Subgroups — C₂₃×#19894 #19167520

Order 5888.

640 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3386	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3359	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3298	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×#1693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1690	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1680	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1679	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1654	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
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	C₂₃×#557	C₂²	1	1	yes	characteristic
18	D₄×Q₈×C₂₃	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	C₂²	1	1	yes	characteristic
24	C₂₃×#498	C₂²	1	1	yes	characteristic
25	C₂₃×#494	C₂²	1	1	yes	characteristic
26	C₂₃×#487	—	2	1	no	—
27	C₂₃×#487	C₂²	1	1	yes	characteristic
28	C₂₃×#487	C₂²	1	1	yes	characteristic
29	C₂₃×#483	—	2	1	no	—
30	C₂₃×#478	—	2	1	no	—
31	C₂₃×#477	—	2	1	no	—
32	C₂₃×#475	—	2	1	no	—
33	C₂₃×#475	C₂²	1	1	yes	characteristic
34	C₂₃×#475	C₂²	1	1	yes	characteristic
35	C₂₃×#475	C₂²	1	1	yes	characteristic
36	C₂₃×#474	C₂²	1	1	yes	characteristic
37	C₂₃×#470	—	2	1	no	—
38	C₂₃×#465	—	2	1	no	—
39	C₂₃×#462	C₂²	1	1	yes	characteristic
40	C₂₃×#461	—	2	1	no	—
41	C₂₃×#461	C₂²	1	1	yes	characteristic
42	C₂₃×#461	C₂²	1	1	yes	characteristic
43	C₂₃×#461	C₂²	1	1	yes	characteristic
44	C₂₃×#452	—	2	1	no	—
45	C₂₃×#451	—	2	1	no	—
46	C₂₃×#448	—	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×#103	—	2	1	no	—
56	C₂₃×#334	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#332	C₂²	1	1	yes	characteristic
59	C₂₃×#332	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	C₂²×Q₈×C₂₃	—	2	1	no	—
70	C₂₃×#137	—	4	4	no	—
71	C₂×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×Q₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
78	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
79	C₂×M₁₆×C₂₃	—	2	1	no	—
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	1	no	—
82	C₂₃×C₄⋊Q₈	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×#125	—	2	1	no	—
89	C₂₃×#125	—	4	1	no	—
90	C₂₃×#123	—	4	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	—	4	1	no	—
95	C₂₃×C₂²⋊Q₈	—	4	1	no	—
96	C₂₃×#121	—	2	1	no	—
97	C₄×Q₈×C₂₃	—	2	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×#106	—	2	1	no	—
111	C₂₃×#106	—	2	1	no	—
112	C₂₃×#106	—	4	1	no	—
113	C₂₃×#106	—	4	1	no	—
114	C₂₃×#106	D₄	1	1	yes	characteristic
115	C₂₃×#106	D₄	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	—	2	1	no	—
117	C₂₃×C₄⋊C₈	—	2	1	no	—
118	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₂₃×#103	—	2	1	no	—
123	C₂₃×#103	—	2	1	no	—
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	—	2	1	no	—
130	C₂₃×#103	—	2	1	no	—
131	C₂₃×#103	—	2	1	no	—
132	C₂₃×#103	—	2	1	no	—
133	C₂₃×#103	D₄	1	1	yes	characteristic
134	C₂₃×#103	D₄	1	1	yes	characteristic
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	—	4	2	no	—
155	C₂×Q₈×C₂₃	—	8	1	no	—
156	C₂×Q₈×C₂₃	—	8	1	no	—
157	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂×D₄×C₂₃	—	4	1	no	—
160	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	4	1	no	—
165	C₂²×C₄×C₂₃	—	4	1	no	—
166	C₂²×C₄×C₂₃	—	4	1	no	—
167	Q₁₆×C₂₃	—	4	2	no	—
168	Q₁₆×C₂₃	—	4	2	no	—
169	Q₁₆×C₂₃	—	8	1	no	—
170	SD₁₆×C₂₃	—	4	1	no	—
171	SD₁₆×C₂₃	—	4	2	no	—
172	SD₁₆×C₂₃	—	4	2	no	—
173	SD₁₆×C₂₃	—	8	1	no	—
174	SD₁₆×C₂₃	SD₁₆	1	4	yes	complemented
175	M₁₆×C₂₃	—	4	2	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	—	2	1	no	—
