Subgroups — C₂₃×#18975 #19166601

Order 5888.

810 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3555	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3248	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3231	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1650	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1643	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1495	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1481	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#563	—	2	2	no	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#557	—	2	1	no	—
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#556	—	2	1	no	—
21	C₂₃×#556	—	2	1	no	—
22	C₂₃×#556	C₂²	1	1	yes	characteristic
23	C₂₃×#554	C₂²	1	1	yes	characteristic
24	C₂₃×#552	—	2	1	no	—
25	C₂₃×#544	—	2	1	no	—
26	C₂₃×#542	—	2	1	no	—
27	C₂₃×#541	—	2	1	no	—
28	C₂₃×#541	—	2	1	no	—
29	C₂₃×#541	—	2	2	no	—
30	C₂₃×#533	C₂²	1	1	yes	characteristic
31	C₂₃×#533	C₂²	1	1	yes	characteristic
32	C₂₃×#533	C₂²	1	1	yes	characteristic
33	C₂₃×#529	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#498	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#485	—	2	2	no	—
39	C₂₃×#479	—	2	1	no	—
40	C₂₃×#479	—	2	2	no	—
41	C₂₃×#475	—	2	1	no	—
42	C₂₃×#475	C₂²	1	1	yes	characteristic
43	C₂₃×#475	C₂²	1	1	yes	characteristic
44	C₂₃×#475	C₂²	1	1	yes	characteristic
45	C₂₃×#474	C₂²	1	1	yes	characteristic
46	C₂₃×#470	—	2	1	no	—
47	C₂₃×#470	—	2	1	no	—
48	C₂₃×#467	—	2	1	no	—
49	C₂₃×#465	—	2	1	no	—
50	C₂₃×#464	—	2	1	no	—
51	C₂₃×#462	—	2	1	no	—
52	C₂₃×#462	C₂²	1	1	yes	characteristic
53	C₂₃×#462	C₂²	1	1	yes	characteristic
54	C₂₃×#462	C₂²	1	1	yes	characteristic
55	C₂₃×#461	C₂²	1	1	yes	characteristic
56	C₂₃×#431	C₂²	1	1	yes	characteristic
57	C₂₃×#423	C₂²	1	1	yes	characteristic
58	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	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×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×Q₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	2	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	1	no	—
82	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×#126	—	2	2	no	—
89	C₂₃×#126	—	2	2	no	—
90	C₂₃×#126	—	2	2	no	—
91	C₂₃×#126	—	2	2	no	—
92	C₂₃×#126	—	2	2	no	—
93	C₂₃×#126	—	2	2	no	—
94	C₂₃×#126	—	2	1	no	—
95	C₂₃×#125	—	2	2	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	2	no	—
98	C₂₃×#125	—	2	2	no	—
99	C₂₃×#125	—	2	1	no	—
100	C₂₃×#125	—	2	1	no	—
101	C₂₃×#125	—	2	2	no	—
102	C₂₃×#125	—	2	1	no	—
103	C₂₃×#125	—	2	1	no	—
104	C₂₃×#125	—	2	1	no	—
105	C₂₃×#125	D₄	1	1	yes	characteristic
106	C₂₃×#124	—	2	2	no	—
107	C₂₃×#124	—	2	1	no	—
108	C₂₃×#124	D₄	1	1	yes	characteristic
109	C₂₃×#123	—	2	2	no	—
110	C₂₃×#123	—	2	1	no	—
111	C₂₃×C₂²⋊Q₈	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	1	no	—
114	C₂₃×C₂²⋊Q₈	—	2	1	no	—
115	C₂₃×C₂²⋊Q₈	—	2	2	no	—
116	C₂₃×C₂²⋊Q₈	—	2	1	no	—
117	C₂₃×C₂²⋊Q₈	—	2	1	no	—
118	C₂₃×C₂²⋊Q₈	—	2	1	no	—
