Subgroups — C₂₃×#18503 #19166129

Order 5888.

922 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3482	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1650	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1649	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1647	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1544	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#557	C₂²	1	1	yes	characteristic
17	C₂₃×#556	—	2	1	no	—
18	C₂₃×#556	C₂²	1	1	yes	characteristic
19	C₂₃×#552	—	2	2	no	—
20	C₂₃×#552	—	2	1	no	—
21	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#548	—	2	1	no	—
23	C₂₃×#547	—	2	1	no	—
24	C₂₃×#547	C₂²	1	1	yes	characteristic
25	C₂₃×#533	—	2	1	no	—
26	C₂₃×#533	C₂²	1	1	yes	characteristic
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#529	—	2	1	no	—
29	C₂₃×#529	—	2	2	no	—
30	C₄×#41×C₂₃	—	2	1	no	—
31	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
32	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
34	C₂₃×#500	C₂²	1	1	yes	characteristic
35	C₂₃×#500	C₂²	1	1	yes	characteristic
36	C₂₃×#499	—	2	2	no	—
37	C₂₃×#498	C₂²	1	1	yes	characteristic
38	C₂₃×#498	C₂²	1	1	yes	characteristic
39	C₂₃×#485	—	2	1	no	—
40	C₂₃×#485	—	2	1	no	—
41	C₂₃×#475	C₂²	1	1	yes	characteristic
42	C₂₃×#475	C₂²	1	1	yes	characteristic
43	C₂₃×#474	C₂²	1	1	yes	characteristic
44	C₂₃×#474	C₂²	1	1	yes	characteristic
45	C₂₃×#462	C₂²	1	1	yes	characteristic
46	C₂₃×#462	C₂²	1	1	yes	characteristic
47	C₂₃×#461	C₂²	1	1	yes	characteristic
48	C₂₃×#461	C₂²	1	1	yes	characteristic
49	C₂₃×#452	—	2	1	no	—
50	C₂₃×#452	—	2	1	no	—
51	C₄×Q₁₆×C₂₃	—	2	1	no	—
52	C₄×Q₁₆×C₂₃	—	2	1	no	—
53	C₄×SD₁₆×C₂₃	—	2	1	no	—
54	C₄×SD₁₆×C₂₃	—	2	1	no	—
55	C₂₃×#431	C₂²	1	1	yes	characteristic
56	C₂₃×#431	C₂²	1	1	yes	characteristic
57	C₂₃×#427	—	2	2	no	—
58	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	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×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×D₄⋊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×#41×C₂₃	D₄	1	1	yes	characteristic
73	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
74	C₂²×Q₈×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	—	2	1	no	—
76	C₂×Q₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂²×C₈×C₂₃	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	1	no	—
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×#126	—	2	2	no	—
87	C₂₃×#126	—	2	2	no	—
88	C₂₃×#126	—	2	2	no	—
89	C₂₃×#125	—	2	2	no	—
90	C₂₃×#125	—	2	2	no	—
91	C₂₃×#125	—	2	1	no	—
92	C₂₃×#125	—	2	1	no	—
93	C₂₃×#125	D₄	1	2	yes	—
94	C₂₃×#124	—	2	2	no	—
95	C₂₃×#124	—	2	1	no	—
96	C₂₃×#123	—	2	2	no	—
97	C₂₃×#123	—	2	2	no	—
98	C₂₃×#123	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	—	2	2	no	—
100	C₂₃×C₂²⋊Q₈	—	2	1	no	—
101	C₂₃×C₂²⋊Q₈	—	2	1	no	—
102	C₂₃×C₂²⋊Q₈	—	2	1	no	—
103	C₂₃×C₂²⋊Q₈	—	2	1	no	—
104	C₂₃×C₂²⋊Q₈	—	2	1	no	—
105	C₂₃×C₂²⋊Q₈	—	2	2	no	—
106	C₂₃×C₂²⋊Q₈	—	2	1	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×#121	—	2	2	no	—
109	C₂₃×#121	—	2	1	no	—
110	C₄×Q₈×C₂₃	—	2	2	no	—
111	C₄×Q₈×C₂₃	—	2	2	no	—
112	C₄×Q₈×C₂₃	—	2	1	no	—
113	C₄×Q₈×C₂₃	—	2	1	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
116	C₄×Q₈×C₂₃	—	2	1	no	—
117	C₄×Q₈×C₂₃	—	2	1	no	—
118	C₄×Q₈×C₂₃	—	2	1	no	—
119	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
120	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
121	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
