Subgroups — C₂₃×#21639 #19169265

Order 5888.

850 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	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2398	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1955	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1923	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1832	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1753	C₂	1	4	yes	maximal, complemented
9	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₂₃	—	2	4	no	—
14	C₂₃×#436	—	2	4	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
20	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#400	C₄	1	2	yes	—
26	C₂₃×#397	C₄	1	2	yes	—
27	C₂₃×#394	C₄	1	2	yes	—
28	C₂₃×#393	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
30	C₂₃×#376	C₂²	1	1	yes	characteristic
31	C₂₃×#336	C₂²	1	4	yes	—
32	C₂₃×#336	C₂²	1	4	yes	—
33	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂×M₁₆×C₂₃	—	2	4	no	—
37	C₂×M₁₆×C₂₃	—	2	8	no	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₂₃	—	2	2	no	—
44	C₄×Q₈×C₂₃	—	2	2	no	—
45	C₄×D₄×C₂₃	—	2	2	no	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
51	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×C₄⋊C₈	—	2	4	no	—
57	C₂₃×C₄⋊C₈	—	2	2	no	—
58	C₂₃×C₄⋊C₈	—	2	2	no	—
59	C₂₃×C₂²⋊C₈	—	2	4	no	—
60	C₂₃×C₂²⋊C₈	—	2	8	no	—
61	C₂₃×#97	—	2	4	no	—
62	C₂₃×#97	—	2	2	no	—
63	C₄×C₈×C₂₃	—	2	4	no	—
64	C₄×C₈×C₂₃	—	2	4	no	—
65	C₄×C₈×C₂₃	D₄	1	4	yes	—
66	C₂₃×#95	C₂×C₄	1	2	yes	—
67	C₂₃×#95	C₂×C₄	1	2	yes	—
68	C₂₃×#95	C₂×C₄	1	4	yes	—
69	C₂₃×#95	C₂×C₄	1	4	yes	—
70	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂×Q₈×C₂₃	—	2	2	no	—
72	C₂×Q₈×C₂₃	—	2	4	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	4	2	no	—
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	4	no	—
78	C₂²×C₄×C₂₃	—	2	4	no	—
79	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₂₃	—	4	8	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	—	2	2	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	4	no	—
93	C₂×C₈×C₂₃	—	2	4	no	—
94	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₂₃	#41	1	4	yes	—
97	C₂×C₈×C₂₃	#41	1	4	yes	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	4	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₂²⋊C₄×C₂₃	—	2	4	no	—
102	C₂²⋊C₄×C₂₃	—	2	8	no	—
103	C₂²⋊C₄×C₂₃	—	4	4	no	—
104	C₄²×C₂₃	—	2	2	no	—
105	C₄²×C₂₃	—	2	2	no	—
106	C₄²×C₂₃	—	2	2	no	—
107	C₄²×C₂₃	—	2	2	no	—
108	C₄²×C₂₃	C₂×D₄	1	2	yes	—
109	C₄²×C₂₃	#41	1	2	yes	—
110	#21639	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₂₃	—	2	2	no	—
112	C₂³×C₂₃	—	2	4	no	—
113	C₂³×C₂₃	—	2	4	no	—
114	C₂³×C₂₃	—	2	2	no	—
115	C₂³×C₂₃	—	2	2	no	—
116	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
117	Q₈×C₂₃	—	2	8	no	—
118	D₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	4	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	2	4	no	—
128	C₂×C₄×C₂₃	—	2	4	no	—
129	C₂×C₄×C₂₃	—	2	4	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	4	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₂₃	#131	1	4	yes	—
136	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
138	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
140	C₈×C₂₃	—	2	4	no	—
141	C₈×C₂₃	—	2	2	no	—
142	C₈×C₂₃	—	4	2	no	—
143	C₈×C₂₃	—	4	2	no	—
144	C₈×C₂₃	#124	1	4	yes	—
145	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂×#436	C₂×C₂₃	1	1	yes	characteristic
147	#2398	C₂×C₂₃	1	1	yes	characteristic
148	#1985	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#1955	C₂×C₂₃	1	1	yes	characteristic
150	#1923	C₂×C₂₃	1	4	yes	complemented
151	#1832	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#1753	C₂×C₂₃	1	4	yes	complemented
153	#1726	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂²×C₂₃	—	4	2	no	—
155	C₂²×C₂₃	—	4	2	no	—
156	C₂²×C₂₃	—	4	4	no	—
157	C₂²×C₂₃	—	4	4	no	—
158	C₂²×C₂₃	—	4	2	no	—
159	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₂₃	C₄×#41	1	2	yes	—
161	C₂²×C₂₃	C₂×#131	1	2	yes	—
162	C₂²×C₂₃	#568	1	2	yes	—
