Subgroups — C₂₃×#24740 #19172366

Order 5888.

1030 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#463	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#463	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2545	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2068	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2067	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2042	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1959	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1879	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1877	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂₃×#463	—	2	4	no	—
28	C₂₃×#463	—	2	4	no	—
29	C₂₃×#461	—	2	4	no	—
30	C₂₃×#440	—	2	4	no	—
31	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
46	C₂₃×#396	—	2	1	no	—
47	C₂₃×#396	—	2	1	no	—
48	C₂₃×#391	C₂²	1	1	yes	characteristic
49	C₂₃×#390	C₂²	1	1	yes	characteristic
50	C₂₃×#389	C₂²	1	1	yes	characteristic
51	C₂₃×#389	C₂²	1	1	yes	characteristic
52	C₂₃×#388	—	2	1	no	—
53	C₂₃×#388	—	2	1	no	—
54	C₂₃×#385	—	2	1	no	—
55	C₂₃×#385	—	2	1	no	—
56	C₂₃×#384	C₂²	1	1	yes	characteristic
57	C₂₃×#380	—	2	1	no	—
58	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#340	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂₃×#336	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	—	2	1	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	2	2	no	—
72	C₂×SD₁₆×C₂₃	—	2	2	no	—
73	C₂×SD₁₆×C₂₃	—	2	2	no	—
74	C₂×SD₁₆×C₂₃	—	2	2	no	—
75	C₂×SD₁₆×C₂₃	D₄	1	4	yes	—
76	C₂×D₈×C₂₃	—	2	2	no	—
77	C₂×D₈×C₂₃	—	2	2	no	—
78	C₂×D₈×C₂₃	—	2	2	no	—
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	—	2	2	no	—
84	C₂₃×C₄⋊Q₈	—	4	1	no	—
85	C₂₃×#124	—	2	2	no	—
86	C₂₃×#124	—	2	2	no	—
87	C₂₃×#124	—	2	2	no	—
88	C₂₃×#124	—	4	1	no	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×D₄×C₂₃	—	2	2	no	—
92	C₄×D₄×C₂₃	—	2	2	no	—
93	C₄×D₄×C₂₃	—	2	2	no	—
94	C₄×D₄×C₂₃	—	4	1	no	—
95	C₄×D₄×C₂₃	—	4	1	no	—
96	C₄×D₄×C₂₃	D₄	1	4	yes	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
105	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
120	C₂×C₄²×C₂₃	—	2	1	no	—
121	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
124	C₂₃×#107	—	2	2	no	—
125	C₂₃×C₄⋊C₈	—	2	2	no	—
126	C₂₃×C₄⋊C₈	—	2	2	no	—
127	C₂₃×C₄⋊C₈	—	2	2	no	—
128	C₂₃×#103	—	2	2	no	—
129	C₂₃×#103	—	2	2	no	—
130	C₂₃×#103	—	2	2	no	—
131	C₂₃×#103	—	2	2	no	—
132	C₂₃×#103	—	2	2	no	—
133	C₂₃×#103	—	4	1	no	—
134	C₂₃×#102	—	2	2	no	—
135	C₂₃×#102	—	2	2	no	—
136	C₂₃×#102	—	2	2	no	—
137	C₂₃×#102	—	4	1	no	—
138	C₂₃×#97	—	2	2	no	—
139	C₂₃×#95	—	2	1	no	—
140	C₂₃×#95	—	2	1	no	—
141	C₂₃×#95	—	2	1	no	—
142	C₂₃×#95	—	2	1	no	—
143	C₂₃×#95	D₄	1	1	yes	characteristic
144	C₂₃×#95	D₄	1	1	yes	characteristic
145	C₂⁴×C₂₃	—	2	1	no	—
146	C₂⁴×C₂₃	—	2	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	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	2	1	no	—
153	C₂×Q₈×C₂₃	—	2	1	no	—
154	C₂×Q₈×C₂₃	—	2	1	no	—
