Subgroups — C₂₃×#17909 #19165535

Order 5888.

626 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3102	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1588	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1586	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1572	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1488	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1482	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#529	C₂²	1	1	yes	characteristic
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#485	C₄	1	1	yes	characteristic
19	C₂₃×#482	C₄	1	1	yes	characteristic
20	C₂₃×#479	C₄	1	1	yes	characteristic
21	C₂₃×#478	C₄	1	1	yes	characteristic
22	C₂₃×#464	C₄	1	1	yes	characteristic
23	C₂₃×#463	C₄	1	1	yes	characteristic
24	C₂₃×#452	C₄	1	1	yes	characteristic
25	C₂₃×#449	C₄	1	1	yes	characteristic
26	C₂₃×#446	C₂²	1	1	yes	characteristic
27	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
32	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#436	C₂²	1	1	yes	characteristic
35	C₂₃×#435	—	2	1	no	—
36	C₈×D₄×C₂₃	—	2	1	no	—
37	C₈×D₄×C₂₃	—	2	1	no	—
38	C₈×D₄×C₂₃	—	2	1	no	—
39	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#433	—	2	1	no	—
43	C₂₃×#432	C₂²	1	1	yes	characteristic
44	C₂₃×#431	—	2	1	no	—
45	C₂₃×#431	C₂²	1	1	yes	characteristic
46	C₂₃×#431	C₂²	1	1	yes	characteristic
47	C₂₃×#431	C₂²	1	1	yes	characteristic
48	C₂₃×#405	—	2	1	no	—
49	C₂×C₄×C₈×C₂₃	—	2	1	no	—
50	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
51	C₂₃×#335	C₂²	1	1	yes	characteristic
52	C₂₃×#335	C₂²	1	1	yes	characteristic
53	C₂₃×#334	C₂²	1	1	yes	characteristic
54	C₂₃×#334	C₂²	1	1	yes	characteristic
55	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×#322	C₂²	1	1	yes	characteristic
64	C₂₃×#322	C₂²	1	1	yes	characteristic
65	C₂₃×#322	C₂²	1	1	yes	characteristic
66	C₂₃×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆×C₂₃	—	2	1	no	—
73	C₂²×C₈×C₂₃	—	2	1	no	—
74	C₂²×C₈×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
78	C₂₃×#126	—	2	1	no	—
79	C₂₃×#125	C₂×C₄	1	1	yes	characteristic
80	C₂₃×#124	C₂×C₄	1	1	yes	characteristic
81	C₂₃×#123	—	2	1	no	—
82	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
83	C₂₃×#121	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₄×D₄×C₂₃	—	2	1	no	—
88	C₄×D₄×C₂₃	—	2	1	no	—
89	C₄×D₄×C₂₃	—	2	1	no	—
90	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×#117	—	2	1	no	—
94	C₂₃×#117	—	2	1	no	—
95	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
99	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
100	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
101	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
102	C₂₃×C₄⋊C₈	—	2	1	no	—
103	C₂₃×C₄⋊C₈	—	2	1	no	—
104	C₂₃×C₄⋊C₈	—	2	1	no	—
105	C₂₃×C₄⋊C₈	—	2	1	no	—
106	C₂₃×C₄⋊C₈	—	2	1	no	—
107	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
108	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
109	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
114	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
115	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
116	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
117	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
118	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
119	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
120	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
121	C₂₃×C₂²⋊C₈	—	2	1	no	—
122	C₂₃×C₂²⋊C₈	—	2	1	no	—
123	C₂₃×C₂²⋊C₈	—	2	1	no	—
124	C₂₃×C₂²⋊C₈	—	2	1	no	—
125	C₂₃×C₂²⋊C₈	—	2	1	no	—
126	C₂₃×C₂²⋊C₈	—	2	1	no	—
127	C₂₃×C₂²⋊C₈	—	2	1	no	—
128	C₂₃×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
129	C₂₃×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
130	C₂₃×#97	—	2	1	no	—
131	C₂₃×#97	—	2	1	no	—
132	C₄×C₈×C₂₃	—	2	1	no	—
133	C₄×C₈×C₂₃	—	2	1	no	—
134	C₄×C₈×C₂₃	—	2	1	no	—
135	C₄×C₈×C₂₃	—	2	1	no	—
136	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
137	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
138	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	#41×C₂₃	—	2	2	no	—
