Subgroups — C₂₃×#17510 #19165136

Order 5888.

970 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic
2	C₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1494	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1488	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1482	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1480	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1476	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#559	—	2	1	no	—
13	C₂₃×#559	C₄	1	1	yes	characteristic
14	C₂₃×#557	C₄	1	1	yes	characteristic
15	C₂₃×#556	—	2	1	no	—
16	C₂₃×#556	C₄	1	1	yes	characteristic
17	C₂₃×#553	—	2	2	no	—
18	C₂₃×#553	C₄	1	2	yes	—
19	C₂₃×#550	C₄	1	1	yes	characteristic
20	C₂₃×#548	C₄	1	2	yes	—
21	C₂₃×#532	—	2	1	no	—
22	C₂₃×#532	C₂²	1	1	yes	characteristic
23	C₂₃×#529	—	2	2	no	—
24	C₂₃×#529	C₂²	1	2	yes	—
25	C₄×#41×C₂₃	—	2	1	no	—
26	C₄×#41×C₂₃	C₂²	1	2	yes	—
27	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₈×Q₈×C₂₃	—	2	1	no	—
30	C₈×D₄×C₂₃	—	2	2	no	—
31	C₈×D₄×C₂₃	—	2	1	no	—
32	C₂₃×#433	—	2	2	no	—
33	C₂₃×#432	C₂²	1	2	yes	—
34	C₂₃×#431	C₂²	1	1	yes	characteristic
35	C₂₃×#431	C₂²	1	2	yes	—
36	C₂₃×#431	C₂²	1	2	yes	—
37	C₂₃×#423	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₃×#408	—	2	1	no	—
40	C₂₃×#405	—	2	1	no	—
41	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
43	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
45	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
46	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
47	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
50	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
52	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₂₃	—	2	1	no	—
54	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
55	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂²×C₈×C₂₃	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₂₃×#127	—	2	1	no	—
64	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
65	C₂₃×#126	—	2	2	no	—
66	C₂₃×#126	—	2	2	no	—
67	C₂₃×#126	—	2	2	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	C₂×C₄	1	2	yes	—
70	C₂₃×#124	—	2	1	no	—
71	C₂₃×#124	—	2	1	no	—
72	C₂₃×#124	—	2	2	no	—
73	C₂₃×#124	—	2	1	no	—
74	C₂₃×#124	C₂×C₄	1	2	yes	—
75	C₂₃×#123	—	2	2	no	—
76	C₂₃×#123	—	2	2	no	—
77	C₂₃×#123	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	—	2	2	no	—
79	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
80	C₂₃×#121	—	2	2	no	—
81	C₂₃×#121	—	2	2	no	—
82	C₂₃×#121	—	2	1	no	—
83	C₂₃×#121	—	2	2	no	—
84	C₂₃×#121	C₂×C₄	1	2	yes	—
85	C₂₃×#121	C₂×C₄	1	2	yes	—
86	C₄×Q₈×C₂₃	—	2	1	no	—
87	C₄×Q₈×C₂₃	—	2	1	no	—
88	C₄×Q₈×C₂₃	—	2	1	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₂₃	—	2	2	no	—
98	C₄×D₄×C₂₃	—	2	2	no	—
99	C₄×D₄×C₂₃	—	2	2	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	2	no	—
103	C₄×D₄×C₂₃	—	2	2	no	—
104	C₄×D₄×C₂₃	—	2	1	no	—
105	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₂₃	D₄	1	2	yes	—
109	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
110	C₄×D₄×C₂₃	C₂³	1	2	yes	—
111	C₂₃×#117	—	2	2	no	—
112	C₂₃×#117	—	2	2	no	—
113	C₂₃×#117	C₂×C₄	1	2	yes	—
114	C₂₃×#117	D₄	1	1	yes	characteristic
115	C₂₃×#117	D₄	1	2	yes	—
116	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	—	2	1	no	—
119	C₂×C₄²×C₂₃	—	2	2	no	—
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
122	C₂₃×C₄⋊C₈	—	2	1	no	—
123	C₂₃×C₄⋊C₈	—	2	1	no	—
124	C₂₃×C₄⋊C₈	—	2	2	no	—
125	C₂₃×C₄⋊C₈	—	2	1	no	—
126	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
127	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
128	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
129	C₂₃×C₂²⋊C₈	—	2	2	no	—
130	C₂₃×C₂²⋊C₈	—	2	2	no	—
131	C₂₃×C₂²⋊C₈	—	2	1	no	—
132	C₂₃×C₂²⋊C₈	—	2	2	no	—
133	C₂₃×#97	—	2	2	no	—
134	C₄×C₈×C₂₃	—	2	2	no	—
135	C₄×C₈×C₂₃	—	2	1	no	—
136	C₄×C₈×C₂₃	—	2	2	no	—
137	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₂₃	C₂³	1	2	yes	—
139	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
140	#41×C₂₃	—	2	2	no	—
141	#41×C₂₃	—	2	2	no	—
142	#41×C₂₃	—	2	4	no	—
143	#41×C₂₃	—	4	2	no	—
144	#41×C₂₃	—	4	2	no	—
145	#41×C₂₃	—	4	2	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	2	no	—
149	C₂×Q₈×C₂₃	—	4	1	no	—
