Subgroups — C₂₃×#18744 #19166370

Order 5888.

852 conjugacy classes of subgroups.

The subgroup lattice has 718 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, complemented
2	C₂₃×#3246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3212	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1650	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1643	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1542	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×#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×#532	C₂²	1	1	yes	characteristic
33	C₂₃×#529	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#495	—	2	1	no	—
36	C₂₃×#492	C₂²	1	1	yes	characteristic
37	C₂₃×#488	—	2	1	no	—
38	C₂₃×#487	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×#468	—	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×#440	—	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×#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×#41×C₂₃	—	2	1	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	2	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×D₈×C₂₃	—	2	1	no	—
79	C₂×M₁₆×C₂₃	—	2	1	no	—
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	2	no	—
82	C₂₃×C₄⋊Q₈	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	—	2	2	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×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×#127	C₂³	1	1	yes	characteristic
92	C₂₃×#126	—	2	2	no	—
93	C₂₃×#126	—	2	2	no	—
94	C₂₃×#126	—	2	2	no	—
95	C₂₃×#126	—	2	2	no	—
96	C₂₃×#126	—	2	2	no	—
97	C₂₃×#126	—	2	2	no	—
98	C₂₃×#126	—	2	1	no	—
99	C₂₃×#125	—	2	2	no	—
100	C₂₃×#125	—	2	2	no	—
101	C₂₃×#125	—	2	2	no	—
102	C₂₃×#125	—	2	2	no	—
103	C₂₃×#125	—	2	1	no	—
104	C₂₃×#125	—	2	1	no	—
105	C₂₃×#125	—	2	2	no	—
106	C₂₃×#125	—	2	1	no	—
107	C₂₃×#125	—	2	1	no	—
108	C₂₃×#125	D₄	1	1	yes	characteristic
109	C₂₃×#124	—	2	2	no	—
110	C₂₃×#124	—	2	1	no	—
111	C₂₃×#124	—	2	1	no	—
112	C₂₃×#124	D₄	1	1	yes	characteristic
113	C₂₃×#123	—	2	2	no	—
114	C₂₃×#123	—	2	1	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	—	2	1	no	—
121	C₂₃×C₂²⋊Q₈	—	2	1	no	—
122	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
123	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
124	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
125	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
126	C₂₃×#121	—	2	1	no	—
127	C₂₃×#121	—	2	1	no	—
128	C₂₃×#121	D₄	1	1	yes	characteristic
129	C₄×Q₈×C₂₃	—	2	2	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	—	2	1	no	—
136	C₄×Q₈×C₂₃	—	2	1	no	—
137	C₄×Q₈×C₂₃	D₄	1	2	yes	—
138	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
139	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
140	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
146	C₂₃×#117	—	2	2	no	—
147	C₂₃×#117	—	2	1	no	—
148	C₂₃×#117	—	2	1	no	—
149	C₂₃×#117	—	2	1	no	—
150	C₂₃×#117	—	2	1	no	—
151	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
153	C₂₃×#107	—	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	2	no	—
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	1	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	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	C₂×Q₈×C₂₃	—	2	2	no	—
179	C₂×Q₈×C₂₃	—	2	1	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
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	1	yes	characteristic
185	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
186	C₂×D₄×C₂₃	—	2	1	no	—
187	C₂×D₄×C₂₃	—	2	1	no	—
188	C₂×D₄×C₂₃	—	2	1	no	—
189	C₂×D₄×C₂₃	—	2	1	no	—
190	C₂×D₄×C₂₃	—	4	1	no	—
191	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	—	2	2	no	—
193	C₂²×C₄×C₂₃	—	2	1	no	—