122	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
123	C₄×D₄×C₂₃	—	2	2	no	—
124	C₄×D₄×C₂₃	—	2	2	no	—
125	C₄×D₄×C₂₃	—	2	1	no	—
126	C₄×D₄×C₂₃	—	2	1	no	—
127	C₄×D₄×C₂₃	—	2	1	no	—
128	C₄×D₄×C₂₃	—	2	1	no	—
129	C₄×D₄×C₂₃	—	2	1	no	—
130	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
131	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
132	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
133	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
134	C₂₃×#117	—	2	2	no	—
135	C₂₃×#117	—	2	2	no	—
136	C₂₃×#117	—	2	1	no	—
137	C₂₃×#117	—	2	1	no	—
138	C₂₃×#117	—	2	1	no	—
139	C₂₃×#117	—	2	1	no	—
140	C₂₃×#117	D₄	1	1	yes	characteristic
141	C₂₃×#117	D₄	1	1	yes	characteristic
142	C₂₃×#117	D₄	1	2	yes	—
143	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₂×C₄²×C₂₃	—	2	2	no	—
146	C₂×C₄²×C₂₃	—	2	1	no	—
147	C₂×C₄²×C₂₃	—	2	1	no	—
148	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
149	C₂₃×#107	—	2	2	no	—
150	C₂₃×#107	—	2	2	no	—
151	C₂₃×#107	—	2	1	no	—
152	C₂₃×#107	—	2	1	no	—
153	C₂₃×#106	—	2	2	no	—
154	C₂₃×#106	—	2	2	no	—
155	C₂₃×#106	—	2	1	no	—
156	C₂₃×#106	—	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	1	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×#103	—	2	1	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×C₂²⋊C₈	—	2	1	no	—
170	C₂₃×C₂²⋊C₈	—	2	1	no	—
171	C₄×C₈×C₂₃	—	2	1	no	—
172	C₄×C₈×C₂₃	—	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	—	2	2	no	—
177	#41×C₂₃	—	2	2	no	—
178	#41×C₂₃	—	4	1	no	—
179	#41×C₂₃	—	4	1	no	—
180	#41×C₂₃	—	4	2	no	—
181	#41×C₂₃	—	4	2	no	—
182	C₂×Q₈×C₂₃	—	2	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	1	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	4	2	no	—
188	C₂×Q₈×C₂₃	—	4	2	no	—
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×D₄×C₂₃	—	2	2	no	—
192	C₂×D₄×C₂₃	—	2	1	no	—
193	C₂×D₄×C₂₃	—	2	1	no	—
194	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₂₃	—	2	2	no	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	1	no	—
199	C₂²×C₄×C₂₃	—	2	1	no	—
200	C₂²×C₄×C₂₃	—	2	1	no	—
201	C₂²×C₄×C₂₃	—	2	1	no	—
202	C₂²×C₄×C₂₃	—	2	1	no	—
203	C₂²×C₄×C₂₃	—	2	1	no	—
204	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	Q₁₆×C₂₃	—	4	2	no	—
208	Q₁₆×C₂₃	—	4	2	no	—
209	SD₁₆×C₂₃	—	4	2	no	—
210	SD₁₆×C₂₃	—	4	2	no	—
211	C₂×C₈×C₂₃	—	2	4	no	—
212	C₂×C₈×C₂₃	—	2	1	no	—
213	C₂×C₈×C₂₃	—	2	1	no	—
214	C₂×C₈×C₂₃	—	2	1	no	—
215	C₂×C₈×C₂₃	—	2	1	no	—
216	C₂×C₈×C₂₃	—	2	1	no	—
217	C₂×C₈×C₂₃	—	2	1	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	2	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	2	no	—
224	C₄⋊C₄×C₂₃	—	2	1	no	—
225	C₄⋊C₄×C₂₃	—	2	2	no	—
226	C₄⋊C₄×C₂₃	—	2	2	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	—	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	—	4	1	no	—
237	C₄⋊C₄×C₂₃	—	4	1	no	—
238	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
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	2	no	—
248	C₂²⋊C₄×C₂₃	—	2	1	no	—
249	C₂²⋊C₄×C₂₃	—	2	1	no	—
250	C₂²⋊C₄×C₂₃	—	2	2	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	C₄²×C₂₃	—	2	2	no	—
257	C₄²×C₂₃	—	2	2	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	—	2	1	no	—
260	C₄²×C₂₃	—	2	1	no	—
261	C₄²×C₂₃	—	2	1	no	—
262	C₄²×C₂₃	—	2	2	no	—
263	C₄²×C₂₃	—	2	1	no	—
264	C₄²×C₂₃	—	2	1	no	—
265	C₄²×C₂₃	—	4	1	no	—
266	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
268	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