119	C₂₃×C₂²⋊Q₈	—	2	1	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	2	yes	—
123	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
124	C₂₃×#121	D₄	1	1	yes	characteristic
125	C₄×Q₈×C₂₃	—	2	2	no	—
126	C₄×Q₈×C₂₃	—	2	1	no	—
127	C₄×Q₈×C₂₃	—	2	1	no	—
128	C₄×Q₈×C₂₃	—	2	1	no	—
129	C₄×Q₈×C₂₃	—	2	1	no	—
130	C₄×Q₈×C₂₃	—	2	1	no	—
131	C₄×Q₈×C₂₃	—	2	1	no	—
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	D₄	1	2	yes	—
136	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
137	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	—	2	1	no	—
141	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
142	C₂₃×#117	—	2	2	no	—
143	C₂₃×#117	—	2	1	no	—
144	C₂₃×#117	—	2	1	no	—
145	C₂₃×#117	—	2	1	no	—
146	C₂₃×#117	—	2	1	no	—
147	C₂₃×#117	—	2	1	no	—
148	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#107	—	2	2	no	—
152	C₂₃×#106	—	2	1	no	—
153	C₂₃×#106	—	2	1	no	—
154	C₂₃×#106	—	2	1	no	—
155	C₂₃×#106	—	2	2	no	—
156	C₂₃×C₄⋊C₈	—	2	1	no	—
157	C₂₃×C₄⋊C₈	—	2	1	no	—
158	C₂₃×C₄⋊C₈	—	2	1	no	—
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	2	no	—
167	C₂₃×#103	—	2	2	no	—
168	C₂₃×#103	—	2	1	no	—
169	C₂₃×#103	—	2	1	no	—
170	C₂₃×#103	—	2	1	no	—
171	C₂₃×#103	—	2	1	no	—
172	C₂₃×#102	—	2	1	no	—
173	C₂₃×C₂²⋊C₈	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₂×Q₈×C₂₃	—	2	2	no	—
177	C₂×Q₈×C₂₃	—	2	1	no	—
178	C₂×Q₈×C₂₃	—	2	1	no	—
179	C₂×Q₈×C₂₃	—	2	1	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
185	C₂×D₄×C₂₃	—	2	1	no	—
186	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	—	2	2	no	—
188	C₂²×C₄×C₂₃	—	2	1	no	—
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	Q₁₆×C₂₃	—	8	1	no	—
193	Q₁₆×C₂₃	—	8	1	no	—
194	Q₁₆×C₂₃	—	8	1	no	—
195	SD₁₆×C₂₃	—	8	1	no	—
196	M₁₆×C₂₃	—	8	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	8	1	no	—
204	C₄⋊C₄×C₂₃	—	2	2	no	—
205	C₄⋊C₄×C₂₃	—	2	2	no	—
206	C₄⋊C₄×C₂₃	—	2	2	no	—
207	C₄⋊C₄×C₂₃	—	2	2	no	—
208	C₄⋊C₄×C₂₃	—	2	2	no	—
209	C₄⋊C₄×C₂₃	—	2	2	no	—
210	C₄⋊C₄×C₂₃	—	2	2	no	—
211	C₄⋊C₄×C₂₃	—	2	2	no	—
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	2	1	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	1	no	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	1	no	—
223	C₄⋊C₄×C₂₃	—	2	1	no	—
224	C₄⋊C₄×C₂₃	—	2	1	no	—
225	C₄⋊C₄×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	—	2	1	no	—
228	C₄⋊C₄×C₂₃	—	2	1	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	4	1	no	—
233	C₄⋊C₄×C₂₃	—	4	1	no	—
234	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
235	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
237	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
239	C₂²⋊C₄×C₂₃	—	2	2	no	—
240	C₂²⋊C₄×C₂₃	—	2	2	no	—
241	C₂²⋊C₄×C₂₃	—	2	2	no	—
242	C₂²⋊C₄×C₂₃	—	2	1	no	—
243	C₂²⋊C₄×C₂₃	—	2	2	no	—
244	C₂²⋊C₄×C₂₃	—	2	1	no	—
245	C₂²⋊C₄×C₂₃	—	2	1	no	—
246	C₂²⋊C₄×C₂₃	—	2	1	no	—
247	C₂²⋊C₄×C₂₃	—	2	1	no	—
248	C₂²⋊C₄×C₂₃	—	2	1	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	no	—
