Subgroups — C₂₃×#19643 #19167269

Order 5888.

686 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3228	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#3131	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1723	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1722	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1720	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1718	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1711	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1708	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂₃×#533	C₂²	1	1	yes	characteristic
18	C₂₃×#533	C₂²	1	1	yes	characteristic
19	C₂₃×#532	C₂²	1	1	yes	characteristic
20	C₂₃×#500	C₂²	1	1	yes	characteristic
21	C₂₃×#500	C₂²	1	1	yes	characteristic
22	C₂₃×#500	C₂²	1	1	yes	characteristic
23	C₂₃×#500	C₂²	1	1	yes	characteristic
24	C₂₃×#498	C₂²	1	1	yes	characteristic
25	C₂₃×#498	C₂²	1	1	yes	characteristic
26	C₂₃×#498	C₂²	1	1	yes	characteristic
27	C₂₃×#494	C₂²	1	1	yes	characteristic
28	C₂₃×#494	C₂²	1	1	yes	characteristic
29	C₂₃×#492	C₂²	1	1	yes	characteristic
30	C₂₃×#488	—	2	2	no	—
31	C₂₃×#487	C₂²	1	1	yes	characteristic
32	C₂₃×#487	C₂²	1	1	yes	characteristic
33	C₂₃×#487	C₂²	1	1	yes	characteristic
34	C₂₃×#487	C₂²	1	1	yes	characteristic
35	C₂₃×#487	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#487	C₂²	1	1	yes	characteristic
39	C₂₃×#487	C₂²	1	1	yes	characteristic
40	C₂₃×#486	C₂²	1	1	yes	characteristic
41	C₂₃×#486	C₂²	1	1	yes	characteristic
42	C₂₃×#486	C₂²	1	1	yes	characteristic
43	C₂₃×#486	C₂²	1	1	yes	characteristic
44	C₂₃×#486	C₂²	1	1	yes	characteristic
45	C₂₃×#485	—	2	2	no	—
46	C₂₃×#482	—	2	2	no	—
47	C₂₃×#479	—	2	2	no	—
48	C₂₃×#478	—	2	2	no	—
49	C₂₃×#470	—	2	2	no	—
50	C₂₃×#468	—	2	2	no	—
51	C₂₃×#467	—	2	2	no	—
52	C₂₃×#441	—	2	2	no	—
53	C₂₃×#440	—	2	2	no	—
54	C₄×Q₁₆×C₂₃	—	2	2	no	—
55	C₂₃×#431	C₂²	1	1	yes	characteristic
56	C₂₃×#416	—	2	2	no	—
57	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#322	C₂²	1	1	yes	characteristic
60	C₂₃×#322	C₂²	1	1	yes	characteristic
61	C₈²×C₂₃	C₂²	1	1	yes	characteristic
62	C₈²×C₂₃	C₂²	1	1	yes	characteristic
63	C₂×#41×C₂₃	—	2	1	no	—
64	C₂×Q₁₆×C₂₃	—	2	2	no	—
65	C₂×Q₁₆×C₂₃	—	2	2	no	—
66	C₂×SD₁₆×C₂₃	—	2	2	no	—
67	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
68	C₂²×C₈×C₂₃	D₄	1	2	yes	—
69	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
71	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
73	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
74	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×#127	C₂³	1	1	yes	characteristic
77	C₂₃×#125	—	2	1	no	—
78	C₂₃×#125	—	2	1	no	—
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×#124	—	2	1	no	—
81	C₂₃×C₂²⋊Q₈	—	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	—	2	1	no	—
87	C₂₃×#121	—	2	1	no	—
88	C₂₃×#121	—	2	1	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₂₃×#117	—	2	1	no	—
98	C₂₃×#117	—	2	1	no	—
99	C₂₃×#117	—	2	1	no	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×#107	—	2	2	no	—
102	C₂₃×#107	—	2	2	no	—
103	C₂₃×#107	—	2	2	no	—
104	C₂₃×#107	—	2	2	no	—
105	C₂₃×#106	—	2	2	no	—
106	C₂₃×#106	—	2	2	no	—
107	C₂₃×#106	—	2	2	no	—
108	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
109	C₂₃×#103	—	2	2	no	—
110	C₂₃×#103	—	2	2	no	—
111	C₂₃×#103	—	2	2	no	—
112	C₂₃×#103	—	2	2	no	—
113	C₂₃×#103	—	2	2	no	—
114	C₂₃×#103	—	2	2	no	—
115	C₂₃×#103	—	2	2	no	—
116	C₂₃×#103	—	2	2	no	—
117	C₂₃×#103	—	2	2	no	—
118	C₂₃×#102	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
