Subgroups — C₂₃×#21623 #19169249

Order 5888.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2331	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2275	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1919	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1776	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#476	C₄	1	8	yes	complemented
21	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#391	—	2	1	no	—
34	C₂₃×#391	—	2	1	no	—
35	C₂₃×#389	C₂²	1	1	yes	characteristic
36	C₂₃×#389	C₂²	1	1	yes	characteristic
37	C₂₃×#387	—	2	1	no	—
38	C₂₃×#387	—	2	1	no	—
39	C₂₃×#382	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
46	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
47	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
48	C₂₃×#376	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#95	—	2	1	no	—
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₂₃	—	2	1	no	—
61	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	—	2	2	no	—
70	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
71	C₄×D₄×C₂₃	—	2	2	no	—
72	C₄×D₄×C₂₃	—	2	2	no	—
73	C₄×D₄×C₂₃	—	2	2	no	—
74	C₄×D₄×C₂₃	—	2	2	no	—
75	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₂₃	Q₈	1	4	yes	complemented
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	—	2	1	no	—
93	C₂×C₄²×C₂₃	—	2	1	no	—
94	C₂×C₄²×C₂₃	—	2	1	no	—
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂₃×#107	—	2	2	no	—
99	C₂₃×#106	—	2	2	no	—
100	C₂₃×#106	C₂×C₄	1	4	yes	complemented
101	C₂₃×#106	C₂×C₄	1	4	yes	complemented
102	C₂₃×C₄⋊C₈	—	2	2	no	—
103	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
104	C₂₃×#102	—	2	2	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×#102	C₂×C₄	1	4	yes	—
107	C₂₃×#102	C₂×C₄	1	4	yes	—
108	C₂₃×#97	—	2	2	no	—
109	C₄×C₈×C₂₃	—	2	2	no	—
110	C₂₃×#95	—	2	1	no	—
111	C₂₃×#95	—	2	1	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	—	2	1	no	—
117	C₂₃×#95	—	2	1	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	D₄	1	1	yes	characteristic
120	C₂₃×#95	D₄	1	1	yes	characteristic
121	C₂₃×#95	D₄	1	1	yes	characteristic
122	C₂₃×#95	D₄	1	1	yes	characteristic
123	C₂⁴×C₂₃	—	2	1	no	—
124	C₂×Q₈×C₂₃	—	2	4	no	—
125	C₂×Q₈×C₂₃	—	8	1	no	—
126	C₂×Q₈×C₂₃	—	8	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
132	C₂×D₄×C₂₃	#41	1	2	yes	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	8	1	no	—
144	C₂²×C₄×C₂₃	—	8	1	no	—
145	C₂²×C₄×C₂₃	—	8	1	no	—
146	C₂²×C₄×C₂₃	—	8	1	no	—
147	C₂²×C₄×C₂₃	—	8	1	no	—
148	C₂²×C₄×C₂₃	—	8	1	no	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
156	C₂×C₈×C₂₃	—	2	1	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
167	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
168	C₄⋊C₄×C₂₃	—	2	4	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	—	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	—	8	1	no	—
179	C₄⋊C₄×C₂₃	—	8	1	no	—
180	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
183	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
184	C₄⋊C₄×C₂₃	#41	1	2	yes	—
185	C₂²⋊C₄×C₂₃	—	2	4	no	—
186	C₂²⋊C₄×C₂₃	—	2	4	no	—
187	C₂²⋊C₄×C₂₃	—	2	4	no	—
188	C₄²×C₂₃	—	2	1	no	—
189	C₄²×C₂₃	—	2	1	no	—
190	C₄²×C₂₃	—	8	1	no	—
191	C₄²×C₂₃	—	8	1	no	—
192	C₄²×C₂₃	—	8	1	no	—
193	C₄²×C₂₃	—	8	1	no	—
194	C₄²×C₂₃	—	8	1	no	—
195	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
196	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
197	C₄²×C₂₃	#41	1	2	yes	—
198	C₄²×C₂₃	#41	1	2	yes	—
199	#21623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	2	2	no	—
202	C₂³×C₂₃	—	2	2	no	—
203	C₂³×C₂₃	—	4	1	no	—
204	C₂³×C₂₃	—	4	1	no	—
205	C₂³×C₂₃	—	4	1	no	—
206	C₂³×C₂₃	—	4	1	no	—
207	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₂₃	—	8	4	no	—
209	D₄×C₂₃	—	2	2	no	—
210	D₄×C₂₃	—	2	2	no	—
211	D₄×C₂₃	—	2	2	no	—
212	D₄×C₂₃	C₄×Q₈	1	4	yes	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	8	4	no	—
249	C₂×C₄×C₂₃	—	8	4	no	—
250	C₂×C₄×C₂₃	—	8	4	no	—
251	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
261	C₈×C₂₃	—	2	2	no	—
262	C₈×C₂₃	—	2	2	no	—
263	C₈×C₂₃	—	4	1	no	—
264	C₈×C₂₃	—	4	1	no	—
265	C₂×#476	C₂×C₂₃	1	1	yes	characteristic
266	#2331	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#2275	C₂×C₂₃	1	1	yes	characteristic
