Subgroups — C₂₃×#19231 #19166857

Order 5888.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 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×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1689	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1685	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1669	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1665	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×#137	—	2	1	no	—
13	C₂×C₂₃×#137	—	2	1	no	—
14	C₂×C₂₃×#131	—	2	1	no	—
15	C₂₃×#532	C₂²	1	1	yes	characteristic
16	C₂₃×#532	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#500	—	2	1	no	—
19	C₂₃×#497	—	2	2	no	—
20	C₂₃×#494	C₂²	1	2	yes	—
21	C₂₃×#492	C₂²	1	2	yes	—
22	C₂₃×#486	—	2	1	no	—
23	C₂₃×#480	—	2	2	no	—
24	C₂₃×#474	—	2	1	no	—
25	C₂₃×#474	C₂²	1	2	yes	—
26	C₂₃×#474	C₂²	1	2	yes	—
27	C₂₃×#474	C₂²	1	2	yes	—
28	C₂₃×#466	—	2	2	no	—
29	C₂₃×#459	—	2	1	no	—
30	C₂₃×#459	C₂²	1	2	yes	—
31	C₂₃×#436	C₂²	1	2	yes	—
32	C₂₃×#435	—	2	2	no	—
33	C₈×D₄×C₂₃	C₂²	1	2	yes	—
34	C₂₃×#431	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
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	2	yes	—
42	C₂₃×#332	C₂²	1	1	yes	characteristic
43	C₂₃×#332	C₂²	1	2	yes	—
44	C₂₃×#332	C₂²	1	1	yes	characteristic
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	C₂₃×#137	—	4	4	no	—
51	C₂₃×#137	—	4	4	no	—
52	C₂×Q₁₆×C₂₃	—	2	2	no	—
53	C₂×Q₁₆×C₂₃	—	2	2	no	—
54	C₂×SD₁₆×C₂₃	—	2	2	no	—
55	C₂×SD₁₆×C₂₃	—	2	2	no	—
56	C₂×D₈×C₂₃	—	2	2	no	—
57	C₂₃×#131	—	4	4	no	—
58	C₂×M₁₆×C₂₃	—	2	2	no	—
59	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₂₃	—	2	1	no	—
62	C₂²×C₈×C₂₃	—	2	1	no	—
63	C₂²×C₈×C₂₃	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	—	2	1	no	—
65	C₂₃×C₄⋊Q₈	—	2	1	no	—
66	C₂₃×C₄⋊Q₈	—	2	1	no	—
67	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₂₃×#127	D₄	1	1	yes	characteristic
71	C₂₃×#127	C₂³	1	1	yes	characteristic
72	C₂₃×#124	—	2	1	no	—
73	C₂₃×#124	—	2	1	no	—
74	C₂₃×#121	—	2	1	no	—
75	C₂₃×#121	—	2	1	no	—
76	C₂₃×#121	—	2	1	no	—
77	C₂₃×#121	D₄	1	2	yes	—
78	C₄×D₄×C₂₃	—	2	1	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	—	2	1	no	—
81	C₄×D₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
85	C₂₃×#107	—	2	2	no	—
86	C₂₃×#107	—	2	2	no	—
87	C₂₃×#107	—	2	2	no	—
88	C₂₃×C₄⋊C₈	—	2	1	no	—
89	C₂₃×C₄⋊C₈	—	2	1	no	—
90	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
91	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
92	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
93	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
94	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
95	C₂₃×#102	—	2	2	no	—
96	C₂₃×#102	—	2	2	no	—
97	C₂₃×#102	—	2	2	no	—
98	C₂₃×#102	—	2	2	no	—
99	C₂₃×C₂²⋊C₈	—	2	1	no	—
100	C₂₃×C₂²⋊C₈	—	2	2	no	—
101	C₂₃×C₂²⋊C₈	—	2	2	no	—
102	C₂₃×#97	D₄	1	2	yes	—
103	C₄×C₈×C₂₃	—	2	1	no	—
104	C₄×C₈×C₂₃	C₂³	1	2	yes	—
105	#41×C₂₃	—	4	2	no	—
106	#41×C₂₃	—	4	2	no	—
107	#41×C₂₃	—	4	2	no	—
108	#41×C₂₃	—	4	2	no	—
109	C₂×Q₈×C₂₃	—	2	1	no	—
110	C₂×Q₈×C₂₃	—	2	1	no	—
111	C₂×Q₈×C₂₃	—	2	2	no	—
112	C₂×Q₈×C₂₃	—	4	1	no	—
113	C₂×Q₈×C₂₃	—	4	2	no	—
114	C₂×Q₈×C₂₃	—	4	2	no	—
115	C₂×D₄×C₂₃	—	2	2	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	—	2	1	no	—
120	C₂×D₄×C₂₃	—	2	1	no	—
121	C₂×D₄×C₂₃	—	4	1	no	—
