Subgroups — C₂₃×#23007 #19170633

Order 5888.

1058 conjugacy classes of subgroups.

The subgroup lattice has 708 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×#442	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2367	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1959	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1935	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1934	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1879	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
25	C₂₃×#463	C₄	1	8	yes	complemented
26	C₂₃×#442	—	2	4	no	—
27	C₄×SD₁₆×C₂₃	—	2	4	no	—
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#391	C₂²	1	1	yes	characteristic
43	C₂₃×#390	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#388	—	2	1	no	—
47	C₂₃×#388	—	2	1	no	—
48	C₂₃×#385	—	2	1	no	—
49	C₂₃×#385	—	2	1	no	—
50	C₂₃×#385	—	2	1	no	—
51	C₂₃×#384	C₂²	1	1	yes	characteristic
52	C₂₃×#380	—	2	1	no	—
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
72	C₂×D₈×C₂₃	—	2	2	no	—
73	C₂×D₈×C₂₃	—	2	2	no	—
74	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
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	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×#124	—	2	2	no	—
80	C₂₃×#124	C₂×C₄	1	4	yes	complemented
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	—	4	1	no	—
83	C₄×D₄×C₂₃	—	2	2	no	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₄×D₄×C₂₃	—	2	2	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	4	1	no	—
88	C₄×D₄×C₂₃	—	4	1	no	—
89	C₄×D₄×C₂₃	—	4	1	no	—
90	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₄²×C₂₃	—	2	1	no	—
113	C₂×C₄²×C₂₃	—	2	1	no	—
114	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂₃×#106	—	2	2	no	—
118	C₂₃×#106	—	2	2	no	—
119	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#103	—	4	1	no	—
122	C₂₃×#103	C₂×C₄	1	4	yes	complemented
123	C₂₃×#102	—	2	2	no	—
124	C₂₃×#102	—	2	2	no	—
125	C₂₃×#102	—	4	1	no	—
126	C₂₃×#102	C₂×C₄	1	4	yes	complemented
127	C₂₃×#102	D₄	1	4	yes	—
128	C₂₃×#97	—	2	2	no	—
129	C₂₃×#97	—	2	2	no	—
130	C₄×C₈×C₂₃	—	2	2	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂₃×#95	—	2	1	no	—
134	C₂₃×#95	—	2	1	no	—
135	C₂₃×#95	—	2	1	no	—
136	C₂₃×#95	D₄	1	1	yes	characteristic
137	C₂₃×#95	D₄	1	1	yes	characteristic
138	C₂⁴×C₂₃	—	2	1	no	—
139	C₂⁴×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
146	C₂×D₄×C₂₃	—	2	1	no	—
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
155	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
156	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₂₃	#41	1	2	yes	—
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	4	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	4	2	no	—
167	C₂²×C₄×C₂₃	—	4	2	no	—
168	C₂²×C₄×C₂₃	—	4	2	no	—
169	C₂²×C₄×C₂₃	—	4	2	no	—
170	C₂²×C₄×C₂₃	—	4	2	no	—
171	C₂²×C₄×C₂₃	—	4	2	no	—
172	C₂²×C₄×C₂₃	—	4	2	no	—
173	C₂²×C₄×C₂₃	—	8	1	no	—
174	C₂²×C₄×C₂₃	—	8	1	no	—
175	C₂²×C₄×C₂₃	—	8	1	no	—
176	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	SD₁₆×C₂₃	—	2	8	no	—
184	D₈×C₂₃	—	2	8	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	—	2	1	no	—
189	C₂×C₈×C₂₃	—	2	1	no	—
190	C₂×C₈×C₂₃	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	1	no	—
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
195	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
196	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
197	C₄⋊C₄×C₂₃	—	2	2	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	1	no	—
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	4	1	no	—
207	C₄⋊C₄×C₂₃	—	4	2	no	—
208	C₄⋊C₄×C₂₃	—	4	2	no	—
