Subgroups — C₂₃×#18664 #19166290

Order 5888.

980 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3036	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1651	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1643	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1627	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1622	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂₃×#554	—	2	1	no	—
14	C₂₃×#554	—	2	2	no	—
15	C₂₃×#554	—	2	2	no	—
16	C₂₃×#554	C₂²	1	2	yes	—
17	D₄×Q₈×C₂₃	—	2	1	no	—
18	D₄×Q₈×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#544	—	2	1	no	—
20	C₂₃×#543	—	2	2	no	—
21	C₂₃×#533	C₂²	1	1	yes	characteristic
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#532	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
25	C₂₃×#518	—	2	1	no	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#498	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#492	C₂²	1	1	yes	characteristic, complemented
29	C₂₃×#487	C₂²	1	1	yes	characteristic, complemented
30	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#476	—	2	1	no	—
32	C₂₃×#475	C₂²	1	2	yes	complemented
33	C₂₃×#465	—	2	2	no	—
34	C₂₃×#465	—	2	2	no	—
35	C₂₃×#463	—	2	2	no	—
36	C₂₃×#461	—	2	1	no	—
37	C₂₃×#461	C₂²	1	2	yes	—
38	C₂₃×#460	C₂²	1	2	yes	complemented
39	C₂₃×#459	C₂²	1	2	yes	—
40	C₄×SD₁₆×C₂₃	—	2	2	no	—
41	C₂₃×#423	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂₃×#417	—	2	1	no	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#333	C₂²	1	2	yes	complemented
46	C₂₃×#332	C₂²	1	2	yes	complemented
47	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
48	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
49	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
50	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₂₃	—	2	1	no	—
52	C₂²×Q₈×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	—	2	1	no	—
55	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
56	C₂×SD₁₆×C₂₃	—	2	2	no	—
57	C₂×SD₁₆×C₂₃	—	2	2	no	—
58	C₂×SD₁₆×C₂₃	—	2	2	no	—
59	C₂×D₈×C₂₃	—	2	2	no	—
60	C₂×M₁₆×C₂₃	—	2	1	no	—
61	C₂²×C₈×C₂₃	—	2	1	no	—
62	C₂²×C₈×C₂₃	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
67	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₂₃×#127	C₂³	1	1	yes	characteristic
71	C₂₃×#125	—	2	2	no	—
72	C₂₃×#125	—	2	2	no	—
73	C₂₃×#125	—	2	1	no	—
74	C₂₃×#125	—	2	1	no	—
75	C₂₃×#124	—	2	1	no	—
76	C₂₃×C₂²⋊Q₈	—	2	4	no	—
77	C₂₃×C₂²⋊Q₈	—	2	4	no	—
78	C₂₃×C₂²⋊Q₈	—	2	4	no	—
79	C₂₃×C₂²⋊Q₈	—	2	4	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	—	2	4	no	—
82	C₂₃×C₂²⋊Q₈	—	2	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	—	2	2	no	—
85	C₂₃×C₂²⋊Q₈	—	2	4	no	—
86	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
87	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
88	C₂₃×#121	—	2	1	no	—
89	C₂₃×#121	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	D₄	1	2	yes	—
93	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
94	C₄×D₄×C₂₃	—	2	2	no	—
95	C₄×D₄×C₂₃	—	2	2	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	—	2	2	no	—
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	D₄	1	2	yes	—
103	C₄×D₄×C₂₃	C₂³	1	2	yes	—
104	C₂₃×#117	—	2	1	no	—
105	C₂₃×#117	—	2	1	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
108	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂₃×#106	—	2	2	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×C₄⋊C₈	—	2	1	no	—
115	C₂₃×C₄⋊C₈	—	2	1	no	—
116	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#102	—	2	2	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#102	—	2	2	no	—
124	C₄×C₈×C₂₃	—	2	1	no	—
125	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₂⁴×C₂₃	—	2	1	no	—
128	#41×C₂₃	—	4	2	no	—
129	#41×C₂₃	—	4	2	no	—
130	C₂×Q₈×C₂₃	—	2	4	no	—
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	8	2	no	—
135	C₂×Q₈×C₂₃	—	8	2	no	—