122	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₂₃	—	2	2	no	—
124	C₂²×C₄×C₂₃	—	4	1	no	—
125	C₂²×C₄×C₂₃	—	4	1	no	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
129	Q₁₆×C₂₃	—	4	4	no	—
130	Q₁₆×C₂₃	—	4	4	no	—
131	SD₁₆×C₂₃	—	4	4	no	—
132	SD₁₆×C₂₃	—	4	4	no	—
133	D₈×C₂₃	—	8	2	no	—
134	M₁₆×C₂₃	—	2	2	no	—
135	M₁₆×C₂₃	—	4	4	no	—
136	M₁₆×C₂₃	D₈	1	4	yes	complemented
137	C₂×C₈×C₂₃	—	2	2	no	—
138	C₂×C₈×C₂₃	—	2	1	no	—
139	C₂×C₈×C₂₃	—	2	1	no	—
140	C₂×C₈×C₂₃	—	2	1	no	—
141	C₂×C₈×C₂₃	—	2	1	no	—
142	C₂×C₈×C₂₃	—	4	2	no	—
143	C₂×C₈×C₂₃	—	4	4	no	—
144	C₂×C₈×C₂₃	—	8	1	no	—
145	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	4	1	no	—
153	C₄⋊C₄×C₂₃	—	4	1	no	—
154	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
155	C₂²⋊C₄×C₂₃	—	2	2	no	—
156	C₂²⋊C₄×C₂₃	—	2	2	no	—
157	C₂²⋊C₄×C₂₃	—	4	1	no	—
158	C₂²⋊C₄×C₂₃	—	4	1	no	—
159	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₂₃	#41	1	1	yes	characteristic
162	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
163	#19231	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₂₃	—	2	2	no	—
165	C₂³×C₂₃	—	4	1	no	—
166	C₂³×C₂₃	#121	1	1	yes	characteristic
167	Q₈×C₂₃	—	4	2	no	—
168	Q₈×C₂₃	—	4	2	no	—
169	Q₈×C₂₃	—	4	1	no	—
170	Q₈×C₂₃	—	4	2	no	—
171	Q₈×C₂₃	—	4	1	no	—
172	Q₈×C₂₃	—	8	2	no	—
173	D₄×C₂₃	—	2	2	no	—
174	D₄×C₂₃	—	2	2	no	—
175	D₄×C₂₃	—	4	4	no	—
176	D₄×C₂₃	—	4	1	no	—
177	D₄×C₂₃	—	4	1	no	—
178	D₄×C₂₃	—	4	1	no	—
179	D₄×C₂₃	—	4	1	no	—
180	D₄×C₂₃	—	4	2	no	—
181	D₄×C₂₃	—	8	1	no	—
182	D₄×C₂₃	—	8	1	no	—
183	D₄×C₂₃	—	8	2	no	—
184	C₂×C₄×C₂₃	—	2	1	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	4	1	no	—
188	C₂×C₄×C₂₃	—	4	2	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	8	2	no	—
193	C₂×C₄×C₂₃	—	8	2	no	—
194	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
195	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
196	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
197	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
198	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
199	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
202	C₈×C₂₃	—	2	2	no	—
203	C₈×C₂₃	—	4	1	no	—
204	C₈×C₂₃	—	4	1	no	—
205	C₈×C₂₃	—	4	1	no	—
206	C₈×C₂₃	—	4	2	no	—
207	C₈×C₂₃	—	4	1	no	—
208	C₈×C₂₃	C₂×D₈	1	4	yes	—
209	#3245	C₂×C₂₃	1	1	yes	characteristic
210	#3226	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#2935	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#1689	C₂×C₂₃	1	2	yes	complemented
214	#1685	C₂×C₂₃	1	2	yes	complemented
215	#1669	C₂×C₂₃	1	2	yes	complemented
216	#1665	C₂×C₂₃	1	2	yes	complemented
217	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#1540	C₂×C₂₃	1	1	yes	characteristic, complemented
219	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
220	C₂²×C₂₃	—	2	1	no	—
221	C₂²×C₂₃	—	2	1	no	—
222	C₂²×C₂₃	—	4	2	no	—
223	C₂²×C₂₃	—	4	2	no	—
224	C₂²×C₂₃	—	4	2	no	—
225	C₂²×C₂₃	—	8	1	no	—
226	C₂²×C₂₃	—	8	1	no	—
227	C₂²×C₂₃	—	8	1	no	—
228	C₂²×C₂₃	#459	1	2	yes	—
229	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
230	C₄×C₂₃	—	2	1	no	—
231	C₄×C₂₃	—	2	1	no	—
232	C₄×C₂₃	—	2	1	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	8	1	no	—
235	C₄×C₂₃	—	8	2	no	—
236	C₄×C₂₃	#459	1	2	yes	—
237	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
