Subgroups — C₂₃×#18581 #19166207

Order 5888.

866 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3523	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3238	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1638	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1637	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1636	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1636	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#557	—	2	1	no	—
13	C₂₃×#557	C₂²	1	1	yes	characteristic
14	C₂₃×#557	C₂²	1	1	yes	characteristic
15	C₂₃×#556	—	2	1	no	—
16	C₂₃×#556	C₂²	1	1	yes	characteristic
17	C₂₃×#552	—	2	2	no	—
18	C₂₃×#552	—	2	1	no	—
19	D₄×Q₈×C₂₃	—	2	1	no	—
20	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#541	—	2	2	no	—
22	C₂₃×#541	—	2	1	no	—
23	C₂₃×#533	—	2	1	no	—
24	C₂₃×#533	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂₃×#529	—	2	1	no	—
27	C₂₃×#529	C₂²	1	1	yes	characteristic
28	C₂₃×#519	—	2	1	no	—
29	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#501	—	2	1	no	—
31	C₂₃×#491	—	2	2	no	—
32	C₂₃×#489	—	2	1	no	—
33	C₂₃×#487	C₂²	1	2	yes	—
34	C₂₃×#487	C₂²	1	2	yes	—
35	C₂₃×#483	—	2	1	no	—
36	C₂₃×#480	—	2	1	no	—
37	C₂₃×#477	—	2	1	no	—
38	C₂₃×#477	—	2	1	no	—
39	C₂₃×#477	C₂²	1	2	yes	—
40	C₂₃×#477	C₂²	1	2	yes	—
41	C₂₃×#477	C₂²	1	2	yes	—
42	C₂₃×#476	C₂²	1	2	yes	—
43	C₂₃×#432	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#107	—	2	1	no	—
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂₃×#419	—	2	2	no	—
47	C₂×C₂₃×#103	—	2	1	no	—
48	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#332	C₂²	1	2	yes	—
50	C₂₃×#332	C₂²	1	2	yes	—
51	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
53	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
54	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
55	C₂²×Q₈×C₂₃	—	2	1	no	—
56	C₂²×Q₈×C₂₃	—	2	1	no	—
57	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	—	2	2	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	—	2	1	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	C₂³	1	2	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₂₃×#126	—	2	2	no	—
71	C₂₃×#126	—	2	2	no	—
72	C₂₃×#126	—	2	2	no	—
73	C₂₃×#126	—	2	2	no	—
74	C₂₃×#126	—	2	1	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	2	no	—
77	C₂₃×#125	—	2	2	no	—
78	C₂₃×#125	—	2	2	no	—
79	C₂₃×#125	—	2	2	no	—
80	C₂₃×#125	—	2	1	no	—
81	C₂₃×#125	—	2	1	no	—
82	C₂₃×#125	D₄	1	1	yes	characteristic
83	C₂₃×#125	D₄	1	2	yes	—
84	C₂₃×#124	—	2	1	no	—
85	C₂₃×#124	D₄	1	1	yes	characteristic
86	C₂₃×#123	—	2	2	no	—
87	C₂₃×#123	—	2	1	no	—
88	C₂₃×C₂²⋊Q₈	—	2	2	no	—
89	C₂₃×C₂²⋊Q₈	—	2	2	no	—
90	C₂₃×C₂²⋊Q₈	—	2	1	no	—
91	C₂₃×C₂²⋊Q₈	—	2	1	no	—
92	C₂₃×C₂²⋊Q₈	—	2	1	no	—
93	C₂₃×C₂²⋊Q₈	—	2	1	no	—
94	C₂₃×C₂²⋊Q₈	—	2	2	no	—
95	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
96	C₂₃×#121	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₂₃	—	2	2	no	—
98	C₄×Q₈×C₂₃	—	2	2	no	—
99	C₄×Q₈×C₂₃	—	2	1	no	—
100	C₄×Q₈×C₂₃	—	2	1	no	—
101	C₄×Q₈×C₂₃	—	2	1	no	—
102	C₄×Q₈×C₂₃	—	2	1	no	—
103	C₄×Q₈×C₂₃	—	2	1	no	—
104	C₄×Q₈×C₂₃	—	2	1	no	—
105	C₄×Q₈×C₂₃	—	2	1	no	—
106	C₄×Q₈×C₂₃	—	2	1	no	—
107	C₄×Q₈×C₂₃	—	2	1	no	—
108	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
109	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
111	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
112	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
113	C₄×D₄×C₂₃	—	2	2	no	—
114	C₄×D₄×C₂₃	—	2	1	no	—
115	C₄×D₄×C₂₃	—	2	1	no	—
116	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
117	C₂₃×#117	—	2	2	no	—
118	C₂₃×#117	—	2	2	no	—
119	C₂₃×#117	—	2	2	no	—
120	C₂₃×#117	—	2	1	no	—
121	C₂₃×#117	—	2	1	no	—
