Subgroups — C₂₃×#18556 #19166182

Order 5888.

954 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3182	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3115	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1649	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1627	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1559	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	—	2	1	no	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#557	—	2	1	no	—
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#554	—	2	2	no	—
21	C₂₃×#554	—	2	1	no	—
22	C₂₃×#554	—	2	1	no	—
23	C₂₃×#554	C₂²	1	1	yes	characteristic
24	D₄×Q₈×C₂₃	—	2	1	no	—
25	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#533	—	2	2	no	—
27	C₂₃×#533	—	2	1	no	—
28	C₂₃×#533	—	2	1	no	—
29	C₂₃×#533	C₂²	1	1	yes	characteristic
30	C₂₃×#532	C₂²	1	1	yes	characteristic
31	C₄×#41×C₂₃	—	2	1	no	—
32	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
34	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#500	C₂²	1	1	yes	characteristic
36	C₂₃×#492	C₂²	1	1	yes	characteristic
37	C₂₃×#487	C₂²	1	1	yes	characteristic
38	C₂₃×#486	C₂²	1	1	yes	characteristic
39	C₂₃×#482	—	2	1	no	—
40	C₂₃×#475	C₂²	1	1	yes	characteristic
41	C₂₃×#474	C₂²	1	1	yes	characteristic
42	C₂₃×#468	—	2	2	no	—
43	C₂₃×#464	—	2	1	no	—
44	C₂₃×#464	—	2	1	no	—
45	C₂₃×#462	C₂²	1	1	yes	characteristic
46	C₂₃×#462	C₂²	1	1	yes	characteristic
47	C₂₃×#460	C₂²	1	1	yes	characteristic
48	C₂₃×#460	C₂²	1	1	yes	characteristic
49	C₂₃×#460	C₂²	1	1	yes	characteristic
50	C₂₃×#459	C₂²	1	1	yes	characteristic
51	C₂₃×#449	—	2	1	no	—
52	C₂₃×#442	—	2	1	no	—
53	C₂₃×#440	—	2	2	no	—
54	C₂₃×#440	—	2	1	no	—
55	C₂₃×#432	C₂²	1	1	yes	characteristic
56	C₂₃×#423	C₂²	1	1	yes	characteristic
57	C₂₃×#417	—	2	1	no	—
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
73	C₂²×Q₈×C₂₃	—	2	1	no	—
74	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
75	C₂₃×#137	—	4	4	no	—
76	C₂×Q₁₆×C₂₃	—	2	1	no	—
77	C₂×Q₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	1	no	—
80	C₂×SD₁₆×C₂₃	—	2	1	no	—
81	C₂×SD₁₆×C₂₃	—	2	2	no	—
82	C₂×D₈×C₂₃	—	2	1	no	—
83	C₂×M₁₆×C₂₃	—	2	1	no	—
84	C₂×M₁₆×C₂₃	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	2	no	—
86	C₂₃×C₄⋊Q₈	—	2	1	no	—
87	C₂₃×C₄⋊Q₈	—	2	2	no	—
88	C₂₃×C₄⋊Q₈	—	2	1	no	—
89	C₂₃×C₄⋊Q₈	—	2	1	no	—
90	C₂₃×C₄⋊Q₈	—	2	1	no	—
91	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
93	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
94	C₂₃×#127	C₂³	1	1	yes	characteristic
95	C₂₃×#125	—	2	2	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	2	no	—
98	C₂₃×#125	—	2	1	no	—
99	C₂₃×#125	—	2	1	no	—
100	C₂₃×#125	—	2	1	no	—
101	C₂₃×#124	—	2	1	no	—
102	C₂₃×C₂²⋊Q₈	—	2	2	no	—
103	C₂₃×C₂²⋊Q₈	—	2	2	no	—
104	C₂₃×C₂²⋊Q₈	—	2	2	no	—
105	C₂₃×C₂²⋊Q₈	—	2	2	no	—
106	C₂₃×C₂²⋊Q₈	—	2	2	no	—
