Subgroups — C₂₃×#18742 #19166368

Order 5888.

908 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3212	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3184	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1642	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1627	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1620	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1485	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#562	—	2	2	no	—
17	C₂₃×#559	—	2	1	no	—
18	C₂₃×#559	C₂²	1	1	yes	characteristic
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#554	—	2	1	no	—
21	C₂₃×#550	C₂²	1	1	yes	characteristic
22	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#548	—	2	1	no	—
24	C₂₃×#547	—	2	1	no	—
25	C₂₃×#542	—	2	1	no	—
26	C₂₃×#542	—	2	1	no	—
27	C₂₃×#540	—	2	1	no	—
28	C₂₃×#539	—	2	2	no	—
29	C₂₃×#536	—	2	1	no	—
30	C₂₃×#533	C₂²	1	1	yes	characteristic
31	C₂₃×#532	C₂²	1	1	yes	characteristic
32	C₂₃×#532	C₂²	1	1	yes	characteristic
33	C₂₃×#529	C₂²	1	1	yes	characteristic
34	C₂₃×#498	C₂²	1	1	yes	characteristic
35	C₂₃×#495	—	2	1	no	—
36	C₂₃×#493	C₂²	1	1	yes	characteristic
37	C₂₃×#488	—	2	1	no	—
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#486	C₂²	1	1	yes	characteristic
40	C₂₃×#476	—	2	1	no	—
41	C₂₃×#475	C₂²	1	1	yes	characteristic
42	C₂₃×#475	C₂²	1	1	yes	characteristic
43	C₂₃×#469	—	2	2	no	—
44	C₂₃×#465	—	2	1	no	—
45	C₂₃×#465	—	2	1	no	—
46	C₂₃×#462	C₂²	1	1	yes	characteristic
47	C₂₃×#461	—	2	1	no	—
48	C₂₃×#461	C₂²	1	1	yes	characteristic
49	C₂₃×#460	C₂²	1	1	yes	characteristic
50	C₂₃×#460	C₂²	1	1	yes	characteristic
51	C₂₃×#460	C₂²	1	1	yes	characteristic
52	C₂₃×#459	C₂²	1	1	yes	characteristic
53	C₂₃×#452	—	2	1	no	—
54	C₂₃×#449	—	2	1	no	—
55	C₂₃×#442	—	2	2	no	—
56	C₂₃×#431	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×#331	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	2	1	no	—
73	C₂×#41×C₂₃	—	2	1	no	—
74	C₂²×Q₈×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×D₈×C₂₃	—	2	2	no	—
81	C₂×D₈×C₂₃	—	2	1	no	—
82	C₂×M₁₆×C₂₃	—	2	1	no	—
83	C₂²×C₈×C₂₃	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₂₃×#127	—	2	1	no	—
91	C₂₃×#127	C₂³	1	1	yes	characteristic
92	C₂₃×#127	C₂³	1	1	yes	characteristic
93	C₂₃×#126	—	2	2	no	—
94	C₂₃×#126	—	2	1	no	—
95	C₂₃×#125	—	2	1	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	1	no	—
98	C₂₃×#125	D₄	1	1	yes	characteristic
99	C₂₃×#124	—	2	2	no	—
100	C₂₃×#124	—	2	2	no	—
101	C₂₃×#124	—	2	1	no	—
102	C₂₃×#124	—	2	1	no	—
103	C₂₃×#124	—	2	1	no	—
104	C₂₃×#124	—	2	1	no	—
105	C₂₃×#124	D₄	1	1	yes	characteristic
106	C₂₃×#123	—	2	2	no	—
107	C₂₃×#123	—	2	2	no	—
108	C₂₃×#123	—	2	2	no	—
109	C₂₃×#123	—	2	2	no	—