194	C₂²×C₄×C₂₃	—	4	1	no	—
195	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
198	Q₁₆×C₂₃	—	8	1	no	—
199	SD₁₆×C₂₃	—	4	4	no	—
200	SD₁₆×C₂₃	—	8	1	no	—
201	SD₁₆×C₂₃	—	8	1	no	—
202	SD₁₆×C₂₃	—	8	1	no	—
203	SD₁₆×C₂₃	—	8	1	no	—
204	D₈×C₂₃	—	8	1	no	—
205	M₁₆×C₂₃	—	4	2	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	2	1	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	2	1	no	—
211	C₂×C₈×C₂₃	—	2	1	no	—
212	C₂×C₈×C₂₃	—	8	1	no	—
213	C₄⋊C₄×C₂₃	—	2	2	no	—
214	C₄⋊C₄×C₂₃	—	2	2	no	—
215	C₄⋊C₄×C₂₃	—	2	2	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	2	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	1	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	2	no	—
229	C₄⋊C₄×C₂₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
240	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
241	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
244	C₂²⋊C₄×C₂₃	—	2	2	no	—
245	C₂²⋊C₄×C₂₃	—	2	2	no	—
246	C₂²⋊C₄×C₂₃	—	2	2	no	—
247	C₂²⋊C₄×C₂₃	—	2	1	no	—
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	no	—
250	C₂²⋊C₄×C₂₃	—	2	1	no	—
251	C₂²⋊C₄×C₂₃	—	2	1	no	—
252	C₂²⋊C₄×C₂₃	—	2	1	no	—
253	C₂²⋊C₄×C₂₃	—	2	1	no	—
254	C₂²⋊C₄×C₂₃	—	2	1	no	—
255	C₂²⋊C₄×C₂₃	—	4	1	no	—
256	C₂²⋊C₄×C₂₃	—	4	1	no	—
257	C₄²×C₂₃	—	2	2	no	—
258	C₄²×C₂₃	—	2	2	no	—
259	C₄²×C₂₃	—	2	2	no	—
260	C₄²×C₂₃	—	2	2	no	—
261	C₄²×C₂₃	—	2	1	no	—
262	C₄²×C₂₃	—	2	1	no	—
263	C₄²×C₂₃	—	2	1	no	—
264	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
268	#18744	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂³×C₂₃	—	2	1	no	—
270	C₂³×C₂₃	—	4	1	no	—
271	C₂³×C₂₃	#120	1	1	yes	characteristic
272	Q₈×C₂₃	—	2	1	no	—
273	Q₈×C₂₃	—	2	1	no	—
274	Q₈×C₂₃	—	2	1	no	—
275	Q₈×C₂₃	—	2	1	no	—
276	Q₈×C₂₃	—	2	1	no	—
277	Q₈×C₂₃	—	2	1	no	—
278	Q₈×C₂₃	—	4	2	no	—
279	Q₈×C₂₃	—	4	1	no	—
280	Q₈×C₂₃	—	4	1	no	—
281	Q₈×C₂₃	—	4	1	no	—
282	Q₈×C₂₃	—	4	1	no	—
283	Q₈×C₂₃	—	8	1	no	—
284	Q₈×C₂₃	—	8	2	no	—
285	D₄×C₂₃	—	2	1	no	—
286	D₄×C₂₃	—	2	1	no	—
287	D₄×C₂₃	—	4	2	no	—
288	D₄×C₂₃	—	4	1	no	—
289	D₄×C₂₃	—	4	1	no	—
290	D₄×C₂₃	—	4	1	no	—
291	D₄×C₂₃	—	4	1	no	—
292	D₄×C₂₃	—	8	1	no	—
293	D₄×C₂₃	—	8	2	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	1	no	—
296	C₂×C₄×C₂₃	—	2	1	no	—
297	C₂×C₄×C₂₃	—	2	1	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	1	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	2	2	no	—
302	C₂×C₄×C₂₃	—	2	1	no	—
303	C₂×C₄×C₂₃	—	2	1	no	—
304	C₂×C₄×C₂₃	—	2	1	no	—
305	C₂×C₄×C₂₃	—	2	2	no	—
306	C₂×C₄×C₂₃	—	2	1	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	4	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	8	2	no	—
312	C₂×C₄×C₂₃	—	8	2	no	—
313	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
314	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
315	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
316	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
319	C₈×C₂₃	—	2	2	no	—
320	C₈×C₂₃	—	4	1	no	—
321	C₈×C₂₃	—	4	1	no	—
322	C₈×C₂₃	—	4	1	no	—
323	C₈×C₂₃	—	4	1	no	—
324	C₈×C₂₃	—	4	1	no	—
325	#3555	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#3246	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#3212	C₂×C₂₃	1	1	yes	characteristic
328	#3182	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1650	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1643	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1628	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1542	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1485	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
340	C₂²×C₂₃	—	2	1	no	—
341	C₂²×C₂₃	—	2	1	no	—
342	C₂²×C₂₃	—	2	1	no	—
343	C₂²×C₂₃	—	4	1	no	—
344	C₂²×C₂₃	—	4	1	no	—