268	#1988	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1919	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1864	C₂×C₂₃	1	1	yes	characteristic
273	#1862	C₂×C₂₃	1	1	yes	characteristic
274	#1804	C₂×C₂₃	1	1	yes	characteristic
275	#1776	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₄×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1768	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1762	C₂×C₂₃	1	1	yes	characteristic
279	C₄×#102	C₂×C₂₃	1	1	yes	characteristic
280	C₂²×C₂₃	—	2	4	no	—
281	C₂²×C₂₃	—	4	2	no	—
282	C₂²×C₂₃	—	4	2	no	—
283	C₂²×C₂₃	—	4	2	no	—
284	C₂²×C₂₃	—	4	2	no	—
285	C₂²×C₂₃	—	4	2	no	—
286	C₂²×C₂₃	—	4	2	no	—
287	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
288	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂²×C₂₃	#525	1	1	yes	characteristic
290	C₂²×C₂₃	#576	1	1	yes	characteristic
291	C₂²×C₂₃	#577	1	1	yes	characteristic
292	C₂²×C₂₃	#577	1	1	yes	characteristic
293	C₂²×C₂₃	#577	1	1	yes	characteristic
294	C₄×C₂₃	—	2	2	no	—
295	C₄×C₂₃	—	2	2	no	—
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	2	no	—
298	C₄×C₂₃	—	4	2	no	—
299	C₄×C₂₃	—	4	1	no	—
300	C₄×C₂₃	—	4	2	no	—
301	C₄×C₂₃	—	4	1	no	—
302	C₄×C₂₃	—	4	2	no	—
303	C₄×C₂₃	—	4	2	no	—
304	C₄×C₂₃	—	4	2	no	—
305	C₄×C₂₃	—	8	1	no	—
306	C₄×C₂₃	—	8	1	no	—
307	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
308	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
309	C₄×C₂₃	#529	1	1	yes	characteristic
310	C₄×C₂₃	#533	1	1	yes	characteristic
311	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
315	#476	C₄×C₂₃	1	8	yes	complemented
316	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
328	#391	—	2	1	no	—
329	#391	—	2	1	no	—
330	#389	C₂²×C₂₃	1	1	yes	characteristic
331	#389	C₂²×C₂₃	1	1	yes	characteristic
332	#387	—	2	1	no	—
333	#387	—	2	1	no	—
334	#382	C₂²×C₂₃	1	1	yes	characteristic
335	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
337	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
338	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
340	C₄×C₂²⋊C₄	—	2	1	no	—
341	C₄×C₂²⋊C₄	—	2	1	no	—
342	C₄×C₂²⋊C₄	—	2	1	no	—
343	#376	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#95	—	2	1	no	—
345	#340	C₂²×C₂₃	1	1	yes	characteristic
346	#340	C₂²×C₂₃	1	1	yes	characteristic
347	#340	C₂²×C₂₃	1	1	yes	characteristic
348	#340	C₂²×C₂₃	1	1	yes	characteristic
349	#340	C₂²×C₂₃	1	1	yes	characteristic
350	#340	C₂²×C₂₃	1	1	yes	characteristic
351	#340	C₂²×C₂₃	1	1	yes	characteristic
352	#340	C₂²×C₂₃	1	1	yes	characteristic
353	#336	C₂²×C₂₃	1	1	yes	characteristic
354	#336	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₂₃	—	4	4	no	—
356	C₂×C₂₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
357	C₂×C₂₃	#2946	1	1	yes	characteristic
358	C₂×C₂₃	#2949	1	1	yes	characteristic
359	C₂×C₂₃	#3081	1	1	yes	characteristic
360	C₂×C₂₃	#3084	1	1	yes	characteristic
361	C₂×C₂₃	#3118	1	1	yes	characteristic
362	C₂×C₂₃	#3125	1	1	yes	characteristic
363	C₂²×Q₈	—	2	1	no	—
364	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	—	2	1	no	—
368	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₄⋊Q₈	—	2	2	no	—
373	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
379	C₄×D₄	Q₈×C₂₃	1	4	yes	complemented
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
401	#107	—	2	2	no	—
402	#106	—	2	2	no	—
403	#106	C₂×C₄×C₂₃	1	4	yes	complemented
404	#106	C₂×C₄×C₂₃	1	4	yes	complemented
405	C₄⋊C₈	—	2	2	no	—
406	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
407	#102	—	2	2	no	—
408	#102	—	2	2	no	—
409	#102	C₂×C₄×C₂₃	1	4	yes	—
410	#102	C₂×C₄×C₂₃	1	4	yes	—
411	#97	—	2	2	no	—
412	C₄×C₈	—	2	2	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	D₄×C₂₃	1	1	yes	characteristic
423	#95	D₄×C₂₃	1	1	yes	characteristic
424	#95	D₄×C₂₃	1	1	yes	characteristic
425	#95	D₄×C₂₃	1	1	yes	characteristic
426	C₂₃	#21623	1	1	yes	characteristic, direct product factor, Sylow
427	C₂⁴	—	2	1	no	—
428	C₂×Q₈	—	2	4	no	—
429	C₂×Q₈	—	8	1	no	—
430	C₂×Q₈	—	8	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
435	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
436	C₂×D₄	#41×C₂₃	1	2	yes	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	8	1	no	—
