Subgroups — C₂₃×#26824 #19174450

Order 5888.

814 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#489	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2451	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2447	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2055	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2015	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1936	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1935	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#489	—	2	4	no	—
23	C₄×SD₁₆×C₂₃	—	2	4	no	—
24	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#398	—	2	1	no	—
39	C₂₃×#397	—	2	1	no	—
40	C₂₃×#397	—	2	1	no	—
41	C₂₃×#395	—	2	1	no	—
42	C₂₃×#391	—	2	1	no	—
43	C₂₃×#391	C₂²	1	1	yes	characteristic
44	C₂₃×#391	C₂²	1	1	yes	characteristic
45	C₂₃×#391	C₂²	1	1	yes	characteristic
46	C₂₃×#391	C₂²	1	1	yes	characteristic
47	C₂₃×#390	C₂²	1	1	yes	characteristic
48	C₂₃×#389	C₂²	1	1	yes	characteristic
49	C₂₃×#389	C₂²	1	1	yes	characteristic
50	C₂₃×#389	C₂²	1	1	yes	characteristic
51	C₂₃×#387	—	2	1	no	—
52	C₂₃×#385	—	2	1	no	—
53	C₂₃×#381	—	2	1	no	—
54	C₄×C₄⋊C₄×C₂₃	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×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂×Q₁₆×C₂₃	—	2	2	no	—
67	C₂×Q₁₆×C₂₃	—	2	2	no	—
68	C₂×Q₁₆×C₂₃	—	4	1	no	—
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	2	2	no	—
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	—	2	2	no	—
77	C₂₃×C₄⋊Q₈	—	4	1	no	—
78	C₂₃×#124	—	2	2	no	—
79	C₂₃×#124	—	4	1	no	—
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	—	4	1	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	—	4	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	—	2	1	no	—
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×#106	—	2	2	no	—
106	C₂₃×#106	—	4	1	no	—
107	C₂₃×C₄⋊C₈	—	2	2	no	—
108	C₂₃×#103	—	2	2	no	—
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×#102	—	2	2	no	—
114	C₂₃×#102	—	2	2	no	—
115	C₂₃×#102	D₄	1	4	yes	—
116	C₂₃×#97	—	2	2	no	—
117	C₂₃×#97	—	2	2	no	—
118	C₄×C₈×C₂₃	—	2	2	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	1	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	D₄	1	1	yes	characteristic
131	C₂₃×#95	D₄	1	1	yes	characteristic
132	C₂⁴×C₂₃	—	2	1	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×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	4	2	no	—
145	C₂×Q₈×C₂₃	—	4	2	no	—
146	C₂×Q₈×C₂₃	—	8	1	no	—
147	C₂×Q₈×C₂₃	#41	1	2	yes	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	4	2	no	—
162	C₂²×C₄×C₂₃	—	4	2	no	—
163	C₂²×C₄×C₂₃	—	8	1	no	—
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₂₃	—	4	4	no	—
172	SD₁₆×C₂₃	—	2	8	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
183	C₂×C₈×C₂₃	#41	1	2	yes	—
184	C₂×C₈×C₂₃	#41	1	2	yes	—
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	4	1	no	—
196	C₄⋊C₄×C₂₃	—	4	2	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	8	1	no	—
199	C₄⋊C₄×C₂₃	—	8	1	no	—
200	C₄⋊C₄×C₂₃	—	8	1	no	—
201	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
202	C₂²⋊C₄×C₂₃	—	4	2	no	—
203	C₂²⋊C₄×C₂₃	—	4	2	no	—
204	C₂²⋊C₄×C₂₃	—	4	2	no	—
205	C₂²⋊C₄×C₂₃	—	4	2	no	—
206	C₂²⋊C₄×C₂₃	—	8	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	4	2	no	—
214	#26824	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₂₃	—	2	2	no	—
216	C₂³×C₂₃	—	4	1	no	—
217	C₂³×C₂₃	—	4	1	no	—
218	C₂³×C₂₃	—	4	1	no	—
219	C₂³×C₂₃	—	4	1	no	—
220	C₂³×C₂₃	—	4	1	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
223	Q₈×C₂₃	—	2	2	no	—
224	Q₈×C₂₃	—	2	2	no	—
225	Q₈×C₂₃	—	4	1	no	—
226	Q₈×C₂₃	—	4	1	no	—
227	Q₈×C₂₃	—	4	1	no	—
228	Q₈×C₂₃	—	4	1	no	—
229	Q₈×C₂₃	—	4	1	no	—
230	Q₈×C₂₃	—	4	1	no	—
231	Q₈×C₂₃	—	8	4	no	—
232	D₄×C₂₃	—	2	2	no	—
233	D₄×C₂₃	—	2	2	no	—
234	D₄×C₂₃	—	4	1	no	—
235	D₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	8	4	no	—
275	C₂×C₄×C₂₃	#137	1	2	yes	—
276	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