155	C₂×Q₈×C₂₃	—	2	1	no	—
156	C₂×Q₈×C₂₃	—	2	1	no	—
157	C₂×Q₈×C₂₃	—	2	1	no	—
158	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	1	no	—
161	C₂×D₄×C₂₃	—	2	1	no	—
162	C₂×D₄×C₂₃	—	2	1	no	—
163	C₂×D₄×C₂₃	—	2	1	no	—
164	C₂×D₄×C₂₃	—	2	1	no	—
165	C₂×D₄×C₂₃	—	2	1	no	—
166	C₂×D₄×C₂₃	—	2	1	no	—
167	C₂×D₄×C₂₃	—	2	1	no	—
168	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
169	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂×D₄×C₂₃	#41	1	2	yes	—
171	C₂²×C₄×C₂₃	—	2	1	no	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	4	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	4	2	no	—
179	C₂²×C₄×C₂₃	—	4	2	no	—
180	C₂²×C₄×C₂₃	—	4	2	no	—
181	C₂²×C₄×C₂₃	—	4	2	no	—
182	C₂²×C₄×C₂₃	—	4	2	no	—
183	C₂²×C₄×C₂₃	—	4	2	no	—
184	C₂²×C₄×C₂₃	—	8	1	no	—
185	C₂²×C₄×C₂₃	—	8	1	no	—
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	SD₁₆×C₂₃	—	2	8	no	—
194	SD₁₆×C₂₃	—	4	4	no	—
195	D₈×C₂₃	—	4	4	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	2	1	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	—	2	1	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
208	C₄⋊C₄×C₂₃	—	2	2	no	—
209	C₄⋊C₄×C₂₃	—	2	1	no	—
210	C₄⋊C₄×C₂₃	—	2	1	no	—
211	C₄⋊C₄×C₂₃	—	2	1	no	—
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	4	1	no	—
217	C₄⋊C₄×C₂₃	—	4	2	no	—
218	C₄⋊C₄×C₂₃	—	4	2	no	—
219	C₄⋊C₄×C₂₃	—	4	2	no	—
220	C₄⋊C₄×C₂₃	—	4	2	no	—
221	C₄⋊C₄×C₂₃	—	4	2	no	—
222	C₄⋊C₄×C₂₃	—	4	2	no	—
223	C₄⋊C₄×C₂₃	—	8	1	no	—
224	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
225	C₂²⋊C₄×C₂₃	—	2	4	no	—
226	C₂²⋊C₄×C₂₃	—	4	2	no	—
227	C₂²⋊C₄×C₂₃	—	4	2	no	—
228	C₂²⋊C₄×C₂₃	—	4	2	no	—
229	C₂²⋊C₄×C₂₃	—	4	2	no	—
230	C₂²⋊C₄×C₂₃	—	4	2	no	—
231	C₂²⋊C₄×C₂₃	—	4	2	no	—
232	C₂²⋊C₄×C₂₃	—	4	2	no	—
233	C₂²⋊C₄×C₂₃	—	4	2	no	—
234	C₄²×C₂₃	—	2	1	no	—
235	C₄²×C₂₃	—	2	1	no	—
236	C₄²×C₂₃	—	2	1	no	—
237	C₄²×C₂₃	—	2	1	no	—
238	C₄²×C₂₃	—	4	2	no	—
239	C₄²×C₂₃	C₂×D₄	1	2	yes	—
240	C₄²×C₂₃	#41	1	2	yes	—
241	#24740	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂³×C₂₃	—	2	2	no	—
243	C₂³×C₂₃	—	2	2	no	—
244	C₂³×C₂₃	—	2	2	no	—
245	C₂³×C₂₃	—	4	1	no	—
246	C₂³×C₂₃	—	4	1	no	—
247	C₂³×C₂₃	—	4	1	no	—
248	C₂³×C₂₃	—	4	1	no	—
249	C₂³×C₂₃	—	4	1	no	—
250	C₂³×C₂₃	—	4	1	no	—
251	C₂³×C₂₃	—	4	1	no	—
252	C₂³×C₂₃	—	4	1	no	—
253	C₂³×C₂₃	—	4	1	no	—
254	C₂³×C₂₃	—	4	1	no	—
255	C₂³×C₂₃	—	4	1	no	—
256	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
257	Q₈×C₂₃	—	2	2	no	—
258	Q₈×C₂₃	—	2	2	no	—
259	Q₈×C₂₃	—	4	1	no	—
260	Q₈×C₂₃	—	4	1	no	—
261	Q₈×C₂₃	—	4	1	no	—
262	Q₈×C₂₃	—	4	1	no	—
263	Q₈×C₂₃	—	4	1	no	—
264	Q₈×C₂₃	—	4	1	no	—
265	D₄×C₂₃	—	2	2	no	—
266	D₄×C₂₃	—	2	2	no	—
267	D₄×C₂₃	—	2	2	no	—
268	D₄×C₂₃	—	4	1	no	—
269	D₄×C₂₃	—	4	1	no	—
270	D₄×C₂₃	—	4	1	no	—
271	D₄×C₂₃	—	4	1	no	—
272	D₄×C₂₃	#121	1	4	yes	—
273	C₂×C₄×C₂₃	—	2	1	no	—
274	C₂×C₄×C₂₃	—	2	1	no	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	1	no	—
277	C₂×C₄×C₂₃	—	2	1	no	—