238	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₄×C₂₃	#577	1	1	yes	characteristic
240	C₄×C₂₃	#578	1	1	yes	characteristic
241	C₂×#137	—	2	1	no	—
242	C₂×#137	—	2	1	no	—
243	C₂×#131	—	2	1	no	—
244	#532	C₂²×C₂₃	1	1	yes	characteristic
245	#532	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
247	#500	—	2	1	no	—
248	#497	—	2	2	no	—
249	#494	C₂²×C₂₃	1	2	yes	—
250	#492	C₂²×C₂₃	1	2	yes	—
251	#486	—	2	1	no	—
252	#480	—	2	2	no	—
253	#474	—	2	1	no	—
254	#474	C₂²×C₂₃	1	2	yes	—
255	#474	C₂²×C₂₃	1	2	yes	—
256	#474	C₂²×C₂₃	1	2	yes	—
257	#466	—	2	2	no	—
258	#459	—	2	1	no	—
259	#459	C₂²×C₂₃	1	2	yes	—
260	#436	C₂²×C₂₃	1	2	yes	—
261	#435	—	2	2	no	—
262	C₈×D₄	C₂²×C₂₃	1	2	yes	—
263	#431	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
266	#408	—	2	1	no	—
267	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
268	#335	C₂²×C₂₃	1	2	yes	—
269	#334	C₂²×C₂₃	1	2	yes	—
270	#332	C₂²×C₂₃	1	2	yes	—
271	#332	C₂²×C₂₃	1	2	yes	—
272	#332	C₂²×C₂₃	1	1	yes	characteristic
273	#332	C₂²×C₂₃	1	1	yes	characteristic
274	#331	C₂²×C₂₃	1	1	yes	characteristic
275	#331	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×C₂₃	—	2	2	no	—
277	C₂×C₂₃	—	8	2	no	—
278	C₂×C₂₃	—	16	1	no	—
279	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
280	C₂×C₂₃	#3063	1	1	yes	characteristic
281	C₂×C₂₃	#3069	1	1	yes	characteristic
282	C₂×#41	—	2	1	no	—
283	C₂×#41	—	2	1	no	—
284	C₂²×Q₈	—	2	1	no	—
285	#137	—	4	4	no	—
286	#137	—	4	4	no	—
287	C₂×Q₁₆	—	2	2	no	—
288	C₂×Q₁₆	—	2	2	no	—
289	C₂×SD₁₆	—	2	2	no	—
290	C₂×SD₁₆	—	2	2	no	—
291	C₂×D₈	—	2	2	no	—
292	#131	—	4	4	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
295	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	—	2	1	no	—
298	C₂²×C₈	—	2	2	no	—
299	C₄⋊Q₈	—	2	1	no	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	—	2	1	no	—
302	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
304	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
305	#127	D₄×C₂₃	1	1	yes	characteristic
306	#127	C₂³×C₂₃	1	1	yes	characteristic
307	#124	—	2	1	no	—
308	#124	—	2	1	no	—
309	#121	—	2	1	no	—
310	#121	—	2	1	no	—
311	#121	—	2	1	no	—
312	#121	D₄×C₂₃	1	2	yes	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	C₂³×C₂₃	1	2	yes	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
320	#107	—	2	2	no	—
321	#107	—	2	2	no	—
322	#107	—	2	2	no	—
323	C₄⋊C₈	—	2	1	no	—
324	C₄⋊C₈	—	2	1	no	—
325	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
326	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
327	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
328	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
330	#102	—	2	2	no	—
331	#102	—	2	2	no	—
332	#102	—	2	2	no	—
333	#102	—	2	2	no	—
334	C₂²⋊C₈	—	2	1	no	—
335	C₂²⋊C₈	—	2	2	no	—
336	C₂²⋊C₈	—	2	2	no	—
337	#97	D₄×C₂₃	1	2	yes	—
338	C₄×C₈	—	2	1	no	—
339	C₄×C₈	C₂³×C₂₃	1	2	yes	—
340	C₂₃	#19231	1	1	yes	characteristic, direct product factor, Sylow
341	#41	—	4	2	no	—
342	#41	—	4	2	no	—
343	#41	—	4	2	no	—
344	#41	—	4	2	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	4	1	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	4	1	no	—
361	C₂²×C₄	—	4	1	no	—
362	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
365	Q₁₆	—	4	4	no	—
366	Q₁₆	—	4	4	no	—
367	SD₁₆	—	4	4	no	—
