Subgroups — C₂₃×#19833 #19167459

Order 5888.

608 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3392	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1717	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1701	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1636	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1621	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1620	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#559	C₂²	1	1	yes	characteristic
17	C₂₃×#557	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	C₂₃×#550	C₂²	1	1	yes	characteristic
20	C₂₃×#493	C₂²	1	1	yes	characteristic
21	C₂₃×#492	—	2	1	no	—
22	C₂₃×#492	C₂²	1	1	yes	characteristic
23	C₂₃×#488	—	2	1	no	—
24	C₂₃×#488	—	2	1	no	—
25	C₂₃×#487	C₂²	1	1	yes	characteristic
26	C₂₃×#486	C₂²	1	1	yes	characteristic
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#486	C₂²	1	1	yes	characteristic
29	C₂₃×#480	—	2	1	no	—
30	C₂₃×#478	—	2	1	no	—
31	C₂₃×#478	—	2	1	no	—
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#477	C₂²	1	1	yes	characteristic
34	C₂₃×#475	C₂²	1	1	yes	characteristic
35	C₂₃×#468	—	2	1	no	—
36	C₂₃×#464	—	2	1	no	—
37	C₂₃×#463	—	2	1	no	—
38	C₂₃×#461	—	2	1	no	—
39	C₂₃×#461	C₂²	1	1	yes	characteristic
40	C₂₃×#460	—	2	1	no	—
41	C₂₃×#460	C₂²	1	1	yes	characteristic
42	C₂₃×#460	C₂²	1	1	yes	characteristic
43	C₂₃×#460	C₂²	1	1	yes	characteristic
44	C₂₃×#459	C₂²	1	1	yes	characteristic
45	C₂₃×#452	—	2	1	no	—
46	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
50	C₄×SD₁₆×C₂₃	—	2	1	no	—
51	C₄×SD₁₆×C₂₃	—	2	1	no	—
52	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
53	C₄×D₈×C₂₃	—	2	1	no	—
54	C₂₃×#436	C₂²	1	1	yes	characteristic
55	C₂₃×#417	—	2	1	no	—
56	C₂₃×#334	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#331	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×Q₈⋊C₈	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×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
69	C₂×SD₁₆×C₂₃	—	2	1	no	—
70	C₂×SD₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
76	C₂×D₈×C₂₃	—	2	1	no	—
77	C₂×D₈×C₂₃	—	4	1	no	—
78	C₂×M₁₆×C₂₃	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×#127	—	2	1	no	—
83	C₂₃×#127	C₂³	1	1	yes	characteristic
84	C₂₃×#127	C₂³	1	1	yes	characteristic
85	C₂₃×#125	—	2	1	no	—
86	C₂₃×#125	—	2	1	no	—
87	C₂₃×#125	—	4	1	no	—
88	C₂₃×#124	—	2	1	no	—
89	C₂₃×#124	—	2	1	no	—
90	C₂₃×#123	—	4	1	no	—
91	C₂₃×C₂²⋊Q₈	—	2	1	no	—
92	C₂₃×C₂²⋊Q₈	—	4	1	no	—
93	C₂₃×#121	—	2	1	no	—
94	C₂₃×#121	—	4	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₂₃	—	2	1	no	—
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#107	—	2	1	no	—
108	C₂₃×#107	D₄	1	1	yes	characteristic
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×#106	D₄	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	—	2	1	no	—
112	C₂₃×C₄⋊C₈	—	2	1	no	—
113	C₂₃×C₄⋊C₈	—	2	1	no	—
114	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₂₃×#103	—	2	1	no	—
119	C₂₃×#103	—	2	1	no	—
120	C₂₃×#103	—	2	1	no	—
121	C₂₃×#103	—	2	1	no	—
122	C₂₃×#103	—	4	1	no	—
123	C₂₃×#103	D₄	1	1	yes	characteristic
124	C₂₃×#103	D₄	1	1	yes	characteristic
125	C₂₃×#103	D₄	1	1	yes	characteristic
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×#102	—	2	1	no	—
128	C₂₃×#102	—	2	1	no	—
129	C₂₃×#102	—	2	1	no	—
130	C₂₃×#102	—	2	1	no	—
131	C₂₃×#102	—	2	1	no	—
132	C₂₃×#102	—	2	1	no	—
133	C₂₃×#102	—	2	1	no	—
134	C₂₃×#102	—	2	1	no	—
135	C₂₃×#102	—	2	1	no	—
136	C₂₃×#102	D₄	1	1	yes	characteristic
137	C₂₃×C₂²⋊C₈	—	2	1	no	—