107	C₂₃×C₂²⋊Q₈	—	2	1	no	—
108	C₂₃×C₂²⋊Q₈	—	2	2	no	—
109	C₂₃×C₂²⋊Q₈	—	2	1	no	—
110	C₂₃×C₂²⋊Q₈	—	2	1	no	—
111	C₂₃×C₂²⋊Q₈	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
113	C₂₃×#121	—	2	1	no	—
114	C₂₃×#121	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	2	no	—
116	C₄×Q₈×C₂₃	—	2	2	no	—
117	C₄×Q₈×C₂₃	—	2	2	no	—
118	C₄×Q₈×C₂₃	—	2	1	no	—
119	C₄×Q₈×C₂₃	—	2	1	no	—
120	C₄×Q₈×C₂₃	—	2	1	no	—
121	C₄×Q₈×C₂₃	—	2	1	no	—
122	C₄×Q₈×C₂₃	—	2	1	no	—
123	C₄×Q₈×C₂₃	—	2	1	no	—
124	C₄×Q₈×C₂₃	—	2	1	no	—
125	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
126	C₄×Q₈×C₂₃	D₄	1	2	yes	—
127	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
128	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
129	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
130	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
131	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
132	C₄×D₄×C₂₃	—	2	2	no	—
133	C₄×D₄×C₂₃	—	2	2	no	—
134	C₄×D₄×C₂₃	—	2	1	no	—
135	C₄×D₄×C₂₃	—	2	1	no	—
136	C₄×D₄×C₂₃	—	2	1	no	—
137	C₄×D₄×C₂₃	—	2	1	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
141	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
142	C₂₃×#117	—	2	2	no	—
143	C₂₃×#117	—	2	1	no	—
144	C₂₃×#117	—	2	1	no	—
145	C₂₃×#117	D₄	1	1	yes	characteristic
146	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
150	C₂×C₄²×C₂₃	—	2	2	no	—
151	C₂×C₄²×C₂₃	—	2	1	no	—
152	C₂×C₄²×C₂₃	—	2	1	no	—
153	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
154	C₂₃×#107	—	2	2	no	—
155	C₂₃×#107	—	2	1	no	—
156	C₂₃×#106	—	2	1	no	—
157	C₂₃×C₄⋊C₈	—	2	1	no	—
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×#103	—	2	1	no	—
163	C₂₃×#103	—	2	2	no	—
164	C₂₃×#103	—	2	1	no	—
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#102	—	2	1	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	1	no	—
169	C₂₃×#102	—	2	1	no	—
170	C₂₃×#102	—	2	2	no	—
171	C₂₃×#102	—	2	1	no	—
172	C₂₃×C₂²⋊C₈	—	2	1	no	—
173	C₂₃×#97	—	2	1	no	—
174	C₂₃×#97	—	2	1	no	—
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
177	#41×C₂₃	—	2	2	no	—
178	#41×C₂₃	—	2	2	no	—
179	#41×C₂₃	—	4	2	no	—
180	#41×C₂₃	—	4	2	no	—
181	C₂×Q₈×C₂₃	—	2	2	no	—
182	C₂×Q₈×C₂₃	—	2	2	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	2	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	4	1	no	—
188	C₂×Q₈×C₂₃	—	4	2	no	—
189	C₂×Q₈×C₂₃	—	4	2	no	—
190	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
194	C₂×D₄×C₂₃	—	2	1	no	—
195	C₂×D₄×C₂₃	—	2	1	no	—
196	C₂×D₄×C₂₃	—	2	1	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	4	1	no	—
199	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₂₃	—	2	1	no	—
201	C₂²×C₄×C₂₃	—	2	2	no	—
202	C₂²×C₄×C₂₃	—	2	1	no	—
203	C₂²×C₄×C₂₃	—	2	2	no	—
204	C₂²×C₄×C₂₃	—	2	1	no	—
205	C₂²×C₄×C₂₃	—	2	1	no	—
206	C₂²×C₄×C₂₃	—	2	1	no	—
207	C₂²×C₄×C₂₃	—	4	1	no	—
