Subgroups — C₂₃×#18950 #19166576

Order 5888.

824 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3246	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3228	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3216	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1653	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1648	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1626	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1626	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1481	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#564	—	2	2	no	—
13	C₂₃×#557	—	2	1	no	—
14	C₂₃×#557	—	2	1	no	—
15	C₂₃×#557	—	2	1	no	—
16	C₂₃×#557	—	2	1	no	—
17	C₂₃×#557	C₂²	1	2	yes	—
18	C₂₃×#557	C₂²	1	2	yes	—
19	C₂₃×#541	—	2	2	no	—
20	C₂₃×#541	—	2	2	no	—
21	C₂₃×#541	—	2	2	no	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#533	C₂²	1	1	yes	characteristic
24	C₂₃×#533	C₂²	1	1	yes	characteristic
25	C₂₃×#532	C₂²	1	1	yes	characteristic
26	C₂₃×#500	C₂²	1	1	yes	characteristic
27	C₂₃×#498	C₂²	1	1	yes	characteristic
28	C₂₃×#497	—	2	1	no	—
29	C₂₃×#489	—	2	1	no	—
30	C₂₃×#486	C₂²	1	1	yes	characteristic
31	C₂₃×#486	C₂²	1	1	yes	characteristic
32	C₂₃×#482	—	2	2	no	—
33	C₂₃×#478	—	2	2	no	—
34	C₂₃×#476	—	2	1	no	—
35	C₂₃×#475	C₂²	1	2	yes	—
36	C₂₃×#465	—	2	2	no	—
37	C₂₃×#462	C₂²	1	2	yes	—
38	C₂₃×#461	—	2	1	no	—
39	C₂₃×#460	C₂²	1	2	yes	—
40	C₂₃×#460	C₂²	1	2	yes	—
41	C₂₃×#452	—	2	2	no	—
42	C₂₃×#432	C₂²	1	1	yes	characteristic
43	C₂₃×#431	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₂₃×#332	C₂²	1	2	yes	—
46	C₂₃×#332	C₂²	1	2	yes	—
47	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₂₃	—	2	1	no	—
52	C₂×Q₁₆×C₂₃	—	2	2	no	—
53	C₂×SD₁₆×C₂₃	—	2	2	no	—
54	C₂×SD₁₆×C₂₃	—	2	2	no	—
55	C₂²×C₈×C₂₃	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	2	no	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	—	2	2	no	—
59	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
63	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₂₃×#127	C₂³	1	1	yes	characteristic
67	C₂₃×#126	—	2	4	no	—
68	C₂₃×#126	—	2	4	no	—
69	C₂₃×#125	—	2	2	no	—
70	C₂₃×#125	—	2	4	no	—
71	C₂₃×#125	—	2	4	no	—
72	C₂₃×#125	—	2	4	no	—
73	C₂₃×#125	—	2	4	no	—
74	C₂₃×#125	—	2	2	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	4	no	—
77	C₂₃×#125	—	2	4	no	—
78	C₂₃×#125	—	2	1	no	—
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×#125	D₄	1	2	yes	—
81	C₂₃×#124	—	2	1	no	—
82	C₂₃×C₂²⋊Q₈	—	2	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
87	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
88	C₂₃×#121	—	2	1	no	—
89	C₂₃×#121	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	—	2	2	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	—	2	2	no	—
95	C₄×Q₈×C₂₃	—	2	2	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×Q₈×C₂₃	—	2	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×Q₈×C₂₃	—	2	1	no	—
100	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
101	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₂₃×#117	—	2	1	no	—
105	C₂₃×#117	—	2	1	no	—
106	C₂₃×#117	D₄	1	2	yes	—
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂₃×#107	—	2	2	no	—