269	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
270	#18503	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂³×C₂₃	—	2	1	no	—
272	C₂³×C₂₃	—	2	1	no	—
273	C₂³×C₂₃	#120	1	1	yes	characteristic
274	Q₈×C₂₃	—	2	1	no	—
275	Q₈×C₂₃	—	2	1	no	—
276	Q₈×C₂₃	—	2	1	no	—
277	Q₈×C₂₃	—	4	4	no	—
278	Q₈×C₂₃	—	4	1	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	—	4	1	no	—
284	Q₈×C₂₃	—	4	1	no	—
285	Q₈×C₂₃	—	4	1	no	—
286	Q₈×C₂₃	#135	1	2	yes	—
287	D₄×C₂₃	—	2	1	no	—
288	D₄×C₂₃	—	2	1	no	—
289	D₄×C₂₃	—	2	1	no	—
290	D₄×C₂₃	—	4	2	no	—
291	D₄×C₂₃	—	4	2	no	—
292	D₄×C₂₃	—	4	4	no	—
293	D₄×C₂₃	#135	1	2	yes	—
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	2	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	2	1	no	—
302	C₂×C₄×C₂₃	—	2	1	no	—
303	C₂×C₄×C₂₃	—	2	2	no	—
304	C₂×C₄×C₂₃	—	2	1	no	—
305	C₂×C₄×C₂₃	—	2	1	no	—
306	C₂×C₄×C₂₃	—	2	2	no	—
307	C₂×C₄×C₂₃	—	4	2	no	—
308	C₂×C₄×C₂₃	—	4	2	no	—
309	C₂×C₄×C₂₃	—	4	4	no	—
310	C₂×C₄×C₂₃	—	4	4	no	—
311	C₂×C₄×C₂₃	—	4	1	no	—
312	C₂×C₄×C₂₃	—	4	1	no	—
313	C₂×C₄×C₂₃	—	4	1	no	—
314	C₂×C₄×C₂₃	—	4	1	no	—
315	C₂×C₄×C₂₃	—	4	2	no	—
316	C₂×C₄×C₂₃	—	4	2	no	—
317	C₂×C₄×C₂₃	—	4	1	no	—
318	C₂×C₄×C₂₃	—	4	1	no	—
319	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
320	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
321	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
322	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
325	C₈×C₂₃	—	2	2	no	—
326	C₈×C₂₃	—	2	2	no	—
327	C₈×C₂₃	—	4	1	no	—
328	C₈×C₂₃	—	4	1	no	—
329	C₈×C₂₃	—	4	1	no	—
330	C₈×C₂₃	—	4	1	no	—
331	#3482	C₂×C₂₃	1	1	yes	characteristic
332	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#3107	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#3107	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1654	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1650	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#1649	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1647	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1544	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#1476	C₂×C₂₃	1	1	yes	characteristic, complemented
346	C₂²×C₂₃	—	2	1	no	—
347	C₂²×C₂₃	—	2	1	no	—
348	C₂²×C₂₃	—	2	2	no	—
349	C₂²×C₂₃	—	2	1	no	—
350	C₂²×C₂₃	—	4	1	no	—
351	C₂²×C₂₃	—	4	1	no	—
352	C₂²×C₂₃	—	4	1	no	—
353	C₂²×C₂₃	—	4	1	no	—
354	C₂²×C₂₃	—	4	1	no	—
355	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
356	C₄×C₂₃	—	2	1	no	—
357	C₄×C₂₃	—	2	1	no	—
358	C₄×C₂₃	—	2	2	no	—
359	C₄×C₂₃	—	2	1	no	—
360	C₄×C₂₃	—	4	1	no	—
361	C₄×C₂₃	—	4	1	no	—
362	C₄×C₂₃	—	4	2	no	—
363	C₄×C₂₃	—	4	2	no	—
364	C₄×C₂₃	—	4	1	no	—
365	C₄×C₂₃	—	4	1	no	—
366	C₄×C₂₃	—	4	1	no	—
367	C₄×C₂₃	—	8	1	no	—
368	C₄×C₂₃	—	8	1	no	—
369	C₄×C₂₃	#456	1	2	yes	—
370	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
371	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
372	C₄×C₂₃	#575	1	1	yes	characteristic
373	C₄×C₂₃	#575	1	1	yes	characteristic
374	#557	C₂²×C₂₃	1	1	yes	characteristic
375	#556	—	2	1	no	—
376	#556	C₂²×C₂₃	1	1	yes	characteristic
377	#552	—	2	2	no	—
378	#552	—	2	1	no	—
379	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
380	#548	—	2	1	no	—
381	#547	—	2	1	no	—
382	#547	C₂²×C₂₃	1	1	yes	characteristic
383	#533	—	2	1	no	—
384	#533	C₂²×C₂₃	1	1	yes	characteristic
385	#533	C₂²×C₂₃	1	1	yes	characteristic
386	#529	—	2	1	no	—
387	#529	—	2	2	no	—
388	C₄×#41	—	2	1	no	—
389	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