280	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
282	C₈×C₂₃	—	2	2	no	—
283	C₈×C₂₃	—	2	2	no	—
284	C₈×C₂₃	—	2	2	no	—
285	C₈×C₂₃	—	4	1	no	—
286	C₂×#489	C₂×C₂₃	1	1	yes	characteristic
287	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
288	#2451	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#2447	C₂×C₂₃	1	1	yes	characteristic
290	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#2055	C₂×C₂₃	1	1	yes	characteristic
293	#2015	C₂×C₂₃	1	1	yes	characteristic
294	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1936	C₂×C₂₃	1	1	yes	characteristic
296	#1935	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1864	C₂×C₂₃	1	1	yes	characteristic
300	#1771	C₂×C₂₃	1	1	yes	characteristic, complemented
301	C₂²×C₂₃	—	4	2	no	—
302	C₂²×C₂₃	—	4	2	no	—
303	C₂²×C₂₃	—	4	2	no	—
304	C₂²×C₂₃	—	8	1	no	—
305	C₂²×C₂₃	—	8	1	no	—
306	C₂²×C₂₃	—	8	1	no	—
307	C₂²×C₂₃	—	8	1	no	—
308	C₂²×C₂₃	#532	1	1	yes	characteristic
309	C₂²×C₂₃	D₄²	1	1	yes	characteristic
310	C₂²×C₂₃	#548	1	1	yes	characteristic
311	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
312	C₂²×C₂₃	#575	1	1	yes	characteristic
313	C₂²×C₂₃	#577	1	1	yes	characteristic
314	C₂²×C₂₃	#577	1	1	yes	characteristic
315	C₄×C₂₃	—	2	2	no	—
316	C₄×C₂₃	—	4	1	no	—
317	C₄×C₂₃	—	4	1	no	—
318	C₄×C₂₃	—	4	2	no	—
319	C₄×C₂₃	—	4	1	no	—
320	C₄×C₂₃	—	4	1	no	—
321	C₄×C₂₃	—	4	1	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	—	8	1	no	—
324	C₄×C₂₃	—	8	1	no	—
325	C₄×C₂₃	—	8	1	no	—
326	C₄×C₂₃	—	8	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	#525	1	1	yes	characteristic
329	C₄×C₂₃	#542	1	1	yes	characteristic
330	C₄×C₂₃	#547	1	1	yes	characteristic
331	C₄×C₂₃	#551	1	1	yes	characteristic
332	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
333	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
338	#489	—	2	4	no	—
339	C₄×SD₁₆	—	2	4	no	—
340	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
354	#398	—	2	1	no	—
355	#397	—	2	1	no	—
356	#397	—	2	1	no	—
357	#395	—	2	1	no	—
358	#391	—	2	1	no	—
359	#391	C₂²×C₂₃	1	1	yes	characteristic
360	#391	C₂²×C₂₃	1	1	yes	characteristic
361	#391	C₂²×C₂₃	1	1	yes	characteristic
362	#391	C₂²×C₂₃	1	1	yes	characteristic
363	#390	C₂²×C₂₃	1	1	yes	characteristic
364	#389	C₂²×C₂₃	1	1	yes	characteristic
365	#389	C₂²×C₂₃	1	1	yes	characteristic
366	#389	C₂²×C₂₃	1	1	yes	characteristic
367	#387	—	2	1	no	—
368	#385	—	2	1	no	—
369	#381	—	2	1	no	—
370	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
371	#340	C₂²×C₂₃	1	1	yes	characteristic
372	#340	C₂²×C₂₃	1	1	yes	characteristic
373	#340	C₂²×C₂₃	1	1	yes	characteristic
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#336	C₂²×C₂₃	1	1	yes	characteristic
376	#336	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₂₃	—	8	2	no	—
378	C₂×C₂₃	#2461	1	1	yes	characteristic
379	C₂×C₂₃	#3172	1	1	yes	characteristic
380	C₂×C₂₃	#3173	1	1	yes	characteristic
381	C₂×C₂₃	#3266	1	1	yes	characteristic
382	C₂×C₂₃	#3267	1	1	yes	characteristic
383	C₂×C₂₃	#3369	1	1	yes	characteristic
384	C₂×C₂₃	#3371	1	1	yes	characteristic
385	C₂²×Q₈	—	2	1	no	—
386	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
389	C₂³×C₄	—	2	1	no	—
390	C₂×Q₁₆	—	2	2	no	—
391	C₂×Q₁₆	—	2	2	no	—
392	C₂×Q₁₆	—	4	1	no	—
393	C₂×SD₁₆	—	2	2	no	—
394	C₂×SD₁₆	—	2	2	no	—
395	C₂×SD₁₆	—	2	2	no	—
396	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄⋊Q₈	—	2	2	no	—
401	C₄⋊Q₈	—	4	1	no	—
402	#124	—	2	2	no	—
403	#124	—	4	1	no	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	—	4	1	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	4	1	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
429	#106	—	2	2	no	—
430	#106	—	4	1	no	—
431	C₄⋊C₈	—	2	2	no	—
432	#103	—	2	2	no	—
433	#103	—	2	2	no	—
434	#103	—	2	2	no	—
435	#103	—	2	2	no	—
436	#103	—	2	2	no	—
437	#102	—	2	2	no	—
438	#102	—	2	2	no	—
439	#102	D₄×C₂₃	1	4	yes	—
440	#97	—	2	2	no	—
441	#97	—	2	2	no	—