209	C₄⋊C₄×C₂₃	—	4	2	no	—
210	C₄⋊C₄×C₂₃	—	4	2	no	—
211	C₄⋊C₄×C₂₃	—	8	1	no	—
212	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
213	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
214	C₂²⋊C₄×C₂₃	—	2	4	no	—
215	C₂²⋊C₄×C₂₃	—	2	4	no	—
216	C₂²⋊C₄×C₂₃	—	2	4	no	—
217	C₂²⋊C₄×C₂₃	—	4	2	no	—
218	C₂²⋊C₄×C₂₃	—	4	2	no	—
219	C₂²⋊C₄×C₂₃	—	4	2	no	—
220	C₂²⋊C₄×C₂₃	—	4	2	no	—
221	C₂²⋊C₄×C₂₃	—	8	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	—	2	1	no	—
225	C₄²×C₂₃	—	2	1	no	—
226	C₄²×C₂₃	—	4	2	no	—
227	C₄²×C₂₃	—	8	1	no	—
228	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
229	C₄²×C₂₃	#41	1	2	yes	—
230	#23007	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂³×C₂₃	—	2	2	no	—
232	C₂³×C₂₃	—	2	2	no	—
233	C₂³×C₂₃	—	2	2	no	—
234	C₂³×C₂₃	—	2	2	no	—
235	C₂³×C₂₃	—	4	1	no	—
236	C₂³×C₂₃	—	4	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	—	4	1	no	—
239	C₂³×C₂₃	—	4	1	no	—
240	C₂³×C₂₃	—	4	1	no	—
241	C₂³×C₂₃	—	4	1	no	—
242	C₂³×C₂₃	—	4	1	no	—
243	C₂³×C₂₃	—	4	1	no	—
244	C₂³×C₂₃	—	4	1	no	—
245	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₂₃	—	2	2	no	—
247	Q₈×C₂₃	—	2	2	no	—
248	Q₈×C₂₃	—	4	1	no	—
249	Q₈×C₂₃	—	4	1	no	—
250	D₄×C₂₃	—	2	2	no	—
251	D₄×C₂₃	—	2	2	no	—
252	D₄×C₂₃	—	2	2	no	—
253	D₄×C₂₃	—	2	2	no	—
254	D₄×C₂₃	—	2	2	no	—
255	D₄×C₂₃	—	4	1	no	—
256	D₄×C₂₃	—	4	1	no	—
257	D₄×C₂₃	C₄×D₄	1	4	yes	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	2	1	no	—
271	C₂×C₄×C₂₃	—	2	1	no	—
272	C₂×C₄×C₂₃	—	2	1	no	—
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	—	4	8	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	4	1	no	—
281	C₂×C₄×C₂₃	—	4	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	4	1	no	—
285	C₂×C₄×C₂₃	—	4	1	no	—
286	C₂×C₄×C₂₃	—	4	1	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	—	8	4	no	—
293	C₂×C₄×C₂₃	—	8	4	no	—
294	C₂×C₄×C₂₃	—	8	4	no	—
295	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
296	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
297	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
298	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
299	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
301	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
303	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
304	C₈×C₂₃	—	2	2	no	—
305	C₈×C₂₃	—	2	2	no	—
306	C₈×C₂₃	—	2	2	no	—
307	C₈×C₂₃	—	4	1	no	—
308	C₂×#463	C₂×C₂₃	1	1	yes	characteristic
309	C₂×#442	C₂×C₂₃	1	1	yes	characteristic
310	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#2367	C₂×C₂₃	1	1	yes	characteristic, complemented
312	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#1959	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#1935	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#1934	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#1879	C₂×C₂₃	1	1	yes	characteristic
318	#1877	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1771	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1762	C₂×C₂₃	1	1	yes	characteristic, complemented
323	C₂²×C₂₃	—	2	4	no	—
324	C₂²×C₂₃	—	4	2	no	—
325	C₂²×C₂₃	—	4	2	no	—
326	C₂²×C₂₃	—	4	2	no	—
327	C₂²×C₂₃	—	4	2	no	—
328	C₂²×C₂₃	—	4	2	no	—
329	C₂²×C₂₃	—	4	2	no	—
330	C₂²×C₂₃	—	4	2	no	—
331	C₂²×C₂₃	—	4	2	no	—
332	C₂²×C₂₃	—	4	2	no	—
333	C₂²×C₂₃	—	8	1	no	—
334	C₂²×C₂₃	—	8	1	no	—
335	C₂²×C₂₃	—	8	1	no	—
336	C₂²×C₂₃	—	8	1	no	—
337	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
338	C₂²×C₂₃	D₄²	1	1	yes	characteristic
339	C₂²×C₂₃	#548	1	1	yes	characteristic
340	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
341	C₂²×C₂₃	#575	1	1	yes	characteristic
