Subgroups — C₂₃×#25445 #19173071

Order 5888.

886 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3063	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3060	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2038	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2007	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1908	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#137	C₂²	1	2	yes	—
13	C₂×C₂₃×#137	C₂²	1	2	yes	—
14	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#131	C₂²	1	2	yes	—
18	C₂₃×#537	C₂²	1	2	yes	—
19	C₂₃×#536	C₂²	1	2	yes	—
20	C₂₃×#497	—	2	2	no	—
21	C₂₃×#495	—	2	1	no	—
22	C₂₃×#494	—	2	1	no	—
23	C₂₃×#473	—	2	4	no	—
24	C₂₃×#470	—	2	1	no	—
25	C₂₃×#470	—	2	2	no	—
26	C₂₃×#469	—	2	1	no	—
27	C₂₃×#468	—	2	2	no	—
28	C₂₃×#467	—	2	2	no	—
29	C₂₃×#465	—	2	2	no	—
30	C₂₃×#465	—	2	2	no	—
31	C₂₃×#444	—	2	4	no	—
32	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
33	C₂₃×#428	C₂²	1	2	yes	—
34	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#104	C₂²	1	2	yes	—
36	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
37	C₂₃×#417	C₂²	1	2	yes	—
38	C₂₃×#416	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#103	C₂²	1	2	yes	—
40	C₂×C₂₃×#101	C₂²	1	2	yes	—
41	C₂₃×#405	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
43	C₂₃×#391	C₂²	1	2	yes	—
44	C₂₃×#343	C₂²	1	2	yes	—
45	C₂₃×#341	C₂²	1	1	yes	characteristic
46	C₂₃×#339	C₂²	1	1	yes	characteristic
47	C₂×#41×C₂₃	C₂³	1	2	yes	—
48	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
50	C₂₃×#137	—	2	4	no	—
51	C₂₃×#137	—	2	4	no	—
52	C₂₃×#137	—	4	2	no	—
53	C₂₃×#137	—	4	2	no	—
54	C₂₃×#135	—	2	2	no	—
55	C₂₃×#135	—	2	2	no	—
56	C₂×Q₁₆×C₂₃	—	2	2	no	—
57	C₂×Q₁₆×C₂₃	—	2	2	no	—
58	C₂×Q₁₆×C₂₃	—	2	2	no	—
59	C₂×Q₁₆×C₂₃	—	2	1	no	—
60	C₂×Q₁₆×C₂₃	—	2	1	no	—
61	C₂×Q₁₆×C₂₃	—	2	2	no	—
62	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
63	C₂×SD₁₆×C₂₃	—	2	2	no	—
64	C₂×SD₁₆×C₂₃	—	2	2	no	—
65	C₂×SD₁₆×C₂₃	D₄	1	2	yes	—
66	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
67	C₂₃×#131	—	2	4	no	—
68	C₂₃×#131	—	2	2	no	—
69	C₂₃×#131	—	2	2	no	—
70	C₂₃×#131	—	4	2	no	—
71	C₂×M₁₆×C₂₃	—	2	2	no	—
72	C₂×M₁₆×C₂₃	—	2	1	no	—
73	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
75	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	2	no	—
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	—	2	2	no	—
81	C₂₃×#124	—	2	2	no	—
82	C₂₃×#123	—	4	2	no	—
83	C₂₃×C₂²⋊Q₈	—	2	2	no	—
84	C₂₃×C₂²⋊Q₈	—	2	2	no	—
85	C₂₃×C₂²⋊Q₈	—	2	2	no	—
86	C₂₃×C₂²⋊Q₈	—	2	2	no	—
87	C₂₃×#121	—	2	2	no	—
88	C₂₃×#121	—	2	2	no	—
89	C₂₃×#121	—	4	2	no	—
90	C₂₃×#117	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×#108	D₄	1	4	yes	complemented
94	C₂₃×#107	—	2	2	no	—
95	C₂₃×#106	—	2	1	no	—