181	C₂×C₈×C₂₃	—	8	1	no	—
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	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	—	4	1	no	—
197	C₄⋊C₄×C₂₃	—	4	1	no	—
198	C₄⋊C₄×C₂₃	—	4	1	no	—
199	C₄⋊C₄×C₂₃	—	4	1	no	—
200	C₄⋊C₄×C₂₃	—	4	1	no	—
201	C₄⋊C₄×C₂₃	—	4	1	no	—
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	1	no	—
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	4	1	no	—
209	C₂²⋊C₄×C₂₃	—	4	1	no	—
210	C₂²⋊C₄×C₂₃	—	4	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	4	1	no	—
214	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
215	#19894	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₂₃	—	2	1	no	—
217	C₂³×C₂₃	—	2	1	no	—
218	Q₈×C₂₃	—	2	1	no	—
219	Q₈×C₂₃	—	2	1	no	—
220	Q₈×C₂₃	—	2	1	no	—
221	Q₈×C₂₃	—	4	1	no	—
222	Q₈×C₂₃	—	4	1	no	—
223	Q₈×C₂₃	—	4	1	no	—
224	Q₈×C₂₃	—	4	1	no	—
225	Q₈×C₂₃	—	4	1	no	—
226	Q₈×C₂₃	—	4	1	no	—
227	Q₈×C₂₃	—	4	1	no	—
228	Q₈×C₂₃	—	4	1	no	—
229	Q₈×C₂₃	—	8	2	no	—
230	Q₈×C₂₃	—	8	2	no	—
231	Q₈×C₂₃	C₂×SD₁₆	1	2	yes	—
232	D₄×C₂₃	—	2	1	no	—
233	D₄×C₂₃	—	2	1	no	—
234	D₄×C₂₃	—	2	1	no	—
235	D₄×C₂₃	—	8	2	no	—
236	D₄×C₂₃	C₂×SD₁₆	1	2	yes	—
237	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	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	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	8	2	no	—
252	C₂×C₄×C₂₃	—	8	2	no	—
253	C₂×C₄×C₂₃	—	8	2	no	—
254	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
257	C₈×C₂₃	—	2	2	no	—
258	C₈×C₂₃	—	2	1	no	—
259	C₈×C₂₃	—	2	1	no	—
260	C₈×C₂₃	—	2	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	C₈×C₂₃	—	4	1	no	—
263	C₈×C₂₃	—	4	1	no	—
264	C₈×C₂₃	—	4	1	no	—
265	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
266	#3386	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#3359	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#3298	C₂×C₂₃	1	1	yes	characteristic
269	#3276	C₂×C₂₃	1	1	yes	characteristic
270	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1693	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1690	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1680	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1679	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1654	C₂×C₂₃	1	1	yes	characteristic
276	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1590	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₂²×C₂₃	—	2	2	no	—
282	C₂²×C₂₃	—	4	1	no	—
283	C₂²×C₂₃	—	4	1	no	—
284	C₂²×C₂₃	—	4	1	no	—
285	C₂²×C₂₃	—	4	1	no	—
286	C₂²×C₂₃	—	4	1	no	—
287	C₂²×C₂₃	D₄²	1	1	yes	characteristic
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	2	1	no	—
290	C₄×C₂₃	—	4	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	—	8	1	no	—
296	C₄×C₂₃	—	8	1	no	—
297	C₄×C₂₃	—	8	1	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	C₂²×SD₁₆	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	C₂×#137	—	2	1	no	—
304	#557	C₂²×C₂₃	1	1	yes	characteristic
305	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
306	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
307	#547	C₂²×C₂₃	1	1	yes	characteristic
308	#501	—	2	1	no	—
309	#500	C₂²×C₂₃	1	1	yes	characteristic
310	#498	C₂²×C₂₃	1	1	yes	characteristic
311	#498	C₂²×C₂₃	1	1	yes	characteristic
312	#494	C₂²×C₂₃	1	1	yes	characteristic
313	#487	—	2	1	no	—
314	#487	C₂²×C₂₃	1	1	yes	characteristic
315	#487	C₂²×C₂₃	1	1	yes	characteristic
316	#483	—	2	1	no	—
317	#478	—	2	1	no	—
318	#477	—	2	1	no	—
319	#475	—	2	1	no	—
320	#475	C₂²×C₂₃	1	1	yes	characteristic
321	#475	C₂²×C₂₃	1	1	yes	characteristic
322	#475	C₂²×C₂₃	1	1	yes	characteristic
