Subgroups — C₂₃×#20821 #19168447

Order 5888.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2314	C₂	1	2	yes	maximal
3	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1836	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1809	C₂	1	2	yes	maximal
7	C₂₃×#1808	C₂	1	2	yes	maximal
8	C₂₃×#1776	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×#107	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₂×C₂₃×#107	C₂²	1	2	yes	—
15	C₂×C₂₃×#107	C₂²	1	2	yes	—
16	C₂×C₂₃×#106	C₂²	1	2	yes	—
17	C₂₃×#405	C₄	1	8	yes	complemented
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#388	—	2	4	no	—
23	C₂₃×#387	—	2	4	no	—
24	C₂₃×#384	C₂²	1	2	yes	—
25	C₂₃×#382	C₂²	1	2	yes	—
26	C₂₃×#380	C₂²	1	4	yes	—
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	—	2	2	no	—
36	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#117	C₂×C₄	1	4	yes	—
42	C₂₃×#117	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
49	C₂×C₄²×C₂₃	—	2	4	no	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	—	2	4	no	—
53	C₂₃×#107	—	2	4	no	—
54	C₂₃×#107	—	2	4	no	—
55	C₂₃×#106	—	2	4	no	—
56	C₂₃×#97	C₂×C₄	1	4	yes	—
57	C₂₃×#97	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
60	C₂₃×#95	—	2	4	no	—
61	C₂₃×#95	—	2	4	no	—
62	C₂₃×#95	—	2	4	no	—
63	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	—	2	4	no	—
69	C₂²×C₄×C₂₃	—	2	4	no	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₂₃	—	2	8	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	2	8	no	—
82	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₂₃	#41	1	2	yes	—
87	C₂×C₈×C₂₃	#41	1	2	yes	—
88	C₂×C₈×C₂₃	#41	1	2	yes	—
89	C₂×C₈×C₂₃	#41	1	2	yes	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	8	4	no	—
95	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
96	C₂²⋊C₄×C₂₃	—	8	4	no	—
97	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
98	C₄²×C₂₃	—	8	4	no	—
99	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
103	#20821	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	1	no	—
105	C₂³×C₂₃	—	2	1	no	—
106	C₂³×C₂₃	—	2	2	no	—
107	C₂³×C₂₃	—	2	2	no	—
108	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	4	4	no	—
123	C₂×C₄×C₂₃	—	4	4	no	—
124	C₂×C₄×C₂₃	—	4	4	no	—
125	C₂×C₄×C₂₃	—	4	4	no	—
126	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	—	2	2	no	—
137	C₈×C₂₃	#117	1	4	yes	—
138	C₈×C₂₃	#117	1	4	yes	—
139	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
140	#2314	C₂×C₂₃	1	2	yes	—
141	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#1845	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#1836	C₂×C₂₃	1	2	yes	complemented
144	#1809	C₂×C₂₃	1	2	yes	—
145	#1808	C₂×C₂₃	1	2	yes	—
146	#1776	C₂×C₂₃	1	2	yes	complemented
147	C₄×#107	C₂×C₂₃	1	2	yes	complemented
148	C₂²×C₂₃	—	2	4	no	—
149	C₂²×C₂₃	—	2	2	no	—
150	C₂²×C₂₃	—	4	2	no	—
151	C₂²×C₂₃	—	4	1	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₂₃	#532	1	1	yes	characteristic
155	C₂²×C₂₃	#532	1	1	yes	characteristic
156	C₂²×C₂₃	#576	1	2	yes	—
157	C₂²×C₂₃	#578	1	2	yes	—
158	C₄×C₂₃	—	2	4	no	—
159	C₄×C₂₃	—	2	4	no	—
160	C₄×C₂₃	—	2	2	no	—
161	C₄×C₂₃	—	8	4	no	—
162	C₄×C₂₃	—	8	4	no	—
163	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
164	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
165	C₄×C₂₃	#533	1	1	yes	characteristic
166	C₄×C₂₃	#533	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×#107	C₂²×C₂₃	1	2	yes	—
171	C₂×#107	C₂²×C₂₃	1	2	yes	—
172	C₂×#107	C₂²×C₂₃	1	2	yes	—
173	C₂×#106	C₂²×C₂₃	1	2	yes	—
174	#405	C₄×C₂₃	1	8	yes	complemented
