Subgroups — C₂₃×#19895 #19167521

Order 5888.

624 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3385	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3276	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1690	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1671	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1643	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1635	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1590	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	—	2	1	no	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#547	C₂²	1	1	yes	characteristic
21	C₂₃×#501	—	2	1	no	—
22	C₂₃×#500	C₂²	1	1	yes	characteristic
23	C₂₃×#498	—	2	1	no	—
24	C₂₃×#498	C₂²	1	1	yes	characteristic
25	C₂₃×#498	C₂²	1	1	yes	characteristic
26	C₂₃×#494	C₂²	1	1	yes	characteristic
27	C₂₃×#487	C₂²	1	1	yes	characteristic
28	C₂₃×#487	C₂²	1	1	yes	characteristic
29	C₂₃×#483	—	2	1	no	—
30	C₂₃×#481	—	2	1	no	—
31	C₂₃×#479	—	2	1	no	—
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#476	—	2	1	no	—
34	C₂₃×#476	C₂²	1	1	yes	characteristic
35	C₂₃×#476	C₂²	1	1	yes	characteristic
36	C₂₃×#476	C₂²	1	1	yes	characteristic
37	C₂₃×#475	—	2	1	no	—
38	C₂₃×#475	C₂²	1	1	yes	characteristic
39	C₂₃×#475	C₂²	1	1	yes	characteristic
40	C₂₃×#474	—	2	1	no	—
41	C₂₃×#470	—	2	1	no	—
42	C₂₃×#465	—	2	1	no	—
43	C₂₃×#462	C₂²	1	1	yes	characteristic
44	C₂₃×#461	C₂²	1	1	yes	characteristic
45	C₂₃×#452	—	2	1	no	—
46	C₂₃×#451	—	2	1	no	—
47	C₂₃×#446	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#440	—	2	1	no	—
50	C₄×SD₁₆×C₂₃	—	2	1	no	—
51	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#436	C₂²	1	1	yes	characteristic
54	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#106	—	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×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#332	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×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₈²×C₂₃	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂²×Q₈×C₂₃	—	2	1	no	—
69	C₂₃×#137	—	4	4	no	—
70	C₂×Q₁₆×C₂₃	—	2	1	no	—
71	C₂×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
75	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
76	C₂×M₁₆×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	4	1	no	—
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×#125	—	2	1	no	—
87	C₂₃×#123	—	4	1	no	—
88	C₂₃×C₂²⋊Q₈	—	2	1	no	—
89	C₂₃×C₂²⋊Q₈	—	2	1	no	—
90	C₂₃×C₂²⋊Q₈	—	2	1	no	—
91	C₂₃×C₂²⋊Q₈	—	4	1	no	—
92	C₂₃×C₂²⋊Q₈	—	4	1	no	—
93	C₂₃×#121	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	1	no	—
100	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#106	—	2	1	no	—
108	C₂₃×#106	—	2	1	no	—
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×#106	—	2	1	no	—
111	C₂₃×#106	—	2	1	no	—
112	C₂₃×#106	—	2	1	no	—
113	C₂₃×#106	—	2	1	no	—
114	C₂₃×#106	—	2	1	no	—
115	C₂₃×#106	—	2	1	no	—
116	C₂₃×#106	D₄	1	1	yes	characteristic
117	C₂₃×#106	D₄	1	1	yes	characteristic
118	C₂₃×C₄⋊C₈	—	2	1	no	—
119	C₂₃×C₄⋊C₈	—	2	1	no	—
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#103	—	2	1	no	—
127	C₂₃×#103	—	2	1	no	—
128	C₂₃×#103	—	2	1	no	—
129	C₂₃×#103	—	4	1	no	—
130	C₂₃×#103	—	4	1	no	—
131	C₂₃×#103	D₄	1	1	yes	characteristic
132	C₂₃×#103	D₄	1	1	yes	characteristic
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×#102	D₄	1	1	yes	characteristic
138	C₂₃×C₂²⋊C₈	—	2	1	no	—
139	C₂₃×C₂²⋊C₈	—	2	1	no	—
140	C₂₃×#97	—	2	1	no	—
141	C₄×C₈×C₂₃	—	2	1	no	—