208	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
211	Q₁₆×C₂₃	—	4	2	no	—
212	Q₁₆×C₂₃	—	4	2	no	—
213	SD₁₆×C₂₃	—	4	4	no	—
214	SD₁₆×C₂₃	—	4	2	no	—
215	SD₁₆×C₂₃	—	4	2	no	—
216	SD₁₆×C₂₃	—	4	2	no	—
217	D₈×C₂₃	—	4	2	no	—
218	M₁₆×C₂₃	—	4	2	no	—
219	M₁₆×C₂₃	—	4	2	no	—
220	C₂×C₈×C₂₃	—	2	1	no	—
221	C₂×C₈×C₂₃	—	2	1	no	—
222	C₂×C₈×C₂₃	—	2	1	no	—
223	C₂×C₈×C₂₃	—	2	1	no	—
224	C₂×C₈×C₂₃	—	2	1	no	—
225	C₂×C₈×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	2	no	—
227	C₄⋊C₄×C₂₃	—	2	2	no	—
228	C₄⋊C₄×C₂₃	—	2	2	no	—
229	C₄⋊C₄×C₂₃	—	2	2	no	—
230	C₄⋊C₄×C₂₃	—	2	2	no	—
231	C₄⋊C₄×C₂₃	—	2	2	no	—
232	C₄⋊C₄×C₂₃	—	2	2	no	—
233	C₄⋊C₄×C₂₃	—	2	2	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	2	2	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	2	1	no	—
239	C₄⋊C₄×C₂₃	—	2	2	no	—
240	C₄⋊C₄×C₂₃	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	1	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	—	2	1	no	—
244	C₄⋊C₄×C₂₃	—	2	1	no	—
245	C₄⋊C₄×C₂₃	—	2	1	no	—
246	C₄⋊C₄×C₂₃	—	2	1	no	—
247	C₄⋊C₄×C₂₃	—	2	1	no	—
248	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
249	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
250	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
251	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
252	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
253	C₂²⋊C₄×C₂₃	—	2	2	no	—
254	C₂²⋊C₄×C₂₃	—	2	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	2	no	—
256	C₂²⋊C₄×C₂₃	—	2	1	no	—
257	C₂²⋊C₄×C₂₃	—	2	1	no	—
258	C₂²⋊C₄×C₂₃	—	2	1	no	—
259	C₂²⋊C₄×C₂₃	—	4	1	no	—
260	C₂²⋊C₄×C₂₃	—	4	1	no	—
261	C₄²×C₂₃	—	2	2	no	—
262	C₄²×C₂₃	—	2	2	no	—
263	C₄²×C₂₃	—	2	1	no	—
264	C₄²×C₂₃	—	2	2	no	—
265	C₄²×C₂₃	—	2	1	no	—
266	C₄²×C₂₃	—	2	1	no	—
267	C₄²×C₂₃	—	2	1	no	—
268	C₄²×C₂₃	—	2	1	no	—
269	C₄²×C₂₃	—	2	1	no	—
270	C₄²×C₂₃	—	2	1	no	—
271	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
272	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
273	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
274	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
275	#18556	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂³×C₂₃	—	2	1	no	—
277	C₂³×C₂₃	—	4	1	no	—
278	C₂³×C₂₃	#120	1	1	yes	characteristic
279	Q₈×C₂₃	—	2	1	no	—
280	Q₈×C₂₃	—	2	1	no	—
281	Q₈×C₂₃	—	2	1	no	—
282	Q₈×C₂₃	—	2	1	no	—
283	Q₈×C₂₃	—	4	2	no	—
284	Q₈×C₂₃	—	4	2	no	—
285	Q₈×C₂₃	—	4	4	no	—
286	Q₈×C₂₃	—	4	1	no	—
287	Q₈×C₂₃	—	4	1	no	—
288	Q₈×C₂₃	—	4	1	no	—
289	Q₈×C₂₃	—	4	1	no	—
290	Q₈×C₂₃	—	8	2	no	—
291	Q₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
292	Q₈×C₂₃	#137	1	2	yes	complemented
293	D₄×C₂₃	—	2	4	no	—
294	D₄×C₂₃	—	2	1	no	—
295	D₄×C₂₃	—	4	2	no	—
296	D₄×C₂₃	—	4	1	no	—
297	D₄×C₂₃	—	4	1	no	—
298	D₄×C₂₃	—	4	1	no	—
299	D₄×C₂₃	—	4	1	no	—
300	D₄×C₂₃	—	8	2	no	—
301	D₄×C₂₃	#137	1	2	yes	complemented