150	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	—	2	2	no	—
154	C₂×D₄×C₂₃	—	2	2	no	—
155	C₂×D₄×C₂₃	—	2	2	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
159	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
160	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	2	no	—
164	C₂²×C₄×C₂₃	—	2	2	no	—
165	C₂²×C₄×C₂₃	—	2	2	no	—
166	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	M₁₆×C₂₃	—	4	2	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	2	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	2	no	—
179	C₂×C₈×C₂₃	—	4	4	no	—
180	C₂×C₈×C₂₃	—	4	2	no	—
181	C₂×C₈×C₂₃	—	4	2	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
195	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
196	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
199	C₂²⋊C₄×C₂₃	—	2	2	no	—
200	C₂²⋊C₄×C₂₃	—	2	2	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	2	no	—
204	C₂²⋊C₄×C₂₃	—	2	2	no	—
205	C₂²⋊C₄×C₂₃	—	2	2	no	—
206	C₂²⋊C₄×C₂₃	—	2	2	no	—
207	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
208	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
210	C₄²×C₂₃	—	2	2	no	—
211	C₄²×C₂₃	—	2	2	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
217	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₂₃	C₂×D₄	1	2	yes	—
219	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₂₃	C₂×D₄	1	2	yes	—
222	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
223	#17510	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	Q₈×C₂₃	—	2	1	no	—
228	Q₈×C₂₃	—	2	2	no	—
229	Q₈×C₂₃	—	2	1	no	—
230	Q₈×C₂₃	—	2	1	no	—
231	Q₈×C₂₃	—	4	2	no	—
232	Q₈×C₂₃	—	4	2	no	—
233	Q₈×C₂₃	#131	1	2	yes	—
234	D₄×C₂₃	—	2	4	no	—
235	D₄×C₂₃	—	2	2	no	—
236	D₄×C₂₃	—	2	2	no	—
237	D₄×C₂₃	—	2	2	no	—
238	D₄×C₂₃	—	4	4	no	—
239	D₄×C₂₃	—	4	4	no	—
240	D₄×C₂₃	—	4	2	no	—
241	D₄×C₂₃	—	8	1	no	—
242	D₄×C₂₃	—	8	2	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	4	2	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	4	no	—
257	C₂×C₄×C₂₃	—	4	2	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	2	no	—
260	C₂×C₄×C₂₃	—	8	2	no	—
261	C₂×C₄×C₂₃	—	8	1	no	—
262	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
263	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
264	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₂₃	#135	1	2	yes	—
269	C₂×C₄×C₂₃	#135	1	2	yes	—
270	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
273	C₈×C₂₃	—	2	2	no	—
274	C₈×C₂₃	—	4	1	no	—
275	C₈×C₂₃	—	4	2	no	—
276	C₈×C₂₃	—	4	1	no	—
277	C₈×C₂₃	—	4	1	no	—
278	C₈×C₂₃	—	4	2	no	—
279	#3480	C₂×C₂₃	1	1	yes	characteristic
280	C₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#2925	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1494	C₂×C₂₃	1	2	yes	complemented
283	#1492	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1488	C₂×C₂₃	1	2	yes	complemented
285	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1482	C₂×C₂₃	1	2	yes	complemented
287	#1480	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1476	C₂×C₂₃	1	2	yes	complemented
289	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
290	C₂²×C₂₃	—	2	1	no	—
291	C₂²×C₂₃	—	2	1	no	—
292	C₂²×C₂₃	—	2	2	no	—
293	C₂²×C₂₃	—	2	2	no	—
294	C₂²×C₂₃	—	2	2	no	—
295	C₂²×C₂₃	—	4	2	no	—
296	C₂²×C₂₃	—	4	2	no	—
297	C₂²×C₂₃	—	4	2	no	—
298	C₂²×C₂₃	#416	1	2	yes	—
299	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
300	C₄×C₂₃	—	2	2	no	—
301	C₄×C₂₃	—	2	1	no	—
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	2	2	no	—
304	C₄×C₂₃	—	2	1	no	—
305	C₄×C₂₃	—	2	2	no	—
306	C₄×C₂₃	—	2	1	no	—
307	C₄×C₂₃	—	4	1	no	—
308	C₄×C₂₃	—	4	2	no	—
309	C₄×C₂₃	—	4	1	no	—
310	C₄×C₂₃	—	4	1	no	—
311	C₄×C₂₃	#416	1	2	yes	—
312	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
313	C₄×C₂₃	#568	1	1	yes	characteristic
314	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
315	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
316	#559	—	2	1	no	—
317	#559	C₄×C₂₃	1	1	yes	characteristic
318	#557	C₄×C₂₃	1	1	yes	characteristic
319	#556	—	2	1	no	—
320	#556	C₄×C₂₃	1	1	yes	characteristic
321	#553	—	2	2	no	—
322	#553	C₄×C₂₃	1	2	yes	—
323	#550	C₄×C₂₃	1	1	yes	characteristic
324	#548	C₄×C₂₃	1	2	yes	—
325	#532	—	2	1	no	—