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	#41×C₂₃	—	4	2	no	—
146	#41×C₂₃	—	8	1	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	1	no	—
151	C₂×Q₈×C₂₃	—	4	1	no	—
152	C₂×Q₈×C₂₃	—	4	1	no	—
153	C₂×Q₈×C₂₃	—	4	2	no	—
154	C₂×Q₈×C₂₃	—	8	1	no	—
155	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄×C₂₃	—	4	1	no	—
158	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	4	1	no	—
163	C₂²×C₄×C₂₃	—	4	1	no	—
164	C₂²×C₄×C₂₃	—	4	1	no	—
165	Q₁₆×C₂₃	—	4	2	no	—
166	Q₁₆×C₂₃	—	4	2	no	—
167	Q₁₆×C₂₃	—	8	1	no	—
168	SD₁₆×C₂₃	—	2	2	no	—
169	SD₁₆×C₂₃	—	4	2	no	—
170	SD₁₆×C₂₃	SD₁₆	1	4	yes	complemented
171	M₁₆×C₂₃	—	4	2	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	4	2	no	—
178	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₄⋊C₄×C₂₃	—	2	1	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	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	—	2	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	—	4	1	no	—
197	C₄⋊C₄×C₂₃	—	4	1	no	—
198	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂²⋊C₄×C₂₃	—	2	1	no	—
203	C₂²⋊C₄×C₂₃	—	2	1	no	—
204	C₂²⋊C₄×C₂₃	—	4	1	no	—
205	C₂²⋊C₄×C₂₃	—	4	1	no	—
206	C₂²⋊C₄×C₂₃	—	4	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	#19895	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×C₂₃	—	2	1	no	—
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	2	1	no	—
216	Q₈×C₂₃	—	2	1	no	—
217	Q₈×C₂₃	—	4	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	Q₈×C₂₃	—	8	2	no	—
222	Q₈×C₂₃	—	8	1	no	—
223	Q₈×C₂₃	—	8	1	no	—
224	Q₈×C₂₃	—	16	1	no	—
225	Q₈×C₂₃	C₂×SD₁₆	1	2	yes	—
226	D₄×C₂₃	—	2	1	no	—
227	D₄×C₂₃	—	2	1	no	—
228	D₄×C₂₃	—	2	1	no	—
229	D₄×C₂₃	—	8	2	no	—
230	D₄×C₂₃	C₂×SD₁₆	1	2	yes	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	8	2	no	—
246	C₂×C₄×C₂₃	—	8	2	no	—
247	C₂×C₄×C₂₃	—	16	1	no	—
248	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
251	C₈×C₂₃	—	2	1	no	—
252	C₈×C₂₃	—	2	1	no	—
253	C₈×C₂₃	—	4	1	no	—
254	C₈×C₂₃	—	4	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	C₈×C₂₃	—	4	1	no	—
258	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
259	C₈×C₂₃	#137	1	2	yes	complemented
260	#3385	C₂×C₂₃	1	1	yes	characteristic, complemented
261	Q₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#3335	C₂×C₂₃	1	1	yes	characteristic
263	#3276	C₂×C₂₃	1	1	yes	characteristic
264	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1695	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1690	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1672	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1671	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1645	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1643	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1635	C₂×C₂₃	1	1	yes	characteristic
273	#1590	C₂×C₂₃	1	1	yes	characteristic, complemented
274	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
275	C₂²×C₂₃	—	2	2	no	—
276	C₂²×C₂₃	—	4	1	no	—
277	C₂²×C₂₃	—	4	1	no	—
278	C₂²×C₂₃	—	4	1	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	4	1	no	—
281	C₂²×C₂₃	D₄²	1	1	yes	characteristic
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	2	1	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	4	1	no	—
288	C₄×C₂₃	—	4	1	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	—	8	1	no	—
291	C₄×C₂₃	—	8	1	no	—
292	C₄×C₂₃	—	8	1	no	—
293	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
294	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
295	C₄×C₂₃	#577	1	1	yes	characteristic
296	C₄×C₂₃	#578	1	1	yes	characteristic