302	C₂×C₄×C₂₃	—	2	4	no	—
303	C₂×C₄×C₂₃	—	2	1	no	—
304	C₂×C₄×C₂₃	—	2	1	no	—
305	C₂×C₄×C₂₃	—	2	1	no	—
306	C₂×C₄×C₂₃	—	2	1	no	—
307	C₂×C₄×C₂₃	—	2	1	no	—
308	C₂×C₄×C₂₃	—	2	1	no	—
309	C₂×C₄×C₂₃	—	2	1	no	—
310	C₂×C₄×C₂₃	—	2	1	no	—
311	C₂×C₄×C₂₃	—	2	2	no	—
312	C₂×C₄×C₂₃	—	2	2	no	—
313	C₂×C₄×C₂₃	—	2	1	no	—
314	C₂×C₄×C₂₃	—	2	1	no	—
315	C₂×C₄×C₂₃	—	2	1	no	—
316	C₂×C₄×C₂₃	—	4	1	no	—
317	C₂×C₄×C₂₃	—	4	2	no	—
318	C₂×C₄×C₂₃	—	4	4	no	—
319	C₂×C₄×C₂₃	—	4	1	no	—
320	C₂×C₄×C₂₃	—	4	1	no	—
321	C₂×C₄×C₂₃	—	4	1	no	—
322	C₂×C₄×C₂₃	—	8	1	no	—
323	C₂×C₄×C₂₃	—	8	2	no	—
324	C₂×C₄×C₂₃	—	8	1	no	—
325	C₂×C₄×C₂₃	—	8	2	no	—
326	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
327	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
328	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
329	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
332	C₈×C₂₃	—	2	2	no	—
333	C₈×C₂₃	—	4	1	no	—
334	C₈×C₂₃	—	4	1	no	—
335	C₈×C₂₃	—	4	1	no	—
336	C₈×C₂₃	—	4	1	no	—
337	C₈×C₂₃	—	4	1	no	—
338	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#3182	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#3115	C₂×C₂₃	1	1	yes	characteristic
341	#3037	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1649	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#1627	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
350	#1559	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#1491	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#1489	C₂×C₂₃	1	1	yes	characteristic, complemented
353	C₂²×C₂₃	—	2	1	no	—
354	C₂²×C₂₃	—	2	2	no	—
355	C₂²×C₂₃	—	2	1	no	—
356	C₂²×C₂₃	—	2	1	no	—
357	C₂²×C₂₃	—	4	1	no	—
358	C₂²×C₂₃	—	4	1	no	—
359	C₂²×C₂₃	—	8	1	no	—
360	C₂²×C₂₃	—	8	1	no	—
361	C₂²×C₂₃	—	8	1	no	—
362	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
363	C₄×C₂₃	—	2	1	no	—
364	C₄×C₂₃	—	2	2	no	—
365	C₄×C₂₃	—	2	1	no	—
366	C₄×C₂₃	—	2	2	no	—
367	C₄×C₂₃	—	2	1	no	—
368	C₄×C₂₃	—	2	1	no	—
369	C₄×C₂₃	—	4	1	no	—
370	C₄×C₂₃	—	4	2	no	—
371	C₄×C₂₃	—	4	1	no	—
372	C₄×C₂₃	—	4	1	no	—
373	C₄×C₂₃	—	4	2	no	—
374	C₄×C₂₃	—	4	1	no	—
375	C₄×C₂₃	—	4	1	no	—
376	C₄×C₂₃	—	8	1	no	—
377	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
378	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
379	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
380	C₄×C₂₃	#575	1	1	yes	characteristic
381	C₂×#137	—	2	1	no	—
382	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
383	#557	—	2	1	no	—
384	#557	C₂²×C₂₃	1	1	yes	characteristic
385	#554	—	2	2	no	—
386	#554	—	2	1	no	—
387	#554	—	2	1	no	—
388	#554	C₂²×C₂₃	1	1	yes	characteristic
389	D₄×Q₈	—	2	1	no	—
390	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
391	#533	—	2	2	no	—
392	#533	—	2	1	no	—
393	#533	—	2	1	no	—
394	#533	C₂²×C₂₃	1	1	yes	characteristic
395	#532	C₂²×C₂₃	1	1	yes	characteristic
