Subgroups — C₂₃×#21617 #19169243

Order 5888.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#478	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1989	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×#1918	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	C₂₃×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×#125	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	C₂²×C₄²×C₂₃	—	2	1	no	—
21	C₂₃×#478	C₄	1	8	yes	complemented
22	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
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×C₄⋊C₈	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×#102	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
34	C₂₃×#389	C₂²	1	1	yes	characteristic
35	C₂₃×#389	C₂²	1	1	yes	characteristic
36	C₂₃×#387	—	2	1	no	—
37	C₂₃×#387	—	2	1	no	—
38	C₂₃×#382	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
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×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
47	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
48	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
49	C₄³×C₂₃	C₂²	1	1	yes	characteristic
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×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
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	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	C₂×C₄	1	4	yes	complemented
70	C₄×D₄×C₂₃	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
75	C₄×D₄×C₂₃	Q₈	1	4	yes	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	4	4	no	—
96	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂₃×#107	—	2	2	no	—
104	C₂₃×#107	C₂×C₄	1	4	yes	complemented
105	C₂₃×#106	—	2	2	no	—
106	C₂₃×#106	C₂×C₄	1	4	yes	complemented
107	C₂₃×C₄⋊C₈	—	2	2	no	—
108	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
109	C₂₃×#102	—	2	2	no	—
110	C₂₃×#102	—	2	2	no	—
111	C₂₃×#102	C₂×C₄	1	4	yes	—
112	C₂₃×#102	C₂×C₄	1	4	yes	—
113	C₂₃×#97	—	2	2	no	—
114	C₂₃×#97	—	2	2	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	—	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	C₂⁴×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
130	C₂×D₄×C₂₃	#41	1	2	yes	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
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	4	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	4	2	no	—
142	C₂²×C₄×C₂₃	—	4	2	no	—
143	C₂²×C₄×C₂₃	—	4	2	no	—
144	C₂²×C₄×C₂₃	—	4	2	no	—
145	C₂²×C₄×C₂₃	—	4	2	no	—
146	C₂²×C₄×C₂₃	—	4	2	no	—
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
165	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₂₃	—	2	4	no	—
167	C₄⋊C₄×C₂₃	—	2	4	no	—
168	C₄⋊C₄×C₂₃	—	2	1	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	—	4	2	no	—
178	C₄⋊C₄×C₂₃	—	4	2	no	—
179	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
180	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
183	C₄⋊C₄×C₂₃	#41	1	2	yes	—
184	C₂²⋊C₄×C₂₃	—	2	4	no	—
185	C₂²⋊C₄×C₂₃	—	2	4	no	—
186	C₂²⋊C₄×C₂₃	—	2	4	no	—
187	C₄²×C₂₃	—	2	2	no	—
188	C₄²×C₂₃	—	2	1	no	—
189	C₄²×C₂₃	—	2	1	no	—
190	C₄²×C₂₃	—	4	1	no	—
191	C₄²×C₂₃	—	4	1	no	—
192	C₄²×C₂₃	—	4	1	no	—
193	C₄²×C₂₃	—	4	2	no	—
194	C₄²×C₂₃	—	4	2	no	—
195	C₄²×C₂₃	—	4	2	no	—
196	C₄²×C₂₃	—	4	2	no	—
197	C₄²×C₂₃	—	4	2	no	—
198	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
199	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
200	C₄²×C₂₃	#41	1	2	yes	—
201	C₄²×C₂₃	#41	1	2	yes	—
202	#21617	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₂₃	—	2	2	no	—
204	C₂³×C₂₃	—	2	2	no	—
205	C₂³×C₂₃	—	2	2	no	—
206	C₂³×C₂₃	—	4	1	no	—
207	C₂³×C₂₃	—	4	1	no	—
208	C₂³×C₂₃	—	4	1	no	—
209	C₂³×C₂₃	—	4	1	no	—
210	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
211	D₄×C₂₃	—	2	2	no	—
212	D₄×C₂₃	—	2	2	no	—
213	D₄×C₂₃	—	2	2	no	—
214	D₄×C₂₃	C₄×Q₈	1	4	yes	—
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	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	4	8	no	—
234	C₂×C₄×C₂₃	—	4	8	no	—
235	C₂×C₄×C₂₃	—	4	8	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	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
254	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