110	C₂₃×#123	—	2	2	no	—
111	C₂₃×#123	—	2	2	no	—
112	C₂₃×#123	—	2	1	no	—
113	C₂₃×C₂²⋊Q₈	—	2	2	no	—
114	C₂₃×C₂²⋊Q₈	—	2	2	no	—
115	C₂₃×C₂²⋊Q₈	—	2	2	no	—
116	C₂₃×C₂²⋊Q₈	—	2	2	no	—
117	C₂₃×C₂²⋊Q₈	—	2	1	no	—
118	C₂₃×C₂²⋊Q₈	—	2	1	no	—
119	C₂₃×C₂²⋊Q₈	—	2	2	no	—
120	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
121	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
122	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
123	C₂₃×#121	—	2	2	no	—
124	C₂₃×#121	—	2	2	no	—
125	C₂₃×#121	—	2	2	no	—
126	C₂₃×#121	—	2	1	no	—
127	C₂₃×#121	—	2	1	no	—
128	C₂₃×#121	—	2	1	no	—
129	C₂₃×#121	—	2	1	no	—
130	C₂₃×#121	D₄	1	2	yes	—
131	C₂₃×#121	D₄	1	1	yes	characteristic
132	C₄×Q₈×C₂₃	—	2	1	no	—
133	C₄×Q₈×C₂₃	—	2	1	no	—
134	C₄×Q₈×C₂₃	—	2	1	no	—
135	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
136	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
137	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×D₄×C₂₃	—	2	2	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	—	2	1	no	—
141	C₄×D₄×C₂₃	—	2	1	no	—
142	C₄×D₄×C₂₃	—	2	1	no	—
143	C₄×D₄×C₂₃	—	2	1	no	—
144	C₄×D₄×C₂₃	—	2	1	no	—
145	C₄×D₄×C₂₃	—	2	1	no	—
146	C₄×D₄×C₂₃	—	2	1	no	—
147	C₄×D₄×C₂₃	D₄	1	2	yes	—
148	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
149	C₂₃×#117	—	2	1	no	—
150	C₂₃×#117	—	2	1	no	—
151	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
153	C₂₃×#106	—	2	2	no	—
154	C₂₃×#106	—	2	1	no	—
155	C₂₃×#106	—	2	1	no	—
156	C₂₃×C₄⋊C₈	—	2	1	no	—
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×#103	—	2	1	no	—
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#102	—	2	2	no	—
166	C₂₃×#102	—	2	2	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	1	no	—
171	C₂₃×C₂²⋊C₈	—	2	1	no	—
172	C₂₃×#97	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	#41×C₂₃	—	8	1	no	—
177	#41×C₂₃	—	8	1	no	—
178	#41×C₂₃	—	8	1	no	—
179	#41×C₂₃	—	8	1	no	—
180	#41×C₂₃	—	8	1	no	—
181	#41×C₂₃	—	8	1	no	—
182	C₂×Q₈×C₂₃	—	2	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	1	no	—
186	C₂×Q₈×C₂₃	—	8	1	no	—
187	C₂×Q₈×C₂₃	—	8	1	no	—
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×D₄×C₂₃	—	2	2	no	—
192	C₂×D₄×C₂₃	—	2	2	no	—
193	C₂×D₄×C₂₃	—	2	1	no	—
194	C₂×D₄×C₂₃	—	2	2	no	—
195	C₂×D₄×C₂₃	—	2	1	no	—
196	C₂×D₄×C₂₃	—	2	1	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	2	1	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	—	4	1	no	—
201	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₂₃	—	2	2	no	—
204	C₂²×C₄×C₂₃	—	2	2	no	—
205	C₂²×C₄×C₂₃	—	2	2	no	—
206	C₂²×C₄×C₂₃	—	2	1	no	—
207	C₂²×C₄×C₂₃	—	2	2	no	—
208	C₂²×C₄×C₂₃	—	4	1	no	—
209	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
212	Q₁₆×C₂₃	—	8	1	no	—