396	C₄×#41	—	2	1	no	—
397	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×C₄×Q₈	—	2	1	no	—
399	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
400	#500	C₂²×C₂₃	1	1	yes	characteristic
401	#492	C₂²×C₂₃	1	1	yes	characteristic
402	#487	C₂²×C₂₃	1	1	yes	characteristic
403	#486	C₂²×C₂₃	1	1	yes	characteristic
404	#482	—	2	1	no	—
405	#475	C₂²×C₂₃	1	1	yes	characteristic
406	#474	C₂²×C₂₃	1	1	yes	characteristic
407	#468	—	2	2	no	—
408	#464	—	2	1	no	—
409	#464	—	2	1	no	—
410	#462	C₂²×C₂₃	1	1	yes	characteristic
411	#462	C₂²×C₂₃	1	1	yes	characteristic
412	#460	C₂²×C₂₃	1	1	yes	characteristic
413	#460	C₂²×C₂₃	1	1	yes	characteristic
414	#460	C₂²×C₂₃	1	1	yes	characteristic
415	#459	C₂²×C₂₃	1	1	yes	characteristic
416	#449	—	2	1	no	—
417	#442	—	2	1	no	—
418	#440	—	2	1	no	—
419	#440	—	2	2	no	—
420	#432	C₂²×C₂₃	1	1	yes	characteristic
421	#423	C₂²×C₂₃	1	1	yes	characteristic
422	#417	—	2	1	no	—
423	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
424	#333	C₂²×C₂₃	1	1	yes	characteristic
425	#333	C₂²×C₂₃	1	1	yes	characteristic
426	#332	C₂²×C₂₃	1	1	yes	characteristic
427	#332	C₂²×C₂₃	1	1	yes	characteristic
428	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
429	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
430	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
431	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
432	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
433	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
434	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
435	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
436	C₂×C₂₃	—	4	1	no	—
437	C₂×C₂₃	—	4	2	no	—
438	C₂×C₂₃	—	16	1	no	—
439	C₂×C₂₃	#3004	1	1	yes	characteristic
440	C₂×C₂₃	#3006	1	1	yes	characteristic
441	C₂×C₂₃	#3018	1	1	yes	characteristic
442	C₂×#41	—	2	1	no	—
443	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
444	C₂²×Q₈	—	2	1	no	—
445	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
446	#137	—	4	4	no	—
447	C₂×Q₁₆	—	2	1	no	—
448	C₂×Q₁₆	—	2	1	no	—
449	C₂×SD₁₆	—	2	1	no	—
450	C₂×SD₁₆	—	2	1	no	—
451	C₂×SD₁₆	—	2	1	no	—
452	C₂×SD₁₆	—	2	2	no	—
453	C₂×D₈	—	2	1	no	—
454	C₂×M₁₆	—	2	1	no	—
455	C₂×M₁₆	—	2	1	no	—
456	C₄⋊Q₈	—	2	2	no	—
457	C₄⋊Q₈	—	2	1	no	—
458	C₄⋊Q₈	—	2	2	no	—
459	C₄⋊Q₈	—	2	1	no	—
460	C₄⋊Q₈	—	2	1	no	—
461	C₄⋊Q₈	—	2	1	no	—
462	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
463	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
464	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
465	#127	C₂³×C₂₃	1	1	yes	characteristic
466	#125	—	2	2	no	—
467	#125	—	2	2	no	—
468	#125	—	2	2	no	—
469	#125	—	2	1	no	—
470	#125	—	2	1	no	—
471	#125	—	2	1	no	—
472	#124	—	2	1	no	—
473	C₂²⋊Q₈	—	2	2	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	2	no	—
476	C₂²⋊Q₈	—	2	2	no	—