163	C₄×C₂₃	—	2	2	no	—
164	C₄×C₂₃	—	2	4	no	—
165	C₄×C₂₃	—	2	2	no	—
166	C₄×C₂₃	—	4	2	no	—
167	C₄×C₂₃	—	4	4	no	—
168	C₄×C₂₃	—	4	2	no	—
169	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
170	C₄×C₂₃	#529	1	2	yes	—
171	C₄×C₂₃	#532	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
173	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
175	C₈×Q₈	—	2	4	no	—
176	#436	—	2	4	no	—
177	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
180	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
181	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
182	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
183	C₂×#97	C₂²×C₂₃	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
185	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
186	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
187	#400	C₄×C₂₃	1	2	yes	—
188	#397	C₄×C₂₃	1	2	yes	—
189	#394	C₄×C₂₃	1	2	yes	—
190	#393	C₄×C₂₃	1	2	yes	—
191	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
192	#376	C₂²×C₂₃	1	1	yes	characteristic
193	#336	C₂²×C₂₃	1	4	yes	—
194	#336	C₂²×C₂₃	1	4	yes	—
195	C₂×C₂₃	—	8	2	no	—
196	C₂×C₂₃	C₄×#124	1	1	yes	characteristic
197	C₂×C₂₃	#2929	1	1	yes	characteristic
198	C₂×C₂₃	#2930	1	1	yes	characteristic
199	C₂×C₂₃	#2969	1	4	yes	—
200	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	8	no	—
205	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
206	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
209	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
218	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
219	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
220	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
222	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	4	no	—
227	C₂²⋊C₈	—	2	8	no	—
228	#97	—	2	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	—	2	4	no	—
232	C₄×C₈	D₄×C₂₃	1	4	yes	—
233	#95	C₂×C₄×C₂₃	1	2	yes	—
234	#95	C₂×C₄×C₂₃	1	2	yes	—
235	#95	C₂×C₄×C₂₃	1	4	yes	—
236	#95	C₂×C₄×C₂₃	1	4	yes	—
237	C₂₃	#21639	1	1	yes	characteristic, direct product factor, Sylow
238	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
254	M₁₆	—	4	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
264	C₂×C₈	#41×C₂₃	1	4	yes	—
265	C₂×C₈	#41×C₂₃	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	C₂×D₄×C₂₃	1	2	yes	—
277	C₄²	#41×C₂₃	1	2	yes	—
278	C₂³	—	2	4	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	4	no	—
283	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	D₄	—	4	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
301	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
302	C₂×C₄	C₂₃×#131	1	4	yes	—
303	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
305	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
307	C₈	—	2	4	no	—
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	C₂₃×#124	1	4	yes	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
318	C₂²	C₄×#41×C₂₃	1	2	yes	—
319	C₂²	C₂×C₂₃×#131	1	2	yes	—
320	C₂²	C₂₃×#568	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
328	C₄	C₂₃×#529	1	2	yes	—
329	C₄	C₂₃×#532	1	1	yes	characteristic
330	C₂	—	8	2	no	—
331	C₂	C₄×C₂₃×#124	1	1	yes	characteristic
332	C₂	C₂₃×#2929	1	1	yes	characteristic
333	C₂	C₂₃×#2930	1	1	yes	characteristic
334	C₂	C₂₃×#2969	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#1985	C₂	semidirect
6. C₂₃×#1923	C₂	semidirect
7. C₂₃×#1832	C₂	semidirect
8. C₂₃×#1753	C₂	semidirect
9. C₂₃×#1726	C₂	semidirect
110. #21639	C₂₃	direct (N × H)
145. C₂×C₈×Q₈	C₂×C₂₃	semidirect
148. #1985	C₂×C₂₃	semidirect
150. #1923	C₂×C₂₃	semidirect
151. #1832	C₂×C₂₃	semidirect
152. #1753	C₂×C₂₃	semidirect
153. #1726	C₂×C₂₃	semidirect
237. C₂₃	#21639	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×C₈×Q₈×C₂₃	2
C₂×C₂₃×#436	2
C₂₃×#2398	2
C₂₃×#1985	2
C₂₃×#1955	2
C₂₃×#1923	2
C₂₃×#1832	2
C₂₃×#1753	2
C₂₃×#1726	2
#21639	23

Subgroups — C23×#21639 #19169265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21639 #19169265