323	#474	C₂²×C₂₃	1	1	yes	characteristic
324	#470	—	2	1	no	—
325	#465	—	2	1	no	—
326	#462	C₂²×C₂₃	1	1	yes	characteristic
327	#461	—	2	1	no	—
328	#461	C₂²×C₂₃	1	1	yes	characteristic
329	#461	C₂²×C₂₃	1	1	yes	characteristic
330	#461	C₂²×C₂₃	1	1	yes	characteristic
331	#452	—	2	1	no	—
332	#451	—	2	1	no	—
333	#448	—	2	1	no	—
334	#446	C₂²×C₂₃	1	1	yes	characteristic
335	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
336	#440	—	2	1	no	—
337	C₄×SD₁₆	—	2	1	no	—
338	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
339	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
340	#436	C₂²×C₂₃	1	1	yes	characteristic
341	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#103	—	2	1	no	—
343	#334	C₂²×C₂₃	1	1	yes	characteristic
344	#334	C₂²×C₂₃	1	1	yes	characteristic
345	#332	C₂²×C₂₃	1	1	yes	characteristic
346	#332	C₂²×C₂₃	1	1	yes	characteristic
347	#332	C₂²×C₂₃	1	1	yes	characteristic
348	#332	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	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
353	C₈²	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×C₂₃	—	4	2	no	—
355	C₂×C₂₃	—	8	1	no	—
356	C₂×C₂₃	#3275	1	1	yes	characteristic
357	C₂×C₂₃	D₄×SD₁₆	1	1	yes	characteristic
358	C₂×C₂₃	#3293	1	1	yes	characteristic
359	C₂×#41	—	2	1	no	—
360	C₂²×Q₈	—	2	1	no	—
361	C₂²×Q₈	—	2	1	no	—
362	#137	—	4	4	no	—
363	C₂×Q₁₆	—	2	1	no	—
364	C₂×Q₁₆	—	2	1	no	—
365	C₂×Q₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	1	no	—
367	C₂×SD₁₆	—	2	1	no	—
368	C₂×SD₁₆	—	2	1	no	—
369	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
370	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
371	C₂×M₁₆	—	2	1	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
377	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
378	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
379	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
380	#125	—	2	1	no	—
381	#125	—	4	1	no	—
382	#123	—	4	1	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	4	1	no	—
387	C₂²⋊Q₈	—	4	1	no	—
388	#121	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
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	C₂³×C₂₃	1	1	yes	characteristic
396	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
397	C₂×C₄⋊C₄	—	2	1	no	—
398	#107	—	2	1	no	—
399	#107	—	2	1	no	—
400	#107	—	2	1	no	—
401	#106	—	2	1	no	—
402	#106	—	2	1	no	—
403	#106	—	2	1	no	—
404	#106	—	4	1	no	—
405	#106	—	4	1	no	—
406	#106	D₄×C₂₃	1	1	yes	characteristic
407	#106	D₄×C₂₃	1	1	yes	characteristic
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	—	2	1	no	—
410	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
412	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
413	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
414	#103	—	2	1	no	—
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	2	1	no	—
418	#103	—	2	1	no	—
419	#103	—	2	1	no	—
420	#103	—	2	1	no	—
421	#103	—	2	1	no	—
422	#103	—	2	1	no	—
423	#103	—	2	1	no	—
424	#103	—	2	1	no	—
425	#103	D₄×C₂₃	1	1	yes	characteristic
426	#103	D₄×C₂₃	1	1	yes	characteristic
427	#102	—	2	1	no	—
428	#102	D₄×C₂₃	1	1	yes	characteristic
429	#102	D₄×C₂₃	1	1	yes	characteristic
430	C₂²⋊C₈	—	2	1	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	#97	—	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	#19894	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	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	4	1	no	—
445	C₂×Q₈	—	4	1	no	—
446	C₂×Q₈	—	4	2	no	—
447	C₂×Q₈	—	4	2	no	—
448	C₂×Q₈	—	8	1	no	—