477	C₂²⋊Q₈	—	2	2	no	—
478	C₂²⋊Q₈	—	2	1	no	—
479	C₂²⋊Q₈	—	2	2	no	—
480	C₂²⋊Q₈	—	2	1	no	—
481	C₂²⋊Q₈	—	2	1	no	—
482	C₂²⋊Q₈	—	2	2	no	—
483	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
484	#121	—	2	1	no	—
485	#121	—	2	1	no	—
486	C₄×Q₈	—	2	2	no	—
487	C₄×Q₈	—	2	2	no	—
488	C₄×Q₈	—	2	2	no	—
489	C₄×Q₈	—	2	1	no	—
490	C₄×Q₈	—	2	1	no	—
491	C₄×Q₈	—	2	1	no	—
492	C₄×Q₈	—	2	1	no	—
493	C₄×Q₈	—	2	1	no	—
494	C₄×Q₈	—	2	1	no	—
495	C₄×Q₈	—	2	1	no	—
496	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
497	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
498	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
499	C₄×Q₈	D₄×C₂₃	1	2	yes	—
500	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
501	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
502	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
503	C₄×D₄	—	2	2	no	—
504	C₄×D₄	—	2	2	no	—
505	C₄×D₄	—	2	1	no	—
506	C₄×D₄	—	2	1	no	—
507	C₄×D₄	—	2	1	no	—
508	C₄×D₄	—	2	1	no	—
509	C₄×D₄	—	2	1	no	—
510	C₄×D₄	—	2	1	no	—
511	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
512	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
513	#117	—	2	2	no	—
514	#117	—	2	1	no	—
515	#117	—	2	1	no	—
516	#117	D₄×C₂₃	1	1	yes	characteristic
517	C₂×C₄⋊C₄	—	2	2	no	—
518	C₂×C₄⋊C₄	—	2	1	no	—
519	C₂×C₄⋊C₄	—	2	1	no	—
520	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
521	C₂×C₄²	—	2	2	no	—
522	C₂×C₄²	—	2	1	no	—
523	C₂×C₄²	—	2	1	no	—
524	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
525	#107	—	2	2	no	—
526	#107	—	2	1	no	—
527	#106	—	2	1	no	—
528	C₄⋊C₈	—	2	1	no	—
529	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
530	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
531	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
532	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
533	#103	—	2	1	no	—
534	#103	—	2	2	no	—
535	#103	—	2	1	no	—
536	#103	—	2	1	no	—
537	#102	—	2	1	no	—
538	#102	—	2	1	no	—
539	#102	—	2	1	no	—
540	#102	—	2	1	no	—
541	#102	—	2	2	no	—
542	#102	—	2	1	no	—
543	C₂²⋊C₈	—	2	1	no	—
544	#97	—	2	1	no	—
545	#97	—	2	1	no	—
546	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
547	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
548	C₂₃	#18556	1	1	yes	characteristic, direct product factor, Sylow
549	#41	—	2	2	no	—
550	#41	—	2	2	no	—
551	#41	—	4	2	no	—
552	#41	—	4	2	no	—
553	C₂×Q₈	—	2	2	no	—
554	C₂×Q₈	—	2	2	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	2	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	—	4	1	no	—
560	C₂×Q₈	—	4	2	no	—
561	C₂×Q₈	—	4	2	no	—
562	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
563	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
564	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
565	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
566	C₂×D₄	—	2	1	no	—