390	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×C₄×Q₈	—	2	1	no	—
392	#500	C₂²×C₂₃	1	1	yes	characteristic
393	#500	C₂²×C₂₃	1	1	yes	characteristic
394	#499	—	2	2	no	—
395	#498	C₂²×C₂₃	1	1	yes	characteristic
396	#498	C₂²×C₂₃	1	1	yes	characteristic
397	#485	—	2	1	no	—
398	#485	—	2	1	no	—
399	#475	C₂²×C₂₃	1	1	yes	characteristic
400	#475	C₂²×C₂₃	1	1	yes	characteristic
401	#474	C₂²×C₂₃	1	1	yes	characteristic
402	#474	C₂²×C₂₃	1	1	yes	characteristic
403	#462	C₂²×C₂₃	1	1	yes	characteristic
404	#462	C₂²×C₂₃	1	1	yes	characteristic
405	#461	C₂²×C₂₃	1	1	yes	characteristic
406	#461	C₂²×C₂₃	1	1	yes	characteristic
407	#452	—	2	1	no	—
408	#452	—	2	1	no	—
409	C₄×Q₁₆	—	2	1	no	—
410	C₄×Q₁₆	—	2	1	no	—
411	C₄×SD₁₆	—	2	1	no	—
412	C₄×SD₁₆	—	2	1	no	—
413	#431	C₂²×C₂₃	1	1	yes	characteristic
414	#431	C₂²×C₂₃	1	1	yes	characteristic
415	#427	—	2	2	no	—
416	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
417	#332	C₂²×C₂₃	1	1	yes	characteristic
418	#332	C₂²×C₂₃	1	1	yes	characteristic
419	#332	C₂²×C₂₃	1	1	yes	characteristic
420	#332	C₂²×C₂₃	1	1	yes	characteristic
421	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
422	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
424	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
425	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
426	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
427	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
428	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
429	C₂×C₂₃	—	4	1	no	—
430	C₂×C₂₃	—	4	2	no	—
431	C₂×C₂₃	—	8	1	no	—
432	C₂×C₂₃	#3002	1	1	yes	characteristic
433	C₂×C₂₃	#3002	1	1	yes	characteristic
434	C₂×C₂₃	C₂×#456	1	1	yes	characteristic
435	C₂×#41	—	2	1	no	—
436	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
437	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
438	C₂²×Q₈	—	2	1	no	—
439	C₂×Q₁₆	—	2	1	no	—
440	C₂×Q₁₆	—	2	1	no	—
441	C₂×SD₁₆	—	2	1	no	—
442	C₂×SD₁₆	—	2	1	no	—
443	C₂²×C₈	—	2	1	no	—
444	C₄⋊Q₈	—	2	1	no	—
445	C₄⋊Q₈	—	2	1	no	—
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	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
450	#126	—	2	2	no	—
451	#126	—	2	2	no	—
452	#126	—	2	2	no	—
453	#125	—	2	2	no	—
454	#125	—	2	2	no	—
455	#125	—	2	1	no	—
456	#125	—	2	1	no	—
457	#125	D₄×C₂₃	1	2	yes	—
458	#124	—	2	2	no	—
459	#124	—	2	1	no	—
460	#123	—	2	2	no	—
461	#123	—	2	2	no	—
462	#123	—	2	2	no	—
463	C₂²⋊Q₈	—	2	2	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	1	no	—
466	C₂²⋊Q₈	—	2	1	no	—
467	C₂²⋊Q₈	—	2	1	no	—
468	C₂²⋊Q₈	—	2	2	no	—
469	C₂²⋊Q₈	—	2	1	no	—
470	C₂²⋊Q₈	—	2	1	no	—
471	C₂²⋊Q₈	—	2	1	no	—
472	#121	—	2	2	no	—
473	#121	—	2	1	no	—
474	C₄×Q₈	—	2	2	no	—
475	C₄×Q₈	—	2	2	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	—	2	1	no	—
481	C₄×Q₈	—	2	1	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
484	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
485	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
486	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
487	C₄×D₄	—	2	2	no	—
488	C₄×D₄	—	2	2	no	—
489	C₄×D₄	—	2	1	no	—
490	C₄×D₄	—	2	1	no	—
491	C₄×D₄	—	2	1	no	—
492	C₄×D₄	—	2	1	no	—
493	C₄×D₄	—	2	1	no	—
494	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
495	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
496	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
497	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
498	#117	—	2	2	no	—
499	#117	—	2	2	no	—