213	SD₁₆×C₂₃	—	8	1	no	—
214	SD₁₆×C₂₃	—	8	1	no	—
215	SD₁₆×C₂₃	—	8	1	no	—
216	SD₁₆×C₂₃	—	8	1	no	—
217	D₈×C₂₃	—	4	4	no	—
218	D₈×C₂₃	—	8	1	no	—
219	M₁₆×C₂₃	—	4	2	no	—
220	C₂×C₈×C₂₃	—	2	1	no	—
221	C₂×C₈×C₂₃	—	2	1	no	—
222	C₂×C₈×C₂₃	—	2	1	no	—
223	C₂×C₈×C₂₃	—	2	1	no	—
224	C₂×C₈×C₂₃	—	2	1	no	—
225	C₂×C₈×C₂₃	—	2	1	no	—
226	C₂×C₈×C₂₃	—	8	1	no	—
227	C₄⋊C₄×C₂₃	—	2	2	no	—
228	C₄⋊C₄×C₂₃	—	2	2	no	—
229	C₄⋊C₄×C₂₃	—	2	2	no	—
230	C₄⋊C₄×C₂₃	—	2	2	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	2	1	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	2	1	no	—
239	C₄⋊C₄×C₂₃	—	2	1	no	—
240	C₄⋊C₄×C₂₃	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	1	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	1	no	—
244	C₄⋊C₄×C₂₃	—	2	1	no	—
245	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
246	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
247	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
248	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
249	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
250	C₂²⋊C₄×C₂₃	—	2	2	no	—
251	C₂²⋊C₄×C₂₃	—	2	2	no	—
252	C₂²⋊C₄×C₂₃	—	2	2	no	—
253	C₂²⋊C₄×C₂₃	—	2	2	no	—
254	C₂²⋊C₄×C₂₃	—	2	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	1	no	—
256	C₂²⋊C₄×C₂₃	—	2	1	no	—
257	C₂²⋊C₄×C₂₃	—	2	1	no	—
258	C₂²⋊C₄×C₂₃	—	2	1	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	2	1	no	—
261	C₂²⋊C₄×C₂₃	—	2	2	no	—
262	C₂²⋊C₄×C₂₃	—	2	2	no	—
263	C₂²⋊C₄×C₂₃	—	2	1	no	—
264	C₂²⋊C₄×C₂₃	—	4	1	no	—
265	C₂²⋊C₄×C₂₃	—	4	1	no	—
266	C₄²×C₂₃	—	2	2	no	—
267	C₄²×C₂₃	—	2	1	no	—
268	C₄²×C₂₃	—	2	1	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
271	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
274	#18742	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂³×C₂₃	—	2	1	no	—
276	C₂³×C₂₃	—	2	2	no	—
277	C₂³×C₂₃	—	4	1	no	—
278	C₂³×C₂₃	#120	1	1	yes	characteristic
279	Q₈×C₂₃	—	2	1	no	—
280	Q₈×C₂₃	—	2	1	no	—
281	Q₈×C₂₃	—	2	1	no	—
282	Q₈×C₂₃	—	2	1	no	—
283	Q₈×C₂₃	—	2	1	no	—
284	Q₈×C₂₃	—	2	1	no	—
285	Q₈×C₂₃	—	4	1	no	—
286	Q₈×C₂₃	—	4	1	no	—
287	Q₈×C₂₃	—	4	1	no	—
288	Q₈×C₂₃	—	4	1	no	—
289	Q₈×C₂₃	—	8	1	no	—
290	Q₈×C₂₃	—	8	2	no	—
291	D₄×C₂₃	—	2	1	no	—
292	D₄×C₂₃	—	2	1	no	—
293	D₄×C₂₃	—	4	2	no	—
294	D₄×C₂₃	—	4	2	no	—
295	D₄×C₂₃	—	4	1	no	—
296	D₄×C₂₃	—	4	1	no	—
297	D₄×C₂₃	—	4	1	no	—
298	D₄×C₂₃	—	4	1	no	—
299	D₄×C₂₃	—	4	1	no	—
300	D₄×C₂₃	—	4	1	no	—
301	D₄×C₂₃	—	4	1	no	—
302	D₄×C₂₃	—	4	1	no	—
303	D₄×C₂₃	—	8	1	no	—
304	D₄×C₂₃	—	8	2	no	—
305	D₄×C₂₃	—	8	2	no	—
306	D₄×C₂₃	—	8	2	no	—
307	D₄×C₂₃	—	8	2	no	—
308	C₂×C₄×C₂₃	—	2	1	no	—
309	C₂×C₄×C₂₃	—	2	1	no	—
310	C₂×C₄×C₂₃	—	2	1	no	—
311	C₂×C₄×C₂₃	—	2	1	no	—
312	C₂×C₄×C₂₃	—	2	1	no	—