138	C₄×C₈×C₂₃	—	2	1	no	—
139	C₄×C₈×C₂₃	—	2	1	no	—
140	C₄×C₈×C₂₃	—	2	1	no	—
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	#41×C₂₃	—	4	2	no	—
145	#41×C₂₃	—	8	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	4	1	no	—
158	C₂×D₄×C₂₃	—	4	1	no	—
159	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	4	1	no	—
162	C₂²×C₄×C₂₃	—	4	1	no	—
163	C₂²×C₄×C₂₃	—	4	1	no	—
164	Q₁₆×C₂₃	—	2	2	no	—
165	SD₁₆×C₂₃	—	4	1	no	—
166	SD₁₆×C₂₃	—	4	2	no	—
167	SD₁₆×C₂₃	—	4	2	no	—
168	SD₁₆×C₂₃	—	8	1	no	—
169	SD₁₆×C₂₃	—	8	1	no	—
170	SD₁₆×C₂₃	—	8	1	no	—
171	SD₁₆×C₂₃	—	8	1	no	—
172	D₈×C₂₃	—	4	2	no	—
173	D₈×C₂₃	—	8	2	no	—
174	M₁₆×C₂₃	—	8	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	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	1	no	—
191	C₄⋊C₄×C₂₃	—	4	1	no	—
192	C₄⋊C₄×C₂₃	—	4	1	no	—
193	C₄⋊C₄×C₂₃	—	4	1	no	—
194	C₄⋊C₄×C₂₃	—	4	1	no	—
195	C₄⋊C₄×C₂₃	—	4	1	no	—
196	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	4	1	no	—
202	C₂²⋊C₄×C₂₃	—	4	1	no	—
203	C₂²⋊C₄×C₂₃	—	4	1	no	—
204	C₂²⋊C₄×C₂₃	—	4	1	no	—
205	C₂²⋊C₄×C₂₃	—	4	1	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	—	4	1	no	—
210	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
211	#19833	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×C₂₃	—	4	1	no	—
214	C₂³×C₂₃	—	4	1	no	—
215	Q₈×C₂₃	—	2	1	no	—
216	Q₈×C₂₃	—	2	1	no	—
217	Q₈×C₂₃	—	2	1	no	—
218	Q₈×C₂₃	—	2	1	no	—
219	Q₈×C₂₃	—	2	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	Q₈×C₂₃	—	4	1	no	—
222	Q₈×C₂₃	—	8	1	no	—
223	Q₈×C₂₃	#135	1	2	yes	—
224	D₄×C₂₃	—	2	1	no	—
225	D₄×C₂₃	—	2	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	4	1	no	—
228	D₄×C₂₃	—	4	1	no	—
229	D₄×C₂₃	—	4	1	no	—
230	D₄×C₂₃	—	4	1	no	—
231	D₄×C₂₃	—	4	1	no	—
232	D₄×C₂₃	—	4	1	no	—
233	D₄×C₂₃	—	4	1	no	—
234	D₄×C₂₃	—	8	2	no	—
235	D₄×C₂₃	—	8	1	no	—
236	D₄×C₂₃	—	8	1	no	—
237	D₄×C₂₃	—	16	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	8	2	no	—
250	C₂×C₄×C₂₃	—	16	1	no	—
251	C₂×C₄×C₂₃	—	16	1	no	—
252	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	2	1	no	—
257	C₈×C₂₃	—	2	1	no	—
258	C₈×C₂₃	—	2	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	C₈×C₂₃	—	4	1	no	—
263	C₈×C₂₃	—	4	1	no	—
264	#3392	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#3365	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#3361	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#3333	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1717	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1701	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1666	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1643	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1636	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1621	C₂×C₂₃	1	1	yes	characteristic
276	#1620	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1583	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1579	C₂×C₂₃	1	1	yes	characteristic, complemented
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	4	1	no	—
281	C₂²×C₂₃	—	4	1	no	—
282	C₂²×C₂₃	—	8	1	no	—
283	C₂²×C₂₃	—	8	1	no	—
284	C₂²×C₂₃	—	8	1	no	—
285	C₂²×C₂₃	—	8	1	no	—
286	C₂²×C₂₃	—	8	1	no	—
287	C₂²×C₂₃	—	8	1	no	—
288	C₂²×C₂₃	D₄²	1	1	yes	characteristic
289	C₄×C₂₃	—	2	2	no	—
290	C₄×C₂₃	—	2	1	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	4	1	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	4	1	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	8	1	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
300	C₄×C₂₃	#575	1	1	yes	characteristic