345	C₂²×C₂₃	—	4	1	no	—
346	C₂²×C₂₃	—	8	1	no	—
347	C₂²×C₂₃	—	8	1	no	—
348	C₂²×C₂₃	—	8	1	no	—
349	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
350	C₄×C₂₃	—	2	1	no	—
351	C₄×C₂₃	—	2	1	no	—
352	C₄×C₂₃	—	2	1	no	—
353	C₄×C₂₃	—	2	1	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	4	2	no	—
356	C₄×C₂₃	—	4	1	no	—
357	C₄×C₂₃	—	4	1	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	4	1	no	—
360	C₄×C₂₃	—	4	2	no	—
361	C₄×C₂₃	—	4	1	no	—
362	C₄×C₂₃	—	4	1	no	—
363	C₄×C₂₃	—	8	1	no	—
364	C₄×C₂₃	#577	1	1	yes	characteristic
365	C₄×C₂₃	#577	1	1	yes	characteristic
366	C₄×C₂₃	#577	1	1	yes	characteristic
367	C₄×C₂₃	#578	1	1	yes	characteristic
368	#563	—	2	2	no	—
369	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
370	#557	—	2	1	no	—
371	#557	C₂²×C₂₃	1	1	yes	characteristic
372	#556	—	2	1	no	—
373	#556	—	2	1	no	—
374	#556	C₂²×C₂₃	1	1	yes	characteristic
375	#554	C₂²×C₂₃	1	1	yes	characteristic
376	#552	—	2	1	no	—
377	#544	—	2	1	no	—
378	#542	—	2	1	no	—
379	#541	—	2	1	no	—
380	#541	—	2	1	no	—
381	#541	—	2	2	no	—
382	#533	C₂²×C₂₃	1	1	yes	characteristic
383	#533	C₂²×C₂₃	1	1	yes	characteristic
384	#532	C₂²×C₂₃	1	1	yes	characteristic
385	#529	C₂²×C₂₃	1	1	yes	characteristic
386	#500	C₂²×C₂₃	1	1	yes	characteristic
387	#495	—	2	1	no	—
388	#492	C₂²×C₂₃	1	1	yes	characteristic
389	#488	—	2	1	no	—
390	#487	C₂²×C₂₃	1	1	yes	characteristic
391	#486	C₂²×C₂₃	1	1	yes	characteristic
392	#476	—	2	1	no	—
393	#475	C₂²×C₂₃	1	1	yes	characteristic
394	#475	C₂²×C₂₃	1	1	yes	characteristic
395	#468	—	2	2	no	—
396	#465	—	2	1	no	—
397	#465	—	2	1	no	—
398	#462	C₂²×C₂₃	1	1	yes	characteristic
399	#461	—	2	1	no	—
400	#461	C₂²×C₂₃	1	1	yes	characteristic
401	#460	C₂²×C₂₃	1	1	yes	characteristic
402	#460	C₂²×C₂₃	1	1	yes	characteristic
403	#460	C₂²×C₂₃	1	1	yes	characteristic
404	#459	C₂²×C₂₃	1	1	yes	characteristic
405	#452	—	2	1	no	—
406	#449	—	2	1	no	—
407	#440	—	2	2	no	—
408	#431	C₂²×C₂₃	1	1	yes	characteristic
409	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
410	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
411	#333	C₂²×C₂₃	1	1	yes	characteristic
412	#333	C₂²×C₂₃	1	1	yes	characteristic
413	#332	C₂²×C₂₃	1	1	yes	characteristic
414	#332	C₂²×C₂₃	1	1	yes	characteristic
415	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
416	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
417	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
418	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
419	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
420	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
421	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
422	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	C₂×C₂₃	—	4	1	no	—
424	C₂×C₂₃	—	8	1	no	—
425	C₂×C₂₃	—	16	1	no	—
426	C₂×C₂₃	#3010	1	1	yes	characteristic
427	C₂×C₂₃	#3013	1	1	yes	characteristic
428	C₂×C₂₃	#3018	1	1	yes	characteristic
429	C₂×#41	—	2	1	no	—
430	C₂×Q₁₆	—	2	1	no	—
431	C₂×SD₁₆	—	2	2	no	—
432	C₂×SD₁₆	—	2	1	no	—
433	C₂×SD₁₆	—	2	1	no	—
434	C₂×SD₁₆	—	2	1	no	—
435	C₂×SD₁₆	—	2	1	no	—
436	C₂×D₈	—	2	1	no	—
437	C₂×M₁₆	—	2	1	no	—
438	C₂²×C₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	2	no	—
440	C₄⋊Q₈	—	2	1	no	—
441	C₄⋊Q₈	—	2	2	no	—
442	C₄⋊Q₈	—	2	1	no	—
443	C₄⋊Q₈	—	2	1	no	—
444	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
445	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
446	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
447	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
449	#127	C₂³×C₂₃	1	1	yes	characteristic
450	#126	—	2	2	no	—
451	#126	—	2	2	no	—
452	#126	—	2	2	no	—
453	#126	—	2	2	no	—
