Subgroups — C₂₃×#19237 #19166863

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3245	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1701	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1699	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1681	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1679	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#131	—	2	1	no	—
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×#492	—	2	1	no	—
17	C₂₃×#489	—	2	2	no	—
18	C₂₃×#487	—	2	1	no	—
19	C₂₃×#487	C₂²	1	2	yes	—
20	C₂₃×#486	C₂²	1	2	yes	—
21	C₂₃×#474	—	2	1	no	—
22	C₂₃×#474	C₂²	1	2	yes	—
23	C₂₃×#466	—	2	2	no	—
24	C₂₃×#461	C₂²	1	2	yes	—
25	C₂₃×#461	C₂²	1	2	yes	—
26	C₂₃×#459	—	2	1	no	—
27	C₂₃×#459	C₂²	1	2	yes	—
28	C₂₃×#448	—	2	2	no	—
29	C₂₃×#436	C₂²	1	2	yes	—
30	C₂₃×#435	—	2	2	no	—
31	C₈×D₄×C₂₃	C₂²	1	2	yes	—
32	C₂₃×#431	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂₃×#417	—	2	1	no	—
36	C₂₃×#417	—	2	1	no	—
37	C₂₃×#408	—	2	1	no	—
38	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#335	C₂²	1	2	yes	—
40	C₂₃×#334	C₂²	1	2	yes	—
41	C₂₃×#332	C₂²	1	1	yes	characteristic
42	C₂₃×#332	C₂²	1	2	yes	—
43	C₂₃×#332	C₂²	1	1	yes	characteristic
44	C₂₃×#332	C₂²	1	2	yes	—
45	C₂₃×#331	C₂²	1	1	yes	characteristic
46	C₂₃×#331	C₂²	1	1	yes	characteristic
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂×#41×C₂₃	—	2	1	no	—
49	C₂²×Q₈×C₂₃	—	2	1	no	—
50	C₂×SD₁₆×C₂₃	—	2	2	no	—
51	C₂×SD₁₆×C₂₃	—	2	2	no	—
52	C₂×D₈×C₂₃	—	2	2	no	—
53	C₂₃×#131	—	4	4	no	—
54	C₂×M₁₆×C₂₃	—	2	2	no	—
55	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₂₃×#127	D₄	1	1	yes	characteristic
67	C₂₃×#127	C₂³	1	1	yes	characteristic
68	C₂₃×#124	—	2	1	no	—
69	C₂₃×#124	—	2	1	no	—
70	C₂₃×#121	—	2	1	no	—
71	C₂₃×#121	—	2	1	no	—
72	C₂₃×#121	—	2	1	no	—
73	C₂₃×#121	D₄	1	2	yes	—
74	C₄×D₄×C₂₃	—	2	1	no	—
75	C₄×D₄×C₂₃	—	2	1	no	—
76	C₄×D₄×C₂₃	—	2	1	no	—
77	C₄×D₄×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂₃×#107	—	2	2	no	—
82	C₂₃×C₄⋊C₈	—	2	1	no	—
83	C₂₃×C₄⋊C₈	—	2	1	no	—
84	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
89	C₂₃×#103	—	2	2	no	—
90	C₂₃×#103	—	2	2	no	—
91	C₂₃×#103	—	2	2	no	—
92	C₂₃×#103	—	2	2	no	—
93	C₂₃×#102	—	2	2	no	—
94	C₂₃×#102	—	2	2	no	—
95	C₂₃×#102	—	2	2	no	—
96	C₂₃×#102	—	2	2	no	—
97	C₂₃×C₂²⋊C₈	—	2	1	no	—
98	C₂₃×C₂²⋊C₈	—	2	2	no	—
99	C₂₃×C₂²⋊C₈	—	2	2	no	—
100	C₂₃×#97	D₄	1	2	yes	—
101	C₄×C₈×C₂₃	—	2	1	no	—
102	C₄×C₈×C₂₃	C₂³	1	2	yes	—
103	#41×C₂₃	—	4	2	no	—
104	#41×C₂₃	—	4	2	no	—
105	#41×C₂₃	—	4	2	no	—
106	#41×C₂₃	—	4	2	no	—
107	C₂×Q₈×C₂₃	—	2	1	no	—
108	C₂×Q₈×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	—	2	2	no	—
110	C₂×Q₈×C₂₃	—	4	1	no	—
111	C₂×Q₈×C₂₃	—	8	1	no	—
112	C₂×Q₈×C₂₃	—	8	1	no	—
113	C₂×D₄×C₂₃	—	2	2	no	—
114	C₂×D₄×C₂₃	—	2	1	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	2	1	no	—
117	C₂×D₄×C₂₃	—	2	1	no	—
118	C₂×D₄×C₂₃	—	2	1	no	—
119	C₂×D₄×C₂₃	—	4	1	no	—
120	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₂₃	—	2	2	no	—
122	C₂²×C₄×C₂₃	—	4	1	no	—
123	C₂²×C₄×C₂₃	—	4	1	no	—
124	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
127	SD₁₆×C₂₃	—	8	2	no	—