301	C₄×C₂₃	#577	1	1	yes	characteristic
302	C₄×C₂₃	#577	1	1	yes	characteristic
303	#559	C₂²×C₂₃	1	1	yes	characteristic
304	#557	C₂²×C₂₃	1	1	yes	characteristic
305	#554	C₂²×C₂₃	1	1	yes	characteristic
306	#550	C₂²×C₂₃	1	1	yes	characteristic
307	#493	C₂²×C₂₃	1	1	yes	characteristic
308	#492	—	2	1	no	—
309	#492	C₂²×C₂₃	1	1	yes	characteristic
310	#488	—	2	1	no	—
311	#488	—	2	1	no	—
312	#487	C₂²×C₂₃	1	1	yes	characteristic
313	#486	C₂²×C₂₃	1	1	yes	characteristic
314	#486	C₂²×C₂₃	1	1	yes	characteristic
315	#486	C₂²×C₂₃	1	1	yes	characteristic
316	#480	—	2	1	no	—
317	#478	—	2	1	no	—
318	#478	—	2	1	no	—
319	#477	C₂²×C₂₃	1	1	yes	characteristic
320	#477	C₂²×C₂₃	1	1	yes	characteristic
321	#475	C₂²×C₂₃	1	1	yes	characteristic
322	#468	—	2	1	no	—
323	#464	—	2	1	no	—
324	#463	—	2	1	no	—
325	#461	—	2	1	no	—
326	#461	C₂²×C₂₃	1	1	yes	characteristic
327	#460	—	2	1	no	—
328	#460	C₂²×C₂₃	1	1	yes	characteristic
329	#460	C₂²×C₂₃	1	1	yes	characteristic
330	#460	C₂²×C₂₃	1	1	yes	characteristic
331	#459	C₂²×C₂₃	1	1	yes	characteristic
332	#452	—	2	1	no	—
333	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
334	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
335	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
336	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
337	C₄×SD₁₆	—	2	1	no	—
338	C₄×SD₁₆	—	2	1	no	—
339	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
340	C₄×D₈	—	2	1	no	—
341	#436	C₂²×C₂₃	1	1	yes	characteristic
342	#417	—	2	1	no	—
343	#334	C₂²×C₂₃	1	1	yes	characteristic
344	#334	C₂²×C₂₃	1	1	yes	characteristic
345	#333	C₂²×C₂₃	1	1	yes	characteristic
346	#332	C₂²×C₂₃	1	1	yes	characteristic
347	#331	C₂²×C₂₃	1	1	yes	characteristic
348	#331	C₂²×C₂₃	1	1	yes	characteristic
349	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
351	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
353	#322	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×C₂₃	—	8	1	no	—
355	C₂×C₂₃	—	16	1	no	—
356	C₂×C₂₃	—	16	1	no	—
357	C₂×C₂₃	#3271	1	1	yes	characteristic
358	C₂×C₂₃	#3282	1	1	yes	characteristic
359	C₂×C₂₃	#3288	1	1	yes	characteristic
360	C₂×#41	—	2	1	no	—
361	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	1	no	—
367	C₂×SD₁₆	—	2	1	no	—
368	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
369	C₂×D₈	—	2	1	no	—
370	C₂×D₈	—	4	1	no	—
371	C₂×M₁₆	—	2	1	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
374	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
375	#127	—	2	1	no	—
376	#127	C₂³×C₂₃	1	1	yes	characteristic
377	#127	C₂³×C₂₃	1	1	yes	characteristic
378	#125	—	2	1	no	—
379	#125	—	2	1	no	—
380	#125	—	4	1	no	—
381	#124	—	2	1	no	—
382	#124	—	2	1	no	—
383	#123	—	4	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	4	1	no	—
386	#121	—	2	1	no	—
387	#121	—	4	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
391	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
392	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	—	2	1	no	—
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄⋊C₄	—	2	1	no	—
399	#107	—	2	1	no	—
400	#107	—	2	1	no	—
401	#107	D₄×C₂₃	1	1	yes	characteristic
402	#106	—	2	1	no	—
403	#106	D₄×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	—	2	1	no	—
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	—	2	1	no	—
407	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
411	#103	—	2	1	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#103	—	4	1	no	—
416	#103	D₄×C₂₃	1	1	yes	characteristic
417	#103	D₄×C₂₃	1	1	yes	characteristic
418	#103	D₄×C₂₃	1	1	yes	characteristic