250	C₂²⋊C₄×C₂₃	—	4	1	no	—
251	C₄²×C₂₃	—	2	2	no	—
252	C₄²×C₂₃	—	2	2	no	—
253	C₄²×C₂₃	—	2	2	no	—
254	C₄²×C₂₃	—	2	2	no	—
255	C₄²×C₂₃	—	2	1	no	—
256	C₄²×C₂₃	—	2	1	no	—
257	C₄²×C₂₃	—	2	1	no	—
258	C₄²×C₂₃	—	4	1	no	—
259	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
262	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
263	#18975	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂³×C₂₃	—	2	1	no	—
265	C₂³×C₂₃	#120	1	1	yes	characteristic
266	Q₈×C₂₃	—	2	1	no	—
267	Q₈×C₂₃	—	2	1	no	—
268	Q₈×C₂₃	—	2	1	no	—
269	Q₈×C₂₃	—	2	1	no	—
270	Q₈×C₂₃	—	2	1	no	—
271	Q₈×C₂₃	—	2	1	no	—
272	Q₈×C₂₃	—	4	2	no	—
273	Q₈×C₂₃	—	4	1	no	—
274	Q₈×C₂₃	—	4	1	no	—
275	Q₈×C₂₃	—	4	1	no	—
276	Q₈×C₂₃	—	4	1	no	—
277	Q₈×C₂₃	—	4	1	no	—
278	Q₈×C₂₃	—	4	1	no	—
279	Q₈×C₂₃	—	8	1	no	—
280	Q₈×C₂₃	—	8	2	no	—
281	D₄×C₂₃	—	2	1	no	—
282	D₄×C₂₃	—	2	1	no	—
283	D₄×C₂₃	—	8	1	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	1	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	2	2	no	—
292	C₂×C₄×C₂₃	—	2	1	no	—
293	C₂×C₄×C₂₃	—	2	1	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	2	no	—
296	C₂×C₄×C₂₃	—	2	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	—	4	1	no	—
302	C₂×C₄×C₂₃	—	8	2	no	—
303	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
304	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
305	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
309	C₈×C₂₃	—	4	1	no	—
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	—	4	1	no	—
312	C₈×C₂₃	—	4	1	no	—
313	C₈×C₂₃	—	4	1	no	—
314	C₈×C₂₃	—	4	1	no	—
315	#3555	C₂×C₂₃	1	1	yes	characteristic
316	#3255	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#3248	C₂×C₂₃	1	1	yes	characteristic
318	#3231	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1650	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1643	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#1495	C₂×C₂₃	1	1	yes	characteristic
329	#1481	C₂×C₂₃	1	1	yes	characteristic, complemented
330	C₂²×C₂₃	—	2	1	no	—
331	C₂²×C₂₃	—	2	1	no	—
332	C₂²×C₂₃	—	2	1	no	—
333	C₂²×C₂₃	—	4	1	no	—
334	C₂²×C₂₃	—	4	1	no	—
335	C₂²×C₂₃	—	4	1	no	—
336	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
337	C₄×C₂₃	—	2	1	no	—
338	C₄×C₂₃	—	2	1	no	—
339	C₄×C₂₃	—	2	1	no	—
340	C₄×C₂₃	—	2	1	no	—
341	C₄×C₂₃	—	4	1	no	—
342	C₄×C₂₃	—	4	1	no	—
343	C₄×C₂₃	—	4	1	no	—
344	C₄×C₂₃	—	4	1	no	—
345	C₄×C₂₃	—	4	2	no	—
346	C₄×C₂₃	—	4	1	no	—
347	C₄×C₂₃	—	4	2	no	—
348	C₄×C₂₃	—	4	1	no	—
349	C₄×C₂₃	—	4	1	no	—
350	C₄×C₂₃	—	8	1	no	—
351	C₄×C₂₃	—	8	1	no	—
352	C₄×C₂₃	#577	1	1	yes	characteristic
353	C₄×C₂₃	#577	1	1	yes	characteristic
354	C₄×C₂₃	#578	1	1	yes	characteristic
355	C₄×C₂₃	#578	1	1	yes	characteristic
356	#563	—	2	2	no	—
357	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
358	#557	—	2	1	no	—
359	#557	C₂²×C₂₃	1	1	yes	characteristic
360	#556	—	2	1	no	—
361	#556	—	2	1	no	—
362	#556	C₂²×C₂₃	1	1	yes	characteristic
363	#554	C₂²×C₂₃	1	1	yes	characteristic
364	#552	—	2	1	no	—