128	SD₁₆×C₂₃	—	8	2	no	—
129	D₈×C₂₃	—	8	2	no	—
130	M₁₆×C₂₃	—	2	2	no	—
131	M₁₆×C₂₃	—	2	2	no	—
132	M₁₆×C₂₃	—	4	4	no	—
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	1	no	—
135	C₂×C₈×C₂₃	—	2	1	no	—
136	C₂×C₈×C₂₃	—	2	1	no	—
137	C₂×C₈×C₂₃	—	2	1	no	—
138	C₂×C₈×C₂₃	—	4	4	no	—
139	C₂×C₈×C₂₃	—	4	2	no	—
140	C₂×C₈×C₂₃	—	8	1	no	—
141	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	4	1	no	—
149	C₄⋊C₄×C₂₃	—	4	1	no	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₂₃	—	2	2	no	—
152	C₂²⋊C₄×C₂₃	—	2	2	no	—
153	C₂²⋊C₄×C₂₃	—	4	1	no	—
154	C₂²⋊C₄×C₂₃	—	4	1	no	—
155	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₂₃	#41	1	1	yes	characteristic
158	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
159	#19237	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₂₃	—	2	2	no	—
161	C₂³×C₂₃	—	4	1	no	—
162	C₂³×C₂₃	#121	1	1	yes	characteristic
163	Q₈×C₂₃	—	4	2	no	—
164	Q₈×C₂₃	—	4	1	no	—
165	Q₈×C₂₃	—	4	1	no	—
166	Q₈×C₂₃	—	4	2	no	—
167	Q₈×C₂₃	—	4	2	no	—
168	Q₈×C₂₃	—	8	2	no	—
169	D₄×C₂₃	—	2	2	no	—
170	D₄×C₂₃	—	2	2	no	—
171	D₄×C₂₃	—	4	4	no	—
172	D₄×C₂₃	—	4	1	no	—
173	D₄×C₂₃	—	4	1	no	—
174	D₄×C₂₃	—	4	1	no	—
175	D₄×C₂₃	—	4	1	no	—
176	D₄×C₂₃	—	4	2	no	—
177	D₄×C₂₃	—	8	1	no	—
178	D₄×C₂₃	—	8	1	no	—
179	D₄×C₂₃	—	8	2	no	—
180	C₂×C₄×C₂₃	—	2	1	no	—
181	C₂×C₄×C₂₃	—	2	1	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	—	8	2	no	—
189	C₂×C₄×C₂₃	—	8	2	no	—
190	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
191	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
192	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
193	C₂×C₄×C₂₃	#135	1	2	yes	—
194	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
198	C₈×C₂₃	—	2	2	no	—
199	C₈×C₂₃	—	2	2	no	—
200	C₈×C₂₃	—	4	1	no	—
201	C₈×C₂₃	—	4	1	no	—
202	C₈×C₂₃	—	4	2	no	—
203	C₈×C₂₃	—	4	1	no	—
204	C₈×C₂₃	—	4	1	no	—
205	#3245	C₂×C₂₃	1	1	yes	characteristic
206	#3206	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#3141	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#2935	C₂×C₂₃	1	1	yes	characteristic, complemented
209	#1701	C₂×C₂₃	1	2	yes	complemented
210	#1699	C₂×C₂₃	1	2	yes	complemented
211	#1681	C₂×C₂₃	1	2	yes	complemented
212	#1679	C₂×C₂₃	1	2	yes	complemented
213	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1540	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂²×C₂₃	—	2	1	no	—
217	C₂²×C₂₃	—	2	1	no	—
218	C₂²×C₂₃	—	4	2	no	—
219	C₂²×C₂₃	—	4	2	no	—
220	C₂²×C₂₃	—	4	2	no	—
221	C₂²×C₂₃	—	8	1	no	—
222	C₂²×C₂₃	—	8	1	no	—
223	C₂²×C₂₃	—	8	1	no	—
224	C₂²×C₂₃	#463	1	2	yes	—
225	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
226	C₄×C₂₃	—	2	1	no	—
227	C₄×C₂₃	—	2	1	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	4	2	no	—
230	C₄×C₂₃	—	8	1	no	—
231	C₄×C₂₃	—	8	2	no	—
232	C₄×C₂₃	#463	1	2	yes	—
233	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
234	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₄×C₂₃	#576	1	1	yes	characteristic
236	C₄×C₂₃	#577	1	1	yes	characteristic
237	C₂×#131	—	2	1	no	—
238	#532	C₂²×C₂₃	1	1	yes	characteristic
239	#532	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