419	#102	—	2	1	no	—
420	#102	—	2	1	no	—
421	#102	—	2	1	no	—
422	#102	—	2	1	no	—
423	#102	—	2	1	no	—
424	#102	—	2	1	no	—
425	#102	—	2	1	no	—
426	#102	—	2	1	no	—
427	#102	—	2	1	no	—
428	#102	—	2	1	no	—
429	#102	D₄×C₂₃	1	1	yes	characteristic
430	C₂²⋊C₈	—	2	1	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	—	2	1	no	—
433	C₄×C₈	—	2	1	no	—
434	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
436	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
437	C₂₃	#19833	1	1	yes	characteristic, direct product factor, Sylow
438	#41	—	4	2	no	—
439	#41	—	8	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	4	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	C₂²×C₄	—	4	1	no	—
458	Q₁₆	—	2	2	no	—
459	SD₁₆	—	4	1	no	—
460	SD₁₆	—	4	2	no	—
461	SD₁₆	—	4	2	no	—
462	SD₁₆	—	8	1	no	—
463	SD₁₆	—	8	1	no	—
464	SD₁₆	—	8	1	no	—
465	SD₁₆	—	8	1	no	—
466	D₈	—	4	2	no	—
467	D₈	—	8	2	no	—
468	M₁₆	—	8	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	C₂²⋊C₄	—	2	1	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	4	1	no	—
504	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
505	C₂³	—	2	1	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	1	no	—
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	Q₈	—	2	1	no	—
511	Q₈	—	2	1	no	—
512	Q₈	—	2	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	8	1	no	—
516	Q₈	C₂₃×#135	1	2	yes	—
517	D₄	—	2	1	no	—
518	D₄	—	2	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	4	1	no	—
527	D₄	—	8	2	no	—
528	D₄	—	8	1	no	—
529	D₄	—	8	1	no	—
530	D₄	—	16	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	8	2	no	—
543	C₂×C₄	—	16	1	no	—
544	C₂×C₄	—	16	1	no	—
545	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
547	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	—	8	1	no	—
565	C₂²	—	8	1	no	—
566	C₂²	D₄²×C₂₃	1	1	yes	characteristic
567	C₄	—	2	2	no	—
568	C₄	—	2	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
578	C₄	C₂₃×#575	1	1	yes	characteristic
579	C₄	C₂₃×#577	1	1	yes	characteristic
580	C₄	C₂₃×#577	1	1	yes	characteristic
581	C₂	—	8	1	no	—
582	C₂	—	16	1	no	—
583	C₂	—	16	1	no	—
584	C₂	C₂₃×#3271	1	1	yes	characteristic
585	C₂	C₂₃×#3282	1	1	yes	characteristic
586	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#1621	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3392	C₂	semidirect
2. C₂₃×#3365	C₂	semidirect
3. C₂₃×#3361	C₂	semidirect
4. C₂₃×#3333	C₂	semidirect
5. C₂₃×#1717	C₂	semidirect
6. C₂₃×#1701	C₂	semidirect
7. C₂₃×#1678	C₂	semidirect
8. C₂₃×#1678	C₂	semidirect
9. C₂₃×#1666	C₂	semidirect
10. C₂₃×#1643	C₂	semidirect
11. C₂₃×#1636	C₂	semidirect
13. C₂₃×#1620	C₂	semidirect
14. C₂₃×#1583	C₂	semidirect
15. C₂₃×#1579	C₂	semidirect
211. #19833	C₂₃	direct (N × H)
264. #3392	C₂×C₂₃	semidirect
265. #3365	C₂×C₂₃	semidirect
266. #3361	C₂×C₂₃	semidirect
267. #3333	C₂×C₂₃	semidirect
268. #1717	C₂×C₂₃	semidirect
269. #1701	C₂×C₂₃	semidirect
270. #1678	C₂×C₂₃	semidirect
271. #1678	C₂×C₂₃	semidirect
272. #1666	C₂×C₂₃	semidirect
273. #1643	C₂×C₂₃	semidirect
274. #1636	C₂×C₂₃	semidirect
276. #1620	C₂×C₂₃	semidirect
277. #1583	C₂×C₂₃	semidirect
278. #1579	C₂×C₂₃	semidirect
437. C₂₃	#19833	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×#3392	2
C₂₃×#3365	2
C₂₃×#3361	2
C₂₃×#3333	2
C₂₃×#1717	2
C₂₃×#1701	2
C₂₃×#1678	2
C₂₃×#1666	2
C₂₃×#1643	2
C₂₃×#1636	2
C₂₃×#1621	2
C₂₃×#1620	2
C₂₃×#1583	2
C₂₃×#1579	2
#19833	23

Subgroups — C23×#19833 #19167459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19833 #19167459