Subgroups — C₂₃×#22581 #19170207

Order 5888.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#419	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2347	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1928	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1927	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1825	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1809	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1787	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1763	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic
16	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂₃×#419	C₄	1	8	yes	complemented
24	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#391	—	2	1	no	—
36	C₂₃×#391	—	2	1	no	—
37	C₂₃×#391	C₂²	1	1	yes	characteristic
38	C₂₃×#391	C₂²	1	1	yes	characteristic
39	C₂₃×#389	C₂²	1	1	yes	characteristic
40	C₂₃×#389	C₂²	1	1	yes	characteristic
41	C₂₃×#388	—	2	1	no	—
42	C₂₃×#388	—	2	1	no	—
43	C₂₃×#383	—	2	1	no	—
44	C₂₃×#380	—	2	1	no	—
45	C₂₃×#380	C₂²	1	1	yes	characteristic
46	C₂₃×#380	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
48	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
51	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₂₃	—	2	1	no	—
61	C₂²×Q₈×C₂₃	—	2	2	no	—
62	C₂²×Q₈×C₂₃	—	2	2	no	—
63	C₂²×Q₈×C₂₃	—	2	2	no	—
64	C₂²×Q₈×C₂₃	—	4	1	no	—
65	C₂²×Q₈×C₂₃	—	4	1	no	—
66	C₂²×Q₈×C₂₃	C₂×C₄	1	4	yes	—
67	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	—	2	1	no	—
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₂₃	—	4	1	no	—
72	C₂×M₁₆×C₂₃	—	4	1	no	—
73	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₂₃	—	2	2	no	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	—	4	1	no	—
81	C₄×Q₈×C₂₃	—	4	1	no	—
82	C₂₃×#117	—	2	2	no	—
83	C₂₃×#117	C₂×C₄	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	—	2	1	no	—
102	C₂×C₄²×C₂₃	—	2	1	no	—
103	C₂×C₄²×C₂₃	—	2	1	no	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×#107	—	4	1	no	—
107	C₂₃×#106	—	4	1	no	—
108	C₂₃×#103	—	2	2	no	—
109	C₂₃×#103	—	2	2	no	—
110	C₂₃×#103	—	4	1	no	—
111	C₂₃×#103	—	4	1	no	—
112	C₂₃×#103	C₂×C₄	1	4	yes	—
113	C₂₃×#103	C₂×C₄	1	4	yes	—
114	C₂₃×#103	C₂×C₄	1	4	yes	—
115	C₂₃×#103	C₂×C₄	1	4	yes	—
116	C₂₃×C₂²⋊C₈	—	2	2	no	—
117	C₂₃×#97	—	4	1	no	—
118	C₄×C₈×C₂₃	—	4	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	1	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	D₄	1	1	yes	characteristic
130	C₂₃×#95	D₄	1	1	yes	characteristic
131	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Q₈×C₂₃	—	2	4	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	8	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	4	2	no	—
143	C₂×Q₈×C₂₃	—	4	2	no	—
144	C₂×Q₈×C₂₃	—	4	4	no	—
145	C₂×Q₈×C₂₃	—	4	4	no	—
146	C₂×Q₈×C₂₃	—	4	4	no	—
147	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
148	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
149	C₂×Q₈×C₂₃	#41	1	2	yes	—
150	C₂×Q₈×C₂₃	#41	1	2	yes	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
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	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	#41	1	2	yes	—
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	M₁₆×C₂₃	—	4	4	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	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
186	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
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	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	4	2	no	—
198	C₄⋊C₄×C₂₃	—	4	2	no	—
199	C₄⋊C₄×C₂₃	—	4	2	no	—
200	C₄⋊C₄×C₂₃	—	4	2	no	—
201	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
202	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	2	no	—
205	C₂²⋊C₄×C₂₃	—	4	1	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₂²⋊C₄×C₂₃	—	8	1	no	—
208	C₄²×C₂₃	—	2	4	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	4	2	no	—
214	C₄²×C₂₃	—	4	2	no	—
215	C₄²×C₂₃	—	8	1	no	—
216	C₄²×C₂₃	—	8	1	no	—
217	#22581	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₂₃	—	2	1	no	—
219	C₂³×C₂₃	—	2	1	no	—
220	C₂³×C₂₃	—	2	1	no	—
221	C₂³×C₂₃	—	2	1	no	—
222	C₂³×C₂₃	—	2	1	no	—
223	C₂³×C₂₃	—	2	1	no	—
224	C₂³×C₂₃	C₄×D₄	1	2	yes	—
225	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₂₃	—	2	2	no	—
227	Q₈×C₂₃	—	2	2	no	—
228	Q₈×C₂₃	—	2	2	no	—
229	Q₈×C₂₃	—	2	2	no	—
230	Q₈×C₂₃	—	2	2	no	—
231	Q₈×C₂₃	—	4	8	no	—
232	Q₈×C₂₃	—	4	1	no	—
233	Q₈×C₂₃	—	4	1	no	—