241	#492	—	2	1	no	—
242	#489	—	2	2	no	—
243	#487	—	2	1	no	—
244	#487	C₂²×C₂₃	1	2	yes	—
245	#486	C₂²×C₂₃	1	2	yes	—
246	#474	—	2	1	no	—
247	#474	C₂²×C₂₃	1	2	yes	—
248	#466	—	2	2	no	—
249	#461	C₂²×C₂₃	1	2	yes	—
250	#461	C₂²×C₂₃	1	2	yes	—
251	#459	—	2	1	no	—
252	#459	C₂²×C₂₃	1	2	yes	—
253	#448	—	2	2	no	—
254	#436	C₂²×C₂₃	1	2	yes	—
255	#435	—	2	2	no	—
256	C₈×D₄	C₂²×C₂₃	1	2	yes	—
257	#431	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
260	#417	—	2	1	no	—
261	#417	—	2	1	no	—
262	#408	—	2	1	no	—
263	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
264	#335	C₂²×C₂₃	1	2	yes	—
265	#334	C₂²×C₂₃	1	2	yes	—
266	#332	C₂²×C₂₃	1	2	yes	—
267	#332	C₂²×C₂₃	1	2	yes	—
268	#332	C₂²×C₂₃	1	1	yes	characteristic
269	#332	C₂²×C₂₃	1	1	yes	characteristic
270	#331	C₂²×C₂₃	1	1	yes	characteristic
271	#331	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₂₃	—	2	2	no	—
273	C₂×C₂₃	—	8	2	no	—
274	C₂×C₂₃	—	16	1	no	—
275	C₂×C₂₃	C₂×#463	1	1	yes	characteristic
276	C₂×C₂₃	#3062	1	1	yes	characteristic
277	C₂×C₂₃	#3069	1	1	yes	characteristic
278	C₂×#41	—	2	1	no	—
279	C₂×#41	—	2	1	no	—
280	C₂²×Q₈	—	2	1	no	—
281	C₂×SD₁₆	—	2	2	no	—
282	C₂×SD₁₆	—	2	2	no	—
283	C₂×D₈	—	2	2	no	—
284	#131	—	4	4	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
287	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	2	2	no	—
291	C₄⋊Q₈	—	2	1	no	—
292	C₄⋊Q₈	—	2	1	no	—
293	C₄⋊Q₈	—	2	1	no	—
294	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
295	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
296	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
297	#127	D₄×C₂₃	1	1	yes	characteristic
298	#127	C₂³×C₂₃	1	1	yes	characteristic
299	#124	—	2	1	no	—
300	#124	—	2	1	no	—
301	#121	—	2	1	no	—
302	#121	—	2	1	no	—
303	#121	—	2	1	no	—
304	#121	D₄×C₂₃	1	2	yes	—
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	C₂³×C₂₃	1	2	yes	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
312	#107	—	2	2	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
317	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
320	#103	—	2	2	no	—
321	#103	—	2	2	no	—
322	#103	—	2	2	no	—
323	#103	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	2	no	—
326	#102	—	2	2	no	—
327	#102	—	2	2	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	C₂²⋊C₈	—	2	2	no	—
330	C₂²⋊C₈	—	2	2	no	—
331	#97	D₄×C₂₃	1	2	yes	—
332	C₄×C₈	—	2	1	no	—
333	C₄×C₈	C₂³×C₂₃	1	2	yes	—
334	C₂₃	#19237	1	1	yes	characteristic, direct product factor, Sylow
335	#41	—	4	2	no	—
336	#41	—	4	2	no	—
337	#41	—	4	2	no	—
338	#41	—	4	2	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	4	1	no	—
343	C₂×Q₈	—	8	1	no	—
344	C₂×Q₈	—	8	1	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	4	1	no	—
352	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	4	1	no	—
355	C₂²×C₄	—	4	1	no	—
356	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
358	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
359	SD₁₆	—	8	2	no	—
360	SD₁₆	—	8	2	no	—
361	D₈	—	8	2	no	—
362	M₁₆	—	2	2	no	—
363	M₁₆	—	2	2	no	—
364	M₁₆	—	4	4	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	4	4	no	—
371	C₂×C₈	—	4	2	no	—