124	C₂₃×#97	D₄	1	2	yes	—
125	C₄×C₈×C₂₃	D₄	1	2	yes	—
126	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
132	#41×C₂₃	—	4	4	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	1	no	—
142	C₂×D₄×C₂₃	—	2	1	no	—
143	C₂×D₄×C₂₃	—	4	1	no	—
144	C₂²×C₄×C₂₃	—	4	1	no	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	Q₁₆×C₂₃	—	4	4	no	—
149	Q₁₆×C₂₃	—	4	4	no	—
150	SD₁₆×C₂₃	—	4	4	no	—
151	M₁₆×C₂₃	—	2	4	no	—
152	C₂×C₈×C₂₃	—	4	2	no	—
153	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
157	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
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	1	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	4	1	no	—
174	C₄⋊C₄×C₂₃	—	4	1	no	—
175	C₄⋊C₄×C₂₃	—	4	1	no	—
176	C₄⋊C₄×C₂₃	—	4	1	no	—
177	C₄⋊C₄×C₂₃	—	4	1	no	—
178	C₄⋊C₄×C₂₃	—	4	1	no	—
179	C₂²⋊C₄×C₂₃	—	4	1	no	—
180	C₂²⋊C₄×C₂₃	—	4	1	no	—
181	C₂²⋊C₄×C₂₃	—	4	1	no	—
182	C₂²⋊C₄×C₂₃	—	4	1	no	—
183	C₂²⋊C₄×C₂₃	—	4	1	no	—
184	C₄²×C₂₃	—	4	1	no	—
185	C₄²×C₂₃	—	4	1	no	—
186	C₄²×C₂₃	—	4	1	no	—
187	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
191	#19643	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₂₃	—	4	1	no	—
193	C₂³×C₂₃	#127	1	1	yes	characteristic
194	Q₈×C₂₃	—	4	2	no	—
195	Q₈×C₂₃	—	4	2	no	—
196	Q₈×C₂₃	—	4	2	no	—
197	Q₈×C₂₃	—	4	2	no	—
198	Q₈×C₂₃	—	4	2	no	—
199	Q₈×C₂₃	—	4	2	no	—
200	Q₈×C₂₃	—	4	2	no	—
201	D₄×C₂₃	—	4	2	no	—
202	D₄×C₂₃	—	4	2	no	—
203	D₄×C₂₃	—	4	2	no	—
204	D₄×C₂₃	—	8	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	—	4	1	no	—
210	C₂×C₄×C₂₃	—	4	1	no	—
211	C₂×C₄×C₂₃	—	4	1	no	—
212	C₂×C₄×C₂₃	—	4	1	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	4	1	no	—
215	C₂×C₄×C₂₃	—	8	2	no	—
216	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
217	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
218	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₈×C₂₃	—	2	2	no	—
223	C₈×C₂₃	—	2	2	no	—
224	C₈×C₂₃	—	2	2	no	—
225	C₈×C₂₃	—	2	2	no	—
226	C₈×C₂₃	—	2	2	no	—
227	C₈×C₂₃	#135	1	4	yes	—
228	#3231	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#3228	C₂×C₂₃	1	1	yes	characteristic
230	#3185	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#3182	C₂×C₂₃	1	1	yes	characteristic
232	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#3131	C₂×C₂₃	1	1	yes	characteristic
234	#1723	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#1722	C₂×C₂₃	1	1	yes	characteristic, complemented
236	#1720	C₂×C₂₃	1	1	yes	characteristic, complemented
237	#1718	C₂×C₂₃	1	1	yes	characteristic, complemented
238	#1711	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#1710	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#1708	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
242	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂²×C₂₃	—	2	1	no	—
244	C₂²×C₂₃	—	2	1	no	—
245	C₂²×C₂₃	—	2	1	no	—
246	C₂²×C₂₃	—	8	1	no	—
247	C₂²×C₂₃	—	8	1	no	—
248	C₂²×C₂₃	—	8	1	no	—
249	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
250	C₄×C₂₃	—	2	1	no	—
251	C₄×C₂₃	—	2	1	no	—
252	C₄×C₂₃	—	2	1	no	—
253	C₄×C₂₃	—	8	1	no	—
254	C₄×C₂₃	—	8	1	no	—
255	C₄×C₂₃	—	8	1	no	—
256	C₄×C₂₃	—	8	1	no	—
257	C₄×C₂₃	—	8	1	no	—
258	C₄×C₂₃	—	8	1	no	—
259	C₄×C₂₃	—	8	1	no	—
260	C₄×C₂₃	C₂×#135	1	2	yes	—
261	C₄×C₂₃	#575	1	2	yes	—
262	C₄×C₂₃	#578	1	2	yes	—
263	#533	C₂²×C₂₃	1	1	yes	characteristic
264	#533	C₂²×C₂₃	1	1	yes	characteristic
265	#533	C₂²×C₂₃	1	1	yes	characteristic
266	#532	C₂²×C₂₃	1	1	yes	characteristic
267	#500	C₂²×C₂₃	1	1	yes	characteristic