500	#117	—	2	1	no	—
501	#117	—	2	1	no	—
502	#117	—	2	1	no	—
503	#117	—	2	1	no	—
504	#117	D₄×C₂₃	1	1	yes	characteristic
505	#117	D₄×C₂₃	1	1	yes	characteristic
506	#117	D₄×C₂₃	1	2	yes	—
507	C₂×C₄⋊C₄	—	2	2	no	—
508	C₂×C₄⋊C₄	—	2	1	no	—
509	C₂×C₄²	—	2	2	no	—
510	C₂×C₄²	—	2	1	no	—
511	C₂×C₄²	—	2	1	no	—
512	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
513	#107	—	2	2	no	—
514	#107	—	2	2	no	—
515	#107	—	2	1	no	—
516	#107	—	2	1	no	—
517	#106	—	2	2	no	—
518	#106	—	2	2	no	—
519	#106	—	2	1	no	—
520	#106	—	2	1	no	—
521	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
522	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
523	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
524	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
525	#103	—	2	1	no	—
526	#103	—	2	1	no	—
527	#103	—	2	1	no	—
528	#103	—	2	1	no	—
529	#103	—	2	1	no	—
530	#103	—	2	1	no	—
531	#102	—	2	1	no	—
532	#102	—	2	1	no	—
533	C₂²⋊C₈	—	2	1	no	—
534	C₂²⋊C₈	—	2	1	no	—
535	C₄×C₈	—	2	1	no	—
536	C₄×C₈	—	2	1	no	—
537	C₄×C₈	—	2	1	no	—
538	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
539	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
540	C₂₃	#18503	1	1	yes	characteristic, direct product factor, Sylow
541	#41	—	2	2	no	—
542	#41	—	2	2	no	—
543	#41	—	4	1	no	—
544	#41	—	4	1	no	—
545	#41	—	4	2	no	—
546	#41	—	4	2	no	—
547	C₂×Q₈	—	2	2	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	—	2	1	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	4	2	no	—
553	C₂×Q₈	—	4	2	no	—
554	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂×D₄	—	2	2	no	—
557	C₂×D₄	—	2	1	no	—
558	C₂×D₄	—	2	1	no	—
559	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
560	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
561	C₂²×C₄	—	2	2	no	—
562	C₂²×C₄	—	2	2	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	1	no	—
569	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
570	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
571	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
572	Q₁₆	—	4	2	no	—
573	Q₁₆	—	4	2	no	—
574	SD₁₆	—	4	2	no	—
575	SD₁₆	—	4	2	no	—
576	C₂×C₈	—	2	4	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	—	2	1	no	—
581	C₂×C₈	—	2	1	no	—
582	C₂×C₈	—	2	1	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	2	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	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	4	1	no	—
602	C₄⋊C₄	—	4	1	no	—
603	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
604	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
605	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
606	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
607	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
608	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	2	no	—
612	C₂²⋊C₄	—	2	2	no	—
613	C₂²⋊C₄	—	2	1	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	1	no	—
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	4	1	no	—
621	C₄²	—	2	2	no	—
622	C₄²	—	2	2	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	2	1	no	—
625	C₄²	—	2	1	no	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	1	no	—
628	C₄²	—	2	1	no	—
629	C₄²	—	2	1	no	—
630	C₄²	—	4	1	no	—
631	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
632	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
633	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
634	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