372	C₂×C₈	—	8	1	no	—
373	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	4	1	no	—
381	C₄⋊C₄	—	4	1	no	—
382	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₂²⋊C₄	—	2	2	no	—
385	C₂²⋊C₄	—	4	1	no	—
386	C₂²⋊C₄	—	4	1	no	—
387	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
388	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
389	C₄²	#41×C₂₃	1	1	yes	characteristic
390	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
391	C₂³	—	2	2	no	—
392	C₂³	—	4	1	no	—
393	C₂³	C₂₃×#121	1	1	yes	characteristic
394	Q₈	—	4	2	no	—
395	Q₈	—	4	1	no	—
396	Q₈	—	4	1	no	—
397	Q₈	—	4	2	no	—
398	Q₈	—	4	2	no	—
399	Q₈	—	8	2	no	—
400	D₄	—	2	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	4	4	no	—
403	D₄	—	4	1	no	—
404	D₄	—	4	1	no	—
405	D₄	—	4	1	no	—
406	D₄	—	4	1	no	—
407	D₄	—	4	2	no	—
408	D₄	—	8	1	no	—
409	D₄	—	8	1	no	—
410	D₄	—	8	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	8	2	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
422	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
423	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
424	C₂×C₄	C₂₃×#135	1	2	yes	—
425	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
426	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
427	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
429	C₈	—	2	2	no	—
430	C₈	—	2	2	no	—
431	C₈	—	4	1	no	—
432	C₈	—	4	1	no	—
433	C₈	—	4	2	no	—
434	C₈	—	4	1	no	—
435	C₈	—	4	1	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	8	1	no	—
442	C₂²	—	8	1	no	—
443	C₂²	—	8	1	no	—
444	C₂²	C₂₃×#463	1	2	yes	—
445	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	4	2	no	—
450	C₄	—	8	1	no	—
451	C₄	—	8	2	no	—
452	C₄	C₂₃×#463	1	2	yes	—
453	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
454	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
455	C₄	C₂₃×#576	1	1	yes	characteristic
456	C₄	C₂₃×#577	1	1	yes	characteristic
457	C₂	—	2	2	no	—
458	C₂	—	8	2	no	—
459	C₂	—	16	1	no	—
460	C₂	C₂×C₂₃×#463	1	1	yes	characteristic
461	C₂	C₂₃×#3062	1	1	yes	characteristic
462	C₂	C₂₃×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3245	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
2. C₂₃×#3206	C₂	semidirect
3. C₂₃×#3141	C₂	semidirect
4. C₂₃×#2935	C₂	semidirect
5. C₂₃×#1701	C₂	semidirect
6. C₂₃×#1699	C₂	semidirect
7. C₂₃×#1681	C₂	semidirect
8. C₂₃×#1679	C₂	semidirect
9. C₂₃×#1565	C₂	semidirect
10. C₂₃×#1540	C₂	semidirect
11. C₂×C₂₃×#332	C₂	semidirect
159. #19237	C₂₃	direct (N × H)
206. #3206	C₂×C₂₃	semidirect
207. #3141	C₂×C₂₃	semidirect
208. #2935	C₂×C₂₃	semidirect
209. #1701	C₂×C₂₃	semidirect
210. #1699	C₂×C₂₃	semidirect
211. #1681	C₂×C₂₃	semidirect
212. #1679	C₂×C₂₃	semidirect
213. #1565	C₂×C₂₃	semidirect
214. #1540	C₂×C₂₃	semidirect
215. C₂×#332	C₂×C₂₃	semidirect
334. C₂₃	#19237	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×#3245	2
C₂₃×#3206	2
C₂₃×#3141	2
C₂₃×#2935	2
C₂₃×#1701	2
C₂₃×#1699	2
C₂₃×#1681	2
C₂₃×#1679	2
C₂₃×#1565	2
C₂₃×#1540	2
C₂×C₂₃×#332	2
#19237	23

Subgroups — C23×#19237 #19166863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19237 #19166863