145	#41×C₂₃	—	2	2	no	—
146	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
159	Q₁₆×C₂₃	—	2	2	no	—
160	SD₁₆×C₂₃	—	2	2	no	—
161	SD₁₆×C₂₃	—	2	2	no	—
162	D₈×C₂₃	—	2	2	no	—
163	M₁₆×C₂₃	—	4	2	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	4	1	no	—
171	C₂×C₈×C₂₃	—	4	1	no	—
172	C₂×C₈×C₂₃	—	4	2	no	—
173	C₂×C₈×C₂₃	—	4	2	no	—
174	C₂×C₈×C₂₃	—	4	2	no	—
175	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	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	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₂₃	—	2	1	no	—
194	C₂²⋊C₄×C₂₃	—	2	1	no	—
195	C₂²⋊C₄×C₂₃	—	2	1	no	—
196	C₂²⋊C₄×C₂₃	—	2	1	no	—
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	2	1	no	—
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	2	1	no	—
202	C₄²×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₂₃	#41	1	1	yes	characteristic
208	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
209	#17909	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₂₃	—	2	1	no	—
211	C₂³×C₂₃	—	2	1	no	—
212	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
213	Q₈×C₂₃	—	2	1	no	—
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	2	1	no	—
216	Q₈×C₂₃	#131	1	2	yes	—
217	D₄×C₂₃	—	2	4	no	—
218	D₄×C₂₃	—	2	1	no	—
219	D₄×C₂₃	—	2	1	no	—
220	D₄×C₂₃	—	2	1	no	—
221	D₄×C₂₃	—	4	2	no	—
222	D₄×C₂₃	#131	1	2	yes	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	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	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	2	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	C₄×D₄	1	1	yes	characteristic
241	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
242	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
243	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
246	C₈×C₂₃	—	2	1	no	—
247	C₈×C₂₃	—	2	1	no	—
248	C₈×C₂₃	—	2	1	no	—
249	C₈×C₂₃	—	2	1	no	—
250	C₈×C₂₃	—	2	1	no	—
251	C₈×C₂₃	—	2	1	no	—
252	C₈×C₂₃	—	2	1	no	—
253	C₈×C₂₃	—	2	1	no	—
254	C₈×C₂₃	—	4	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	#3102	C₂×C₂₃	1	1	yes	characteristic
261	#2966	C₂×C₂₃	1	1	yes	characteristic, complemented
262	C₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1588	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1587	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1586	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1585	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1580	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1579	C₂×C₂₃	1	1	yes	characteristic, complemented
269	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
270	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1572	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1488	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1482	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1452	C₂×C₂₃	1	1	yes	characteristic, complemented
275	C₂²×C₂₃	—	2	1	no	—
276	C₂²×C₂₃	—	2	2	no	—
277	C₂²×C₂₃	—	2	1	no	—
278	C₂²×C₂₃	—	2	1	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	4	1	no	—
281	C₂²×C₂₃	—	4	1	no	—
282	C₂²×C₂₃	—	4	1	no	—
283	C₂²×C₂₃	—	4	1	no	—
284	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
285	C₄×C₂₃	—	2	1	no	—
286	C₄×C₂₃	—	2	2	no	—
287	C₄×C₂₃	—	2	1	no	—
288	C₄×C₂₃	—	2	1	no	—
289	C₄×C₂₃	—	2	1	no	—
290	C₄×C₂₃	—	4	1	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	4	1	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
296	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
297	C₄×C₂₃	#575	1	1	yes	characteristic
298	C₄×C₂₃	#575	1	1	yes	characteristic
299	#529	C₂²×C₂₃	1	1	yes	characteristic
300	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
301	#485	C₄×C₂₃	1	1	yes	characteristic
302	#482	C₄×C₂₃	1	1	yes	characteristic
303	#479	C₄×C₂₃	1	1	yes	characteristic
304	#478	C₄×C₂₃	1	1	yes	characteristic
305	#464	C₄×C₂₃	1	1	yes	characteristic
306	#463	C₄×C₂₃	1	1	yes	characteristic
307	#452	C₄×C₂₃	1	1	yes	characteristic
308	#449	C₄×C₂₃	1	1	yes	characteristic
309	#446	C₂²×C₂₃	1	1	yes	characteristic