278	C₂×C₄×C₂₃	—	2	1	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	2	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	1	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	4	8	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	—	4	1	no	—
302	C₂×C₄×C₂₃	—	4	1	no	—
303	C₂×C₄×C₂₃	—	4	1	no	—
304	C₂×C₄×C₂₃	—	4	1	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	—	8	4	no	—
308	C₂×C₄×C₂₃	—	8	4	no	—
309	C₂×C₄×C₂₃	#121	1	2	yes	—
310	C₂×C₄×C₂₃	#121	1	2	yes	—
311	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
312	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
315	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
316	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
317	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
318	C₈×C₂₃	—	2	2	no	—
319	C₈×C₂₃	—	4	1	no	—
320	C₈×C₂₃	—	4	1	no	—
321	C₈×C₂₃	—	4	1	no	—
322	C₂×#463	C₂×C₂₃	1	1	yes	characteristic
323	C₂×#463	C₂×C₂₃	1	1	yes	characteristic
324	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
325	C₂×#440	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#2545	C₂×C₂₃	1	1	yes	characteristic, complemented
327	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#2068	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#2067	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#2042	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1959	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1940	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1879	C₂×C₂₃	1	1	yes	characteristic
335	#1877	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
337	C₂²×C₂₃	—	2	4	no	—
338	C₂²×C₂₃	—	4	2	no	—
339	C₂²×C₂₃	—	4	2	no	—
340	C₂²×C₂₃	—	4	2	no	—
341	C₂²×C₂₃	—	4	2	no	—
342	C₂²×C₂₃	—	4	2	no	—
343	C₂²×C₂₃	—	4	2	no	—
344	C₂²×C₂₃	—	8	1	no	—
345	C₂²×C₂₃	—	8	1	no	—
346	C₂²×C₂₃	—	8	1	no	—
347	C₂²×C₂₃	—	8	1	no	—
348	C₂²×C₂₃	—	8	1	no	—
349	C₂²×C₂₃	—	8	1	no	—
350	C₂²×C₂₃	—	8	1	no	—
351	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
352	C₂²×C₂₃	D₄²	1	1	yes	characteristic
353	C₂²×C₂₃	#548	1	1	yes	characteristic
354	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
355	C₂²×C₂₃	#575	1	1	yes	characteristic
356	C₂²×C₂₃	#577	1	1	yes	characteristic
357	C₂²×C₂₃	#578	1	1	yes	characteristic
358	C₄×C₂₃	—	2	2	no	—
359	C₄×C₂₃	—	2	2	no	—
360	C₄×C₂₃	—	2	2	no	—
361	C₄×C₂₃	—	4	2	no	—
362	C₄×C₂₃	—	4	1	no	—
363	C₄×C₂₃	—	4	1	no	—
364	C₄×C₂₃	—	4	1	no	—
365	C₄×C₂₃	—	4	2	no	—
366	C₄×C₂₃	—	8	1	no	—
367	C₄×C₂₃	—	8	1	no	—
368	C₄×C₂₃	—	8	1	no	—
369	C₄×C₂₃	—	8	1	no	—
370	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
371	C₄×C₂₃	#537	1	1	yes	characteristic
372	C₄×C₂₃	#547	1	1	yes	characteristic
373	C₄×C₂₃	#553	1	1	yes	characteristic
374	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
375	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
376	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
384	C₂²×C₄⋊C₄	—	2	1	no	—
385	#463	—	2	4	no	—
386	#463	—	2	4	no	—
387	#461	—	2	4	no	—
388	#440	—	2	4	no	—