297	C₂×#137	—	2	1	no	—
298	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
299	#554	C₂²×C₂₃	1	1	yes	characteristic
300	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
301	#547	C₂²×C₂₃	1	1	yes	characteristic
302	#501	—	2	1	no	—
303	#500	C₂²×C₂₃	1	1	yes	characteristic
304	#498	—	2	1	no	—
305	#498	C₂²×C₂₃	1	1	yes	characteristic
306	#498	C₂²×C₂₃	1	1	yes	characteristic
307	#494	C₂²×C₂₃	1	1	yes	characteristic
308	#487	C₂²×C₂₃	1	1	yes	characteristic
309	#487	C₂²×C₂₃	1	1	yes	characteristic
310	#483	—	2	1	no	—
311	#481	—	2	1	no	—
312	#479	—	2	1	no	—
313	#477	C₂²×C₂₃	1	1	yes	characteristic
314	#476	—	2	1	no	—
315	#476	C₂²×C₂₃	1	1	yes	characteristic
316	#476	C₂²×C₂₃	1	1	yes	characteristic
317	#476	C₂²×C₂₃	1	1	yes	characteristic
318	#475	—	2	1	no	—
319	#475	C₂²×C₂₃	1	1	yes	characteristic
320	#475	C₂²×C₂₃	1	1	yes	characteristic
321	#474	—	2	1	no	—
322	#470	—	2	1	no	—
323	#465	—	2	1	no	—
324	#462	C₂²×C₂₃	1	1	yes	characteristic
325	#461	C₂²×C₂₃	1	1	yes	characteristic
326	#452	—	2	1	no	—
327	#451	—	2	1	no	—
328	#446	C₂²×C₂₃	1	1	yes	characteristic
329	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
330	#440	—	2	1	no	—
331	C₄×SD₁₆	—	2	1	no	—
332	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
333	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
334	#436	C₂²×C₂₃	1	1	yes	characteristic
335	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#106	—	2	1	no	—
337	#334	C₂²×C₂₃	1	1	yes	characteristic
338	#334	C₂²×C₂₃	1	1	yes	characteristic
339	#333	C₂²×C₂₃	1	1	yes	characteristic
340	#333	C₂²×C₂₃	1	1	yes	characteristic
341	#332	C₂²×C₂₃	1	1	yes	characteristic
342	#332	C₂²×C₂₃	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
346	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
347	C₈²	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₂₃	—	4	2	no	—
349	C₂×C₂₃	—	8	1	no	—
350	C₂×C₂₃	#3276	1	1	yes	characteristic
351	C₂×C₂₃	D₄×SD₁₆	1	1	yes	characteristic
352	C₂×C₂₃	#3290	1	1	yes	characteristic
353	C₂×#41	—	2	1	no	—
354	C₂²×Q₈	—	2	1	no	—
355	#137	—	4	4	no	—
356	C₂×Q₁₆	—	2	1	no	—
357	C₂×Q₁₆	—	2	1	no	—
358	C₂×Q₁₆	—	2	1	no	—
359	C₂×SD₁₆	—	2	1	no	—
360	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
361	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
362	C₂×M₁₆	—	2	1	no	—
363	C₂²×C₈	—	2	1	no	—
364	C₄⋊Q₈	—	2	1	no	—
365	C₄⋊Q₈	—	2	1	no	—
366	C₄⋊Q₈	—	2	1	no	—
367	C₄⋊Q₈	—	4	1	no	—
368	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
370	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
372	#125	—	2	1	no	—
373	#123	—	4	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	4	1	no	—
378	C₂²⋊Q₈	—	4	1	no	—
379	#121	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	#107	—	2	1	no	—
391	#107	—	2	1	no	—
392	#107	—	2	1	no	—
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	#106	—	2	1	no	—
396	#106	—	2	1	no	—
397	#106	—	2	1	no	—
398	#106	—	2	1	no	—
399	#106	—	2	1	no	—
400	#106	—	2	1	no	—
401	#106	—	2	1	no	—
402	#106	D₄×C₂₃	1	1	yes	characteristic
403	#106	D₄×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	—	2	1	no	—
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
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	#103	—	2	1	no	—
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	—	4	1	no	—
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	D₄×C₂₃	1	1	yes	characteristic
423	#102	D₄×C₂₃	1	1	yes	characteristic
424	C₂²⋊C₈	—	2	1	no	—
425	C₂²⋊C₈	—	2	1	no	—
426	#97	—	2	1	no	—
427	C₄×C₈	—	2	1	no	—