257	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₂²×Q₈	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	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
263	C₈×C₂₃	—	2	2	no	—
264	C₈×C₂₃	—	2	2	no	—
265	C₈×C₂₃	—	4	1	no	—
266	C₈×C₂₃	—	4	1	no	—
267	C₂×#478	C₂×C₂₃	1	1	yes	characteristic
268	C₄×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
269	C₄²×D₄	C₂×C₂₃	1	1	yes	characteristic
270	#1989	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1918	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1864	C₂×C₂₃	1	1	yes	characteristic
275	#1862	C₂×C₂₃	1	1	yes	characteristic
276	#1804	C₂×C₂₃	1	1	yes	characteristic
277	#1776	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1776	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1762	C₂×C₂₃	1	1	yes	characteristic
281	#1762	C₂×C₂₃	1	1	yes	characteristic
282	C₂²×C₂₃	—	2	4	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	—	4	2	no	—
288	C₂²×C₂₃	—	4	2	no	—
289	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
290	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
291	C₂²×C₂₃	#525	1	1	yes	characteristic
292	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂²×C₂₃	#575	1	1	yes	characteristic
294	C₂²×C₂₃	#575	1	1	yes	characteristic
295	C₂²×C₂₃	#575	1	1	yes	characteristic
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	2	no	—
298	C₄×C₂₃	—	2	2	no	—
299	C₄×C₂₃	—	2	2	no	—
300	C₄×C₂₃	—	2	2	no	—
301	C₄×C₂₃	—	4	2	no	—
302	C₄×C₂₃	—	4	1	no	—
303	C₄×C₂₃	—	4	2	no	—
304	C₄×C₂₃	—	4	2	no	—
305	C₄×C₂₃	—	4	2	no	—
306	C₄×C₂₃	—	4	2	no	—
307	C₄×C₂₃	—	8	1	no	—
308	C₄×C₂₃	—	8	1	no	—
309	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
310	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
311	C₄×C₂₃	#529	1	1	yes	characteristic
312	C₄×C₂₃	#533	1	1	yes	characteristic
313	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
317	C₂²×C₄²	—	2	1	no	—
318	#478	C₄×C₂₃	1	8	yes	complemented
319	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
331	#389	C₂²×C₂₃	1	1	yes	characteristic
332	#389	C₂²×C₂₃	1	1	yes	characteristic
333	#387	—	2	1	no	—
334	#387	—	2	1	no	—
335	#382	C₂²×C₂₃	1	1	yes	characteristic
336	C₄×C₄⋊C₄	—	2	1	no	—
337	C₄×C₄⋊C₄	—	2	1	no	—
338	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
340	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
341	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
342	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
343	C₄×C₂²⋊C₄	—	2	1	no	—
344	C₄×C₂²⋊C₄	—	2	1	no	—
345	C₄×C₂²⋊C₄	—	2	1	no	—
346	C₄³	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	#340	C₂²×C₂₃	1	1	yes	characteristic
354	#340	C₂²×C₂₃	1	1	yes	characteristic
355	#336	C₂²×C₂₃	1	1	yes	characteristic
356	#336	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₂₃	—	4	4	no	—
358	C₂×C₂₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
359	C₂×C₂₃	#2943	1	1	yes	characteristic
360	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
361	C₂×C₂₃	#3077	1	1	yes	characteristic
362	C₂×C₂₃	#3080	1	1	yes	characteristic
363	C₂×C₂₃	#3111	1	1	yes	characteristic
364	C₂×C₂₃	#3114	1	1	yes	characteristic
365	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	—	2	1	no	—
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	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
373	#125	—	2	2	no	—
374	#125	C₂×C₄×C₂₃	1	4	yes	complemented
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
380	C₄×D₄	Q₈×C₂₃	1	4	yes	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
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×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	—	2	1	no	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	—	4	4	no	—
401	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
402	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
403	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
404	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
408	#107	—	2	2	no	—
409	#107	C₂×C₄×C₂₃	1	4	yes	complemented
410	#106	—	2	2	no	—
411	#106	C₂×C₄×C₂₃	1	4	yes	complemented
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
414	#102	—	2	2	no	—
415	#102	—	2	2	no	—
416	#102	C₂×C₄×C₂₃	1	4	yes	—
417	#102	C₂×C₄×C₂₃	1	4	yes	—
418	#97	—	2	2	no	—