234	Q₈×C₂₃	#117	1	4	yes	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	2	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	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	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	1	no	—
258	C₂×C₄×C₂₃	—	4	8	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
278	C₂×C₄×C₂₃	#137	1	2	yes	—
279	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
280	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
282	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
285	C₈×C₂₃	—	2	2	no	—
286	C₈×C₂₃	—	2	2	no	—
287	C₈×C₂₃	—	4	1	no	—
288	C₈×C₂₃	—	4	1	no	—
289	C₂×#419	C₂×C₂₃	1	1	yes	characteristic
290	#2347	C₂×C₂₃	1	1	yes	characteristic
291	Q₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
292	#1935	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1928	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1927	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1825	C₂×C₂₃	1	1	yes	characteristic
299	#1809	C₂×C₂₃	1	1	yes	characteristic
300	#1787	C₂×C₂₃	1	1	yes	characteristic
301	#1763	C₂×C₂₃	1	1	yes	characteristic, complemented
302	C₄×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1753	C₂×C₂₃	1	1	yes	characteristic
304	C₂²×C₂₃	—	2	2	no	—
305	C₂²×C₂₃	—	2	2	no	—
306	C₂²×C₂₃	—	2	2	no	—
307	C₂²×C₂₃	—	4	1	no	—
308	C₂²×C₂₃	—	4	1	no	—
309	C₂²×C₂₃	—	4	1	no	—
310	C₂²×C₂₃	—	4	1	no	—
311	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
312	C₂²×C₂₃	#546	1	1	yes	characteristic
313	C₂²×C₂₃	#546	1	1	yes	characteristic
314	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
315	C₂²×C₂₃	#575	1	1	yes	characteristic
316	C₂²×C₂₃	#577	1	1	yes	characteristic
317	C₂²×C₂₃	#578	1	1	yes	characteristic
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	2	4	no	—
320	C₄×C₂₃	—	2	2	no	—
321	C₄×C₂₃	—	4	1	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	—	4	2	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	—	8	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
332	C₄×C₂₃	#518	1	1	yes	characteristic
333	C₄×C₂₃	#551	1	1	yes	characteristic
334	C₄×C₂₃	#551	1	1	yes	characteristic
335	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
336	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
342	#419	C₄×C₂₃	1	8	yes	complemented
343	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
354	#391	—	2	1	no	—
355	#391	—	2	1	no	—
356	#391	C₂²×C₂₃	1	1	yes	characteristic
357	#391	C₂²×C₂₃	1	1	yes	characteristic
358	#389	C₂²×C₂₃	1	1	yes	characteristic
359	#389	C₂²×C₂₃	1	1	yes	characteristic
360	#388	—	2	1	no	—
361	#388	—	2	1	no	—
362	#383	—	2	1	no	—
363	#380	—	2	1	no	—
364	#380	C₂²×C₂₃	1	1	yes	characteristic
365	#380	C₂²×C₂₃	1	1	yes	characteristic
366	C₄×C₄⋊C₄	—	2	1	no	—
367	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
368	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
369	C₄×C₂²⋊C₄	—	2	1	no	—
370	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
371	#340	C₂²×C₂₃	1	1	yes	characteristic
372	#340	C₂²×C₂₃	1	1	yes	characteristic
373	#340	C₂²×C₂₃	1	1	yes	characteristic
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#336	C₂²×C₂₃	1	1	yes	characteristic
378	#336	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×C₂₃	—	4	2	no	—
380	C₂×C₂₃	#2342	1	1	yes	characteristic
381	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
382	C₂×C₂₃	#2947	1	1	yes	characteristic
383	C₂×C₂₃	#3313	1	1	yes	characteristic
384	C₂×C₂₃	#3320	1	1	yes	characteristic
385	C₂×C₂₃	#3326	1	1	yes	characteristic
386	C₂×C₂₃	#3328	1	1	yes	characteristic
387	C₂²×Q₈	—	2	1	no	—
388	C₂²×Q₈	—	2	2	no	—
389	C₂²×Q₈	—	2	2	no	—
390	C₂²×Q₈	—	2	2	no	—
391	C₂²×Q₈	—	4	1	no	—
392	C₂²×Q₈	—	4	1	no	—
393	C₂²×Q₈	C₂×C₄×C₂₃	1	4	yes	—
394	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
395	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
396	C₂³×C₄	—	2	1	no	—
397	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×M₁₆	—	4	1	no	—
399	C₂×M₁₆	—	4	1	no	—
400	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
401	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄×Q₈	—	2	2	no	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	4	1	no	—
408	C₄×Q₈	—	4	1	no	—
409	#117	—	2	2	no	—
410	#117	C₂×C₄×C₂₃	1	4	yes	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	—	2	1	no	—
431	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
433	#107	—	4	1	no	—
434	#106	—	4	1	no	—
435	#103	—	2	2	no	—
436	#103	—	2	2	no	—