136	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	2	no	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	1	no	—
142	C₂×D₄×C₂₃	—	2	1	no	—
143	C₂×D₄×C₂₃	—	4	1	no	—
144	C₂×D₄×C₂₃	SD₁₆	1	4	yes	complemented
145	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	4	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	2	no	—
151	C₂²×C₄×C₂₃	—	4	1	no	—
152	C₂²×C₄×C₂₃	—	8	1	no	—
153	C₂²×C₄×C₂₃	—	8	1	no	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	SD₁₆×C₂₃	—	4	4	no	—
158	SD₁₆×C₂₃	—	4	4	no	—
159	SD₁₆×C₂₃	—	8	2	no	—
160	D₈×C₂₃	—	4	4	no	—
161	M₁₆×C₂₃	—	4	2	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	2	no	—
165	C₂×C₈×C₂₃	—	2	2	no	—
166	C₂×C₈×C₂₃	—	4	2	no	—
167	C₂×C₈×C₂₃	—	8	1	no	—
168	C₄⋊C₄×C₂₃	—	2	4	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	4	no	—
171	C₄⋊C₄×C₂₃	—	2	2	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
187	C₂²⋊C₄×C₂₃	—	2	4	no	—
188	C₂²⋊C₄×C₂₃	—	2	4	no	—
189	C₂²⋊C₄×C₂₃	—	2	2	no	—
190	C₂²⋊C₄×C₂₃	—	2	2	no	—
191	C₂²⋊C₄×C₂₃	—	2	2	no	—
192	C₂²⋊C₄×C₂₃	—	2	2	no	—
193	C₂²⋊C₄×C₂₃	—	4	1	no	—
194	C₂²⋊C₄×C₂₃	—	4	1	no	—
195	C₄²×C₂₃	—	2	2	no	—
196	C₄²×C₂₃	—	2	2	no	—
197	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
201	#18664	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₂₃	—	2	2	no	—
203	C₂³×C₂₃	—	2	4	no	—
204	C₂³×C₂₃	—	4	1	no	—
205	C₂³×C₂₃	—	8	1	no	—
206	C₂³×C₂₃	—	8	1	no	—
207	C₂³×C₂₃	#120	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	2	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	4	2	no	—
211	Q₈×C₂₃	—	4	1	no	—
212	Q₈×C₂₃	—	4	1	no	—
213	Q₈×C₂₃	—	4	1	no	—
214	Q₈×C₂₃	—	4	1	no	—
215	Q₈×C₂₃	—	8	1	no	—
216	Q₈×C₂₃	—	8	4	no	—
217	D₄×C₂₃	—	2	2	no	—
218	D₄×C₂₃	—	2	4	no	—
219	D₄×C₂₃	—	4	2	no	—
220	D₄×C₂₃	—	4	1	no	—
221	D₄×C₂₃	—	4	1	no	—
222	D₄×C₂₃	—	4	1	no	—
223	D₄×C₂₃	—	4	1	no	—
224	D₄×C₂₃	—	8	2	no	—
225	D₄×C₂₃	C₂×SD₁₆	1	4	yes	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	2	no	—
235	C₂×C₄×C₂₃	—	4	2	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	8	2	no	—
238	C₂×C₄×C₂₃	—	8	1	no	—
239	C₂×C₄×C₂₃	—	8	4	no	—
240	C₂×C₄×C₂₃	—	8	4	no	—
241	C₂×C₄×C₂₃	—	8	2	no	—
242	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
243	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
244	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
246	C₂×C₄×C₂₃	#137	1	2	yes	—
247	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₈×C₂₃	—	2	2	no	—
251	C₈×C₂₃	—	4	1	no	—
252	C₈×C₂₃	—	4	2	no	—
253	C₈×C₂₃	—	4	2	no	—
254	#3501	C₂×C₂₃	1	1	yes	characteristic, complemented
255	#3246	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#3036	C₂×C₂₃	1	1	yes	characteristic
258	#1651	C₂×C₂₃	1	2	yes	complemented
259	#1643	C₂×C₂₃	1	2	yes	complemented
260	#1627	C₂×C₂₃	1	2	yes	complemented
261	#1622	C₂×C₂₃	1	2	yes	complemented
262	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
264	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
265	C₂²×C₂₃	—	2	1	no	—
266	C₂²×C₂₃	—	2	1	no	—
267	C₂²×C₂₃	—	2	4	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	8	4	no	—
271	C₂²×C₂₃	—	8	1	no	—
272	C₂²×C₂₃	—	8	1	no	—
273	C₂²×C₂₃	—	8	1	no	—
274	C₂²×C₂₃	#448	1	2	yes	—
275	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
276	C₄×C₂₃	—	2	1	no	—
277	C₄×C₂₃	—	2	1	no	—
278	C₄×C₂₃	—	2	1	no	—
279	C₄×C₂₃	—	4	2	no	—
280	C₄×C₂₃	—	4	2	no	—
281	C₄×C₂₃	—	4	2	no	—
282	C₄×C₂₃	—	4	2	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	8	1	no	—
285	C₄×C₂₃	#448	1	2	yes	—
286	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
287	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
288	C₄×C₂₃	#576	1	1	yes	characteristic
289	C₄×C₂₃	#577	1	1	yes	characteristic
290	C₂×AGL(1,ℤ/8)	—	2	1	no	—
291	#554	—	2	1	no	—
292	#554	—	2	2	no	—
293	#554	—	2	2	no	—
294	#554	C₂²×C₂₃	1	2	yes	—
295	D₄×Q₈	—	2	1	no	—
296	D₄×Q₈	C₂²×C₂₃	1	2	yes	—
297	#544	—	2	1	no	—
298	#543	—	2	2	no	—
299	#533	C₂²×C₂₃	1	1	yes	characteristic
300	#533	C₂²×C₂₃	1	1	yes	characteristic