268	#500	C₂²×C₂₃	1	1	yes	characteristic
269	#500	C₂²×C₂₃	1	1	yes	characteristic
270	#500	C₂²×C₂₃	1	1	yes	characteristic
271	#498	C₂²×C₂₃	1	1	yes	characteristic
272	#498	C₂²×C₂₃	1	1	yes	characteristic
273	#498	C₂²×C₂₃	1	1	yes	characteristic
274	#494	C₂²×C₂₃	1	1	yes	characteristic
275	#494	C₂²×C₂₃	1	1	yes	characteristic
276	#492	C₂²×C₂₃	1	1	yes	characteristic
277	#488	—	2	2	no	—
278	#487	C₂²×C₂₃	1	1	yes	characteristic
279	#487	C₂²×C₂₃	1	1	yes	characteristic
280	#487	C₂²×C₂₃	1	1	yes	characteristic
281	#487	C₂²×C₂₃	1	1	yes	characteristic
282	#487	C₂²×C₂₃	1	1	yes	characteristic
283	#487	C₂²×C₂₃	1	1	yes	characteristic
284	#487	C₂²×C₂₃	1	1	yes	characteristic
285	#487	C₂²×C₂₃	1	1	yes	characteristic
286	#487	C₂²×C₂₃	1	1	yes	characteristic
287	#486	C₂²×C₂₃	1	1	yes	characteristic
288	#486	C₂²×C₂₃	1	1	yes	characteristic
289	#486	C₂²×C₂₃	1	1	yes	characteristic
290	#486	C₂²×C₂₃	1	1	yes	characteristic
291	#486	C₂²×C₂₃	1	1	yes	characteristic
292	#485	—	2	2	no	—
293	#482	—	2	2	no	—
294	#479	—	2	2	no	—
295	#478	—	2	2	no	—
296	#470	—	2	2	no	—
297	#468	—	2	2	no	—
298	#467	—	2	2	no	—
299	#441	—	2	2	no	—
300	#440	—	2	2	no	—
301	C₄×Q₁₆	—	2	2	no	—
302	#431	C₂²×C₂₃	1	1	yes	characteristic
303	#416	—	2	2	no	—
304	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
306	#322	C₂²×C₂₃	1	1	yes	characteristic
307	#322	C₂²×C₂₃	1	1	yes	characteristic
308	C₈²	C₂²×C₂₃	1	1	yes	characteristic
309	C₈²	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×C₂₃	—	4	1	no	—
311	C₂×C₂₃	—	16	1	no	—
312	C₂×C₂₃	#3147	1	1	yes	characteristic
313	C₂×C₂₃	#3153	1	1	yes	characteristic
314	C₂×C₂₃	#3156	1	1	yes	characteristic
315	C₂×#41	—	2	1	no	—
316	C₂×Q₁₆	—	2	2	no	—
317	C₂×Q₁₆	—	2	2	no	—
318	C₂×SD₁₆	—	2	2	no	—
319	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
320	C₂²×C₈	D₄×C₂₃	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
322	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
323	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
324	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
325	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
326	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
327	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
328	#127	C₂³×C₂₃	1	1	yes	characteristic
329	#125	—	2	1	no	—
330	#125	—	2	1	no	—
331	#125	—	2	1	no	—
332	#124	—	2	1	no	—
333	C₂²⋊Q₈	—	2	1	no	—
334	C₂²⋊Q₈	—	2	1	no	—
335	C₂²⋊Q₈	—	2	1	no	—
336	C₂²⋊Q₈	—	2	1	no	—
337	C₂²⋊Q₈	—	2	1	no	—
338	C₂²⋊Q₈	—	2	1	no	—
339	#121	—	2	1	no	—
340	#121	—	2	1	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×Q₈	—	2	1	no	—
343	C₄×Q₈	—	2	1	no	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	—	2	1	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	—	2	1	no	—
349	#117	—	2	1	no	—
350	#117	—	2	1	no	—
351	#117	—	2	1	no	—
352	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
353	#107	—	2	2	no	—
354	#107	—	2	2	no	—
355	#107	—	2	2	no	—
356	#107	—	2	2	no	—
357	#106	—	2	2	no	—
358	#106	—	2	2	no	—
359	#106	—	2	2	no	—
360	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
361	#103	—	2	2	no	—
362	#103	—	2	2	no	—
363	#103	—	2	2	no	—
364	#103	—	2	2	no	—
365	#103	—	2	2	no	—
366	#103	—	2	2	no	—
367	#103	—	2	2	no	—
368	#103	—	2	2	no	—
369	#103	—	2	2	no	—
370	#102	—	2	2	no	—
371	#102	—	2	2	no	—
372	#102	—	2	2	no	—
373	#102	—	2	2	no	—
374	#102	—	2	2	no	—
375	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
376	#97	D₄×C₂₃	1	2	yes	—
377	C₄×C₈	D₄×C₂₃	1	2	yes	—
378	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