96	C₂₃×#106	—	2	1	no	—
97	C₂₃×#106	—	2	2	no	—
98	C₂₃×#104	—	2	4	no	—
99	C₂₃×#104	—	2	2	no	—
100	C₂₃×#103	—	2	2	no	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#103	—	2	2	no	—
103	C₂₃×#103	D₄	1	2	yes	—
104	C₂₃×#102	—	2	2	no	—
105	C₂₃×#102	D₄	1	2	yes	—
106	C₂₃×#101	—	2	4	no	—
107	C₂₃×#97	—	2	2	no	—
108	C₂₃×#97	—	2	1	no	—
109	C₄×C₈×C₂₃	—	2	1	no	—
110	C₂₃×#95	—	4	2	no	—
111	#41×C₂₃	—	2	2	no	—
112	#41×C₂₃	—	2	1	no	—
113	#41×C₂₃	—	2	1	no	—
114	#41×C₂₃	—	2	1	no	—
115	#41×C₂₃	—	2	2	no	—
116	#41×C₂₃	—	2	2	no	—
117	#41×C₂₃	—	2	2	no	—
118	#41×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	—	2	2	no	—
120	C₂×Q₈×C₂₃	—	2	2	no	—
121	C₂×Q₈×C₂₃	—	2	2	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	2	2	no	—
124	C₂×Q₈×C₂₃	—	2	2	no	—
125	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
126	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	4	2	no	—
130	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₂₃	—	2	2	no	—
139	Q₁₆×C₂₃	—	4	4	no	—
140	Q₁₆×C₂₃	—	4	4	no	—
141	Q₁₆×C₂₃	—	4	2	no	—
142	Q₁₆×C₂₃	—	4	2	no	—
143	SD₁₆×C₂₃	—	2	4	no	—
144	SD₁₆×C₂₃	—	4	4	no	—
145	SD₁₆×C₂₃	—	4	4	no	—
146	D₈×C₂₃	—	2	2	no	—
147	M₁₆×C₂₃	—	2	2	no	—
148	M₁₆×C₂₃	—	2	1	no	—
149	M₁₆×C₂₃	—	2	1	no	—
150	M₁₆×C₂₃	—	2	1	no	—
151	M₁₆×C₂₃	—	4	4	no	—
152	M₁₆×C₂₃	—	4	2	no	—
153	M₁₆×C₂₃	C₂×D₄	1	4	yes	—
154	M₁₆×C₂₃	#41	1	2	yes	—
155	C₂×C₈×C₂₃	—	2	2	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	—	4	4	no	—
161	C₂×C₈×C₂₃	—	4	2	no	—
162	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
166	C₂×C₈×C₂₃	#41	1	2	yes	—
167	C₄⋊C₄×C₂₃	—	2	2	no	—
168	C₄⋊C₄×C₂₃	—	2	2	no	—
169	C₄⋊C₄×C₂₃	—	4	2	no	—
170	C₄⋊C₄×C₂₃	—	4	2	no	—
171	C₄⋊C₄×C₂₃	—	4	2	no	—
172	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
173	C₂²⋊C₄×C₂₃	—	2	1	no	—
174	C₂²⋊C₄×C₂₃	—	4	2	no	—
175	C₂²⋊C₄×C₂₃	—	4	2	no	—
176	C₂²⋊C₄×C₂₃	—	4	2	no	—
177	C₂²⋊C₄×C₂₃	—	4	2	no	—
178	C₄²×C₂₃	—	2	1	no	—
179	C₄²×C₂₃	—	2	1	no	—
180	C₄²×C₂₃	—	2	1	no	—
181	#25445	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₂₃	—	2	2	no	—
183	C₂³×C₂₃	—	2	1	no	—
184	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
185	Q₈×C₂₃	—	2	2	no	—
186	Q₈×C₂₃	—	2	1	no	—
187	Q₈×C₂₃	—	2	1	no	—
188	Q₈×C₂₃	—	2	2	no	—
189	Q₈×C₂₃	—	4	2	no	—
190	Q₈×C₂₃	—	4	2	no	—
191	Q₈×C₂₃	—	4	2	no	—
192	Q₈×C₂₃	—	4	2	no	—
193	D₄×C₂₃	—	2	2	no	—
194	D₄×C₂₃	—	2	1	no	—
195	D₄×C₂₃	—	2	1	no	—
196	D₄×C₂₃	—	2	2	no	—
197	D₄×C₂₃	—	4	1	no	—
198	D₄×C₂₃	—	4	2	no	—
199	D₄×C₂₃	—	4	1	no	—
200	D₄×C₂₃	—	4	2	no	—
201	D₄×C₂₃	—	16	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	4	1	no	—
206	C₂×C₄×C₂₃	—	4	2	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	—	4	2	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	4	2	no	—
