Subgroups — C₂₃×#19627 #19167253

Order 5888.

784 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3205	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1720	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1717	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1711	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1709	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂₃×#498	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#494	C₂²	1	2	yes	complemented
18	C₂₃×#493	C₂²	1	2	yes	—
19	C₂₃×#492	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#490	—	2	2	no	—
21	C₂₃×#489	—	2	2	no	—
22	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#487	C₂²	1	2	yes	complemented
24	C₂₃×#487	C₂²	1	2	yes	complemented
25	C₂₃×#486	—	2	2	no	—
26	C₂₃×#486	C₂²	1	2	yes	—
27	C₂₃×#486	C₂²	1	2	yes	—
28	C₂₃×#486	C₂²	1	2	yes	—
29	C₂₃×#486	C₂²	1	2	yes	—
30	C₂₃×#486	C₂²	1	2	yes	—
31	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#486	C₂²	1	2	yes	complemented
33	C₂₃×#465	—	2	2	no	—
34	C₂₃×#462	—	2	2	no	—
35	C₂₃×#459	—	2	2	no	—
36	C₂₃×#451	—	2	2	no	—
37	C₂₃×#447	—	2	2	no	—
38	C₄×SD₁₆×C₂₃	—	2	2	no	—
39	C₂₃×#431	C₂²	1	1	yes	characteristic
40	C₂₃×#418	—	2	2	no	—
41	C₂₃×#417	—	2	2	no	—
42	C₂×C₂₃×#102	—	2	2	no	—
43	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₂₃	C₂²	1	2	yes	complemented
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	2	1	no	—
51	C₂²×D₄×C₂₃	—	2	1	no	—
52	C₂×Q₁₆×C₂₃	—	2	4	no	—
53	C₂×SD₁₆×C₂₃	—	2	2	no	—
54	C₂×D₈×C₂₃	—	2	4	no	—
55	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
56	C₂²×C₈×C₂₃	D₄	1	2	yes	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₂₃×#127	—	2	1	no	—
65	C₂₃×#127	—	2	1	no	—
66	C₂₃×#127	—	2	1	no	—
67	C₂₃×#127	C₂³	1	2	yes	—
68	C₂₃×#127	C₂³	1	1	yes	characteristic
69	C₂₃×#125	—	2	1	no	—
70	C₂₃×#124	—	2	1	no	—
71	C₂₃×C₂²⋊Q₈	—	2	1	no	—
72	C₂₃×C₂²⋊Q₈	—	2	1	no	—
73	C₂₃×#121	—	2	1	no	—
74	C₂₃×#121	—	2	1	no	—
75	C₄×Q₈×C₂₃	—	2	1	no	—
76	C₄×Q₈×C₂₃	—	2	1	no	—
77	C₄×D₄×C₂₃	—	2	1	no	—
78	C₄×D₄×C₂₃	—	2	1	no	—
79	C₂₃×#117	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×#106	—	2	2	no	—
84	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
85	C₂₃×#103	—	2	4	no	—
86	C₂₃×#103	—	2	2	no	—
87	C₂₃×#103	—	2	4	no	—
88	C₂₃×#102	—	2	4	no	—
89	C₂₃×#102	—	2	4	no	—
90	C₂₃×#102	—	2	4	no	—
91	C₂₃×#102	—	2	4	no	—
92	C₂₃×#102	—	2	4	no	—
93	C₂₃×#102	—	2	2	no	—
94	C₂₃×#102	—	2	4	no	—
95	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
96	C₂₃×#97	D₄	1	2	yes	—
97	C₄×C₈×C₂₃	D₄	1	2	yes	—
98	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₂₃	C₂³	1	2	yes	—
100	C₄×C₈×C₂₃	C₂³	1	2	yes	—
101	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
102	C₂⁴×C₂₃	—	4	1	no	—
103	#41×C₂₃	—	4	4	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	1	no	—
108	C₂×Q₈×C₂₃	—	4	1	no	—
109	C₂×Q₈×C₂₃	—	8	2	no	—
110	C₂×D₄×C₂₃	—	2	1	no	—
111	C₂×D₄×C₂₃	—	2	2	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	2	1	no	—
114	C₂×D₄×C₂₃	—	2	2	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	4	4	no	—
117	C₂×D₄×C₂₃	—	4	1	no	—
118	C₂×D₄×C₂₃	—	4	1	no	—
119	C₂×D₄×C₂₃	—	4	1	no	—
120	C₂×D₄×C₂₃	—	4	1	no	—