567	C₂×D₄	—	2	1	no	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	—	2	1	no	—
570	C₂×D₄	—	4	1	no	—
571	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
572	C₂²×C₄	—	2	1	no	—
573	C₂²×C₄	—	2	2	no	—
574	C₂²×C₄	—	2	1	no	—
575	C₂²×C₄	—	2	2	no	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	2	1	no	—
578	C₂²×C₄	—	2	1	no	—
579	C₂²×C₄	—	4	1	no	—
580	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
582	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
583	Q₁₆	—	4	2	no	—
584	Q₁₆	—	4	2	no	—
585	SD₁₆	—	4	4	no	—
586	SD₁₆	—	4	2	no	—
587	SD₁₆	—	4	2	no	—
588	SD₁₆	—	4	2	no	—
589	D₈	—	4	2	no	—
590	M₁₆	—	4	2	no	—
591	M₁₆	—	4	2	no	—
592	C₂×C₈	—	2	1	no	—
593	C₂×C₈	—	2	1	no	—
594	C₂×C₈	—	2	1	no	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	—	2	1	no	—
597	C₂×C₈	—	2	1	no	—
598	C₄⋊C₄	—	2	2	no	—
599	C₄⋊C₄	—	2	2	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	2	2	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
621	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
622	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
623	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
624	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
625	C₂²⋊C₄	—	2	2	no	—
626	C₂²⋊C₄	—	2	2	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	1	no	—
629	C₂²⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	2	1	no	—
631	C₂²⋊C₄	—	4	1	no	—
632	C₂²⋊C₄	—	4	1	no	—
633	C₄²	—	2	2	no	—
634	C₄²	—	2	2	no	—
635	C₄²	—	2	1	no	—
636	C₄²	—	2	2	no	—
637	C₄²	—	2	1	no	—
638	C₄²	—	2	1	no	—
639	C₄²	—	2	1	no	—
640	C₄²	—	2	1	no	—
641	C₄²	—	2	1	no	—
642	C₄²	—	2	1	no	—
643	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
644	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
645	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
646	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
647	C₂³	—	2	1	no	—
648	C₂³	—	4	1	no	—
649	C₂³	C₂₃×#120	1	1	yes	characteristic
650	Q₈	—	2	1	no	—
651	Q₈	—	2	1	no	—
652	Q₈	—	2	1	no	—
653	Q₈	—	2	1	no	—
654	Q₈	—	4	2	no	—
655	Q₈	—	4	2	no	—
656	Q₈	—	4	4	no	—
657	Q₈	—	4	1	no	—
658	Q₈	—	4	1	no	—
659	Q₈	—	4	1	no	—
660	Q₈	—	4	1	no	—
661	Q₈	—	8	2	no	—
662	Q₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
663	Q₈	C₂₃×#137	1	2	yes	complemented
664	D₄	—	2	4	no	—
665	D₄	—	2	1	no	—
666	D₄	—	4	2	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	—	4	1	no	—
671	D₄	—	8	2	no	—
672	D₄	C₂₃×#137	1	2	yes	complemented
673	C₂×C₄	—	2	4	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	2	1	no	—
678	C₂×C₄	—	2	1	no	—
679	C₂×C₄	—	2	1	no	—
680	C₂×C₄	—	2	1	no	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	4	1	no	—
688	C₂×C₄	—	4	2	no	—
689	C₂×C₄	—	4	4	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	8	1	no	—
694	C₂×C₄	—	8	2	no	—
695	C₂×C₄	—	8	1	no	—