379	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₂₃	#19643	1	1	yes	characteristic, direct product factor, Sylow
385	#41	—	4	4	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	4	1	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
401	Q₁₆	—	4	4	no	—
402	Q₁₆	—	4	4	no	—
403	SD₁₆	—	4	4	no	—
404	M₁₆	—	2	4	no	—
405	C₂×C₈	—	4	2	no	—
406	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
407	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
408	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
409	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
410	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
411	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
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	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	4	1	no	—
427	C₄⋊C₄	—	4	1	no	—
428	C₄⋊C₄	—	4	1	no	—
429	C₄⋊C₄	—	4	1	no	—
430	C₄⋊C₄	—	4	1	no	—
431	C₄⋊C₄	—	4	1	no	—
432	C₂²⋊C₄	—	4	1	no	—
433	C₂²⋊C₄	—	4	1	no	—
434	C₂²⋊C₄	—	4	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₂²⋊C₄	—	4	1	no	—
437	C₄²	—	4	1	no	—
438	C₄²	—	4	1	no	—
439	C₄²	—	4	1	no	—
440	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
444	C₂³	—	4	1	no	—
445	C₂³	C₂₃×#127	1	1	yes	characteristic
446	Q₈	—	4	2	no	—
447	Q₈	—	4	2	no	—
448	Q₈	—	4	2	no	—
449	Q₈	—	4	2	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	D₄	—	4	2	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	8	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	8	2	no	—
468	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
469	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
470	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	2	no	—
476	C₈	—	2	2	no	—
477	C₈	—	2	2	no	—
478	C₈	—	2	2	no	—
479	C₈	C₂₃×#135	1	4	yes	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	8	1	no	—
484	C₂²	—	8	1	no	—
485	C₂²	—	8	1	no	—
486	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	8	1	no	—
491	C₄	—	8	1	no	—
492	C₄	—	8	1	no	—
493	C₄	—	8	1	no	—
494	C₄	—	8	1	no	—
495	C₄	—	8	1	no	—
496	C₄	—	8	1	no	—
497	C₄	C₂×C₂₃×#135	1	2	yes	—
498	C₄	C₂₃×#575	1	2	yes	—
499	C₄	C₂₃×#578	1	2	yes	—
500	C₂	—	4	1	no	—
501	C₂	—	16	1	no	—
502	C₂	C₂₃×#3147	1	1	yes	characteristic
503	C₂	C₂₃×#3153	1	1	yes	characteristic
504	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
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×#3231	C₂	semidirect
3. C₂₃×#3185	C₂	semidirect
5. C₂₃×#3170	C₂	semidirect
7. C₂₃×#1723	C₂	semidirect
8. C₂₃×#1722	C₂	semidirect
9. C₂₃×#1720	C₂	semidirect
10. C₂₃×#1718	C₂	semidirect
11. C₂₃×#1711	C₂	semidirect
12. C₂₃×#1710	C₂	semidirect
13. C₂₃×#1708	C₂	semidirect
14. C₂₃×#1707	C₂	semidirect
15. C₈×M₁₆×C₂₃	C₂	semidirect
191. #19643	C₂₃	direct (N × H)
228. #3231	C₂×C₂₃	semidirect
230. #3185	C₂×C₂₃	semidirect
232. #3170	C₂×C₂₃	semidirect
234. #1723	C₂×C₂₃	semidirect
235. #1722	C₂×C₂₃	semidirect
236. #1720	C₂×C₂₃	semidirect
237. #1718	C₂×C₂₃	semidirect
238. #1711	C₂×C₂₃	semidirect
239. #1710	C₂×C₂₃	semidirect
240. #1708	C₂×C₂₃	semidirect
241. #1707	C₂×C₂₃	semidirect
242. C₈×M₁₆	C₂×C₂₃	semidirect
384. C₂₃	#19643	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×#3231	2
C₂₃×#3228	2
C₂₃×#3185	2
C₂₃×#3182	2
C₂₃×#3170	2
C₂₃×#3131	2
C₂₃×#1723	2
C₂₃×#1722	2
C₂₃×#1720	2
C₂₃×#1718	2
C₂₃×#1711	2
C₂₃×#1710	2
C₂₃×#1708	2
C₂₃×#1707	2
C₈×M₁₆×C₂₃	2
#19643	23

Subgroups — C23×#19643 #19167269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19643 #19167269