109	C₂₃×#106	—	2	2	no	—
110	C₂₃×#106	—	2	2	no	—
111	C₂₃×C₄⋊C₈	—	2	1	no	—
112	C₂₃×C₄⋊C₈	—	2	1	no	—
113	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
114	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
115	C₂₃×#103	—	2	2	no	—
116	C₂₃×#103	—	2	2	no	—
117	C₂₃×#102	—	2	2	no	—
118	C₂₃×#102	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×C₂²⋊C₈	—	2	1	no	—
122	C₂₃×C₂²⋊C₈	—	2	1	no	—
123	C₂₃×#97	—	2	1	no	—
124	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
126	#41×C₂₃	—	8	2	no	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	—	2	1	no	—
129	C₂×Q₈×C₂₃	—	2	1	no	—
130	C₂×Q₈×C₂₃	—	2	1	no	—
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
133	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×C₂₃	—	4	1	no	—
138	C₂²×C₄×C₂₃	—	4	1	no	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	Q₁₆×C₂₃	—	8	2	no	—
143	SD₁₆×C₂₃	—	8	2	no	—
144	SD₁₆×C₂₃	—	8	2	no	—
145	C₂×C₈×C₂₃	—	2	1	no	—
146	C₂×C₈×C₂₃	—	2	1	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	8	1	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	4	no	—
155	C₄⋊C₄×C₂₃	—	2	4	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	—	2	2	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	2	no	—
169	C₄⋊C₄×C₂₃	—	2	2	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
176	C₂²⋊C₄×C₂₃	—	2	2	no	—
177	C₂²⋊C₄×C₂₃	—	2	2	no	—
178	C₂²⋊C₄×C₂₃	—	2	2	no	—
179	C₂²⋊C₄×C₂₃	—	4	1	no	—
180	C₂²⋊C₄×C₂₃	—	4	1	no	—
181	C₄²×C₂₃	—	2	4	no	—
182	C₄²×C₂₃	—	2	4	no	—
183	C₄²×C₂₃	—	2	2	no	—
184	C₄²×C₂₃	—	2	2	no	—
185	C₄²×C₂₃	—	2	2	no	—
186	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
190	#18950	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₂₃	—	4	1	no	—
192	C₂³×C₂₃	#120	1	1	yes	characteristic
193	Q₈×C₂₃	—	2	2	no	—
194	Q₈×C₂₃	—	2	2	no	—
195	Q₈×C₂₃	—	2	2	no	—
196	Q₈×C₂₃	—	2	2	no	—
197	Q₈×C₂₃	—	4	2	no	—
198	Q₈×C₂₃	—	4	2	no	—
199	Q₈×C₂₃	—	4	2	no	—
200	Q₈×C₂₃	—	4	1	no	—
201	Q₈×C₂₃	—	4	1	no	—
202	Q₈×C₂₃	—	4	2	no	—
203	D₄×C₂₃	—	4	2	no	—
204	D₄×C₂₃	—	4	1	no	—
205	D₄×C₂₃	—	4	2	no	—
206	D₄×C₂₃	—	4	1	no	—
207	D₄×C₂₃	—	8	2	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	2	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	4	1	no	—
218	C₂×C₄×C₂₃	—	4	1	no	—
219	C₂×C₄×C₂₃	—	8	2	no	—
220	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
221	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
222	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₈×C₂₃	—	4	1	no	—
227	C₈×C₂₃	—	4	1	no	—
228	C₈×C₂₃	—	4	2	no	—
229	C₈×C₂₃	—	4	2	no	—
230	#3556	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#3246	C₂×C₂₃	1	1	yes	characteristic
232	#3228	C₂×C₂₃	1	1	yes	characteristic
233	#3216	C₂×C₂₃	1	1	yes	characteristic
234	#1653	C₂×C₂₃	1	2	yes	complemented
235	#1648	C₂×C₂₃	1	2	yes	complemented
236	#1626	C₂×C₂₃	1	2	yes	complemented
237	#1626	C₂×C₂₃	1	2	yes	complemented
238	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#1497	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#1481	C₂×C₂₃	1	1	yes	characteristic, complemented
241	C₂²×C₂₃	—	2	1	no	—
242	C₂²×C₂₃	—	2	1	no	—
243	C₂²×C₂₃	—	2	1	no	—
244	C₂²×C₂₃	—	8	1	no	—
245	C₂²×C₂₃	—	8	1	no	—
246	C₂²×C₂₃	—	8	1	no	—