442	C₄×C₈	—	2	2	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	#95	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	D₄×C₂₃	1	1	yes	characteristic
455	#95	D₄×C₂₃	1	1	yes	characteristic
456	C₂₃	#26824	1	1	yes	characteristic, direct product factor, Sylow
457	C₂⁴	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	4	2	no	—
470	C₂×Q₈	—	4	2	no	—
471	C₂×Q₈	—	8	1	no	—
472	C₂×Q₈	#41×C₂₃	1	2	yes	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	—	8	1	no	—
489	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
496	Q₁₆	—	4	4	no	—
497	SD₁₆	—	2	8	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
508	C₂×C₈	#41×C₂₃	1	2	yes	—
509	C₂×C₈	#41×C₂₃	1	2	yes	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	4	1	no	—
521	C₄⋊C₄	—	4	2	no	—
522	C₄⋊C₄	—	4	2	no	—
523	C₄⋊C₄	—	8	1	no	—
524	C₄⋊C₄	—	8	1	no	—
525	C₄⋊C₄	—	8	1	no	—
526	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
527	C₂²⋊C₄	—	4	2	no	—
528	C₂²⋊C₄	—	4	2	no	—
529	C₂²⋊C₄	—	4	2	no	—
530	C₂²⋊C₄	—	4	2	no	—
531	C₂²⋊C₄	—	8	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	4	2	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	—	4	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	—	4	1	no	—
546	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
547	Q₈	—	2	2	no	—
548	Q₈	—	2	2	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	4	1	no	—
551	Q₈	—	4	1	no	—
552	Q₈	—	4	1	no	—
553	Q₈	—	4	1	no	—
554	Q₈	—	4	1	no	—
555	Q₈	—	8	4	no	—
556	D₄	—	2	2	no	—
557	D₄	—	2	2	no	—
558	D₄	—	4	1	no	—
559	D₄	—	4	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	8	4	no	—
599	C₂×C₄	C₂₃×#137	1	2	yes	—
600	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
602	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
603	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
604	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
605	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
606	C₈	—	2	2	no	—
607	C₈	—	2	2	no	—
608	C₈	—	2	2	no	—
609	C₈	—	4	1	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	1	no	—
616	C₂²	—	8	1	no	—
617	C₂²	C₂₃×#532	1	1	yes	characteristic
618	C₂²	D₄²×C₂₃	1	1	yes	characteristic
619	C₂²	C₂₃×#548	1	1	yes	characteristic
620	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
621	C₂²	C₂₃×#575	1	1	yes	characteristic
622	C₂²	C₂₃×#577	1	1	yes	characteristic
623	C₂²	C₂₃×#577	1	1	yes	characteristic
624	C₄	—	2	2	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	2	no	—
632	C₄	—	8	1	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	1	no	—
635	C₄	—	8	1	no	—
636	C₄	—	8	1	no	—
637	C₄	C₂₃×#525	1	1	yes	characteristic
638	C₄	C₂₃×#542	1	1	yes	characteristic
639	C₄	C₂₃×#547	1	1	yes	characteristic
640	C₄	C₂₃×#551	1	1	yes	characteristic
641	C₂	—	8	2	no	—
642	C₂	C₂₃×#2461	1	1	yes	characteristic
643	C₂	C₂₃×#3172	1	1	yes	characteristic
644	C₂	C₂₃×#3173	1	1	yes	characteristic
645	C₂	C₂₃×#3266	1	1	yes	characteristic
646	C₂	C₂₃×#3267	1	1	yes	characteristic
647	C₂	C₂₃×#3369	1	1	yes	characteristic
648	C₂	C₂₃×#3371	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2451	C₂	semidirect
5. C₂₃×#2066	C₂	semidirect
6. C₂₃×#2058	C₂	semidirect
9. C₂₃×#1971	C₂	semidirect
11. C₂₃×#1935	C₂	semidirect
12. C₂₃×#1933	C₂	semidirect
13. C₂₃×#1878	C₂	semidirect
15. C₂₃×#1771	C₂	semidirect
214. #26824	C₂₃	direct (N × H)
288. #2451	C₂×C₂₃	semidirect
290. #2066	C₂×C₂₃	semidirect
291. #2058	C₂×C₂₃	semidirect
294. #1971	C₂×C₂₃	semidirect
296. #1935	C₂×C₂₃	semidirect
297. #1933	C₂×C₂₃	semidirect
298. #1878	C₂×C₂₃	semidirect
300. #1771	C₂×C₂₃	semidirect
456. C₂₃	#26824	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×#489	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#2451	2
C₂₃×#2447	2
C₂₃×#2066	2
C₂₃×#2058	2
C₂₃×#2055	2
C₂₃×#2015	2
C₂₃×#1971	2
C₂₃×#1936	2
C₂₃×#1935	2
C₂₃×#1933	2
C₂₃×#1878	2
C₂₃×#1864	2
C₂₃×#1771	2
#26824	23

Subgroups — C23×#26824 #19174450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26824 #19174450