365	#544	—	2	1	no	—
366	#542	—	2	1	no	—
367	#541	—	2	1	no	—
368	#541	—	2	1	no	—
369	#541	—	2	2	no	—
370	#533	C₂²×C₂₃	1	1	yes	characteristic
371	#533	C₂²×C₂₃	1	1	yes	characteristic
372	#533	C₂²×C₂₃	1	1	yes	characteristic
373	#529	C₂²×C₂₃	1	1	yes	characteristic
374	#500	C₂²×C₂₃	1	1	yes	characteristic
375	#498	C₂²×C₂₃	1	1	yes	characteristic
376	#487	C₂²×C₂₃	1	1	yes	characteristic
377	#487	C₂²×C₂₃	1	1	yes	characteristic
378	#485	—	2	2	no	—
379	#479	—	2	1	no	—
380	#479	—	2	2	no	—
381	#475	—	2	1	no	—
382	#475	C₂²×C₂₃	1	1	yes	characteristic
383	#475	C₂²×C₂₃	1	1	yes	characteristic
384	#475	C₂²×C₂₃	1	1	yes	characteristic
385	#474	C₂²×C₂₃	1	1	yes	characteristic
386	#470	—	2	1	no	—
387	#470	—	2	1	no	—
388	#467	—	2	1	no	—
389	#465	—	2	1	no	—
390	#464	—	2	1	no	—
391	#462	—	2	1	no	—
392	#462	C₂²×C₂₃	1	1	yes	characteristic
393	#462	C₂²×C₂₃	1	1	yes	characteristic
394	#462	C₂²×C₂₃	1	1	yes	characteristic
395	#461	C₂²×C₂₃	1	1	yes	characteristic
396	#431	C₂²×C₂₃	1	1	yes	characteristic
397	#423	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
399	#333	C₂²×C₂₃	1	1	yes	characteristic
400	#333	C₂²×C₂₃	1	1	yes	characteristic
401	#332	C₂²×C₂₃	1	1	yes	characteristic
402	#332	C₂²×C₂₃	1	1	yes	characteristic
403	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
404	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
405	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
406	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
407	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
408	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
409	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
410	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
411	C₂×C₂₃	—	4	1	no	—
412	C₂×C₂₃	—	8	1	no	—
413	C₂×C₂₃	#3013	1	1	yes	characteristic
414	C₂×C₂₃	#3013	1	1	yes	characteristic
415	C₂×C₂₃	#3022	1	1	yes	characteristic
416	C₂×Q₁₆	—	2	1	no	—
417	C₂×Q₁₆	—	2	1	no	—
418	C₂×Q₁₆	—	2	1	no	—
419	C₂×SD₁₆	—	2	1	no	—
420	C₂×M₁₆	—	2	1	no	—
421	C₂²×C₈	—	2	1	no	—
422	C₄⋊Q₈	—	2	2	no	—
423	C₄⋊Q₈	—	2	1	no	—
424	C₄⋊Q₈	—	2	2	no	—
425	C₄⋊Q₈	—	2	1	no	—
426	C₄⋊Q₈	—	2	1	no	—
427	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
428	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
429	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
433	#126	—	2	2	no	—
434	#126	—	2	2	no	—
435	#126	—	2	2	no	—
436	#126	—	2	2	no	—
437	#126	—	2	2	no	—
438	#126	—	2	2	no	—
439	#126	—	2	1	no	—
440	#125	—	2	2	no	—
441	#125	—	2	2	no	—
442	#125	—	2	2	no	—
443	#125	—	2	2	no	—
444	#125	—	2	1	no	—
445	#125	—	2	2	no	—
446	#125	—	2	1	no	—
447	#125	—	2	1	no	—
448	#125	—	2	1	no	—
449	#125	—	2	1	no	—
450	#125	D₄×C₂₃	1	1	yes	characteristic
451	#124	—	2	2	no	—
452	#124	—	2	1	no	—
453	#124	D₄×C₂₃	1	1	yes	characteristic
454	#123	—	2	2	no	—
455	#123	—	2	1	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	2	no	—
461	C₂²⋊Q₈	—	2	1	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