448	C₂²×C₄	—	8	1	no	—
449	C₂²×C₄	—	8	1	no	—
450	C₂²×C₄	—	8	1	no	—
451	C₂²×C₄	—	8	1	no	—
452	C₂²×C₄	—	8	1	no	—
453	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
471	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	8	1	no	—
483	C₄⋊C₄	—	8	1	no	—
484	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
485	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
486	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
487	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
488	C₄⋊C₄	#41×C₂₃	1	2	yes	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	8	1	no	—
495	C₄²	—	8	1	no	—
496	C₄²	—	8	1	no	—
497	C₄²	—	8	1	no	—
498	C₄²	—	8	1	no	—
499	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
500	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
501	C₄²	#41×C₂₃	1	2	yes	—
502	C₄²	#41×C₂₃	1	2	yes	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
511	Q₈	—	8	4	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	2	no	—
515	D₄	C₄×Q₈×C₂₃	1	4	yes	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	8	4	no	—
552	C₂×C₄	—	8	4	no	—
553	C₂×C₄	—	8	4	no	—
554	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
555	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
556	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
557	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
558	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
559	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
560	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
561	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
562	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
564	C₈	—	2	2	no	—
565	C₈	—	2	2	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	2	no	—
575	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
576	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
577	C₂²	C₂₃×#525	1	1	yes	characteristic
578	C₂²	C₂₃×#576	1	1	yes	characteristic
579	C₂²	C₂₃×#577	1	1	yes	characteristic
580	C₂²	C₂₃×#577	1	1	yes	characteristic
581	C₂²	C₂₃×#577	1	1	yes	characteristic
582	C₄	—	2	2	no	—
583	C₄	—	2	2	no	—
584	C₄	—	2	2	no	—
585	C₄	—	2	2	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	1	no	—
593	C₄	—	8	1	no	—
594	C₄	—	8	1	no	—
595	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
596	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
597	C₄	C₂₃×#529	1	1	yes	characteristic
598	C₄	C₂₃×#533	1	1	yes	characteristic
599	C₂	—	4	4	no	—
600	C₂	C₄×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
601	C₂	C₂₃×#2946	1	1	yes	characteristic
602	C₂	C₂₃×#2949	1	1	yes	characteristic
603	C₂	C₂₃×#3081	1	1	yes	characteristic
604	C₂	C₂₃×#3084	1	1	yes	characteristic
605	C₂	C₂₃×#3118	1	1	yes	characteristic
606	C₂	C₂₃×#3125	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2331	C₂	semidirect
4. C₂₃×#1988	C₂	semidirect
5. C₂₃×#1949	C₂	semidirect
6. C₂₃×#1920	C₂	semidirect
7. C₂₃×#1919	C₂	semidirect
11. C₂₃×#1776	C₂	semidirect
12. C₄×C₂₃×#106	C₂	semidirect
13. C₂₃×#1768	C₂	semidirect
20. C₂₃×#476	C₄	semidirect
70. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
76. C₄×D₄×C₂₃	Q₈	semidirect
100. C₂₃×#106	C₂×C₄	semidirect
101. C₂₃×#106	C₂×C₄	semidirect
103. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
199. #21623	C₂₃	direct (N × H)
266. #2331	C₂×C₂₃	semidirect
268. #1988	C₂×C₂₃	semidirect
269. #1949	C₂×C₂₃	semidirect
270. #1920	C₂×C₂₃	semidirect
271. #1919	C₂×C₂₃	semidirect
275. #1776	C₂×C₂₃	semidirect
276. C₄×#106	C₂×C₂₃	semidirect
277. #1768	C₂×C₂₃	semidirect
315. #476	C₄×C₂₃	semidirect
373. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
379. C₄×D₄	Q₈×C₂₃	semidirect
403. #106	C₂×C₄×C₂₃	semidirect
404. #106	C₂×C₄×C₂₃	semidirect
406. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
426. C₂₃	#21623	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#476	2
C₂₃×#2331	2
C₂₃×#2275	2
C₂₃×#1988	2
C₂₃×#1949	2
C₂₃×#1920	2
C₂₃×#1919	2
C₂₃×#1864	2
C₂₃×#1862	2
C₂₃×#1804	2
C₂₃×#1776	2
C₄×C₂₃×#106	2
C₂₃×#1768	2
C₂₃×#1762	2
C₄×C₂₃×#102	2
#21623	23

Subgroups — C23×#21623 #19169249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21623 #19169249