Subgroups — C₂₃×#19748 #19167374

Order 5888.

792 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3184	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
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×#1719	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1715	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1714	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	—	2	2	no	—
17	C₂₃×#533	C₂²	1	1	yes	characteristic
18	C₂₃×#532	C₂²	1	1	yes	characteristic
19	C₂₃×#532	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂₃×#501	—	2	2	no	—
22	C₂₃×#500	C₂²	1	1	yes	characteristic
23	C₂₃×#500	C₂²	1	1	yes	characteristic
24	C₂₃×#500	C₂²	1	1	yes	characteristic
25	C₂₃×#500	C₂²	1	1	yes	characteristic
26	C₂₃×#498	C₂²	1	1	yes	characteristic
27	C₂₃×#498	C₂²	1	1	yes	characteristic
28	C₂₃×#498	C₂²	1	1	yes	characteristic
29	C₂₃×#497	—	2	2	no	—
30	C₂₃×#494	C₂²	1	1	yes	characteristic
31	C₂₃×#494	C₂²	1	1	yes	characteristic
32	C₂₃×#493	C₂²	1	1	yes	characteristic
33	C₂₃×#492	C₂²	1	1	yes	characteristic
34	C₂₃×#492	C₂²	1	1	yes	characteristic
35	C₂₃×#492	C₂²	1	1	yes	characteristic
36	C₂₃×#492	C₂²	1	1	yes	characteristic
37	C₂₃×#489	—	2	2	no	—
38	C₂₃×#487	C₂²	1	1	yes	characteristic
39	C₂₃×#487	C₂²	1	1	yes	characteristic
40	C₂₃×#487	C₂²	1	1	yes	characteristic
41	C₂₃×#487	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×#486	C₂²	1	1	yes	characteristic
46	C₂₃×#486	C₂²	1	1	yes	characteristic
47	C₂₃×#486	C₂²	1	1	yes	characteristic
48	C₂₃×#469	—	2	2	no	—
49	C₂₃×#468	—	2	2	no	—
50	C₂₃×#468	—	2	2	no	—
51	C₂₃×#442	—	2	2	no	—
52	C₂₃×#440	—	2	2	no	—
53	C₂₃×#440	—	2	2	no	—
54	C₂₃×#428	—	2	2	no	—
55	C₂₃×#417	—	2	2	no	—
56	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
57	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#322	C₂²	1	1	yes	characteristic
60	C₂₃×#322	C₂²	1	1	yes	characteristic
61	C₂₃×#322	C₂²	1	1	yes	characteristic
62	C₈²×C₂₃	C₂²	1	1	yes	characteristic
63	C₂×#41×C₂₃	—	2	1	no	—
64	C₂×#41×C₂₃	—	2	1	no	—
65	C₂²×Q₈×C₂₃	—	2	1	no	—
66	C₂₃×#137	—	4	8	no	—
67	C₂×Q₁₆×C₂₃	—	2	2	no	—
68	C₂×Q₁₆×C₂₃	—	2	2	no	—
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	2	2	no	—
72	C₂×SD₁₆×C₂₃	—	2	2	no	—
73	C₂×D₈×C₂₃	—	2	2	no	—
74	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
75	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
76	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
77	C₂₃×C₄⋊Q₈	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×#127	C₂³	1	1	yes	characteristic
87	C₂₃×#127	C₂³	1	1	yes	characteristic
88	C₂₃×#125	—	2	1	no	—
89	C₂₃×#124	—	2	1	no	—
90	C₂₃×#124	—	2	1	no	—
91	C₂₃×C₂²⋊Q₈	—	2	1	no	—
92	C₂₃×C₂²⋊Q₈	—	2	1	no	—
93	C₂₃×#121	—	2	1	no	—
94	C₂₃×#121	—	2	1	no	—
95	C₂₃×#121	—	2	1	no	—
96	C₂₃×#121	—	2	1	no	—
97	C₄×Q₈×C₂₃	—	2	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₂₃×#117	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂₃×#107	—	2	2	no	—
108	C₂₃×#107	—	2	2	no	—
109	C₂₃×#107	—	2	2	no	—
110	C₂₃×#107	—	2	2	no	—
111	C₂₃×#106	—	2	2	no	—
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×#106	—	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×#102	—	2	2	no	—
124	C₂₃×#97	D₄	1	2	yes	—
125	C₂₃×#97	D₄	1	2	yes	—
126	C₂₃×#97	D₄	1	2	yes	—
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	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
133	#41×C₂₃	—	4	4	no	—
134	#41×C₂₃	—	4	4	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×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	4	4	no	—
142	C₂×Q₈×C₂₃	—	4	1	no	—
143	C₂×D₄×C₂₃	—	2	1	no	—