389	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
392	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
394	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
397	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
399	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
400	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
401	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
402	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
404	#396	—	2	1	no	—
405	#396	—	2	1	no	—
406	#391	C₂²×C₂₃	1	1	yes	characteristic
407	#390	C₂²×C₂₃	1	1	yes	characteristic
408	#389	C₂²×C₂₃	1	1	yes	characteristic
409	#389	C₂²×C₂₃	1	1	yes	characteristic
410	#388	—	2	1	no	—
411	#388	—	2	1	no	—
412	#385	—	2	1	no	—
413	#385	—	2	1	no	—
414	#384	C₂²×C₂₃	1	1	yes	characteristic
415	#380	—	2	1	no	—
416	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
417	#340	C₂²×C₂₃	1	1	yes	characteristic
418	#340	C₂²×C₂₃	1	1	yes	characteristic
419	#336	C₂²×C₂₃	1	1	yes	characteristic
420	#336	C₂²×C₂₃	1	1	yes	characteristic
421	C₂×C₂₃	—	4	4	no	—
422	C₂×C₂₃	—	16	1	no	—
423	C₂×C₂₃	#2416	1	1	yes	characteristic
424	C₂×C₂₃	#3036	1	1	yes	characteristic
425	C₂×C₂₃	#3046	1	1	yes	characteristic
426	C₂×C₂₃	#3265	1	1	yes	characteristic
427	C₂×C₂₃	#3269	1	1	yes	characteristic
428	C₂×C₂₃	#3367	1	1	yes	characteristic
429	C₂×C₂₃	#3376	1	1	yes	characteristic
430	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
433	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
434	C₂³×C₄	—	2	1	no	—
435	C₂³×C₄	—	2	1	no	—
436	C₂³×C₄	—	2	1	no	—
437	C₂×SD₁₆	—	2	2	no	—
438	C₂×SD₁₆	—	2	2	no	—
439	C₂×SD₁₆	—	2	2	no	—
440	C₂×SD₁₆	—	2	2	no	—
441	C₂×SD₁₆	—	2	2	no	—
442	C₂×SD₁₆	D₄×C₂₃	1	4	yes	—
443	C₂×D₈	—	2	2	no	—
444	C₂×D₈	—	2	2	no	—
445	C₂×D₈	—	2	2	no	—
446	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
447	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₄⋊Q₈	—	2	2	no	—
451	C₄⋊Q₈	—	4	1	no	—
452	#124	—	2	2	no	—
453	#124	—	2	2	no	—
454	#124	—	2	2	no	—
455	#124	—	4	1	no	—
456	C₄×Q₈	—	2	2	no	—
457	C₄×Q₈	—	2	2	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	2	no	—
460	C₄×D₄	—	2	2	no	—
461	C₄×D₄	—	4	1	no	—
462	C₄×D₄	—	4	1	no	—
463	C₄×D₄	D₄×C₂₃	1	4	yes	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄⋊C₄	—	2	1	no	—
468	C₂×C₄⋊C₄	—	2	1	no	—
469	C₂×C₄⋊C₄	—	2	1	no	—
470	C₂×C₄⋊C₄	—	2	1	no	—
471	C₂×C₄⋊C₄	—	4	4	no	—
472	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
473	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
475	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
476	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
477	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₂²⋊C₄	—	2	1	no	—
480	C₂×C₂²⋊C₄	—	2	1	no	—
481	C₂×C₂²⋊C₄	—	2	1	no	—
482	C₂×C₂²⋊C₄	—	2	1	no	—
483	C₂×C₂²⋊C₄	—	2	1	no	—
484	C₂×C₂²⋊C₄	—	2	1	no	—
485	C₂×C₂²⋊C₄	—	2	1	no	—
486	C₂×C₂²⋊C₄	—	2	1	no	—
487	C₂×C₄²	—	2	1	no	—
488	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