454	#126	—	2	2	no	—
455	#126	—	2	2	no	—
456	#126	—	2	1	no	—
457	#125	—	2	2	no	—
458	#125	—	2	2	no	—
459	#125	—	2	2	no	—
460	#125	—	2	2	no	—
461	#125	—	2	1	no	—
462	#125	—	2	2	no	—
463	#125	—	2	1	no	—
464	#125	—	2	1	no	—
465	#125	—	2	1	no	—
466	#125	D₄×C₂₃	1	1	yes	characteristic
467	#124	—	2	2	no	—
468	#124	—	2	1	no	—
469	#124	—	2	1	no	—
470	#124	D₄×C₂₃	1	1	yes	characteristic
471	#123	—	2	2	no	—
472	#123	—	2	1	no	—
473	C₂²⋊Q₈	—	2	2	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	1	no	—
476	C₂²⋊Q₈	—	2	1	no	—
477	C₂²⋊Q₈	—	2	2	no	—
478	C₂²⋊Q₈	—	2	1	no	—
479	C₂²⋊Q₈	—	2	1	no	—
480	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
481	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
482	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
483	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
484	#121	—	2	1	no	—
485	#121	—	2	1	no	—
486	#121	D₄×C₂₃	1	1	yes	characteristic
487	C₄×Q₈	—	2	2	no	—
488	C₄×Q₈	—	2	1	no	—
489	C₄×Q₈	—	2	1	no	—
490	C₄×Q₈	—	2	1	no	—
491	C₄×Q₈	—	2	1	no	—
492	C₄×Q₈	—	2	1	no	—
493	C₄×Q₈	—	2	1	no	—
494	C₄×Q₈	—	2	1	no	—
495	C₄×Q₈	D₄×C₂₃	1	2	yes	—
496	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
497	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
498	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
499	C₄×D₄	—	2	1	no	—
500	C₄×D₄	—	2	1	no	—
501	C₄×D₄	—	2	1	no	—
502	C₄×D₄	—	2	1	no	—
503	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
504	#117	—	2	2	no	—
505	#117	—	2	1	no	—
506	#117	—	2	1	no	—
507	#117	—	2	1	no	—
508	#117	—	2	1	no	—
509	C₂×C₄⋊C₄	—	2	1	no	—
510	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
511	#107	—	2	2	no	—
512	#106	—	2	1	no	—
513	#106	—	2	1	no	—
514	C₄⋊C₈	—	2	1	no	—
515	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
516	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
517	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
518	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
519	#103	—	2	2	no	—
520	#103	—	2	1	no	—
521	#103	—	2	1	no	—
522	#103	—	2	1	no	—
523	#103	—	2	1	no	—
524	#102	—	2	2	no	—
525	#102	—	2	1	no	—
526	#102	—	2	1	no	—
527	#102	—	2	1	no	—
528	#102	—	2	1	no	—
529	C₂²⋊C₈	—	2	1	no	—
530	#97	—	2	1	no	—
531	C₄×C₈	—	2	1	no	—
532	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
533	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
534	C₂₃	#18744	1	1	yes	characteristic, direct product factor, Sylow
535	#41	—	8	1	no	—
536	#41	—	8	1	no	—
537	C₂×Q₈	—	2	2	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
542	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	1	no	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	—	4	1	no	—
550	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	4	1	no	—
554	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	Q₁₆	—	8	1	no	—
558	SD₁₆	—	4	4	no	—
559	SD₁₆	—	8	1	no	—
560	SD₁₆	—	8	1	no	—
561	SD₁₆	—	8	1	no	—
562	SD₁₆	—	8	1	no	—
563	D₈	—	8	1	no	—
564	M₁₆	—	4	2	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	2	1	no	—
570	C₂×C₈	—	2	1	no	—
571	C₂×C₈	—	8	1	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	2	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	1	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
599	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
600	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
601	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
602	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	1	no	—
607	C₂²⋊C₄	—	2	2	no	—
608	C₂²⋊C₄	—	2	1	no	—
609	C₂²⋊C₄	—	2	1	no	—
610	C₂²⋊C₄	—	2	1	no	—
611	C₂²⋊C₄	—	2	1	no	—
612	C₂²⋊C₄	—	2	1	no	—
613	C₂²⋊C₄	—	2	1	no	—
614	C₂²⋊C₄	—	4	1	no	—
615	C₂²⋊C₄	—	4	1	no	—