313	C₂×C₄×C₂₃	—	2	1	no	—
314	C₂×C₄×C₂₃	—	2	1	no	—
315	C₂×C₄×C₂₃	—	2	1	no	—
316	C₂×C₄×C₂₃	—	2	1	no	—
317	C₂×C₄×C₂₃	—	2	2	no	—
318	C₂×C₄×C₂₃	—	2	1	no	—
319	C₂×C₄×C₂₃	—	2	1	no	—
320	C₂×C₄×C₂₃	—	4	1	no	—
321	C₂×C₄×C₂₃	—	4	1	no	—
322	C₂×C₄×C₂₃	—	4	1	no	—
323	C₂×C₄×C₂₃	—	4	1	no	—
324	C₂×C₄×C₂₃	—	8	2	no	—
325	C₂×C₄×C₂₃	—	8	2	no	—
326	C₂×C₄×C₂₃	—	8	2	no	—
327	C₂×C₄×C₂₃	—	8	2	no	—
328	C₂×C₄×C₂₃	—	8	2	no	—
329	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
330	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
331	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
332	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
333	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
335	C₈×C₂₃	—	2	2	no	—
336	C₈×C₂₃	—	4	1	no	—
337	C₈×C₂₃	—	4	1	no	—
338	C₈×C₂₃	—	4	1	no	—
339	C₈×C₂₃	—	4	1	no	—
340	C₈×C₂₃	—	4	1	no	—
341	#3554	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#3246	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#3212	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#3184	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#1642	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#1627	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#1620	C₂×C₂₃	1	1	yes	characteristic, complemented
353	#1540	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
356	C₂²×C₂₃	—	2	1	no	—
357	C₂²×C₂₃	—	2	1	no	—
358	C₂²×C₂₃	—	2	1	no	—
359	C₂²×C₂₃	—	4	1	no	—
360	C₂²×C₂₃	—	4	2	no	—
361	C₂²×C₂₃	—	4	1	no	—
362	C₂²×C₂₃	—	4	2	no	—
363	C₂²×C₂₃	—	4	2	no	—
364	C₂²×C₂₃	—	4	1	no	—
365	C₂²×C₂₃	—	8	1	no	—
366	C₂²×C₂₃	—	8	1	no	—
367	C₂²×C₂₃	—	8	1	no	—
368	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
369	C₄×C₂₃	—	2	1	no	—
370	C₄×C₂₃	—	2	1	no	—
371	C₄×C₂₃	—	2	1	no	—
372	C₄×C₂₃	—	2	1	no	—
373	C₄×C₂₃	—	4	1	no	—
374	C₄×C₂₃	—	4	1	no	—
375	C₄×C₂₃	—	4	1	no	—
376	C₄×C₂₃	—	4	2	no	—
377	C₄×C₂₃	—	4	1	no	—
378	C₄×C₂₃	—	4	1	no	—
379	C₄×C₂₃	—	4	1	no	—
380	C₄×C₂₃	—	4	1	no	—
381	C₄×C₂₃	—	8	1	no	—
382	C₄×C₂₃	#576	1	1	yes	characteristic
383	C₄×C₂₃	#577	1	1	yes	characteristic
384	C₄×C₂₃	#577	1	1	yes	characteristic
385	C₄×C₂₃	#577	1	1	yes	characteristic
386	#562	—	2	2	no	—
387	#559	—	2	1	no	—
388	#559	C₂²×C₂₃	1	1	yes	characteristic
389	#557	C₂²×C₂₃	1	1	yes	characteristic
390	#554	—	2	1	no	—
391	#550	C₂²×C₂₃	1	1	yes	characteristic
392	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
393	#548	—	2	1	no	—
394	#547	—	2	1	no	—
395	#542	—	2	1	no	—
396	#542	—	2	1	no	—
397	#540	—	2	1	no	—
398	#539	—	2	2	no	—
399	#536	—	2	1	no	—
400	#533	C₂²×C₂₃	1	1	yes	characteristic
401	#532	C₂²×C₂₃	1	1	yes	characteristic
402	#532	C₂²×C₂₃	1	1	yes	characteristic
403	#529	C₂²×C₂₃	1	1	yes	characteristic