301	#532	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×C₂²⋊Q₈	—	2	2	no	—
303	#518	—	2	1	no	—
304	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
305	#498	C₂²×C₂₃	1	1	yes	characteristic, complemented
306	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
307	#487	C₂²×C₂₃	1	1	yes	characteristic, complemented
308	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
309	#476	—	2	1	no	—
310	#475	C₂²×C₂₃	1	2	yes	complemented
311	#465	—	2	2	no	—
312	#465	—	2	2	no	—
313	#463	—	2	2	no	—
314	#461	—	2	1	no	—
315	#461	C₂²×C₂₃	1	2	yes	—
316	#460	C₂²×C₂₃	1	2	yes	complemented
317	#459	C₂²×C₂₃	1	2	yes	—
318	C₄×SD₁₆	—	2	2	no	—
319	#423	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
321	#417	—	2	1	no	—
322	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
323	#333	C₂²×C₂₃	1	2	yes	complemented
324	#332	C₂²×C₂₃	1	2	yes	complemented
325	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
326	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
327	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
328	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
329	C₂×C₂₃	—	2	2	no	—
330	C₂×C₂₃	—	4	4	no	—
331	C₂×C₂₃	—	16	1	no	—
332	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
333	C₂×C₂₃	#3012	1	1	yes	characteristic
334	C₂×C₂₃	#3018	1	1	yes	characteristic
335	C₂×#41	—	2	1	no	—
336	C₂²×Q₈	—	2	2	no	—
337	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
338	C₂³×C₄	—	2	1	no	—
339	AGL(1,ℤ/8)	—	4	4	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂×D₈	—	2	2	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂²×C₈	—	2	1	no	—
346	C₂²×C₈	—	2	1	no	—
347	C₄⋊Q₈	—	2	2	no	—
348	C₄⋊Q₈	—	2	2	no	—
349	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
350	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
354	#127	C₂³×C₂₃	1	1	yes	characteristic
355	#125	—	2	2	no	—
356	#125	—	2	2	no	—
357	#125	—	2	1	no	—
358	#125	—	2	1	no	—
359	#124	—	2	1	no	—
360	C₂²⋊Q₈	—	2	4	no	—
361	C₂²⋊Q₈	—	2	4	no	—
362	C₂²⋊Q₈	—	2	4	no	—
363	C₂²⋊Q₈	—	2	4	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	4	no	—
370	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
371	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
372	#121	—	2	1	no	—
373	#121	—	2	1	no	—
374	C₄×Q₈	—	2	1	no	—
375	C₄×Q₈	—	2	1	no	—
376	C₄×Q₈	D₄×C₂₃	1	2	yes	—
377	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	—	2	2	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	D₄×C₂₃	1	2	yes	—
387	C₄×D₄	C₂³×C₂₃	1	2	yes	—
388	#117	—	2	1	no	—
389	#117	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	4	no	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	4	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
396	#106	—	2	2	no	—
397	#106	—	2	2	no	—
398	C₄⋊C₈	—	2	1	no	—
399	C₄⋊C₈	—	2	1	no	—
400	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
401	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
402	#103	—	2	2	no	—
403	#103	—	2	2	no	—
404	#103	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	C₄×C₈	—	2	1	no	—
409	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₂₃	#18664	1	1	yes	characteristic, direct product factor, Sylow
412	C₂⁴	—	2	1	no	—
413	#41	—	4	2	no	—
414	#41	—	4	2	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	8	2	no	—
420	C₂×Q₈	—	8	2	no	—
421	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
422	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	4	1	no	—
429	C₂×D₄	SD₁₆×C₂₃	1	4	yes	complemented
430	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	4	1	no	—
437	C₂²×C₄	—	8	1	no	—
438	C₂²×C₄	—	8	1	no	—
439	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	SD₁₆	—	4	4	no	—
443	SD₁₆	—	4	4	no	—
444	SD₁₆	—	8	2	no	—
445	D₈	—	4	4	no	—
446	M₁₆	—	4	2	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	4	2	no	—
452	C₂×C₈	—	8	1	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