437	#103	—	4	1	no	—
438	#103	—	4	1	no	—
439	#103	C₂×C₄×C₂₃	1	4	yes	—
440	#103	C₂×C₄×C₂₃	1	4	yes	—
441	#103	C₂×C₄×C₂₃	1	4	yes	—
442	#103	C₂×C₄×C₂₃	1	4	yes	—
443	C₂²⋊C₈	—	2	2	no	—
444	#97	—	4	1	no	—
445	C₄×C₈	—	4	1	no	—
446	#95	—	2	1	no	—
447	#95	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	D₄×C₂₃	1	1	yes	characteristic
457	#95	D₄×C₂₃	1	1	yes	characteristic
458	C₂₃	#22581	1	1	yes	characteristic, direct product factor, Sylow
459	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₂×Q₈	—	2	4	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	8	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	4	2	no	—
471	C₂×Q₈	—	4	2	no	—
472	C₂×Q₈	—	4	4	no	—
473	C₂×Q₈	—	4	4	no	—
474	C₂×Q₈	—	4	4	no	—
475	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
476	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
477	C₂×Q₈	#41×C₂₃	1	2	yes	—
478	C₂×Q₈	#41×C₂₃	1	2	yes	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
495	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₂²×C₄	#41×C₂₃	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
502	M₁₆	—	4	4	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
514	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	4	2	no	—
526	C₄⋊C₄	—	4	2	no	—
527	C₄⋊C₄	—	4	2	no	—
528	C₄⋊C₄	—	4	2	no	—
529	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
530	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	4	1	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₂²⋊C₄	—	8	1	no	—
536	C₄²	—	2	4	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	4	2	no	—
542	C₄²	—	4	2	no	—
543	C₄²	—	8	1	no	—
544	C₄²	—	8	1	no	—
545	C₂³	—	2	1	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	C₄×D₄×C₂₃	1	2	yes	—
552	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
553	Q₈	—	2	2	no	—
554	Q₈	—	2	2	no	—
555	Q₈	—	2	2	no	—
556	Q₈	—	2	2	no	—
557	Q₈	—	2	2	no	—
558	Q₈	—	4	8	no	—
559	Q₈	—	4	1	no	—
560	Q₈	—	4	1	no	—
561	Q₈	C₂₃×#117	1	4	yes	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
605	C₂×C₄	C₂₃×#137	1	2	yes	—
606	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
608	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
609	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
610	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
611	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
612	C₈	—	2	2	no	—
613	C₈	—	2	2	no	—
614	C₈	—	4	1	no	—
615	C₈	—	4	1	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
624	C₂²	C₂₃×#546	1	1	yes	characteristic
625	C₂²	C₂₃×#546	1	1	yes	characteristic
626	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
627	C₂²	C₂₃×#575	1	1	yes	characteristic
628	C₂²	C₂₃×#577	1	1	yes	characteristic
629	C₂²	C₂₃×#578	1	1	yes	characteristic
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	4	no	—
633	C₄	—	4	2	no	—
634	C₄	—	4	2	no	—
635	C₄	—	4	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	2	no	—
639	C₄	—	8	1	no	—
640	C₄	—	8	1	no	—
641	C₄	—	8	1	no	—
642	C₄	—	8	1	no	—
643	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
644	C₄	C₂₃×#518	1	1	yes	characteristic
645	C₄	C₂₃×#551	1	1	yes	characteristic
646	C₄	C₂₃×#551	1	1	yes	characteristic
647	C₂	—	4	2	no	—
648	C₂	C₂₃×#2342	1	1	yes	characteristic
649	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
650	C₂	C₂₃×#2947	1	1	yes	characteristic
651	C₂	C₂₃×#3313	1	1	yes	characteristic
652	C₂	C₂₃×#3320	1	1	yes	characteristic
653	C₂	C₂₃×#3326	1	1	yes	characteristic
654	C₂	C₂₃×#3328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1935	C₂	semidirect
5. C₂₃×#1933	C₂	semidirect
6. C₂₃×#1928	C₂	semidirect
7. C₂₃×#1927	C₂	semidirect
8. C₂₃×#1865	C₂	semidirect
9. C₂₃×#1863	C₂	semidirect
13. C₂₃×#1763	C₂	semidirect
14. C₄×C₂₃×#103	C₂	semidirect
23. C₂₃×#419	C₄	semidirect
217. #22581	C₂₃	direct (N × H)
292. #1935	C₂×C₂₃	semidirect
293. #1933	C₂×C₂₃	semidirect
294. #1928	C₂×C₂₃	semidirect
295. #1927	C₂×C₂₃	semidirect
296. #1865	C₂×C₂₃	semidirect
297. #1863	C₂×C₂₃	semidirect
301. #1763	C₂×C₂₃	semidirect
302. C₄×#103	C₂×C₂₃	semidirect
342. #419	C₄×C₂₃	semidirect
458. C₂₃	#22581	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#419	2
C₂₃×#2347	2
Q₈×C₂²⋊C₄×C₂₃	2
C₂₃×#1935	2
C₂₃×#1933	2
C₂₃×#1928	2
C₂₃×#1927	2
C₂₃×#1865	2
C₂₃×#1863	2
C₂₃×#1825	2
C₂₃×#1809	2
C₂₃×#1787	2
C₂₃×#1763	2
C₄×C₂₃×#103	2
C₂₃×#1753	2
#22581	23

Subgroups — C23×#22581 #19170207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22581 #19170207