247	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
248	C₄×C₂₃	—	2	1	no	—
249	C₄×C₂₃	—	2	1	no	—
250	C₄×C₂₃	—	2	1	no	—
251	C₄×C₂₃	—	2	1	no	—
252	C₄×C₂₃	—	4	2	no	—
253	C₄×C₂₃	—	4	2	no	—
254	C₄×C₂₃	—	4	2	no	—
255	C₄×C₂₃	—	4	2	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	4	2	no	—
258	C₄×C₂₃	—	8	1	no	—
259	C₄×C₂₃	#577	1	2	yes	—
260	C₄×C₂₃	#577	1	1	yes	characteristic
261	C₄×C₂₃	#578	1	1	yes	characteristic
262	#564	—	2	2	no	—
263	#557	—	2	1	no	—
264	#557	—	2	1	no	—
265	#557	—	2	1	no	—
266	#557	—	2	1	no	—
267	#557	C₂²×C₂₃	1	2	yes	—
268	#557	C₂²×C₂₃	1	2	yes	—
269	#541	—	2	2	no	—
270	#541	—	2	2	no	—
271	#541	—	2	2	no	—
272	#533	C₂²×C₂₃	1	1	yes	characteristic
273	#533	C₂²×C₂₃	1	1	yes	characteristic
274	#533	C₂²×C₂₃	1	1	yes	characteristic
275	#532	C₂²×C₂₃	1	1	yes	characteristic
276	#500	C₂²×C₂₃	1	1	yes	characteristic
277	#498	C₂²×C₂₃	1	1	yes	characteristic
278	#497	—	2	1	no	—
279	#489	—	2	1	no	—
280	#486	C₂²×C₂₃	1	1	yes	characteristic
281	#486	C₂²×C₂₃	1	1	yes	characteristic
282	#482	—	2	2	no	—
283	#478	—	2	2	no	—
284	#476	—	2	1	no	—
285	#475	C₂²×C₂₃	1	2	yes	—
286	#465	—	2	2	no	—
287	#462	C₂²×C₂₃	1	2	yes	—
288	#461	—	2	1	no	—
289	#460	C₂²×C₂₃	1	2	yes	—
290	#460	C₂²×C₂₃	1	2	yes	—
291	#452	—	2	2	no	—
292	#432	C₂²×C₂₃	1	1	yes	characteristic
293	#431	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
295	#332	C₂²×C₂₃	1	2	yes	—
296	#332	C₂²×C₂₃	1	2	yes	—
297	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
298	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
299	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
300	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
301	C₂×C₂₃	—	4	1	no	—
302	C₂×C₂₃	—	16	1	no	—
303	C₂×C₂₃	#3010	1	1	yes	characteristic
304	C₂×C₂₃	#3013	1	1	yes	characteristic
305	C₂×C₂₃	#3020	1	1	yes	characteristic
306	C₂×#41	—	2	1	no	—
307	C₂×Q₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×SD₁₆	—	2	2	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
315	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
316	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
317	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
318	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
319	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
320	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
321	#127	C₂³×C₂₃	1	1	yes	characteristic
322	#126	—	2	4	no	—
323	#126	—	2	4	no	—
324	#125	—	2	2	no	—
325	#125	—	2	4	no	—
326	#125	—	2	4	no	—
327	#125	—	2	4	no	—
328	#125	—	2	4	no	—
329	#125	—	2	2	no	—
330	#125	—	2	2	no	—
331	#125	—	2	4	no	—
332	#125	—	2	4	no	—
333	#125	—	2	1	no	—
334	#125	—	2	1	no	—
335	#125	D₄×C₂₃	1	2	yes	—
336	#124	—	2	1	no	—
337	C₂²⋊Q₈	—	2	1	no	—
338	C₂²⋊Q₈	—	2	1	no	—
339	C₂²⋊Q₈	—	2	1	no	—
340	C₂²⋊Q₈	—	2	1	no	—
341	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
342	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
343	#121	—	2	1	no	—
344	#121	—	2	1	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
356	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
357	C₄×D₄	—	2	1	no	—
358	C₄×D₄	—	2	1	no	—
359	#117	—	2	1	no	—
360	#117	—	2	1	no	—
361	#117	D₄×C₂₃	1	2	yes	—