310	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
311	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
313	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
314	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
315	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
316	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
317	#436	C₂²×C₂₃	1	1	yes	characteristic
318	#435	—	2	1	no	—
319	C₈×D₄	—	2	1	no	—
320	C₈×D₄	—	2	1	no	—
321	C₈×D₄	—	2	1	no	—
322	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
323	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
324	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
325	#433	—	2	1	no	—
326	#432	C₂²×C₂₃	1	1	yes	characteristic
327	#431	—	2	1	no	—
328	#431	C₂²×C₂₃	1	1	yes	characteristic
329	#431	C₂²×C₂₃	1	1	yes	characteristic
330	#431	C₂²×C₂₃	1	1	yes	characteristic
331	#405	—	2	1	no	—
332	C₂×C₄×C₈	—	2	1	no	—
333	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
334	#335	C₂²×C₂₃	1	1	yes	characteristic
335	#335	C₂²×C₂₃	1	1	yes	characteristic
336	#334	C₂²×C₂₃	1	1	yes	characteristic
337	#334	C₂²×C₂₃	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
346	#322	C₂²×C₂₃	1	1	yes	characteristic
347	#322	C₂²×C₂₃	1	1	yes	characteristic
348	#322	C₂²×C₂₃	1	1	yes	characteristic
349	#322	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×C₂₃	—	4	1	no	—
351	C₂×C₂₃	—	4	2	no	—
352	C₂×C₂₃	—	8	1	no	—
353	C₂×C₂₃	#2929	1	1	yes	characteristic
354	C₂×C₂₃	#2943	1	1	yes	characteristic
355	C₂×C₂₃	#2955	1	1	yes	characteristic
356	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
357	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
358	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
359	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
360	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
361	C₂×M₁₆	—	2	1	no	—
362	C₂²×C₈	—	2	1	no	—
363	C₂²×C₈	—	2	1	no	—
364	C₂²×C₈	—	2	1	no	—
365	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
366	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
367	#126	—	2	1	no	—
368	#125	C₂×C₄×C₂₃	1	1	yes	characteristic
369	#124	C₂×C₄×C₂₃	1	1	yes	characteristic
370	#123	—	2	1	no	—
371	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
372	#121	C₂×C₄×C₂₃	1	1	yes	characteristic
373	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
374	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
375	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
380	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
381	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
387	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
388	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
389	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
390	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	—	2	1	no	—
393	C₄⋊C₈	—	2	1	no	—
394	C₄⋊C₈	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
397	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
398	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
402	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
403	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
404	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
405	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
406	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
407	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
408	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
409	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
410	C₂²⋊C₈	—	2	1	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	1	no	—
413	C₂²⋊C₈	—	2	1	no	—
414	C₂²⋊C₈	—	2	1	no	—
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
418	C₂²⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
419	#97	—	2	1	no	—
420	#97	—	2	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	—	2	1	no	—
423	C₄×C₈	—	2	1	no	—
424	C₄×C₈	—	2	1	no	—
425	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
426	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
427	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