489	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
490	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
491	#107	—	2	2	no	—
492	C₄⋊C₈	—	2	2	no	—
493	C₄⋊C₈	—	2	2	no	—
494	C₄⋊C₈	—	2	2	no	—
495	#103	—	2	2	no	—
496	#103	—	2	2	no	—
497	#103	—	2	2	no	—
498	#103	—	2	2	no	—
499	#103	—	2	2	no	—
500	#103	—	4	1	no	—
501	#102	—	2	2	no	—
502	#102	—	2	2	no	—
503	#102	—	2	2	no	—
504	#102	—	4	1	no	—
505	#97	—	2	2	no	—
506	#95	—	2	1	no	—
507	#95	—	2	1	no	—
508	#95	—	2	1	no	—
509	#95	—	2	1	no	—
510	#95	D₄×C₂₃	1	1	yes	characteristic
511	#95	D₄×C₂₃	1	1	yes	characteristic
512	C₂₃	#24740	1	1	yes	characteristic, direct product factor, Sylow
513	C₂⁴	—	2	1	no	—
514	C₂⁴	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
537	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
538	C₂×D₄	#41×C₂₃	1	2	yes	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	4	2	no	—
547	C₂²×C₄	—	4	2	no	—
548	C₂²×C₄	—	4	2	no	—
549	C₂²×C₄	—	4	2	no	—
550	C₂²×C₄	—	4	2	no	—
551	C₂²×C₄	—	4	2	no	—
552	C₂²×C₄	—	8	1	no	—
553	C₂²×C₄	—	8	1	no	—
554	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
559	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
560	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
561	SD₁₆	—	2	8	no	—
562	SD₁₆	—	4	4	no	—
563	D₈	—	4	4	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	2	1	no	—
570	C₂×C₈	—	2	1	no	—
571	C₂×C₈	—	2	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	4	1	no	—
585	C₄⋊C₄	—	4	2	no	—
586	C₄⋊C₄	—	4	2	no	—
587	C₄⋊C₄	—	4	2	no	—
588	C₄⋊C₄	—	4	2	no	—
589	C₄⋊C₄	—	4	2	no	—
590	C₄⋊C₄	—	4	2	no	—
591	C₄⋊C₄	—	8	1	no	—
592	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
593	C₂²⋊C₄	—	2	4	no	—
594	C₂²⋊C₄	—	4	2	no	—
595	C₂²⋊C₄	—	4	2	no	—
596	C₂²⋊C₄	—	4	2	no	—
597	C₂²⋊C₄	—	4	2	no	—
598	C₂²⋊C₄	—	4	2	no	—
599	C₂²⋊C₄	—	4	2	no	—
600	C₂²⋊C₄	—	4	2	no	—
601	C₂²⋊C₄	—	4	2	no	—
602	C₄²	—	2	1	no	—
603	C₄²	—	2	1	no	—
604	C₄²	—	2	1	no	—
605	C₄²	—	2	1	no	—
606	C₄²	—	4	2	no	—
607	C₄²	C₂×D₄×C₂₃	1	2	yes	—
608	C₄²	#41×C₂₃	1	2	yes	—
609	C₂³	—	2	2	no	—
610	C₂³	—	2	2	no	—
611	C₂³	—	2	2	no	—
612	C₂³	—	4	1	no	—
613	C₂³	—	4	1	no	—
614	C₂³	—	4	1	no	—
615	C₂³	—	4	1	no	—
616	C₂³	—	4	1	no	—
617	C₂³	—	4	1	no	—
618	C₂³	—	4	1	no	—
619	C₂³	—	4	1	no	—
620	C₂³	—	4	1	no	—
621	C₂³	—	4	1	no	—
622	C₂³	—	4	1	no	—
623	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
624	Q₈	—	2	2	no	—
625	Q₈	—	2	2	no	—
626	Q₈	—	4	1	no	—
627	Q₈	—	4	1	no	—
628	Q₈	—	4	1	no	—
629	Q₈	—	4	1	no	—
630	Q₈	—	4	1	no	—
631	Q₈	—	4	1	no	—
632	D₄	—	2	2	no	—
633	D₄	—	2	2	no	—
634	D₄	—	2	2	no	—
635	D₄	—	4	1	no	—
636	D₄	—	4	1	no	—
637	D₄	—	4	1	no	—
638	D₄	—	4	1	no	—
639	D₄	C₂₃×#121	1	4	yes	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	4	8	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	8	4	no	—
675	C₂×C₄	—	8	4	no	—
676	C₂×C₄	C₂₃×#121	1	2	yes	—
677	C₂×C₄	C₂₃×#121	1	2	yes	—
678	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