470	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
471	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	2	2	no	—
477	C₂²⋊C₄	—	2	2	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
483	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
485	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
486	C₂³	—	2	2	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	4	1	no	—
489	C₂³	—	8	1	no	—
490	C₂³	—	8	1	no	—
491	C₂³	C₂₃×#120	1	1	yes	characteristic
492	Q₈	—	2	2	no	—
493	Q₈	—	2	2	no	—
494	Q₈	—	4	2	no	—
495	Q₈	—	4	1	no	—
496	Q₈	—	4	1	no	—
497	Q₈	—	4	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	8	1	no	—
500	Q₈	—	8	4	no	—
501	D₄	—	2	2	no	—
502	D₄	—	2	4	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	1	no	—
506	D₄	—	4	1	no	—
507	D₄	—	4	1	no	—
508	D₄	—	8	2	no	—
509	D₄	C₂×SD₁₆×C₂₃	1	4	yes	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	8	2	no	—
522	C₂×C₄	—	8	1	no	—
523	C₂×C₄	—	8	4	no	—
524	C₂×C₄	—	8	4	no	—
525	C₂×C₄	—	8	2	no	—
526	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
527	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
528	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
529	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
530	C₂×C₄	C₂₃×#137	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
533	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
534	C₈	—	2	2	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	2	no	—
537	C₈	—	4	2	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	8	4	no	—
544	C₂²	—	8	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	—	8	1	no	—
547	C₂²	C₂₃×#448	1	2	yes	—
548	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	2	no	—
556	C₄	—	4	2	no	—
557	C₄	—	8	1	no	—
558	C₄	C₂₃×#448	1	2	yes	—
559	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
560	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
561	C₄	C₂₃×#576	1	1	yes	characteristic
562	C₄	C₂₃×#577	1	1	yes	characteristic
563	C₂	—	2	2	no	—
564	C₂	—	4	4	no	—
565	C₂	—	16	1	no	—
566	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
567	C₂	C₂₃×#3012	1	1	yes	characteristic
568	C₂	C₂₃×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3036	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×#3501	C₂	semidirect
2. C₂₃×#3246	C₂	semidirect
3. C₂₃×#3167	C₂	semidirect
5. C₂₃×#1651	C₂	semidirect
6. C₂₃×#1643	C₂	semidirect
7. C₂₃×#1627	C₂	semidirect
8. C₂₃×#1622	C₂	semidirect
9. C₂₃×#1550	C₂	semidirect
10. C₂₃×#1493	C₂	semidirect
11. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
27. C₂₃×#498	C₂²	semidirect
28. C₂₃×#492	C₂²	semidirect
29. C₂₃×#487	C₂²	semidirect
30. C₂₃×#486	C₂²	semidirect
32. C₂₃×#475	C₂²	semidirect
38. C₂₃×#460	C₂²	semidirect
45. C₂₃×#333	C₂²	semidirect
46. C₂₃×#332	C₂²	semidirect
47. C₂₃×Q₈⋊C₈	C₂²	semidirect
48. C₂₃×Q₈⋊C₈	C₂²	semidirect
144. C₂×D₄×C₂₃	SD₁₆	semidirect
201. #18664	C₂₃	direct (N × H)
254. #3501	C₂×C₂₃	semidirect
255. #3246	C₂×C₂₃	semidirect
256. #3167	C₂×C₂₃	semidirect
258. #1651	C₂×C₂₃	semidirect
259. #1643	C₂×C₂₃	semidirect
260. #1627	C₂×C₂₃	semidirect
261. #1622	C₂×C₂₃	semidirect
262. #1550	C₂×C₂₃	semidirect
263. #1493	C₂×C₂₃	semidirect
264. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
305. #498	C₂²×C₂₃	semidirect
306. #492	C₂²×C₂₃	semidirect
307. #487	C₂²×C₂₃	semidirect
308. #486	C₂²×C₂₃	semidirect
310. #475	C₂²×C₂₃	semidirect
316. #460	C₂²×C₂₃	semidirect
323. #333	C₂²×C₂₃	semidirect
324. #332	C₂²×C₂₃	semidirect
325. Q₈⋊C₈	C₂²×C₂₃	semidirect
326. Q₈⋊C₈	C₂²×C₂₃	semidirect
411. C₂₃	#18664	direct (N × H)
429. C₂×D₄	SD₁₆×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3501	2
C₂₃×#3246	2
C₂₃×#3167	2
C₂₃×#3036	2
C₂₃×#1651	2
C₂₃×#1643	2
C₂₃×#1627	2
C₂₃×#1622	2
C₂₃×#1550	2
C₂₃×#1493	2
C₂×C₂₃×D₄⋊C₈	2
#18664	23

Subgroups — C23×#18664 #19166290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18664 #19166290