635	C₂³	—	2	1	no	—
636	C₂³	—	2	1	no	—
637	C₂³	C₂₃×#120	1	1	yes	characteristic
638	Q₈	—	2	1	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	4	4	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	—	4	1	no	—
645	Q₈	—	4	1	no	—
646	Q₈	—	4	1	no	—
647	Q₈	—	4	1	no	—
648	Q₈	—	4	1	no	—
649	Q₈	—	4	1	no	—
650	Q₈	C₂₃×#135	1	2	yes	—
651	D₄	—	2	1	no	—
652	D₄	—	2	1	no	—
653	D₄	—	2	1	no	—
654	D₄	—	4	2	no	—
655	D₄	—	4	2	no	—
656	D₄	—	4	4	no	—
657	D₄	C₂₃×#135	1	2	yes	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	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	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	4	2	no	—
672	C₂×C₄	—	4	2	no	—
673	C₂×C₄	—	4	4	no	—
674	C₂×C₄	—	4	4	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	2	no	—
680	C₂×C₄	—	4	2	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
684	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
685	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
686	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
687	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
688	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
689	C₈	—	2	2	no	—
690	C₈	—	2	2	no	—
691	C₈	—	4	1	no	—
692	C₈	—	4	1	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₂²	—	2	1	no	—
696	C₂²	—	2	1	no	—
697	C₂²	—	2	2	no	—
698	C₂²	—	2	1	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	—	4	1	no	—
704	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
705	C₄	—	2	1	no	—
706	C₄	—	2	1	no	—
707	C₄	—	2	2	no	—
708	C₄	—	2	1	no	—
709	C₄	—	4	1	no	—
710	C₄	—	4	1	no	—
711	C₄	—	4	2	no	—
712	C₄	—	4	2	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	1	no	—
715	C₄	—	4	1	no	—
716	C₄	—	8	1	no	—
717	C₄	—	8	1	no	—
718	C₄	C₂₃×#456	1	2	yes	—
719	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
720	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
721	C₄	C₂₃×#575	1	1	yes	characteristic
722	C₄	C₂₃×#575	1	1	yes	characteristic
723	C₂	—	4	1	no	—
724	C₂	—	4	2	no	—
725	C₂	—	8	1	no	—
726	C₂	C₂₃×#3002	1	1	yes	characteristic
727	C₂	C₂₃×#3002	1	1	yes	characteristic
728	C₂	C₂×C₂₃×#456	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#120
prime radical	C₂₃×#3482	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂₃×#3160	C₂	semidirect
3. C₂₃×#3107	C₂	semidirect
4. C₂₃×#3107	C₂	semidirect
5. C₂₃×#1654	C₂	semidirect
6. C₂₃×#1653	C₂	semidirect
7. C₂₃×#1652	C₂	semidirect
8. C₂₃×#1651	C₂	semidirect
9. C₂₃×#1650	C₂	semidirect
10. C₂₃×#1649	C₂	semidirect
11. C₂₃×#1648	C₂	semidirect
12. C₂₃×#1647	C₂	semidirect
13. C₂₃×#1544	C₂	semidirect
14. C₂₃×#1476	C₂	semidirect
15. C₂₃×#1476	C₂	semidirect
270. #18503	C₂₃	direct (N × H)
332. #3160	C₂×C₂₃	semidirect
333. #3107	C₂×C₂₃	semidirect
334. #3107	C₂×C₂₃	semidirect
335. #1654	C₂×C₂₃	semidirect
336. #1653	C₂×C₂₃	semidirect
337. #1652	C₂×C₂₃	semidirect
338. #1651	C₂×C₂₃	semidirect
339. #1650	C₂×C₂₃	semidirect
340. #1649	C₂×C₂₃	semidirect
341. #1648	C₂×C₂₃	semidirect
342. #1647	C₂×C₂₃	semidirect
343. #1544	C₂×C₂₃	semidirect
344. #1476	C₂×C₂₃	semidirect
345. #1476	C₂×C₂₃	semidirect
540. C₂₃	#18503	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×#3482	2
C₂₃×#3160	2
C₂₃×#3107	2
C₂₃×#1654	2
C₂₃×#1653	2
C₂₃×#1652	2
C₂₃×#1651	2
C₂₃×#1650	2
C₂₃×#1649	2
C₂₃×#1648	2
C₂₃×#1647	2
C₂₃×#1544	2
C₂₃×#1476	2
#18503	23

Subgroups — C23×#18503 #19166129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18503 #19166129