122	C₂₃×#117	D₄	1	2	yes	—
123	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
124	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
125	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
126	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	—	2	1	no	—
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂₃×#107	—	2	2	no	—
130	C₂₃×#107	—	2	2	no	—
131	C₂₃×#107	—	2	2	no	—
132	C₂₃×#106	—	2	2	no	—
133	C₂₃×C₄⋊C₈	—	2	1	no	—
134	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
135	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
136	C₂₃×#103	—	2	2	no	—
137	C₂₃×#103	—	2	2	no	—
138	C₂₃×#103	—	2	2	no	—
139	C₂₃×#103	—	2	4	no	—
140	C₂₃×#103	—	2	4	no	—
141	C₂₃×#102	—	2	2	no	—
142	C₂₃×C₂²⋊C₈	—	2	1	no	—
143	C₂₃×#97	—	2	1	no	—
144	C₂₃×#97	—	2	1	no	—
145	C₄×C₈×C₂₃	C₂³	1	2	yes	—
146	C₂×Q₈×C₂₃	—	2	2	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	2	no	—
151	C₂×Q₈×C₂₃	—	2	2	no	—
152	C₂×Q₈×C₂₃	—	4	1	no	—
153	C₂×Q₈×C₂₃	—	4	4	no	—
154	C₂×Q₈×C₂₃	—	4	2	no	—
155	C₂×Q₈×C₂₃	—	4	2	no	—
156	C₂×Q₈×C₂₃	—	4	1	no	—
157	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂×D₄×C₂₃	—	2	1	no	—
161	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	—	2	2	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	4	1	no	—
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	M₁₆×C₂₃	—	4	2	no	—
171	C₂×C₈×C₂₃	—	2	2	no	—
172	C₂×C₈×C₂₃	—	2	2	no	—
173	C₂×C₈×C₂₃	—	2	2	no	—
174	C₂×C₈×C₂₃	—	4	2	no	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₄⋊C₄×C₂₃	—	2	2	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	2	no	—
197	C₄⋊C₄×C₂₃	—	2	2	no	—
198	C₄⋊C₄×C₂₃	—	4	1	no	—
199	C₄⋊C₄×C₂₃	—	4	1	no	—
200	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
202	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₂₃	—	2	2	no	—
207	C₂²⋊C₄×C₂₃	—	2	2	no	—
208	C₂²⋊C₄×C₂₃	—	2	2	no	—
209	C₂²⋊C₄×C₂₃	—	2	1	no	—
210	C₂²⋊C₄×C₂₃	—	2	2	no	—
211	C₂²⋊C₄×C₂₃	—	2	1	no	—
212	C₂²⋊C₄×C₂₃	—	2	1	no	—
213	C₂²⋊C₄×C₂₃	—	2	1	no	—
214	C₂²⋊C₄×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	2	2	no	—
216	C₄²×C₂₃	—	2	2	no	—
217	C₄²×C₂₃	—	2	2	no	—
218	C₄²×C₂₃	—	2	1	no	—
219	C₄²×C₂₃	—	2	1	no	—
220	C₄²×C₂₃	—	2	2	no	—
221	C₄²×C₂₃	—	2	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
228	#18581	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	#120	1	1	yes	characteristic
231	Q₈×C₂₃	—	2	1	no	—
232	Q₈×C₂₃	—	2	1	no	—
233	Q₈×C₂₃	—	2	1	no	—
234	Q₈×C₂₃	—	2	2	no	—
235	Q₈×C₂₃	—	4	2	no	—
236	Q₈×C₂₃	—	4	4	no	—
237	Q₈×C₂₃	—	4	4	no	—
238	Q₈×C₂₃	—	4	1	no	—
239	Q₈×C₂₃	—	4	1	no	—
240	Q₈×C₂₃	—	4	2	no	—
241	Q₈×C₂₃	—	8	2	no	—
242	Q₈×C₂₃	#137	1	2	yes	—
243	D₄×C₂₃	—	2	2	no	—
244	D₄×C₂₃	—	4	2	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	4	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	2	no	—
264	C₂×C₄×C₂₃	—	8	2	no	—
265	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
266	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
267	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
268	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
271	C₈×C₂₃	—	4	2	no	—
272	C₈×C₂₃	—	4	2	no	—
273	C₈×C₂₃	—	4	2	no	—
274	#3523	C₂×C₂₃	1	1	yes	characteristic
275	#3238	C₂×C₂₃	1	1	yes	characteristic
276	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1638	C₂×C₂₃	1	2	yes	complemented
279	#1637	C₂×C₂₃	1	2	yes	complemented
280	#1636	C₂×C₂₃	1	2	yes	complemented
281	#1636	C₂×C₂₃	1	2	yes	complemented
282	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1491	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1483	C₂×C₂₃	1	1	yes	characteristic
285	C₂²×C₂₃	—	2	1	no	—
286	C₂²×C₂₃	—	2	1	no	—
287	C₂²×C₂₃	—	4	1	no	—