326	#532	C₂²×C₂₃	1	1	yes	characteristic
327	#529	—	2	2	no	—
328	#529	C₂²×C₂₃	1	2	yes	—
329	C₄×#41	—	2	1	no	—
330	C₄×#41	C₂²×C₂₃	1	2	yes	—
331	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₈×Q₈	—	2	1	no	—
334	C₈×D₄	—	2	2	no	—
335	C₈×D₄	—	2	1	no	—
336	#433	—	2	2	no	—
337	#432	C₂²×C₂₃	1	2	yes	—
338	#431	C₂²×C₂₃	1	2	yes	—
339	#431	C₂²×C₂₃	1	1	yes	characteristic
340	#431	C₂²×C₂₃	1	2	yes	—
341	#423	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	#408	—	2	1	no	—
344	#405	—	2	1	no	—
345	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
347	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
349	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
351	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
352	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
353	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
354	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
355	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
356	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
357	C₂×C₂₃	—	2	2	no	—
358	C₂×C₂₃	—	4	2	no	—
359	C₂×C₂₃	—	8	2	no	—
360	C₂×C₂₃	#2880	1	1	yes	characteristic
361	C₂×C₂₃	C₂×#416	1	1	yes	characteristic
362	C₂×C₂₃	#2901	1	1	yes	characteristic
363	C₂×#41	—	2	1	no	—
364	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
365	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
366	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
367	C₂×M₁₆	—	2	1	no	—
368	C₂²×C₈	—	2	1	no	—
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	—	2	1	no	—
371	C₄⋊Q₈	—	2	1	no	—
372	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
373	#127	—	2	1	no	—
374	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
375	#126	—	2	2	no	—
376	#126	—	2	2	no	—
377	#126	—	2	2	no	—
378	#125	—	2	2	no	—
379	#125	C₂×C₄×C₂₃	1	2	yes	—
380	#124	—	2	1	no	—
381	#124	—	2	1	no	—
382	#124	—	2	2	no	—
383	#124	—	2	1	no	—
384	#124	C₂×C₄×C₂₃	1	2	yes	—
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	#123	—	2	2	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
390	#121	—	2	2	no	—
391	#121	—	2	2	no	—
392	#121	—	2	1	no	—
393	#121	—	2	2	no	—
394	#121	C₂×C₄×C₂₃	1	2	yes	—
395	#121	C₂×C₄×C₂₃	1	2	yes	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
401	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
402	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
403	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
404	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
405	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
416	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
417	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
418	C₄×D₄	D₄×C₂₃	1	2	yes	—
419	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
420	C₄×D₄	C₂³×C₂₃	1	2	yes	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	C₂×C₄×C₂₃	1	2	yes	—
424	#117	D₄×C₂₃	1	1	yes	characteristic
425	#117	D₄×C₂₃	1	2	yes	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	2	no	—
430	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
432	C₄⋊C₈	—	2	1	no	—
433	C₄⋊C₈	—	2	1	no	—
434	C₄⋊C₈	—	2	2	no	—
435	C₄⋊C₈	—	2	1	no	—
436	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
437	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
438	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
439	C₂²⋊C₈	—	2	2	no	—
440	C₂²⋊C₈	—	2	2	no	—
441	C₂²⋊C₈	—	2	1	no	—
442	C₂²⋊C₈	—	2	2	no	—
443	#97	—	2	2	no	—
444	C₄×C₈	—	2	2	no	—
445	C₄×C₈	—	2	1	no	—
446	C₄×C₈	—	2	2	no	—
447	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₂₃	1	2	yes	—
449	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₂₃	#17510	1	1	yes	characteristic, direct product factor, Sylow
451	#41	—	2	2	no	—
452	#41	—	2	2	no	—
453	#41	—	2	4	no	—
454	#41	—	4	2	no	—
455	#41	—	4	2	no	—
456	#41	—	4	2	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	4	1	no	—
461	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
462	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
463	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
470	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
471	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