404	#498	C₂²×C₂₃	1	1	yes	characteristic
405	#495	—	2	1	no	—
406	#493	C₂²×C₂₃	1	1	yes	characteristic
407	#488	—	2	1	no	—
408	#486	C₂²×C₂₃	1	1	yes	characteristic
409	#486	C₂²×C₂₃	1	1	yes	characteristic
410	#476	—	2	1	no	—
411	#475	C₂²×C₂₃	1	1	yes	characteristic
412	#475	C₂²×C₂₃	1	1	yes	characteristic
413	#469	—	2	2	no	—
414	#465	—	2	1	no	—
415	#465	—	2	1	no	—
416	#462	C₂²×C₂₃	1	1	yes	characteristic
417	#461	—	2	1	no	—
418	#461	C₂²×C₂₃	1	1	yes	characteristic
419	#460	C₂²×C₂₃	1	1	yes	characteristic
420	#460	C₂²×C₂₃	1	1	yes	characteristic
421	#460	C₂²×C₂₃	1	1	yes	characteristic
422	#459	C₂²×C₂₃	1	1	yes	characteristic
423	#452	—	2	1	no	—
424	#449	—	2	1	no	—
425	#442	—	2	2	no	—
426	#431	C₂²×C₂₃	1	1	yes	characteristic
427	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
428	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
429	#332	C₂²×C₂₃	1	1	yes	characteristic
430	#332	C₂²×C₂₃	1	1	yes	characteristic
431	#331	C₂²×C₂₃	1	1	yes	characteristic
432	#331	C₂²×C₂₃	1	1	yes	characteristic
433	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
434	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
435	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
436	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
437	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
438	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
439	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
440	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
441	C₂×C₂₃	—	4	1	no	—
442	C₂×C₂₃	—	8	1	no	—
443	C₂×C₂₃	—	8	2	no	—
444	C₂×C₂₃	—	16	1	no	—
445	C₂×C₂₃	#3010	1	1	yes	characteristic
446	C₂×C₂₃	#3012	1	1	yes	characteristic
447	C₂×C₂₃	#3018	1	1	yes	characteristic
448	C₂×#41	—	2	1	no	—
449	C₂×#41	—	2	1	no	—
450	C₂×#41	—	2	1	no	—
451	C₂²×Q₈	—	2	1	no	—
452	C₂×Q₁₆	—	2	1	no	—
453	C₂×SD₁₆	—	2	1	no	—
454	C₂×SD₁₆	—	2	1	no	—
455	C₂×SD₁₆	—	2	1	no	—
456	C₂×SD₁₆	—	2	1	no	—
457	C₂×D₈	—	2	2	no	—
458	C₂×D₈	—	2	1	no	—
459	C₂×M₁₆	—	2	1	no	—
460	C₂²×C₈	—	2	1	no	—
461	C₄⋊Q₈	—	2	1	no	—
462	C₄⋊Q₈	—	2	1	no	—
463	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
464	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
465	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
466	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
467	#127	—	2	1	no	—
468	#127	C₂³×C₂₃	1	1	yes	characteristic
469	#127	C₂³×C₂₃	1	1	yes	characteristic
470	#126	—	2	2	no	—
471	#126	—	2	1	no	—
472	#125	—	2	2	no	—
473	#125	—	2	1	no	—
474	#125	—	2	1	no	—
475	#125	D₄×C₂₃	1	1	yes	characteristic
476	#124	—	2	2	no	—
477	#124	—	2	2	no	—
478	#124	—	2	1	no	—
479	#124	—	2	1	no	—
480	#124	—	2	1	no	—
481	#124	—	2	1	no	—