121	C₂×D₄×C₂₃	—	4	4	no	—
122	C₂×D₄×C₂₃	—	8	2	no	—
123	C₂²×C₄×C₂₃	—	4	1	no	—
124	C₂²×C₄×C₂₃	—	4	1	no	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₂₃	—	8	4	no	—
129	SD₁₆×C₂₃	—	4	4	no	—
130	D₈×C₂₃	—	8	4	no	—
131	M₁₆×C₂₃	—	2	4	no	—
132	C₂×C₈×C₂₃	—	4	2	no	—
133	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
135	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	2	1	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	4	1	no	—
146	C₄⋊C₄×C₂₃	—	4	1	no	—
147	C₄⋊C₄×C₂₃	—	4	1	no	—
148	C₄⋊C₄×C₂₃	—	4	1	no	—
149	C₂²⋊C₄×C₂₃	—	4	1	no	—
150	C₂²⋊C₄×C₂₃	—	4	1	no	—
151	C₂²⋊C₄×C₂₃	—	4	1	no	—
152	C₄²×C₂₃	—	4	1	no	—
153	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
157	#19627	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₂₃	—	4	1	no	—
159	C₂³×C₂₃	—	4	2	no	—
160	C₂³×C₂₃	—	8	2	no	—
161	C₂³×C₂₃	—	8	2	no	—
162	C₂³×C₂₃	—	8	2	no	—
163	C₂³×C₂₃	#127	1	1	yes	characteristic
164	Q₈×C₂₃	—	4	2	no	—
165	Q₈×C₂₃	—	4	2	no	—
166	Q₈×C₂₃	—	4	2	no	—
167	Q₈×C₂₃	—	4	4	no	—
168	Q₈×C₂₃	—	8	2	no	—
169	D₄×C₂₃	—	4	4	no	—
170	D₄×C₂₃	—	4	2	no	—
171	D₄×C₂₃	—	4	2	no	—
172	D₄×C₂₃	—	4	4	no	—
173	D₄×C₂₃	—	4	2	no	—
174	D₄×C₂₃	—	4	4	no	—
175	D₄×C₂₃	—	8	2	no	—
176	D₄×C₂₃	—	8	2	no	—
177	D₄×C₂₃	—	8	2	no	—
178	D₄×C₂₃	—	8	2	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	8	2	no	—
186	C₂×C₄×C₂₃	—	8	2	no	—
187	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
188	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
189	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
190	C₂×C₄×C₂₃	#135	1	4	yes	—
191	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
194	C₈×C₂₃	—	2	4	no	—
195	C₈×C₂₃	—	2	2	no	—
196	C₈×C₂₃	—	2	2	no	—
197	C₈×C₂₃	—	2	4	no	—
198	#3209	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#3205	C₂×C₂₃	1	1	yes	characteristic
200	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
201	#3141	C₂×C₂₃	1	1	yes	characteristic, complemented
202	#3138	C₂×C₂₃	1	1	yes	characteristic, complemented
203	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#1720	C₂×C₂₃	1	2	yes	complemented
205	#1717	C₂×C₂₃	1	2	yes	complemented
206	#1711	C₂×C₂₃	1	2	yes	complemented
207	#1709	C₂×C₂₃	1	2	yes	complemented
208	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
209	C₂²×C₂₃	—	2	1	no	—
210	C₂²×C₂₃	—	2	1	no	—
211	C₂²×C₂₃	—	8	1	no	—
212	C₂²×C₂₃	—	8	2	no	—
213	C₂²×C₂₃	—	8	2	no	—
214	C₂²×C₂₃	—	8	1	no	—
215	C₂²×C₂₃	—	8	2	no	—
216	C₂²×C₂₃	—	8	1	no	—
217	C₂²×C₂₃	—	16	4	no	—
218	C₂²×C₂₃	#495	1	2	yes	—
219	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
220	C₄×C₂₃	—	2	1	no	—
221	C₄×C₂₃	—	2	1	no	—
222	C₄×C₂₃	—	8	1	no	—
223	C₄×C₂₃	—	8	1	no	—
224	C₄×C₂₃	—	8	2	no	—
225	C₄×C₂₃	—	8	1	no	—
226	C₄×C₂₃	#495	1	2	yes	—
227	C₄×C₂₃	C₂×#135	1	2	yes	—
228	C₄×C₂₃	#575	1	2	yes	—
229	C₄×C₂₃	#576	1	2	yes	—
230	#533	C₂²×C₂₃	1	1	yes	characteristic
231	#532	C₂²×C₂₃	1	1	yes	characteristic
232	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
234	#498	C₂²×C₂₃	1	1	yes	characteristic, complemented
235	#494	C₂²×C₂₃	1	2	yes	complemented
236	#493	C₂²×C₂₃	1	2	yes	—
237	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
238	#490	—	2	2	no	—
239	#489	—	2	2	no	—