212	C₂×C₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	4	2	no	—
215	C₂×C₄×C₂₃	—	8	2	no	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
222	C₈×C₂₃	—	2	2	no	—
223	C₈×C₂₃	—	2	4	no	—
224	C₈×C₂₃	—	2	2	no	—
225	C₈×C₂₃	—	4	1	no	—
226	C₈×C₂₃	#121	1	2	yes	—
227	C₈×C₂₃	#121	1	2	yes	—
228	#3156	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂×#473	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#3063	C₂×C₂₃	1	2	yes	complemented
231	#3060	C₂×C₂₃	1	1	yes	characteristic
232	C₂×#444	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#2038	C₂×C₂₃	1	2	yes	complemented
234	#2007	C₂×C₂₃	1	2	yes	complemented
235	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
236	#1951	C₂×C₂₃	1	1	yes	characteristic, complemented
237	#1908	C₂×C₂₃	1	2	yes	complemented
238	#1765	C₂×C₂₃	1	1	yes	characteristic, complemented
239	C₂²×C₂₃	—	2	2	no	—
240	C₂²×C₂₃	—	4	1	no	—
241	C₂²×C₂₃	—	4	2	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	—	4	2	no	—
244	C₂²×C₂₃	—	4	2	no	—
245	C₂²×C₂₃	#540	1	1	yes	characteristic
246	C₂²×C₂₃	D₄²	1	1	yes	characteristic
247	C₂²×C₂₃	#550	1	1	yes	characteristic
248	C₄×C₂₃	—	4	1	no	—
249	C₄×C₂₃	—	4	2	no	—
250	C₄×C₂₃	—	4	2	no	—
251	C₄×C₂₃	—	8	1	no	—
252	C₄×C₂₃	—	8	2	no	—
253	C₄×C₂₃	—	8	2	no	—
254	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
255	C₄×C₂₃	#532	1	1	yes	characteristic
256	C₄×C₂₃	D₄²	1	1	yes	characteristic
257	C₄×C₂₃	#546	1	1	yes	characteristic
258	C₂×#137	C₂²×C₂₃	1	2	yes	—
259	C₂×#137	C₂²×C₂₃	1	2	yes	—
260	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
261	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×#131	C₂²×C₂₃	1	2	yes	—
264	#537	C₂²×C₂₃	1	2	yes	—
265	#536	C₂²×C₂₃	1	2	yes	—
266	#497	—	2	2	no	—
267	#495	—	2	1	no	—
268	#494	—	2	1	no	—
269	#473	—	2	4	no	—
270	#470	—	2	1	no	—
271	#470	—	2	2	no	—
272	#469	—	2	1	no	—
273	#468	—	2	2	no	—
274	#467	—	2	2	no	—
275	#465	—	2	2	no	—
276	#465	—	2	2	no	—
277	#444	—	2	4	no	—
278	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
279	#428	C₂²×C₂₃	1	2	yes	—
280	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×#104	C₂²×C₂₃	1	2	yes	—
282	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
283	#417	C₂²×C₂₃	1	2	yes	—
284	#416	C₂²×C₂₃	1	1	yes	characteristic
285	C₂×#103	C₂²×C₂₃	1	2	yes	—
286	C₂×#101	C₂²×C₂₃	1	2	yes	—
287	#405	C₂²×C₂₃	1	1	yes	characteristic
288	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
289	#391	C₂²×C₂₃	1	2	yes	—
290	#343	C₂²×C₂₃	1	2	yes	—
291	#341	C₂²×C₂₃	1	1	yes	characteristic
292	#339	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×C₂₃	—	2	1	no	—
294	C₂×C₂₃	—	2	1	no	—
295	C₂×C₂₃	—	8	1	no	—
296	C₂×C₂₃	—	8	2	no	—
297	C₂×C₂₃	#2428	1	1	yes	characteristic
298	C₂×C₂₃	#3289	1	2	yes	—
299	C₂×#41	C₂³×C₂₃	1	2	yes	—