482	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
483	M₁₆	—	4	2	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	4	4	no	—
491	C₂×C₈	—	4	2	no	—
492	C₂×C₈	—	4	2	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
504	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
505	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
506	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
507	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
508	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
509	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
519	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
520	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
521	C₄²	—	2	2	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
528	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₂₃	1	2	yes	—
530	C₄²	C₂×D₄×C₂₃	1	2	yes	—
531	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
532	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
533	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
534	C₂³	—	2	2	no	—
535	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
536	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
537	Q₈	—	2	1	no	—
538	Q₈	—	2	2	no	—
539	Q₈	—	2	1	no	—
540	Q₈	—	2	1	no	—
541	Q₈	—	4	2	no	—
542	Q₈	—	4	2	no	—
543	Q₈	C₂₃×#131	1	2	yes	—
544	D₄	—	2	4	no	—
545	D₄	—	2	2	no	—
546	D₄	—	2	2	no	—
547	D₄	—	2	2	no	—
548	D₄	—	4	4	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	2	no	—
551	D₄	—	8	1	no	—
552	D₄	—	8	2	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	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	—	4	2	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	4	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	8	2	no	—
571	C₂×C₄	—	8	1	no	—
572	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
573	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
574	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
575	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
578	C₂×C₄	C₂₃×#135	1	2	yes	—
579	C₂×C₄	C₂₃×#135	1	2	yes	—
580	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
581	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
583	C₈	—	2	2	no	—
584	C₈	—	4	2	no	—
585	C₈	—	4	1	no	—
586	C₈	—	4	1	no	—
587	C₈	—	4	1	no	—
588	C₈	—	4	2	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	C₂₃×#416	1	2	yes	—
598	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
599	C₄	—	2	2	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	1	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	1	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	2	no	—
608	C₄	—	4	1	no	—
609	C₄	—	4	1	no	—
610	C₄	C₂₃×#416	1	2	yes	—
611	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
612	C₄	C₂₃×#568	1	1	yes	characteristic
613	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
614	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
615	C₂	—	2	2	no	—
616	C₂	—	4	2	no	—
617	C₂	—	8	2	no	—
618	C₂	C₂₃×#2880	1	1	yes	characteristic
619	C₂	C₂×C₂₃×#416	1	1	yes	characteristic
620	C₂	C₂₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
prime radical	C₂₃×#3480	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×#41×C₂₃	C₂	semidirect
3. C₂₃×#2925	C₂	semidirect
4. C₂₃×#1494	C₂	semidirect
5. C₂₃×#1492	C₂	semidirect
6. C₂₃×#1488	C₂	semidirect
7. C₂₃×#1487	C₂	semidirect
8. C₂₃×#1482	C₂	semidirect
9. C₂₃×#1480	C₂	semidirect
10. C₂₃×#1476	C₂	semidirect
11. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
223. #17510	C₂₃	direct (N × H)
280. C₈×#41	C₂×C₂₃	semidirect
281. #2925	C₂×C₂₃	semidirect
282. #1494	C₂×C₂₃	semidirect
283. #1492	C₂×C₂₃	semidirect
284. #1488	C₂×C₂₃	semidirect
285. #1487	C₂×C₂₃	semidirect
286. #1482	C₂×C₂₃	semidirect
287. #1480	C₂×C₂₃	semidirect
288. #1476	C₂×C₂₃	semidirect
289. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
450. C₂₃	#17510	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×#3480	2
C₈×#41×C₂₃	2
C₂₃×#2925	2
C₂₃×#1494	2
C₂₃×#1492	2
C₂₃×#1488	2
C₂₃×#1487	2
C₂₃×#1482	2
C₂₃×#1480	2
C₂₃×#1476	2
C₂×C₂₃×Q₈⋊C₈	2
#17510	23

Subgroups — C23×#17510 #19165136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17510 #19165136