449	C₂×Q₈	—	8	1	no	—
450	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
451	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	4	1	no	—
458	C₂²×C₄	—	4	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	Q₁₆	—	4	2	no	—
461	Q₁₆	—	4	2	no	—
462	Q₁₆	—	8	1	no	—
463	SD₁₆	—	4	1	no	—
464	SD₁₆	—	4	2	no	—
465	SD₁₆	—	4	2	no	—
466	SD₁₆	—	8	1	no	—
467	SD₁₆	SD₁₆×C₂₃	1	4	yes	complemented
468	M₁₆	—	4	2	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	—	8	1	no	—
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	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₄⋊C₄	—	4	1	no	—
495	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	4	1	no	—
502	C₂²⋊C₄	—	4	1	no	—
503	C₂²⋊C₄	—	4	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	4	1	no	—
507	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
508	C₂³	—	2	1	no	—
509	C₂³	—	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	—	4	1	no	—
516	Q₈	—	4	1	no	—
517	Q₈	—	4	1	no	—
518	Q₈	—	4	1	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	8	2	no	—
522	Q₈	—	8	2	no	—
523	Q₈	C₂×SD₁₆×C₂₃	1	2	yes	—
524	D₄	—	2	1	no	—
525	D₄	—	2	1	no	—
526	D₄	—	2	1	no	—
527	D₄	—	8	2	no	—
528	D₄	C₂×SD₁₆×C₂₃	1	2	yes	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	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	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	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	—	4	1	no	—
543	C₂×C₄	—	8	2	no	—
544	C₂×C₄	—	8	2	no	—
545	C₂×C₄	—	8	2	no	—
546	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
548	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
549	C₈	—	2	2	no	—
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	2	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
558	C₂²	—	2	2	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	D₄²×C₂₃	1	1	yes	characteristic
565	C₄	—	2	2	no	—
566	C₄	—	2	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	8	1	no	—
573	C₄	—	8	1	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	1	no	—
576	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
577	C₄	C₂₃×#575	1	1	yes	characteristic
578	C₄	C₂₃×#577	1	1	yes	characteristic
579	C₄	C₂₃×#577	1	1	yes	characteristic
580	C₂	—	4	2	no	—
581	C₂	—	8	1	no	—
582	C₂	C₂₃×#3275	1	1	yes	characteristic
583	C₂	D₄×SD₁₆×C₂₃	1	1	yes	characteristic
584	C₂	C₂₃×#3293	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×#3386	C₂	semidirect
2. C₂₃×#3359	C₂	semidirect
5. C₂₃×#1707	C₂	semidirect
6. C₂₃×#1693	C₂	semidirect
7. C₂₃×#1690	C₂	semidirect
8. C₂₃×#1680	C₂	semidirect
9. C₂₃×#1679	C₂	semidirect
11. C₂₃×#1653	C₂	semidirect
12. C₂₃×#1651	C₂	semidirect
13. C₂₃×#1651	C₂	semidirect
14. C₂₃×#1590	C₂	semidirect
15. C₈×SD₁₆×C₂₃	C₂	semidirect
174. SD₁₆×C₂₃	SD₁₆	semidirect
215. #19894	C₂₃	direct (N × H)
266. #3386	C₂×C₂₃	semidirect
267. #3359	C₂×C₂₃	semidirect
270. #1707	C₂×C₂₃	semidirect
271. #1693	C₂×C₂₃	semidirect
272. #1690	C₂×C₂₃	semidirect
273. #1680	C₂×C₂₃	semidirect
274. #1679	C₂×C₂₃	semidirect
276. #1653	C₂×C₂₃	semidirect
277. #1651	C₂×C₂₃	semidirect
278. #1651	C₂×C₂₃	semidirect
279. #1590	C₂×C₂₃	semidirect
280. C₈×SD₁₆	C₂×C₂₃	semidirect
437. C₂₃	#19894	direct (N × H)
467. SD₁₆	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×#3386	2
C₂₃×#3359	2
C₂₃×#3298	2
C₂₃×#3276	2
C₂₃×#1707	2
C₂₃×#1693	2
C₂₃×#1690	2
C₂₃×#1680	2
C₂₃×#1679	2
C₂₃×#1654	2
C₂₃×#1653	2
C₂₃×#1651	2
C₂₃×#1590	2
C₈×SD₁₆×C₂₃	2
#19894	23

Subgroups — C23×#19894 #19167520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19894 #19167520