₂₃×#99	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	4	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂₃×#349	1	2	yes	—
332	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
333	C₂²	C₂₃×#504	1	2	yes	—
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	—	4	1	no	—
340	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
341	C₄	C₂₃×#413	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	2	2	no	—
344	C₂	—	4	1	no	—
345	C₂	—	8	1	no	—
346	C₂	C₂₃×#1459	1	1	yes	characteristic
347	C₂	C₂₃×#2113	1	1	yes	characteristic
348	C₂	C₂×C₂₃×#349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₂₃	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ 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×#1322	C₂	semidirect
5. C₂₃×#1314	C₂	semidirect
104. #11467	C₂₃	direct (N × H)
146. #2112	C₂×C₂₃	semidirect
147. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
148. #1322	C₂×C₂₃	semidirect
149. #1314	C₂×C₂₃	semidirect
232. C₂₃	#11467	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#435	2
C₂₃×#2112	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂₃×#1322	2
C₂₃×#1314	2
#11467	23

Subgroups — C23×#11467 #19159093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11467 #19159093