679	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
680	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
681	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
682	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
683	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
684	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
685	C₈	—	2	2	no	—
686	C₈	—	4	1	no	—
687	C₈	—	4	1	no	—
688	C₈	—	4	1	no	—
689	C₂²	—	2	4	no	—
690	C₂²	—	4	2	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	2	no	—
693	C₂²	—	4	2	no	—
694	C₂²	—	4	2	no	—
695	C₂²	—	4	2	no	—
696	C₂²	—	8	1	no	—
697	C₂²	—	8	1	no	—
698	C₂²	—	8	1	no	—
699	C₂²	—	8	1	no	—
700	C₂²	—	8	1	no	—
701	C₂²	—	8	1	no	—
702	C₂²	—	8	1	no	—
703	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
704	C₂²	D₄²×C₂₃	1	1	yes	characteristic
705	C₂²	C₂₃×#548	1	1	yes	characteristic
706	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
707	C₂²	C₂₃×#575	1	1	yes	characteristic
708	C₂²	C₂₃×#577	1	1	yes	characteristic
709	C₂²	C₂₃×#578	1	1	yes	characteristic
710	C₄	—	2	2	no	—
711	C₄	—	2	2	no	—
712	C₄	—	2	2	no	—
713	C₄	—	4	2	no	—
714	C₄	—	4	1	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	2	no	—
718	C₄	—	8	1	no	—
719	C₄	—	8	1	no	—
720	C₄	—	8	1	no	—
721	C₄	—	8	1	no	—
722	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
723	C₄	C₂₃×#537	1	1	yes	characteristic
724	C₄	C₂₃×#547	1	1	yes	characteristic
725	C₄	C₂₃×#553	1	1	yes	characteristic
726	C₂	—	4	4	no	—
727	C₂	—	16	1	no	—
728	C₂	C₂₃×#2416	1	1	yes	characteristic
729	C₂	C₂₃×#3036	1	1	yes	characteristic
730	C₂	C₂₃×#3046	1	1	yes	characteristic
731	C₂	C₂₃×#3265	1	1	yes	characteristic
732	C₂	C₂₃×#3269	1	1	yes	characteristic
733	C₂	C₂₃×#3367	1	1	yes	characteristic
734	C₂	C₂₃×#3376	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#461	C₂	semidirect
4. C₂×C₂₃×#440	C₂	semidirect
5. C₂₃×#2545	C₂	semidirect
6. D₄×C₄⋊C₄×C₂₃	C₂	semidirect
7. C₂₃×#2068	C₂	semidirect
8. C₂₃×#2067	C₂	semidirect
9. C₂₃×#2043	C₂	semidirect
10. C₂₃×#2042	C₂	semidirect
11. C₂₃×#1959	C₂	semidirect
12. C₂₃×#1940	C₂	semidirect
14. C₂₃×#1877	C₂	semidirect
15. C₂₃×#1803	C₂	semidirect
241. #24740	C₂₃	direct (N × H)
324. C₂×#461	C₂×C₂₃	semidirect
325. C₂×#440	C₂×C₂₃	semidirect
326. #2545	C₂×C₂₃	semidirect
327. D₄×C₄⋊C₄	C₂×C₂₃	semidirect
328. #2068	C₂×C₂₃	semidirect
329. #2067	C₂×C₂₃	semidirect
330. #2043	C₂×C₂₃	semidirect
331. #2042	C₂×C₂₃	semidirect
332. #1959	C₂×C₂₃	semidirect
333. #1940	C₂×C₂₃	semidirect
335. #1877	C₂×C₂₃	semidirect
336. #1803	C₂×C₂₃	semidirect
512. C₂₃	#24740	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×#463	2
C₂×C₂₃×#461	2
C₂×C₂₃×#440	2
C₂₃×#2545	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#2068	2
C₂₃×#2067	2
C₂₃×#2043	2
C₂₃×#2042	2
C₂₃×#1959	2
C₂₃×#1940	2
C₂₃×#1879	2
C₂₃×#1877	2
C₂₃×#1803	2
#24740	23

Subgroups — C23×#24740 #19172366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24740 #19172366