616	C₄²	—	2	2	no	—
617	C₄²	—	2	2	no	—
618	C₄²	—	2	2	no	—
619	C₄²	—	2	2	no	—
620	C₄²	—	2	1	no	—
621	C₄²	—	2	1	no	—
622	C₄²	—	2	1	no	—
623	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
624	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
625	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
626	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
627	C₂³	—	2	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	C₂₃×#120	1	1	yes	characteristic
630	Q₈	—	2	1	no	—
631	Q₈	—	2	1	no	—
632	Q₈	—	2	1	no	—
633	Q₈	—	2	1	no	—
634	Q₈	—	2	1	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	4	2	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	4	1	no	—
639	Q₈	—	4	1	no	—
640	Q₈	—	4	1	no	—
641	Q₈	—	8	1	no	—
642	Q₈	—	8	2	no	—
643	D₄	—	2	1	no	—
644	D₄	—	2	1	no	—
645	D₄	—	4	2	no	—
646	D₄	—	4	1	no	—
647	D₄	—	4	1	no	—
648	D₄	—	4	1	no	—
649	D₄	—	4	1	no	—
650	D₄	—	8	1	no	—
651	D₄	—	8	2	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	1	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	8	2	no	—
670	C₂×C₄	—	8	2	no	—
671	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
672	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
673	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
675	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
676	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
677	C₈	—	2	2	no	—
678	C₈	—	4	1	no	—
679	C₈	—	4	1	no	—
680	C₈	—	4	1	no	—
681	C₈	—	4	1	no	—
682	C₈	—	4	1	no	—
683	C₂²	—	2	1	no	—
684	C₂²	—	2	1	no	—
685	C₂²	—	2	1	no	—
686	C₂²	—	4	1	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	8	1	no	—
690	C₂²	—	8	1	no	—
691	C₂²	—	8	1	no	—
692	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
693	C₄	—	2	1	no	—
694	C₄	—	2	1	no	—
695	C₄	—	2	1	no	—
696	C₄	—	2	1	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	2	no	—
699	C₄	—	4	1	no	—
700	C₄	—	4	1	no	—
701	C₄	—	4	1	no	—
702	C₄	—	4	1	no	—
703	C₄	—	4	2	no	—
704	C₄	—	4	1	no	—
705	C₄	—	4	1	no	—
706	C₄	—	8	1	no	—
707	C₄	C₂₃×#577	1	1	yes	characteristic
708	C₄	C₂₃×#577	1	1	yes	characteristic
709	C₄	C₂₃×#577	1	1	yes	characteristic
710	C₄	C₂₃×#578	1	1	yes	characteristic
711	C₂	—	4	1	no	—
712	C₂	—	8	1	no	—
713	C₂	—	16	1	no	—
714	C₂	C₂₃×#3010	1	1	yes	characteristic
715	C₂	C₂₃×#3013	1	1	yes	characteristic
716	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3212	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
1. C₂₃×#3555	C₂	semidirect
2. C₂₃×#3246	C₂	semidirect
4. C₂₃×#3182	C₂	semidirect
5. C₂₃×#1650	C₂	semidirect
6. C₂₃×#1648	C₂	semidirect
7. C₂₃×#1645	C₂	semidirect
8. C₂₃×#1643	C₂	semidirect
9. C₂₃×#1628	C₂	semidirect
10. C₂₃×#1626	C₂	semidirect
11. C₂₃×#1623	C₂	semidirect
12. C₂₃×#1622	C₂	semidirect
13. C₂₃×#1542	C₂	semidirect
14. C₂₃×#1485	C₂	semidirect
15. C₂₃×#1483	C₂	semidirect
268. #18744	C₂₃	direct (N × H)
325. #3555	C₂×C₂₃	semidirect
326. #3246	C₂×C₂₃	semidirect
328. #3182	C₂×C₂₃	semidirect
329. #1650	C₂×C₂₃	semidirect
330. #1648	C₂×C₂₃	semidirect
331. #1645	C₂×C₂₃	semidirect
332. #1643	C₂×C₂₃	semidirect
333. #1628	C₂×C₂₃	semidirect
334. #1626	C₂×C₂₃	semidirect
335. #1623	C₂×C₂₃	semidirect
336. #1622	C₂×C₂₃	semidirect
337. #1542	C₂×C₂₃	semidirect
338. #1485	C₂×C₂₃	semidirect
339. #1483	C₂×C₂₃	semidirect
534. C₂₃	#18744	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×#3246	2
C₂₃×#3212	2
C₂₃×#3182	2
C₂₃×#1650	2
C₂₃×#1648	2
C₂₃×#1645	2
C₂₃×#1643	2
C₂₃×#1628	2
C₂₃×#1626	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1542	2
C₂₃×#1485	2
C₂₃×#1483	2
#18744	23

Subgroups — C23×#18744 #19166370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18744 #19166370