466	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
467	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
468	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
469	#121	D₄×C₂₃	1	1	yes	characteristic
470	C₄×Q₈	—	2	2	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×Q₈	—	2	1	no	—
478	C₄×Q₈	—	2	1	no	—
479	C₄×Q₈	—	2	1	no	—
480	C₄×Q₈	D₄×C₂₃	1	2	yes	—
481	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
483	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
484	C₄×D₄	—	2	1	no	—
485	C₄×D₄	—	2	1	no	—
486	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
487	#117	—	2	2	no	—
488	#117	—	2	1	no	—
489	#117	—	2	1	no	—
490	#117	—	2	1	no	—
491	#117	—	2	1	no	—
492	#117	—	2	1	no	—
493	C₂×C₄⋊C₄	—	2	1	no	—
494	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
495	#107	—	2	1	no	—
496	#107	—	2	2	no	—
497	#106	—	2	1	no	—
498	#106	—	2	1	no	—
499	#106	—	2	1	no	—
500	#106	—	2	2	no	—
501	C₄⋊C₈	—	2	1	no	—
502	C₄⋊C₈	—	2	1	no	—
503	C₄⋊C₈	—	2	1	no	—
504	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
505	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
506	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
507	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
508	#103	—	2	1	no	—
509	#103	—	2	1	no	—
510	#103	—	2	1	no	—
511	#103	—	2	2	no	—
512	#103	—	2	2	no	—
513	#103	—	2	1	no	—
514	#103	—	2	1	no	—
515	#103	—	2	1	no	—
516	#103	—	2	1	no	—
517	#102	—	2	1	no	—
518	C₂²⋊C₈	—	2	1	no	—
519	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
520	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
521	C₂₃	#18975	1	1	yes	characteristic, direct product factor, Sylow
522	C₂×Q₈	—	2	2	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	1	no	—
527	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
529	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
530	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
537	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
538	Q₁₆	—	8	1	no	—
539	Q₁₆	—	8	1	no	—
540	Q₁₆	—	8	1	no	—
541	SD₁₆	—	8	1	no	—
542	M₁₆	—	8	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	8	1	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	4	1	no	—
579	C₄⋊C₄	—	4	1	no	—
580	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
582	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
583	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
584	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
585	C₂²⋊C₄	—	2	2	no	—
586	C₂²⋊C₄	—	2	2	no	—
587	C₂²⋊C₄	—	2	2	no	—
588	C₂²⋊C₄	—	2	1	no	—
589	C₂²⋊C₄	—	2	2	no	—
590	C₂²⋊C₄	—	2	1	no	—
591	C₂²⋊C₄	—	2	1	no	—
592	C₂²⋊C₄	—	2	1	no	—
593	C₂²⋊C₄	—	2	1	no	—
594	C₂²⋊C₄	—	2	1	no	—
595	C₂²⋊C₄	—	2	1	no	—
596	C₂²⋊C₄	—	4	1	no	—
597	C₄²	—	2	2	no	—
598	C₄²	—	2	2	no	—
599	C₄²	—	2	2	no	—
600	C₄²	—	2	2	no	—
601	C₄²	—	2	1	no	—
602	C₄²	—	2	1	no	—
603	C₄²	—	2	1	no	—
604	C₄²	—	4	1	no	—
605	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