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	—	2	1	no	—
146	C₂×D₄×C₂₃	—	2	1	no	—
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	4	1	no	—
150	C₂×D₄×C₂₃	—	4	1	no	—
151	C₂²×C₄×C₂₃	—	4	1	no	—
152	C₂²×C₄×C₂₃	—	4	1	no	—
153	C₂²×C₄×C₂₃	—	4	1	no	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	Q₁₆×C₂₃	—	4	4	no	—
158	Q₁₆×C₂₃	—	4	4	no	—
159	SD₁₆×C₂₃	—	4	4	no	—
160	SD₁₆×C₂₃	—	4	4	no	—
161	SD₁₆×C₂₃	—	4	4	no	—
162	SD₁₆×C₂₃	—	4	4	no	—
163	D₈×C₂₃	—	4	4	no	—
164	M₁₆×C₂₃	—	2	4	no	—
165	M₁₆×C₂₃	—	2	4	no	—
166	M₁₆×C₂₃	—	4	2	no	—
167	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	4	1	no	—
186	C₄⋊C₄×C₂₃	—	4	1	no	—
187	C₄⋊C₄×C₂₃	—	4	1	no	—
188	C₄⋊C₄×C₂₃	—	4	1	no	—
189	C₂²⋊C₄×C₂₃	—	4	1	no	—
190	C₂²⋊C₄×C₂₃	—	4	1	no	—
191	C₂²⋊C₄×C₂₃	—	4	1	no	—
192	C₂²⋊C₄×C₂₃	—	4	1	no	—
193	C₂²⋊C₄×C₂₃	—	4	1	no	—
194	C₄²×C₂₃	—	4	1	no	—
195	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
199	#19748	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₂₃	—	4	1	no	—
201	C₂³×C₂₃	—	4	1	no	—
202	C₂³×C₂₃	#127	1	1	yes	characteristic
203	Q₈×C₂₃	—	4	2	no	—
204	Q₈×C₂₃	—	4	2	no	—
205	Q₈×C₂₃	—	4	2	no	—
206	Q₈×C₂₃	—	4	2	no	—
207	Q₈×C₂₃	—	4	2	no	—
208	Q₈×C₂₃	—	4	2	no	—
209	Q₈×C₂₃	—	8	2	no	—
210	D₄×C₂₃	—	4	2	no	—
211	D₄×C₂₃	—	4	2	no	—
212	D₄×C₂₃	—	4	2	no	—
213	D₄×C₂₃	—	4	2	no	—
214	D₄×C₂₃	—	4	2	no	—
215	D₄×C₂₃	—	4	2	no	—
216	D₄×C₂₃	—	8	2	no	—
217	D₄×C₂₃	—	8	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	4	1	no	—
222	C₂×C₄×C₂₃	—	4	1	no	—
223	C₂×C₄×C₂₃	—	4	1	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	8	2	no	—
228	C₂×C₄×C₂₃	—	8	2	no	—
229	C₂×C₄×C₂₃	—	8	2	no	—
230	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
231	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
232	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
236	C₈×C₂₃	—	2	2	no	—
237	C₈×C₂₃	—	2	2	no	—
238	C₈×C₂₃	—	2	2	no	—
239	C₈×C₂₃	—	2	2	no	—
240	C₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
241	C₈×C₂₃	#137	1	4	yes	complemented
242	#3184	C₂×C₂₃	1	1	yes	characteristic
243	#3182	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#3182	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#3165	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#3141	C₂×C₂₃	1	1	yes	characteristic, complemented
248	#1723	C₂×C₂₃	1	1	yes	characteristic, complemented
249	#1722	C₂×C₂₃	1	1	yes	characteristic, complemented
250	#1720	C₂×C₂₃	1	1	yes	characteristic, complemented
251	#1719	C₂×C₂₃	1	1	yes	characteristic, complemented
252	#1715	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#1714	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
255	#1706	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#1455	C₂×C₂₃	1	1	yes	characteristic, complemented
257	C₂²×C₂₃	—	2	1	no	—
258	C₂²×C₂₃	—	2	1	no	—
259	C₂²×C₂₃	—	2	1	no	—
260	C₂²×C₂₃	—	8	1	no	—
261	C₂²×C₂₃	—	8	1	no	—
262	C₂²×C₂₃	—	8	1	no	—
263	C₂²×C₂₃	—	8	1	no	—
264	C₂²×C₂₃	—	8	1	no	—
265	C₂²×C₂₃	—	8	1	no	—
266	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
267	C₄×C₂₃	—	2	1	no	—
268	C₄×C₂₃	—	2	1	no	—
269	C₄×C₂₃	—	2	1	no	—
270	C₄×C₂₃	—	8	1	no	—
271	C₄×C₂₃	—	8	1	no	—
272	C₄×C₂₃	—	8	1	no	—
273	C₄×C₂₃	—	8	1	no	—
274	C₄×C₂₃	—	8	1	no	—
275	C₄×C₂₃	—	8	1	no	—
276	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
277	C₄×C₂₃	C₂×#137	1	2	yes	—
278	C₄×C₂₃	#575	1	2	yes	—
279	C₂×#137	—	2	2	no	—
280	#533	C₂²×C₂₃	1	1	yes	characteristic
281	#532	C₂²×C₂₃	1	1	yes	characteristic
282	#532	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
284	#501	—	2	2	no	—