288	C₂²×C₂₃	—	4	1	no	—
289	C₂²×C₂₃	—	4	1	no	—
290	C₂²×C₂₃	#454	1	2	yes	—
291	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
292	C₄×C₂₃	—	2	1	no	—
293	C₄×C₂₃	—	2	2	no	—
294	C₄×C₂₃	—	2	1	no	—
295	C₄×C₂₃	—	2	1	no	—
296	C₄×C₂₃	—	2	1	no	—
297	C₄×C₂₃	—	4	1	no	—
298	C₄×C₂₃	—	4	2	no	—
299	C₄×C₂₃	—	4	1	no	—
300	C₄×C₂₃	—	4	1	no	—
301	C₄×C₂₃	—	4	2	no	—
302	C₄×C₂₃	—	4	1	no	—
303	C₄×C₂₃	—	4	1	no	—
304	C₄×C₂₃	—	4	1	no	—
305	C₄×C₂₃	—	8	2	no	—
306	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
307	C₄×C₂₃	#575	1	1	yes	characteristic
308	C₄×C₂₃	#578	1	2	yes	—
309	#557	—	2	1	no	—
310	#557	C₂²×C₂₃	1	1	yes	characteristic
311	#557	C₂²×C₂₃	1	1	yes	characteristic
312	#556	—	2	1	no	—
313	#556	C₂²×C₂₃	1	1	yes	characteristic
314	#552	—	2	2	no	—
315	#552	—	2	1	no	—
316	D₄×Q₈	—	2	1	no	—
317	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
318	#541	—	2	2	no	—
319	#541	—	2	1	no	—
320	#533	—	2	1	no	—
321	#533	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	#529	—	2	1	no	—
324	#529	C₂²×C₂₃	1	1	yes	characteristic
325	#519	—	2	1	no	—
326	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
327	#501	—	2	1	no	—
328	#491	—	2	2	no	—
329	#489	—	2	1	no	—
330	#487	C₂²×C₂₃	1	2	yes	—
331	#487	C₂²×C₂₃	1	2	yes	—
332	#483	—	2	1	no	—
333	#480	—	2	1	no	—
334	#477	—	2	1	no	—
335	#477	—	2	1	no	—
336	#477	C₂²×C₂₃	1	2	yes	—
337	#477	C₂²×C₂₃	1	2	yes	—
338	#477	C₂²×C₂₃	1	2	yes	—
339	#476	C₂²×C₂₃	1	2	yes	—
340	#432	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#107	—	2	1	no	—
342	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	#419	—	2	2	no	—
344	C₂×#103	—	2	1	no	—
345	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
346	#332	C₂²×C₂₃	1	2	yes	—
347	#332	C₂²×C₂₃	1	2	yes	—
348	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
349	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
350	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
351	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
352	C₂×C₂₃	—	2	2	no	—
353	C₂×C₂₃	—	8	1	no	—
354	C₂×C₂₃	#3007	1	1	yes	characteristic
355	C₂×C₂₃	#3011	1	1	yes	characteristic
356	C₂×C₂₃	C₂×#454	1	1	yes	characteristic
357	C₂²×Q₈	—	2	1	no	—
358	C₂²×Q₈	—	2	1	no	—
359	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
360	C₂×M₁₆	—	2	1	no	—
361	C₂²×C₈	—	2	1	no	—
362	C₄⋊Q₈	—	2	2	no	—
363	C₄⋊Q₈	—	2	1	no	—
364	C₄⋊Q₈	—	2	1	no	—
365	C₄⋊Q₈	—	2	1	no	—
366	C₄⋊Q₈	—	2	1	no	—
367	C₄⋊Q₈	—	2	1	no	—
368	C₄⋊Q₈	—	2	1	no	—
369	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
370	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
372	#126	—	2	2	no	—
373	#126	—	2	2	no	—
374	#126	—	2	2	no	—
375	#126	—	2	2	no	—
376	#126	—	2	1	no	—
377	#125	—	2	2	no	—
378	#125	—	2	2	no	—
379	#125	—	2	2	no	—
380	#125	—	2	1	no	—
381	#125	—	2	2	no	—
382	#125	—	2	2	no	—
383	#125	—	2	1	no	—
384	#125	D₄×C₂₃	1	1	yes	characteristic
385	#125	D₄×C₂₃	1	2	yes	—
386	#124	—	2	1	no	—
387	#124	D₄×C₂₃	1	1	yes	characteristic
388	#123	—	2	2	no	—
389	#123	—	2	1	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	—	2	2	no	—
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	—	2	1	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
398	#121	D₄×C₂₃	1	1	yes	characteristic
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	—	2	1	no	—
406	C₄×Q₈	—	2	1	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	—	2	1	no	—
410	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
411	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
412	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
413	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
414	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