240	#487	C₂²×C₂₃	1	2	yes	complemented
241	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
242	#487	C₂²×C₂₃	1	2	yes	complemented
243	#486	—	2	2	no	—
244	#486	C₂²×C₂₃	1	2	yes	—
245	#486	C₂²×C₂₃	1	2	yes	—
246	#486	C₂²×C₂₃	1	2	yes	—
247	#486	C₂²×C₂₃	1	2	yes	—
248	#486	C₂²×C₂₃	1	2	yes	—
249	#486	C₂²×C₂₃	1	2	yes	complemented
250	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
251	#465	—	2	2	no	—
252	#462	—	2	2	no	—
253	#459	—	2	2	no	—
254	#451	—	2	2	no	—
255	#447	—	2	2	no	—
256	C₄×SD₁₆	—	2	2	no	—
257	#431	C₂²×C₂₃	1	1	yes	characteristic
258	#418	—	2	2	no	—
259	#417	—	2	2	no	—
260	C₂×#102	—	2	2	no	—
261	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
263	#322	C₂²×C₂₃	1	2	yes	complemented
264	C₈²	C₂²×C₂₃	1	2	yes	complemented
265	C₂×C₂₃	—	2	2	no	—
266	C₂×C₂₃	—	16	1	no	—
267	C₂×C₂₃	—	16	2	no	—
268	C₂×C₂₃	C₂×#495	1	1	yes	characteristic
269	C₂×C₂₃	#3153	1	1	yes	characteristic
270	C₂×C₂₃	#3155	1	1	yes	characteristic
271	C₂×#41	—	2	1	no	—
272	C₂²×Q₈	—	2	1	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂×Q₁₆	—	2	4	no	—
277	C₂×SD₁₆	—	2	2	no	—
278	C₂×D₈	—	2	4	no	—
279	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
280	C₂²×C₈	D₄×C₂₃	1	2	yes	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
285	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
286	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
288	#127	—	2	1	no	—
289	#127	—	2	1	no	—
290	#127	—	2	1	no	—
291	#127	C₂³×C₂₃	1	2	yes	—
292	#127	C₂³×C₂₃	1	1	yes	characteristic
293	#125	—	2	1	no	—
294	#124	—	2	1	no	—
295	C₂²⋊Q₈	—	2	1	no	—
296	C₂²⋊Q₈	—	2	1	no	—
297	#121	—	2	1	no	—
298	#121	—	2	1	no	—
299	C₄×Q₈	—	2	1	no	—
300	C₄×Q₈	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	—	2	1	no	—
303	#117	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
307	#106	—	2	2	no	—
308	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
309	#103	—	2	4	no	—
310	#103	—	2	2	no	—
311	#103	—	2	4	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	#102	—	2	4	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	2	2	no	—
318	#102	—	2	4	no	—
319	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
320	#97	D₄×C₂₃	1	2	yes	—
321	C₄×C₈	D₄×C₂₃	1	2	yes	—
322	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
323	C₄×C₈	C₂³×C₂₃	1	2	yes	—
324	C₄×C₈	C₂³×C₂₃	1	2	yes	—
325	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
326	C₂₃	#19627	1	1	yes	characteristic, direct product factor, Sylow
327	C₂⁴	—	4	1	no	—
328	#41	—	4	4	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	4	1	no	—
334	C₂×Q₈	—	8	2	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	4	4	no	—
342	C₂×D₄	—	4	1	no	—
343	C₂×D₄	—	4	1	no	—
344	C₂×D₄	—	4	1	no	—
345	C₂×D₄	—	4	1	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	—	8	2	no	—
348	C₂²×C₄	—	4	1	no	—
349	C₂²×C₄	—	4	1	no	—
350	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
353	Q₁₆	—	8	4	no	—
354	SD₁₆	—	4	4	no	—
355	D₈	—	8	4	no	—
356	M₁₆	—	2	4	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
361	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	4	1	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₂²⋊C₄	—	4	1	no	—
375	C₂²⋊C₄	—	4	1	no	—