606	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
607	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
608	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
609	C₂³	—	2	1	no	—
610	C₂³	C₂₃×#120	1	1	yes	characteristic
611	Q₈	—	2	1	no	—
612	Q₈	—	2	1	no	—
613	Q₈	—	2	1	no	—
614	Q₈	—	2	1	no	—
615	Q₈	—	2	1	no	—
616	Q₈	—	2	1	no	—
617	Q₈	—	4	2	no	—
618	Q₈	—	4	1	no	—
619	Q₈	—	4	1	no	—
620	Q₈	—	4	1	no	—
621	Q₈	—	4	1	no	—
622	Q₈	—	4	1	no	—
623	Q₈	—	4	1	no	—
624	Q₈	—	8	1	no	—
625	Q₈	—	8	2	no	—
626	D₄	—	2	1	no	—
627	D₄	—	2	1	no	—
628	D₄	—	8	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	8	2	no	—
648	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
649	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
650	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
651	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
652	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
654	C₈	—	4	1	no	—
655	C₈	—	4	1	no	—
656	C₈	—	4	1	no	—
657	C₈	—	4	1	no	—
658	C₈	—	4	1	no	—
659	C₈	—	4	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	4	1	no	—
664	C₂²	—	4	1	no	—
665	C₂²	—	4	1	no	—
666	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
667	C₄	—	2	1	no	—
668	C₄	—	2	1	no	—
669	C₄	—	2	1	no	—
670	C₄	—	2	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	1	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	2	no	—
678	C₄	—	4	1	no	—
679	C₄	—	4	1	no	—
680	C₄	—	8	1	no	—
681	C₄	—	8	1	no	—
682	C₄	C₂₃×#577	1	1	yes	characteristic
683	C₄	C₂₃×#577	1	1	yes	characteristic
684	C₄	C₂₃×#578	1	1	yes	characteristic
685	C₄	C₂₃×#578	1	1	yes	characteristic
686	C₂	—	4	1	no	—
687	C₂	—	8	1	no	—
688	C₂	C₂₃×#3013	1	1	yes	characteristic
689	C₂	C₂₃×#3013	1	1	yes	characteristic
690	C₂	C₂₃×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#121	D₄
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
2. C₂₃×#3255	C₂	semidirect
4. C₂₃×#3231	C₂	semidirect
5. C₂₃×#1653	C₂	semidirect
6. C₂₃×#1652	C₂	semidirect
7. C₂₃×#1650	C₂	semidirect
8. C₂₃×#1648	C₂	semidirect
9. C₂₃×#1645	C₂	semidirect
10. C₂₃×#1645	C₂	semidirect
11. C₂₃×#1644	C₂	semidirect
12. C₂₃×#1643	C₂	semidirect
13. C₂₃×#1550	C₂	semidirect
15. C₂₃×#1481	C₂	semidirect
263. #18975	C₂₃	direct (N × H)
316. #3255	C₂×C₂₃	semidirect
318. #3231	C₂×C₂₃	semidirect
319. #1653	C₂×C₂₃	semidirect
320. #1652	C₂×C₂₃	semidirect
321. #1650	C₂×C₂₃	semidirect
322. #1648	C₂×C₂₃	semidirect
323. #1645	C₂×C₂₃	semidirect
324. #1645	C₂×C₂₃	semidirect
325. #1644	C₂×C₂₃	semidirect
326. #1643	C₂×C₂₃	semidirect
327. #1550	C₂×C₂₃	semidirect
329. #1481	C₂×C₂₃	semidirect
521. C₂₃	#18975	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×#3555	2
C₂₃×#3255	2
C₂₃×#3248	2
C₂₃×#3231	2
C₂₃×#1653	2
C₂₃×#1652	2
C₂₃×#1650	2
C₂₃×#1648	2
C₂₃×#1645	2
C₂₃×#1644	2
C₂₃×#1643	2
C₂₃×#1550	2
C₂₃×#1495	2
C₂₃×#1481	2
#18975	23

Subgroups — C23×#18975 #19166601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18975 #19166601