482	#124	D₄×C₂₃	1	1	yes	characteristic
483	#123	—	2	2	no	—
484	#123	—	2	2	no	—
485	#123	—	2	2	no	—
486	#123	—	2	2	no	—
487	#123	—	2	2	no	—
488	#123	—	2	2	no	—
489	#123	—	2	1	no	—
490	C₂²⋊Q₈	—	2	2	no	—
491	C₂²⋊Q₈	—	2	2	no	—
492	C₂²⋊Q₈	—	2	2	no	—
493	C₂²⋊Q₈	—	2	2	no	—
494	C₂²⋊Q₈	—	2	1	no	—
495	C₂²⋊Q₈	—	2	2	no	—
496	C₂²⋊Q₈	—	2	1	no	—
497	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
498	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
499	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
500	#121	—	2	2	no	—
501	#121	—	2	2	no	—
502	#121	—	2	2	no	—
503	#121	—	2	1	no	—
504	#121	—	2	1	no	—
505	#121	—	2	1	no	—
506	#121	—	2	1	no	—
507	#121	D₄×C₂₃	1	1	yes	characteristic
508	#121	D₄×C₂₃	1	2	yes	—
509	C₄×Q₈	—	2	1	no	—
510	C₄×Q₈	—	2	1	no	—
511	C₄×Q₈	—	2	1	no	—
512	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
513	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
514	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
515	C₄×D₄	—	2	2	no	—
516	C₄×D₄	—	2	1	no	—
517	C₄×D₄	—	2	1	no	—
518	C₄×D₄	—	2	1	no	—
519	C₄×D₄	—	2	1	no	—
520	C₄×D₄	—	2	1	no	—
521	C₄×D₄	—	2	1	no	—
522	C₄×D₄	—	2	1	no	—
523	C₄×D₄	—	2	1	no	—
524	C₄×D₄	D₄×C₂₃	1	2	yes	—
525	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
526	#117	—	2	1	no	—
527	#117	—	2	1	no	—
528	C₂×C₄⋊C₄	—	2	2	no	—
529	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
530	#106	—	2	2	no	—
531	#106	—	2	1	no	—
532	#106	—	2	1	no	—
533	C₄⋊C₈	—	2	1	no	—
534	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
535	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
536	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
537	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
538	#103	—	2	1	no	—
539	#103	—	2	1	no	—
540	#103	—	2	1	no	—
541	#103	—	2	1	no	—
542	#102	—	2	2	no	—
543	#102	—	2	2	no	—
544	#102	—	2	1	no	—
545	#102	—	2	1	no	—
546	#102	—	2	1	no	—
547	#102	—	2	1	no	—
548	C₂²⋊C₈	—	2	1	no	—
549	#97	—	2	1	no	—
550	C₄×C₈	—	2	1	no	—
551	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
552	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
553	C₂₃	#18742	1	1	yes	characteristic, direct product factor, Sylow
554	#41	—	8	1	no	—
555	#41	—	8	1	no	—
556	#41	—	8	1	no	—
557	#41	—	8	1	no	—
558	#41	—	8	1	no	—
559	#41	—	8	1	no	—
560	C₂×Q₈	—	2	2	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	8	1	no	—
565	C₂×Q₈	—	8	1	no	—
566	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
568	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
569	C₂×D₄	—	2	2	no	—
570	C₂×D₄	—	2	2	no	—
571	C₂×D₄	—	2	1	no	—
572	C₂×D₄	—	2	2	no	—
573	C₂×D₄	—	2	1	no	—
574	C₂×D₄	—	2	1	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	1	no	—