428	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₂₃	#19895	1	1	yes	characteristic, direct product factor, Sylow
432	#41	—	4	2	no	—
433	#41	—	8	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	4	1	no	—
439	C₂×Q₈	—	4	1	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	8	1	no	—
442	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
446	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	—	4	1	no	—
452	Q₁₆	—	4	2	no	—
453	Q₁₆	—	4	2	no	—
454	Q₁₆	—	8	1	no	—
455	SD₁₆	—	2	2	no	—
456	SD₁₆	—	4	2	no	—
457	SD₁₆	SD₁₆×C₂₃	1	4	yes	complemented
458	M₁₆	—	4	2	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	4	2	no	—
465	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
486	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₂²⋊C₄	—	4	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	2	1	no	—
502	Q₈	—	2	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	8	2	no	—
508	Q₈	—	8	1	no	—
509	Q₈	—	8	1	no	—
510	Q₈	—	16	1	no	—
511	Q₈	C₂×SD₁₆×C₂₃	1	2	yes	—
512	D₄	—	2	1	no	—
513	D₄	—	2	1	no	—
514	D₄	—	2	1	no	—
515	D₄	—	8	2	no	—
516	D₄	C₂×SD₁₆×C₂₃	1	2	yes	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	8	2	no	—
532	C₂×C₄	—	8	2	no	—
533	C₂×C₄	—	16	1	no	—
534	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
536	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
537	C₈	—	2	1	no	—
538	C₈	—	2	1	no	—
539	C₈	—	4	1	no	—
540	C₈	—	4	1	no	—
541	C₈	—	4	1	no	—
542	C₈	—	4	1	no	—
543	C₈	—	4	1	no	—
544	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
545	C₈	C₂₃×#137	1	2	yes	complemented
546	C₂²	—	2	2	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	D₄²×C₂₃	1	1	yes	characteristic
553	C₄	—	2	2	no	—
554	C₄	—	2	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	8	1	no	—
561	C₄	—	8	1	no	—
562	C₄	—	8	1	no	—
563	C₄	—	8	1	no	—
564	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
565	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
566	C₄	C₂₃×#577	1	1	yes	characteristic
567	C₄	C₂₃×#578	1	1	yes	characteristic
568	C₂	—	4	2	no	—
569	C₂	—	8	1	no	—
570	C₂	C₂₃×#3276	1	1	yes	characteristic
571	C₂	D₄×SD₁₆×C₂₃	1	1	yes	characteristic
572	C₂	C₂₃×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#547	C₂²
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×#3385	C₂	semidirect
2. Q₈×SD₁₆×C₂₃	C₂	semidirect
5. C₂₃×#1707	C₂	semidirect
6. C₂₃×#1695	C₂	semidirect
7. C₂₃×#1690	C₂	semidirect
8. C₂₃×#1672	C₂	semidirect
9. C₂₃×#1671	C₂	semidirect
10. C₂₃×#1645	C₂	semidirect
11. C₂₃×#1643	C₂	semidirect
12. C₂₃×#1637	C₂	semidirect
14. C₂₃×#1590	C₂	semidirect
15. C₈×SD₁₆×C₂₃	C₂	semidirect
170. SD₁₆×C₂₃	SD₁₆	semidirect
211. #19895	C₂₃	direct (N × H)
259. C₈×C₂₃	#137	semidirect
260. #3385	C₂×C₂₃	semidirect
261. Q₈×SD₁₆	C₂×C₂₃	semidirect
264. #1707	C₂×C₂₃	semidirect
265. #1695	C₂×C₂₃	semidirect
266. #1690	C₂×C₂₃	semidirect
267. #1672	C₂×C₂₃	semidirect
268. #1671	C₂×C₂₃	semidirect
269. #1645	C₂×C₂₃	semidirect
270. #1643	C₂×C₂₃	semidirect
271. #1637	C₂×C₂₃	semidirect
273. #1590	C₂×C₂₃	semidirect
274. C₈×SD₁₆	C₂×C₂₃	semidirect
431. C₂₃	#19895	direct (N × H)
457. SD₁₆	SD₁₆×C₂₃	semidirect
545. C₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3385	2
Q₈×SD₁₆×C₂₃	2
C₂₃×#3335	2
C₂₃×#3276	2
C₂₃×#1707	2
C₂₃×#1695	2
C₂₃×#1690	2
C₂₃×#1672	2
C₂₃×#1671	2
C₂₃×#1645	2
C₂₃×#1643	2
C₂₃×#1637	2
C₂₃×#1635	2
C₂₃×#1590	2
C₈×SD₁₆×C₂₃	2
#19895	23

Subgroups — C23×#19895 #19167521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19895 #19167521