428	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₂₃	#17909	1	1	yes	characteristic, direct product factor, Sylow
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
437	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
441	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
448	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
449	Q₁₆	—	2	2	no	—
450	SD₁₆	—	2	2	no	—
451	SD₁₆	—	2	2	no	—
452	D₈	—	2	2	no	—
453	M₁₆	—	4	2	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	4	1	no	—
461	C₂×C₈	—	4	1	no	—
462	C₂×C₈	—	4	2	no	—
463	C₂×C₈	—	4	2	no	—
464	C₂×C₈	—	4	2	no	—
465	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
466	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
467	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
468	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
469	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
472	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	C₂²×C₄×C₂₃	1	1	yes	characteristic
480	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
481	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
482	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
483	C₂²⋊C₄	—	2	1	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	2	1	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
496	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₄²	#41×C₂₃	1	1	yes	characteristic
498	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
502	Q₈	—	2	1	no	—
503	Q₈	—	2	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	C₂₃×#131	1	2	yes	—
506	D₄	—	2	4	no	—
507	D₄	—	2	1	no	—
508	D₄	—	2	1	no	—
509	D₄	—	2	1	no	—
510	D₄	—	4	2	no	—
511	D₄	C₂₃×#131	1	2	yes	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
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	—	4	1	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
530	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
533	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
534	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
535	C₈	—	2	1	no	—
536	C₈	—	2	1	no	—
537	C₈	—	2	1	no	—
538	C₈	—	2	1	no	—
539	C₈	—	2	1	no	—
540	C₈	—	2	1	no	—
541	C₈	—	2	1	no	—
542	C₈	—	2	1	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
559	C₄	—	2	1	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	1	no	—
562	C₄	—	2	1	no	—
563	C₄	—	2	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
570	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
571	C₄	C₂₃×#575	1	1	yes	characteristic
572	C₄	C₂₃×#575	1	1	yes	characteristic
573	C₂	—	4	1	no	—
574	C₂	—	4	2	no	—
575	C₂	—	8	1	no	—
576	C₂	C₂₃×#2929	1	1	yes	characteristic
577	C₂	C₂₃×#2943	1	1	yes	characteristic
578	C₂	C₂₃×#2955	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	C₂×C₄×D₄
prime radical	C₂₃×#449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2966	C₂	semidirect
3. C₈×#41×C₂₃	C₂	semidirect
4. C₂₃×#1588	C₂	semidirect
5. C₂₃×#1587	C₂	semidirect
6. C₂₃×#1586	C₂	semidirect
7. C₂₃×#1585	C₂	semidirect
8. C₂₃×#1580	C₂	semidirect
9. C₂₃×#1579	C₂	semidirect
10. C₂₃×SD₁₆⋊C₈	C₂	semidirect
11. C₂₃×SD₁₆⋊C₈	C₂	semidirect
12. C₂₃×#1572	C₂	semidirect
13. C₂₃×#1488	C₂	semidirect
14. C₂₃×#1482	C₂	semidirect
15. C₂₃×#1452	C₂	semidirect
209. #17909	C₂₃	direct (N × H)
261. #2966	C₂×C₂₃	semidirect
262. C₈×#41	C₂×C₂₃	semidirect
263. #1588	C₂×C₂₃	semidirect
264. #1587	C₂×C₂₃	semidirect
265. #1586	C₂×C₂₃	semidirect
266. #1585	C₂×C₂₃	semidirect
267. #1580	C₂×C₂₃	semidirect
268. #1579	C₂×C₂₃	semidirect
269. SD₁₆⋊C₈	C₂×C₂₃	semidirect
270. SD₁₆⋊C₈	C₂×C₂₃	semidirect
271. #1572	C₂×C₂₃	semidirect
272. #1488	C₂×C₂₃	semidirect
273. #1482	C₂×C₂₃	semidirect
274. #1452	C₂×C₂₃	semidirect
433. C₂₃	#17909	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×#3102	2
C₂₃×#2966	2
C₈×#41×C₂₃	2
C₂₃×#1588	2
C₂₃×#1587	2
C₂₃×#1586	2
C₂₃×#1585	2
C₂₃×#1580	2
C₂₃×#1579	2
C₂₃×SD₁₆⋊C₈	2
C₂₃×#1572	2
C₂₃×#1488	2
C₂₃×#1482	2
C₂₃×#1452	2
#17909	23

Subgroups — C23×#17909 #19165535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17909 #19165535