578	C₂×D₄	—	4	1	no	—
579	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
580	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
581	C₂²×C₄	—	2	2	no	—
582	C₂²×C₄	—	2	2	no	—
583	C₂²×C₄	—	2	2	no	—
584	C₂²×C₄	—	2	1	no	—
585	C₂²×C₄	—	2	2	no	—
586	C₂²×C₄	—	4	1	no	—
587	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
588	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
589	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
590	Q₁₆	—	8	1	no	—
591	SD₁₆	—	8	1	no	—
592	SD₁₆	—	8	1	no	—
593	SD₁₆	—	8	1	no	—
594	SD₁₆	—	8	1	no	—
595	D₈	—	4	4	no	—
596	D₈	—	8	1	no	—
597	M₁₆	—	4	2	no	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	8	1	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	2	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
624	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
625	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
626	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
627	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
628	C₂²⋊C₄	—	2	2	no	—
629	C₂²⋊C₄	—	2	2	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	2	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	1	no	—
634	C₂²⋊C₄	—	2	1	no	—
635	C₂²⋊C₄	—	2	1	no	—
636	C₂²⋊C₄	—	2	1	no	—
637	C₂²⋊C₄	—	2	1	no	—
638	C₂²⋊C₄	—	2	1	no	—
639	C₂²⋊C₄	—	2	2	no	—
640	C₂²⋊C₄	—	2	2	no	—
641	C₂²⋊C₄	—	2	1	no	—
642	C₂²⋊C₄	—	4	1	no	—
643	C₂²⋊C₄	—	4	1	no	—
644	C₄²	—	2	2	no	—
645	C₄²	—	2	1	no	—
646	C₄²	—	2	1	no	—
647	C₄²	—	2	1	no	—
648	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
649	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
650	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
651	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
652	C₂³	—	2	1	no	—
653	C₂³	—	2	2	no	—
654	C₂³	—	4	1	no	—
655	C₂³	C₂₃×#120	1	1	yes	characteristic
656	Q₈	—	2	1	no	—
657	Q₈	—	2	1	no	—
658	Q₈	—	2	1	no	—
659	Q₈	—	2	1	no	—
660	Q₈	—	2	1	no	—
661	Q₈	—	2	1	no	—
662	Q₈	—	4	1	no	—
663	Q₈	—	4	1	no	—
664	Q₈	—	4	1	no	—
665	Q₈	—	4	1	no	—
666	Q₈	—	8	1	no	—
667	Q₈	—	8	2	no	—
668	D₄	—	2	1	no	—
669	D₄	—	2	1	no	—
670	D₄	—	4	2	no	—
671	D₄	—	4	2	no	—
672	D₄	—	4	1	no	—
673	D₄	—	4	1	no	—
674	D₄	—	4	1	no	—
675	D₄	—	4	1	no	—
676	D₄	—	4	1	no	—
677	D₄	—	4	1	no	—
678	D₄	—	4	1	no	—
679	D₄	—	4	1	no	—
680	D₄	—	8	2	no	—
681	D₄	—	8	1	no	—
682	D₄	—	8	2	no	—
683	D₄	—	8	2	no	—
684	D₄	—	8	2	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	1	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	2	1	no	—
697	C₂×C₄	—	4	1	no	—
698	C₂×C₄	—	4	1	no	—
699	C₂×C₄	—	4	1	no	—
700	C₂×C₄	—	4	1	no	—
701	C₂×C₄	—	8	2	no	—
702	C₂×C₄	—	8	2	no	—
703	C₂×C₄	—	8	2	no	—
704	C₂×C₄	—	8	2	no	—