285	#500	C₂²×C₂₃	1	1	yes	characteristic
286	#500	C₂²×C₂₃	1	1	yes	characteristic
287	#500	C₂²×C₂₃	1	1	yes	characteristic
288	#500	C₂²×C₂₃	1	1	yes	characteristic
289	#498	C₂²×C₂₃	1	1	yes	characteristic
290	#498	C₂²×C₂₃	1	1	yes	characteristic
291	#498	C₂²×C₂₃	1	1	yes	characteristic
292	#497	—	2	2	no	—
293	#494	C₂²×C₂₃	1	1	yes	characteristic
294	#494	C₂²×C₂₃	1	1	yes	characteristic
295	#493	C₂²×C₂₃	1	1	yes	characteristic
296	#492	C₂²×C₂₃	1	1	yes	characteristic
297	#492	C₂²×C₂₃	1	1	yes	characteristic
298	#492	C₂²×C₂₃	1	1	yes	characteristic
299	#492	C₂²×C₂₃	1	1	yes	characteristic
300	#489	—	2	2	no	—
301	#487	C₂²×C₂₃	1	1	yes	characteristic
302	#487	C₂²×C₂₃	1	1	yes	characteristic
303	#487	C₂²×C₂₃	1	1	yes	characteristic
304	#487	C₂²×C₂₃	1	1	yes	characteristic
305	#486	C₂²×C₂₃	1	1	yes	characteristic
306	#486	C₂²×C₂₃	1	1	yes	characteristic
307	#486	C₂²×C₂₃	1	1	yes	characteristic
308	#486	C₂²×C₂₃	1	1	yes	characteristic
309	#486	C₂²×C₂₃	1	1	yes	characteristic
310	#486	C₂²×C₂₃	1	1	yes	characteristic
311	#469	—	2	2	no	—
312	#468	—	2	2	no	—
313	#468	—	2	2	no	—
314	#442	—	2	2	no	—
315	#440	—	2	2	no	—
316	#440	—	2	2	no	—
317	#428	—	2	2	no	—
318	#417	—	2	2	no	—
319	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
320	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
321	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
322	#322	C₂²×C₂₃	1	1	yes	characteristic
323	#322	C₂²×C₂₃	1	1	yes	characteristic
324	#322	C₂²×C₂₃	1	1	yes	characteristic
325	C₈²	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×C₂₃	—	4	1	no	—
327	C₂×C₂₃	—	16	1	no	—
328	C₂×C₂₃	—	16	1	no	—
329	C₂×C₂₃	#3151	1	1	yes	characteristic
330	C₂×C₂₃	#3152	1	1	yes	characteristic
331	C₂×C₂₃	#3153	1	1	yes	characteristic
332	C₂×#41	—	2	1	no	—
333	C₂×#41	—	2	1	no	—
334	C₂²×Q₈	—	2	1	no	—
335	#137	—	4	8	no	—
336	C₂×Q₁₆	—	2	2	no	—
337	C₂×Q₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×D₈	—	2	2	no	—
343	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
344	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
345	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
346	C₄⋊Q₈	—	2	1	no	—
347	C₄⋊Q₈	—	2	1	no	—
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
355	#127	C₂³×C₂₃	1	1	yes	characteristic
356	#127	C₂³×C₂₃	1	1	yes	characteristic
357	#125	—	2	1	no	—
358	#124	—	2	1	no	—
359	#124	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	#121	—	2	1	no	—
363	#121	—	2	1	no	—
364	#121	—	2	1	no	—
365	#121	—	2	1	no	—
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	#117	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
376	#107	—	2	2	no	—
377	#107	—	2	2	no	—
378	#107	—	2	2	no	—
379	#107	—	2	2	no	—
380	#106	—	2	2	no	—
381	#106	—	2	2	no	—
382	#106	—	2	2	no	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#97	D₄×C₂₃	1	2	yes	—
394	#97	D₄×C₂₃	1	2	yes	—
395	#97	D₄×C₂₃	1	2	yes	—
396	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₂₃	#19748	1	1	yes	characteristic, direct product factor, Sylow
403	#41	—	4	4	no	—
404	#41	—	4	4	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	4	4	no	—
412	C₂×Q₈	—	4	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	4	1	no	—
420	C₂×D₄	—	4	1	no	—
421	C₂²×C₄	—	4	1	no	—
422	C₂²×C₄	—	4	1	no	—
423	C₂²×C₄	—	4	1	no	—
424	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
427	Q₁₆	—	4	4	no	—
428	Q₁₆	—	4	4	no	—
429	SD₁₆	—	4	4	no	—
430	SD₁₆	—	4	4	no	—
431	SD₁₆	—	4	4	no	—
432	SD₁₆	—	4	4	no	—
433	D₈	—	4	4	no	—
434	M₁₆	—	2	4	no	—
435	M₁₆	—	2	4	no	—
436	M₁₆	—	4	2	no	—