300	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
301	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
302	#137	—	2	4	no	—
303	#137	—	2	4	no	—
304	#137	—	4	2	no	—
305	#137	—	4	2	no	—
306	#135	—	2	2	no	—
307	#135	—	2	2	no	—
308	C₂×Q₁₆	—	2	2	no	—
309	C₂×Q₁₆	—	2	2	no	—
310	C₂×Q₁₆	—	2	2	no	—
311	C₂×Q₁₆	—	2	1	no	—
312	C₂×Q₁₆	—	2	1	no	—
313	C₂×Q₁₆	—	2	2	no	—
314	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
315	C₂×SD₁₆	—	2	2	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	D₄×C₂₃	1	2	yes	—
318	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
319	#131	—	2	4	no	—
320	#131	—	2	2	no	—
321	#131	—	2	2	no	—
322	#131	—	4	2	no	—
323	C₂×M₁₆	—	2	2	no	—
324	C₂×M₁₆	—	2	1	no	—
325	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
326	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
327	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
328	C₂²×C₈	—	2	1	no	—
329	C₂²×C₈	—	2	2	no	—
330	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
332	C₄⋊Q₈	—	2	2	no	—
333	#124	—	2	2	no	—
334	#123	—	4	2	no	—
335	C₂²⋊Q₈	—	2	2	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₂²⋊Q₈	—	2	2	no	—
339	#121	—	2	2	no	—
340	#121	—	2	2	no	—
341	#121	—	4	2	no	—
342	#117	C₂³×C₂₃	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	#108	D₄×C₂₃	1	4	yes	complemented
346	#107	—	2	2	no	—
347	#106	—	2	1	no	—
348	#106	—	2	1	no	—
349	#106	—	2	2	no	—
350	#104	—	2	4	no	—
351	#104	—	2	2	no	—
352	#103	—	2	2	no	—
353	#103	—	2	2	no	—
354	#103	—	2	2	no	—
355	#103	D₄×C₂₃	1	2	yes	—
356	#102	—	2	2	no	—
357	#102	D₄×C₂₃	1	2	yes	—
358	#101	—	2	4	no	—
359	#97	—	2	2	no	—
360	#97	—	2	1	no	—
361	C₄×C₈	—	2	1	no	—
362	#95	—	4	2	no	—
363	C₂₃	#25445	1	1	yes	characteristic, direct product factor, Sylow
364	#41	—	2	2	no	—
365	#41	—	2	1	no	—
366	#41	—	2	1	no	—
367	#41	—	2	1	no	—
368	#41	—	2	2	no	—
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	#41	—	2	1	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
379	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
384	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
391	Q₁₆	—	2	2	no	—
392	Q₁₆	—	4	4	no	—
393	Q₁₆	—	4	4	no	—
394	Q₁₆	—	4	2	no	—
395	Q₁₆	—	4	2	no	—
396	SD₁₆	—	2	4	no	—
397	SD₁₆	—	4	4	no	—
398	SD₁₆	—	4	4	no	—
399	D₈	—	2	2	no	—
400	M₁₆	—	2	2	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	1	no	—
403	M₁₆	—	2	1	no	—
404	M₁₆	—	4	4	no	—
405	M₁₆	—	4	2	no	—
406	M₁₆	C₂×D₄×C₂₃	1	4	yes	—
407	M₁₆	#41×C₂₃	1	2	yes	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	4	4	no	—
414	C₂×C₈	—	4	2	no	—
415	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
416	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
417	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