175	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
179	#388	—	2	4	no	—
180	#387	—	2	4	no	—
181	#384	C₂²×C₂₃	1	2	yes	—
182	#382	C₂²×C₂₃	1	2	yes	—
183	#380	C₂²×C₂₃	1	4	yes	—
184	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
185	#340	C₂²×C₂₃	1	4	yes	—
186	#340	C₂²×C₂₃	1	4	yes	—
187	C₂×C₂₃	—	4	2	no	—
188	C₂×C₂₃	#2285	1	1	yes	characteristic
189	C₂×C₂₃	#2913	1	2	yes	—
190	C₂×C₂₃	#3175	1	2	yes	—
191	C₂×C₂₃	#3176	1	1	yes	characteristic
192	C₂×C₂₃	#3177	1	1	yes	characteristic
193	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
200	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
204	#117	C₂×C₄×C₂₃	1	4	yes	—
205	#117	Q₈×C₂₃	1	4	yes	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
209	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
215	#107	—	2	4	no	—
216	#107	—	2	4	no	—
217	#107	—	2	4	no	—
218	#106	—	2	4	no	—
219	#97	C₂×C₄×C₂₃	1	4	yes	—
220	#97	C₂×C₄×C₂₃	1	4	yes	—
221	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
222	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
223	#95	—	2	4	no	—
224	#95	—	2	4	no	—
225	#95	—	2	4	no	—
226	C₂₃	#20821	1	1	yes	characteristic, direct product factor, Sylow
227	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
238	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
240	M₁₆	—	2	8	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
249	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
250	C₂×C₈	#41×C₂₃	1	2	yes	—
251	C₂×C₈	#41×C₂₃	1	2	yes	—
252	C₂×C₈	#41×C₂₃	1	2	yes	—
253	C₂×C₈	#41×C₂₃	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	8	4	no	—
259	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
260	C₂²⋊C₄	—	8	4	no	—
261	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
262	C₄²	—	8	4	no	—
263	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
264	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
265	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
266	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
272	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
292	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	C₂₃×#117	1	4	yes	—
301	C₈	C₂₃×#117	1	4	yes	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
308	C₂²	C₂₃×#532	1	1	yes	characteristic
309	C₂²	C₂₃×#532	1	1	yes	characteristic
310	C₂²	C₂₃×#576	1	2	yes	—
311	C₂²	C₂₃×#578	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	2	no	—
315	C₄	—	8	4	no	—
316	C₄	—	8	4	no	—
317	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
318	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
319	C₄	C₂₃×#533	1	1	yes	characteristic
320	C₄	C₂₃×#533	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₂₃×#2285	1	1	yes	characteristic
323	C₂	C₂₃×#2913	1	2	yes	—
324	C₂	C₂₃×#3175	1	2	yes	—
325	C₂	C₂₃×#3176	1	1	yes	characteristic
326	C₂	C₂₃×#3177	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1846	C₂	semidirect
4. C₂₃×#1845	C₂	semidirect
5. C₂₃×#1836	C₂	semidirect
8. C₂₃×#1776	C₂	semidirect
9. C₄×C₂₃×#107	C₂	semidirect
17. C₂₃×#405	C₄	semidirect
103. #20821	C₂₃	direct (N × H)
141. #1846	C₂×C₂₃	semidirect
142. #1845	C₂×C₂₃	semidirect
143. #1836	C₂×C₂₃	semidirect
146. #1776	C₂×C₂₃	semidirect
147. C₄×#107	C₂×C₂₃	semidirect
174. #405	C₄×C₂₃	semidirect
226. C₂₃	#20821	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×#405	2
C₂₃×#2314	2
C₂₃×#1846	2
C₂₃×#1845	2
C₂₃×#1836	2
C₂₃×#1809	2
C₂₃×#1808	2
C₂₃×#1776	2
C₄×C₂₃×#107	2
#20821	23

Subgroups — C23×#20821 #19168447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20821 #19168447