362	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
363	#107	—	2	2	no	—
364	#106	—	2	2	no	—
365	#106	—	2	2	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
369	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
370	#103	—	2	2	no	—
371	#103	—	2	2	no	—
372	#102	—	2	2	no	—
373	#102	—	2	2	no	—
374	#102	—	2	2	no	—
375	#102	—	2	2	no	—
376	C₂²⋊C₈	—	2	1	no	—
377	C₂²⋊C₈	—	2	1	no	—
378	#97	—	2	1	no	—
379	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₂₃	#18950	1	1	yes	characteristic, direct product factor, Sylow
382	#41	—	8	2	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
389	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	Q₁₆	—	8	2	no	—
399	SD₁₆	—	8	2	no	—
400	SD₁₆	—	8	2	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	8	1	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
429	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
430	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
431	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₂²⋊C₄	—	4	1	no	—
437	C₄²	—	2	4	no	—
438	C₄²	—	2	4	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	2	no	—
442	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
445	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
446	C₂³	—	4	1	no	—
447	C₂³	C₂₃×#120	1	1	yes	characteristic
448	Q₈	—	2	2	no	—
449	Q₈	—	2	2	no	—
450	Q₈	—	2	2	no	—
451	Q₈	—	2	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	—	4	1	no	—
456	Q₈	—	4	1	no	—
457	Q₈	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	1	no	—
462	D₄	—	8	2	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	8	2	no	—
475	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
476	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
477	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
481	C₈	—	4	1	no	—
482	C₈	—	4	1	no	—
483	C₈	—	4	2	no	—
484	C₈	—	4	2	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	1	no	—
491	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	2	1	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	8	1	no	—
503	C₄	C₂₃×#577	1	2	yes	—
504	C₄	C₂₃×#577	1	1	yes	characteristic
505	C₄	C₂₃×#578	1	1	yes	characteristic
506	C₂	—	4	1	no	—
507	C₂	—	16	1	no	—
508	C₂	C₂₃×#3010	1	1	yes	characteristic
509	C₂	C₂₃×#3013	1	1	yes	characteristic
510	C₂	C₂₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
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×#3556	C₂	semidirect
5. C₂₃×#1653	C₂	semidirect
6. C₂₃×#1648	C₂	semidirect
7. C₂₃×#1626	C₂	semidirect
8. C₂₃×#1626	C₂	semidirect
9. C₂₃×#1552	C₂	semidirect
10. C₂₃×#1497	C₂	semidirect
11. C₂₃×#1481	C₂	semidirect
190. #18950	C₂₃	direct (N × H)
230. #3556	C₂×C₂₃	semidirect
234. #1653	C₂×C₂₃	semidirect
235. #1648	C₂×C₂₃	semidirect
236. #1626	C₂×C₂₃	semidirect
237. #1626	C₂×C₂₃	semidirect
238. #1552	C₂×C₂₃	semidirect
239. #1497	C₂×C₂₃	semidirect
240. #1481	C₂×C₂₃	semidirect
381. C₂₃	#18950	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×#3556	2
C₂₃×#3246	2
C₂₃×#3228	2
C₂₃×#3216	2
C₂₃×#1653	2
C₂₃×#1648	2
C₂₃×#1626	2
C₂₃×#1552	2
C₂₃×#1497	2
C₂₃×#1481	2
#18950	23

Subgroups — C23×#18950 #19166576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18950 #19166576