376	C₂²⋊C₄	—	4	1	no	—
377	C₄²	—	4	1	no	—
378	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
380	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
381	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
382	C₂³	—	4	1	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	8	2	no	—
385	C₂³	—	8	2	no	—
386	C₂³	—	8	2	no	—
387	C₂³	C₂₃×#127	1	1	yes	characteristic
388	Q₈	—	4	2	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	4	4	no	—
392	Q₈	—	8	2	no	—
393	D₄	—	4	4	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	4	no	—
399	D₄	—	8	2	no	—
400	D₄	—	8	2	no	—
401	D₄	—	8	2	no	—
402	D₄	—	8	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	8	2	no	—
411	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
412	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
413	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
414	C₂×C₄	C₂₃×#135	1	4	yes	—
415	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
417	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	2	4	no	—
422	C₂²	—	2	1	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	1	no	—
430	C₂²	—	16	4	no	—
431	C₂²	C₂₃×#495	1	2	yes	—
432	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	1	no	—
437	C₄	—	8	2	no	—
438	C₄	—	8	1	no	—
439	C₄	C₂₃×#495	1	2	yes	—
440	C₄	C₂×C₂₃×#135	1	2	yes	—
441	C₄	C₂₃×#575	1	2	yes	—
442	C₄	C₂₃×#576	1	2	yes	—
443	C₂	—	2	2	no	—
444	C₂	—	16	2	no	—
445	C₂	—	16	1	no	—
446	C₂	C₂×C₂₃×#495	1	1	yes	characteristic
447	C₂	C₂₃×#3153	1	1	yes	characteristic
448	C₂	C₂₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3205	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3209	C₂	semidirect
3. C₂₃×#3167	C₂	semidirect
4. C₂₃×#3141	C₂	semidirect
5. C₂₃×#3138	C₂	semidirect
6. C₂×C₂₃×#486	C₂	semidirect
7. C₂₃×#1720	C₂	semidirect
8. C₂₃×#1717	C₂	semidirect
9. C₂₃×#1711	C₂	semidirect
10. C₂₃×#1709	C₂	semidirect
11. C₈×M₁₆×C₂₃	C₂	semidirect
16. C₂₃×#498	C₂²	semidirect
17. C₂₃×#494	C₂²	semidirect
19. C₂₃×#492	C₂²	semidirect
22. C₂₃×#487	C₂²	semidirect
23. C₂₃×#487	C₂²	semidirect
24. C₂₃×#487	C₂²	semidirect
31. C₂₃×#486	C₂²	semidirect
32. C₂₃×#486	C₂²	semidirect
45. C₂₃×#322	C₂²	semidirect
46. C₈²×C₂₃	C₂²	semidirect
157. #19627	C₂₃	direct (N × H)
198. #3209	C₂×C₂₃	semidirect
200. #3167	C₂×C₂₃	semidirect
201. #3141	C₂×C₂₃	semidirect
202. #3138	C₂×C₂₃	semidirect
203. C₂×#486	C₂×C₂₃	semidirect
204. #1720	C₂×C₂₃	semidirect
205. #1717	C₂×C₂₃	semidirect
206. #1711	C₂×C₂₃	semidirect
207. #1709	C₂×C₂₃	semidirect
208. C₈×M₁₆	C₂×C₂₃	semidirect
234. #498	C₂²×C₂₃	semidirect
235. #494	C₂²×C₂₃	semidirect
237. #492	C₂²×C₂₃	semidirect
240. #487	C₂²×C₂₃	semidirect
241. #487	C₂²×C₂₃	semidirect
242. #487	C₂²×C₂₃	semidirect
249. #486	C₂²×C₂₃	semidirect
250. #486	C₂²×C₂₃	semidirect
263. #322	C₂²×C₂₃	semidirect
264. C₈²	C₂²×C₂₃	semidirect
326. C₂₃	#19627	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×#3209	2
C₂₃×#3205	2
C₂₃×#3167	2
C₂₃×#3141	2
C₂₃×#3138	2
C₂×C₂₃×#486	2
C₂₃×#1720	2
C₂₃×#1717	2
C₂₃×#1711	2
C₂₃×#1709	2
C₈×M₁₆×C₂₃	2
#19627	23

Subgroups — C23×#19627 #19167253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19627 #19167253