437	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
438	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
439	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
440	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
441	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
442	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	4	1	no	—
456	C₄⋊C₄	—	4	1	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₄⋊C₄	—	4	1	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₂²⋊C₄	—	4	1	no	—
461	C₂²⋊C₄	—	4	1	no	—
462	C₂²⋊C₄	—	4	1	no	—
463	C₂²⋊C₄	—	4	1	no	—
464	C₄²	—	4	1	no	—
465	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
467	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
468	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
469	C₂³	—	4	1	no	—
470	C₂³	—	4	1	no	—
471	C₂³	C₂₃×#127	1	1	yes	characteristic
472	Q₈	—	4	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	2	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	8	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	2	no	—
484	D₄	—	4	2	no	—
485	D₄	—	8	2	no	—
486	D₄	—	8	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	8	2	no	—
497	C₂×C₄	—	8	2	no	—
498	C₂×C₄	—	8	2	no	—
499	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
500	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
501	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
505	C₈	—	2	2	no	—
506	C₈	—	2	2	no	—
507	C₈	—	2	2	no	—
508	C₈	—	2	2	no	—
509	C₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
510	C₈	C₂₃×#137	1	4	yes	complemented
511	C₂²	—	2	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	8	1	no	—
515	C₂²	—	8	1	no	—
516	C₂²	—	8	1	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	—	8	1	no	—
520	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	1	no	—
524	C₄	—	8	1	no	—
525	C₄	—	8	1	no	—
526	C₄	—	8	1	no	—
527	C₄	—	8	1	no	—
528	C₄	—	8	1	no	—
529	C₄	—	8	1	no	—
530	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
531	C₄	C₂×C₂₃×#137	1	2	yes	—
532	C₄	C₂₃×#575	1	2	yes	—
533	C₂	—	4	1	no	—
534	C₂	—	16	1	no	—
535	C₂	—	16	1	no	—
536	C₂	C₂₃×#3151	1	1	yes	characteristic
537	C₂	C₂₃×#3152	1	1	yes	characteristic
538	C₂	C₂₃×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3184	C₂
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×#3182	C₂	semidirect
3. C₂₃×#3182	C₂	semidirect
4. C₂₃×#3165	C₂	semidirect
5. C₂₃×#3152	C₂	semidirect
6. C₂₃×#3141	C₂	semidirect
7. C₂₃×#1723	C₂	semidirect
8. C₂₃×#1722	C₂	semidirect
9. C₂₃×#1720	C₂	semidirect
10. C₂₃×#1719	C₂	semidirect
11. C₂₃×#1715	C₂	semidirect
12. C₂₃×#1714	C₂	semidirect
13. C₂₃×#1707	C₂	semidirect
14. C₂₃×#1706	C₂	semidirect
15. C₂₃×#1455	C₂	semidirect
199. #19748	C₂₃	direct (N × H)
241. C₈×C₂₃	#137	semidirect
243. #3182	C₂×C₂₃	semidirect
244. #3182	C₂×C₂₃	semidirect
245. #3165	C₂×C₂₃	semidirect
246. #3152	C₂×C₂₃	semidirect
247. #3141	C₂×C₂₃	semidirect
248. #1723	C₂×C₂₃	semidirect
249. #1722	C₂×C₂₃	semidirect
250. #1720	C₂×C₂₃	semidirect
251. #1719	C₂×C₂₃	semidirect
252. #1715	C₂×C₂₃	semidirect
253. #1714	C₂×C₂₃	semidirect
254. #1707	C₂×C₂₃	semidirect
255. #1706	C₂×C₂₃	semidirect
256. #1455	C₂×C₂₃	semidirect
402. C₂₃	#19748	direct (N × H)
510. C₈	C₂₃×#137	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3184	2
C₂₃×#3182	2
C₂₃×#3165	2
C₂₃×#3152	2
C₂₃×#3141	2
C₂₃×#1723	2
C₂₃×#1722	2
C₂₃×#1720	2
C₂₃×#1719	2
C₂₃×#1715	2
C₂₃×#1714	2
C₂₃×#1707	2
C₂₃×#1706	2
C₂₃×#1455	2
#19748	23

Subgroups — C23×#19748 #19167374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19748 #19167374