705	C₂×C₄	—	8	2	no	—
706	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
707	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
708	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
709	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
710	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
711	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
712	C₈	—	2	2	no	—
713	C₈	—	4	1	no	—
714	C₈	—	4	1	no	—
715	C₈	—	4	1	no	—
716	C₈	—	4	1	no	—
717	C₈	—	4	1	no	—
718	C₂²	—	2	1	no	—
719	C₂²	—	2	1	no	—
720	C₂²	—	2	1	no	—
721	C₂²	—	4	1	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	4	2	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	1	no	—
727	C₂²	—	8	1	no	—
728	C₂²	—	8	1	no	—
729	C₂²	—	8	1	no	—
730	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
731	C₄	—	2	1	no	—
732	C₄	—	2	1	no	—
733	C₄	—	2	1	no	—
734	C₄	—	2	1	no	—
735	C₄	—	4	1	no	—
736	C₄	—	4	1	no	—
737	C₄	—	4	1	no	—
738	C₄	—	4	2	no	—
739	C₄	—	4	1	no	—
740	C₄	—	4	1	no	—
741	C₄	—	4	1	no	—
742	C₄	—	4	1	no	—
743	C₄	—	8	1	no	—
744	C₄	C₂₃×#576	1	1	yes	characteristic
745	C₄	C₂₃×#577	1	1	yes	characteristic
746	C₄	C₂₃×#577	1	1	yes	characteristic
747	C₄	C₂₃×#577	1	1	yes	characteristic
748	C₂	—	4	1	no	—
749	C₂	—	8	1	no	—
750	C₂	—	8	2	no	—
751	C₂	—	16	1	no	—
752	C₂	C₂₃×#3010	1	1	yes	characteristic
753	C₂	C₂₃×#3012	1	1	yes	characteristic
754	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3554	C₂	semidirect
2. C₂₃×#3246	C₂	semidirect
3. C₂₃×#3212	C₂	semidirect
4. C₂₃×#3184	C₂	semidirect
5. C₂₃×#1653	C₂	semidirect
6. C₂₃×#1651	C₂	semidirect
7. C₂₃×#1642	C₂	semidirect
8. C₂₃×#1640	C₂	semidirect
9. C₂₃×#1627	C₂	semidirect
10. C₂₃×#1626	C₂	semidirect
11. C₂₃×#1621	C₂	semidirect
12. C₂₃×#1620	C₂	semidirect
13. C₂₃×#1540	C₂	semidirect
14. C₂₃×#1485	C₂	semidirect
15. C₂₃×#1483	C₂	semidirect
274. #18742	C₂₃	direct (N × H)
341. #3554	C₂×C₂₃	semidirect
342. #3246	C₂×C₂₃	semidirect
343. #3212	C₂×C₂₃	semidirect
344. #3184	C₂×C₂₃	semidirect
345. #1653	C₂×C₂₃	semidirect
346. #1651	C₂×C₂₃	semidirect
347. #1642	C₂×C₂₃	semidirect
348. #1640	C₂×C₂₃	semidirect
349. #1627	C₂×C₂₃	semidirect
350. #1626	C₂×C₂₃	semidirect
351. #1621	C₂×C₂₃	semidirect
352. #1620	C₂×C₂₃	semidirect
353. #1540	C₂×C₂₃	semidirect
354. #1485	C₂×C₂₃	semidirect
355. #1483	C₂×C₂₃	semidirect
553. C₂₃	#18742	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×#3554	2
C₂₃×#3246	2
C₂₃×#3212	2
C₂₃×#3184	2
C₂₃×#1653	2
C₂₃×#1651	2
C₂₃×#1642	2
C₂₃×#1640	2
C₂₃×#1627	2
C₂₃×#1626	2
C₂₃×#1621	2
C₂₃×#1620	2
C₂₃×#1540	2
C₂₃×#1485	2
C₂₃×#1483	2
#18742	23

Subgroups — C23×#18742 #19166368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18742 #19166368