419	#117	—	2	2	no	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	#117	—	2	1	no	—
423	#117	—	2	1	no	—
424	#117	D₄×C₂₃	1	2	yes	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
431	#107	—	2	2	no	—
432	#107	—	2	2	no	—
433	#107	—	2	2	no	—
434	#106	—	2	2	no	—
435	C₄⋊C₈	—	2	1	no	—
436	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
437	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
438	#103	—	2	2	no	—
439	#103	—	2	2	no	—
440	#103	—	2	2	no	—
441	#103	—	2	4	no	—
442	#103	—	2	4	no	—
443	#102	—	2	2	no	—
444	C₂²⋊C₈	—	2	1	no	—
445	#97	—	2	1	no	—
446	#97	—	2	1	no	—
447	C₄×C₈	C₂³×C₂₃	1	2	yes	—
448	C₂₃	#18581	1	1	yes	characteristic, direct product factor, Sylow
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	4	1	no	—
456	C₂×Q₈	—	4	4	no	—
457	C₂×Q₈	—	4	2	no	—
458	C₂×Q₈	—	4	2	no	—
459	C₂×Q₈	—	4	1	no	—
460	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	4	1	no	—
470	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
473	M₁₆	—	4	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	4	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	4	1	no	—
502	C₄⋊C₄	—	4	1	no	—
503	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
504	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	1	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	1	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
528	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
530	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
531	C₂³	—	2	1	no	—
532	C₂³	C₂₃×#120	1	1	yes	characteristic
533	Q₈	—	2	1	no	—
534	Q₈	—	2	1	no	—
535	Q₈	—	2	1	no	—
536	Q₈	—	2	2	no	—
537	Q₈	—	4	2	no	—
538	Q₈	—	4	4	no	—
539	Q₈	—	4	4	no	—
540	Q₈	—	4	1	no	—
541	Q₈	—	4	1	no	—
542	Q₈	—	4	2	no	—
543	Q₈	—	8	2	no	—
544	Q₈	C₂₃×#137	1	2	yes	—
545	D₄	—	2	2	no	—
546	D₄	—	4	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	8	2	no	—
567	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
568	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
569	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
570	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
571	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
573	C₈	—	4	2	no	—
574	C₈	—	4	2	no	—
575	C₈	—	4	2	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	C₂₃×#454	1	2	yes	—
582	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
583	C₄	—	2	1	no	—
584	C₄	—	2	2	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	8	2	no	—
597	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
598	C₄	C₂₃×#575	1	1	yes	characteristic
599	C₄	C₂₃×#578	1	2	yes	—
600	C₂	—	2	2	no	—
601	C₂	—	8	1	no	—
602	C₂	C₂₃×#3007	1	1	yes	characteristic
603	C₂	C₂₃×#3011	1	1	yes	characteristic
604	C₂	C₂×C₂₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#121	D₄
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
3. C₂₃×#3139	C₂	semidirect
4. C₂×C₂₃×#477	C₂	semidirect
5. C₂₃×#1638	C₂	semidirect
6. C₂₃×#1637	C₂	semidirect
7. C₂₃×#1636	C₂	semidirect
8. C₂₃×#1636	C₂	semidirect
9. C₂₃×#1552	C₂	semidirect
10. C₂₃×#1491	C₂	semidirect
228. #18581	C₂₃	direct (N × H)
276. #3139	C₂×C₂₃	semidirect
277. C₂×#477	C₂×C₂₃	semidirect
278. #1638	C₂×C₂₃	semidirect
279. #1637	C₂×C₂₃	semidirect
280. #1636	C₂×C₂₃	semidirect
281. #1636	C₂×C₂₃	semidirect
282. #1552	C₂×C₂₃	semidirect
283. #1491	C₂×C₂₃	semidirect
448. C₂₃	#18581	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×#3523	2
C₂₃×#3238	2
C₂₃×#3139	2
C₂×C₂₃×#477	2
C₂₃×#1638	2
C₂₃×#1637	2
C₂₃×#1636	2
C₂₃×#1552	2
C₂₃×#1491	2
C₂₃×#1483	2
#18581	23

Subgroups — C23×#18581 #19166207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18581 #19166207