368	SD₁₆	—	4	4	no	—
369	D₈	—	8	2	no	—
370	M₁₆	—	2	2	no	—
371	M₁₆	—	4	4	no	—
372	M₁₆	D₈×C₂₃	1	4	yes	complemented
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	4	2	no	—
379	C₂×C₈	—	4	4	no	—
380	C₂×C₈	—	8	1	no	—
381	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	4	1	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	2	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	1	no	—
395	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₄²	#41×C₂₃	1	1	yes	characteristic
398	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
399	C₂³	—	2	2	no	—
400	C₂³	—	4	1	no	—
401	C₂³	C₂₃×#121	1	1	yes	characteristic
402	Q₈	—	4	2	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	4	1	no	—
405	Q₈	—	4	2	no	—
406	Q₈	—	4	1	no	—
407	Q₈	—	8	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	2	2	no	—
410	D₄	—	4	4	no	—
411	D₄	—	4	1	no	—
412	D₄	—	4	1	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	1	no	—
415	D₄	—	4	2	no	—
416	D₄	—	8	1	no	—
417	D₄	—	8	1	no	—
418	D₄	—	8	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	8	2	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
430	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
431	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
432	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
433	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
434	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
437	C₈	—	2	2	no	—
438	C₈	—	4	1	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	2	no	—
441	C₈	—	4	1	no	—
442	C₈	—	4	1	no	—
443	C₈	C₂×D₈×C₂₃	1	4	yes	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	8	1	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	C₂₃×#459	1	2	yes	—
453	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	4	2	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	2	no	—
460	C₄	C₂₃×#459	1	2	yes	—
461	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
462	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
463	C₄	C₂₃×#577	1	1	yes	characteristic
464	C₄	C₂₃×#578	1	1	yes	characteristic
465	C₂	—	2	2	no	—
466	C₂	—	8	2	no	—
467	C₂	—	16	1	no	—
468	C₂	C₂×C₂₃×#459	1	1	yes	characteristic
469	C₂	C₂₃×#3063	1	1	yes	characteristic
470	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×#3226	C₂	semidirect
3. C₂₃×#3152	C₂	semidirect
4. C₂₃×#2935	C₂	semidirect
5. C₂₃×#1689	C₂	semidirect
6. C₂₃×#1685	C₂	semidirect
7. C₂₃×#1669	C₂	semidirect
8. C₂₃×#1665	C₂	semidirect
9. C₂₃×#1565	C₂	semidirect
10. C₂₃×#1540	C₂	semidirect
11. C₂×C₂₃×#332	C₂	semidirect
136. M₁₆×C₂₃	D₈	semidirect
163. #19231	C₂₃	direct (N × H)
210. #3226	C₂×C₂₃	semidirect
211. #3152	C₂×C₂₃	semidirect
212. #2935	C₂×C₂₃	semidirect
213. #1689	C₂×C₂₃	semidirect
214. #1685	C₂×C₂₃	semidirect
215. #1669	C₂×C₂₃	semidirect
216. #1665	C₂×C₂₃	semidirect
217. #1565	C₂×C₂₃	semidirect
218. #1540	C₂×C₂₃	semidirect
219. C₂×#332	C₂×C₂₃	semidirect
340. C₂₃	#19231	direct (N × H)
372. M₁₆	D₈×C₂₃	semidirect

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×#3226	2
C₂₃×#3152	2
C₂₃×#2935	2
C₂₃×#1689	2
C₂₃×#1685	2
C₂₃×#1669	2
C₂₃×#1665	2
C₂₃×#1565	2
C₂₃×#1540	2
C₂×C₂₃×#332	2
#19231	23

Subgroups — C23×#19231 #19166857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19231 #19166857