418	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
419	C₂×C₈	#41×C₂₃	1	2	yes	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	—	4	2	no	—
425	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
426	C₂²⋊C₄	—	2	1	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₂²⋊C₄	—	4	2	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	1	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	1	no	—
436	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
437	Q₈	—	2	2	no	—
438	Q₈	—	2	1	no	—
439	Q₈	—	2	1	no	—
440	Q₈	—	2	2	no	—
441	Q₈	—	4	2	no	—
442	Q₈	—	4	2	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	1	no	—
447	D₄	—	2	1	no	—
448	D₄	—	2	2	no	—
449	D₄	—	4	1	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	1	no	—
452	D₄	—	4	2	no	—
453	D₄	—	16	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	8	2	no	—
468	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
473	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	4	no	—
476	C₈	—	2	2	no	—
477	C₈	—	4	1	no	—
478	C₈	C₂₃×#121	1	2	yes	—
479	C₈	C₂₃×#121	1	2	yes	—
480	C₂²	—	2	2	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	C₂₃×#540	1	1	yes	characteristic
487	C₂²	D₄²×C₂₃	1	1	yes	characteristic
488	C₂²	C₂₃×#550	1	1	yes	characteristic
489	C₄	—	4	1	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	1	no	—
493	C₄	—	8	2	no	—
494	C₄	—	8	2	no	—
495	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
496	C₄	C₂₃×#532	1	1	yes	characteristic
497	C₄	D₄²×C₂₃	1	1	yes	characteristic
498	C₄	C₂₃×#546	1	1	yes	characteristic
499	C₂	—	2	1	no	—
500	C₂	—	2	1	no	—
501	C₂	—	8	1	no	—
502	C₂	—	8	2	no	—
503	C₂	C₂₃×#2428	1	1	yes	characteristic
504	C₂	C₂₃×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3156	C₂	semidirect
2. C₂×C₂₃×#473	C₂	semidirect
3. C₂₃×#3063	C₂	semidirect
5. C₂×C₂₃×#444	C₂	semidirect
6. C₂₃×#2038	C₂	semidirect
7. C₂₃×#2007	C₂	semidirect
8. C₂₃×#1971	C₂	semidirect
9. C₂₃×#1951	C₂	semidirect
10. C₂₃×#1908	C₂	semidirect
11. C₂₃×#1765	C₂	semidirect
93. C₂₃×#108	D₄	semidirect
181. #25445	C₂₃	direct (N × H)
228. #3156	C₂×C₂₃	semidirect
229. C₂×#473	C₂×C₂₃	semidirect
230. #3063	C₂×C₂₃	semidirect
232. C₂×#444	C₂×C₂₃	semidirect
233. #2038	C₂×C₂₃	semidirect
234. #2007	C₂×C₂₃	semidirect
235. #1971	C₂×C₂₃	semidirect
236. #1951	C₂×C₂₃	semidirect
237. #1908	C₂×C₂₃	semidirect
238. #1765	C₂×C₂₃	semidirect
345. #108	D₄×C₂₃	semidirect
363. C₂₃	#25445	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×#3156	2
C₂×C₂₃×#473	2
C₂₃×#3063	2
C₂₃×#3060	2
C₂×C₂₃×#444	2
C₂₃×#2038	2
C₂₃×#2007	2
C₂₃×#1971	2
C₂₃×#1951	2
C₂₃×#1908	2
C₂₃×#1765	2
#25445	23

Subgroups — C23×#25445 #19173071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25445 #19173071