696	C₂×C₄	—	8	2	no	—
697	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
698	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
699	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
700	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
701	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
702	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
703	C₈	—	2	2	no	—
704	C₈	—	4	1	no	—
705	C₈	—	4	1	no	—
706	C₈	—	4	1	no	—
707	C₈	—	4	1	no	—
708	C₈	—	4	1	no	—
709	C₂²	—	2	1	no	—
710	C₂²	—	2	2	no	—
711	C₂²	—	2	1	no	—
712	C₂²	—	2	1	no	—
713	C₂²	—	4	1	no	—
714	C₂²	—	4	1	no	—
715	C₂²	—	8	1	no	—
716	C₂²	—	8	1	no	—
717	C₂²	—	8	1	no	—
718	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
719	C₄	—	2	1	no	—
720	C₄	—	2	2	no	—
721	C₄	—	2	1	no	—
722	C₄	—	2	2	no	—
723	C₄	—	2	1	no	—
724	C₄	—	2	1	no	—
725	C₄	—	4	1	no	—
726	C₄	—	4	2	no	—
727	C₄	—	4	1	no	—
728	C₄	—	4	1	no	—
729	C₄	—	4	2	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	8	1	no	—
733	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
734	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
735	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
736	C₄	C₂₃×#575	1	1	yes	characteristic
737	C₂	—	4	1	no	—
738	C₂	—	4	2	no	—
739	C₂	—	16	1	no	—
740	C₂	C₂₃×#3004	1	1	yes	characteristic
741	C₂	C₂₃×#3006	1	1	yes	characteristic
742	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×#3115	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. Q₈×#41×C₂₃	C₂	semidirect
2. C₂₃×#3182	C₂	semidirect
4. C₂₃×#3037	C₂	semidirect
5. C₂₃×#1649	C₂	semidirect
6. C₂₃×#1648	C₂	semidirect
7. C₂₃×#1645	C₂	semidirect
8. C₂₃×#1644	C₂	semidirect
9. C₂₃×#1627	C₂	semidirect
10. C₂₃×#1626	C₂	semidirect
11. C₂₃×#1623	C₂	semidirect
12. C₂₃×#1622	C₂	semidirect
13. C₂₃×#1559	C₂	semidirect
14. C₂₃×#1491	C₂	semidirect
15. C₂₃×#1489	C₂	semidirect
275. #18556	C₂₃	direct (N × H)
292. Q₈×C₂₃	#137	semidirect
301. D₄×C₂₃	#137	semidirect
338. Q₈×#41	C₂×C₂₃	semidirect
339. #3182	C₂×C₂₃	semidirect
341. #3037	C₂×C₂₃	semidirect
342. #1649	C₂×C₂₃	semidirect
343. #1648	C₂×C₂₃	semidirect
344. #1645	C₂×C₂₃	semidirect
345. #1644	C₂×C₂₃	semidirect
346. #1627	C₂×C₂₃	semidirect
347. #1626	C₂×C₂₃	semidirect
348. #1623	C₂×C₂₃	semidirect
349. #1622	C₂×C₂₃	semidirect
350. #1559	C₂×C₂₃	semidirect
351. #1491	C₂×C₂₃	semidirect
352. #1489	C₂×C₂₃	semidirect
548. C₂₃	#18556	direct (N × H)
663. Q₈	C₂₃×#137	semidirect
672. D₄	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₂₃	2
C₂₃×#3182	2
C₂₃×#3115	2
C₂₃×#3037	2
C₂₃×#1649	2
C₂₃×#1648	2
C₂₃×#1645	2
C₂₃×#1644	2
C₂₃×#1627	2
C₂₃×#1626	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1559	2
C₂₃×#1491	2
C₂₃×#1489	2
#18556	23

Subgroups — C23×#18556 #19166182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18556 #19166182