419	#97	—	2	2	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	C₂₃	#21617	1	1	yes	characteristic, direct product factor, Sylow
430	C₂⁴	—	2	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	4	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	1	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	—	4	2	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	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	4	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	—	2	1	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	—	4	2	no	—
485	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
486	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
487	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
488	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
489	C₄⋊C₄	#41×C₂₃	1	2	yes	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	1	no	—
499	C₄²	—	4	2	no	—
500	C₄²	—	4	2	no	—
501	C₄²	—	4	2	no	—
502	C₄²	—	4	2	no	—
503	C₄²	—	4	2	no	—
504	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
505	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
506	C₄²	#41×C₂₃	1	2	yes	—
507	C₄²	#41×C₂₃	1	2	yes	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	1	no	—
515	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	—	2	2	no	—
519	D₄	C₄×Q₈×C₂₃	1	4	yes	—
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	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	4	8	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	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
558	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
559	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
560	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
561	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
562	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
566	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	4	2	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
580	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
581	C₂²	C₂₃×#525	1	1	yes	characteristic
582	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
583	C₂²	C₂₃×#575	1	1	yes	characteristic
584	C₂²	C₂₃×#575	1	1	yes	characteristic
585	C₂²	C₂₃×#575	1	1	yes	characteristic
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	2	no	—
597	C₄	—	8	1	no	—
598	C₄	—	8	1	no	—
599	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
600	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
601	C₄	C₂₃×#529	1	1	yes	characteristic
602	C₄	C₂₃×#533	1	1	yes	characteristic
603	C₂	—	4	4	no	—
604	C₂	C₄×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
605	C₂	C₂₃×#2943	1	1	yes	characteristic
606	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
607	C₂	C₂₃×#3077	1	1	yes	characteristic
608	C₂	C₂₃×#3080	1	1	yes	characteristic
609	C₂	C₂₃×#3111	1	1	yes	characteristic
610	C₂	C₂₃×#3114	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 ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×#125	C₂	semidirect
4. C₂₃×#1989	C₂	semidirect
5. C₂₃×#1949	C₂	semidirect
6. C₂₃×#1920	C₂	semidirect
7. C₂₃×#1918	C₂	semidirect
11. C₂₃×#1776	C₂	semidirect
12. C₂₃×#1776	C₂	semidirect
13. C₂₃×#1767	C₂	semidirect
21. C₂₃×#478	C₄	semidirect
69. C₂₃×#125	C₂×C₄	semidirect
104. C₂₃×#107	C₂×C₄	semidirect
106. C₂₃×#106	C₂×C₄	semidirect
108. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
202. #21617	C₂₃	direct (N × H)
268. C₄×#125	C₂×C₂₃	semidirect
270. #1989	C₂×C₂₃	semidirect
271. #1949	C₂×C₂₃	semidirect
272. #1920	C₂×C₂₃	semidirect
273. #1918	C₂×C₂₃	semidirect
277. #1776	C₂×C₂₃	semidirect
278. #1776	C₂×C₂₃	semidirect
279. #1767	C₂×C₂₃	semidirect
318. #478	C₄×C₂₃	semidirect
374. #125	C₂×C₄×C₂₃	semidirect
409. #107	C₂×C₄×C₂₃	semidirect
411. #106	C₂×C₄×C₂₃	semidirect
413. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
429. C₂₃	#21617	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×#478	2
C₄×C₂₃×#125	2
C₄²×D₄×C₂₃	2
C₂₃×#1989	2
C₂₃×#1949	2
C₂₃×#1920	2
C₂₃×#1918	2
C₂₃×#1864	2
C₂₃×#1862	2
C₂₃×#1804	2
C₂₃×#1776	2
C₂₃×#1767	2
C₂₃×#1762	2
#21617	23

Subgroups — C23×#21617 #19169243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21617 #19169243