342	C₂²×C₂₃	#577	1	1	yes	characteristic
343	C₂²×C₂₃	#577	1	1	yes	characteristic
344	C₄×C₂₃	—	2	2	no	—
345	C₄×C₂₃	—	2	2	no	—
346	C₄×C₂₃	—	2	2	no	—
347	C₄×C₂₃	—	4	2	no	—
348	C₄×C₂₃	—	4	2	no	—
349	C₄×C₂₃	—	4	1	no	—
350	C₄×C₂₃	—	4	1	no	—
351	C₄×C₂₃	—	4	1	no	—
352	C₄×C₂₃	—	4	2	no	—
353	C₄×C₂₃	—	8	1	no	—
354	C₄×C₂₃	—	8	1	no	—
355	C₄×C₂₃	—	8	1	no	—
356	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
357	C₄×C₂₃	#520	1	1	yes	characteristic
358	C₄×C₂₃	#547	1	1	yes	characteristic
359	C₄×C₂₃	#553	1	1	yes	characteristic
360	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
361	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
368	C₂²×C₄⋊C₄	—	2	1	no	—
369	#463	C₄×C₂₃	1	8	yes	complemented
370	#442	—	2	4	no	—
371	C₄×SD₁₆	—	2	4	no	—
372	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
386	#391	C₂²×C₂₃	1	1	yes	characteristic
387	#390	C₂²×C₂₃	1	1	yes	characteristic
388	#389	C₂²×C₂₃	1	1	yes	characteristic
389	#389	C₂²×C₂₃	1	1	yes	characteristic
390	#388	—	2	1	no	—
391	#388	—	2	1	no	—
392	#385	—	2	1	no	—
393	#385	—	2	1	no	—
394	#385	—	2	1	no	—
395	#384	C₂²×C₂₃	1	1	yes	characteristic
396	#380	—	2	1	no	—
397	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
398	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
399	C₄×C₂²⋊C₄	—	2	1	no	—
400	#340	C₂²×C₂₃	1	1	yes	characteristic
401	#340	C₂²×C₂₃	1	1	yes	characteristic
402	#340	C₂²×C₂₃	1	1	yes	characteristic
403	#340	C₂²×C₂₃	1	1	yes	characteristic
404	#336	C₂²×C₂₃	1	1	yes	characteristic
405	#336	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×C₂₃	—	4	4	no	—
407	C₂×C₂₃	—	8	2	no	—
408	C₂×C₂₃	#2361	1	1	yes	characteristic
409	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
410	C₂×C₂₃	#2952	1	1	yes	characteristic
411	C₂×C₂₃	#3265	1	1	yes	characteristic
412	C₂×C₂₃	#3267	1	1	yes	characteristic
413	C₂×C₂₃	#3373	1	1	yes	characteristic
414	C₂×C₂₃	#3376	1	1	yes	characteristic
415	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
418	C₂³×C₄	—	2	1	no	—
419	C₂³×C₄	—	2	1	no	—
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	—	2	1	no	—
422	C₂×SD₁₆	—	2	2	no	—
423	C₂×SD₁₆	—	2	2	no	—
424	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
425	C₂×D₈	—	2	2	no	—
426	C₂×D₈	—	2	2	no	—
427	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
428	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	#124	—	2	2	no	—
433	#124	C₂×C₄×C₂₃	1	4	yes	complemented
434	C₄×Q₈	—	2	2	no	—
435	C₄×Q₈	—	4	1	no	—
436	C₄×D₄	—	2	2	no	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	4	1	no	—
441	C₄×D₄	—	4	1	no	—
442	C₄×D₄	—	4	1	no	—
443	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
444	C₂×C₄⋊C₄	—	2	1	no	—
445	C₂×C₄⋊C₄	—	2	1	no	—
446	C₂×C₄⋊C₄	—	2	1	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	—	4	4	no	—
450	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
453	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
455	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	—	2	1	no	—
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	1	no	—
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₂²⋊C₄	—	2	1	no	—
465	C₂×C₄²	—	2	1	no	—
466	C₂×C₄²	—	2	1	no	—
467	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
470	#106	—	2	2	no	—
471	#106	—	2	2	no	—
472	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
473	#103	—	2	2	no	—
474	#103	—	4	1	no	—
475	#103	C₂×C₄×C₂₃	1	4	yes	complemented
476	#102	—	2	2	no	—
477	#102	—	2	2	no	—
478	#102	—	4	1	no	—
479	#102	C₂×C₄×C₂₃	1	4	yes	complemented
480	#102	D₄×C₂₃	1	4	yes	—
481	#97	—	2	2	no	—
482	#97	—	2	2	no	—
483	C₄×C₈	—	2	2	no	—
484	#95	—	2	1	no	—
485	#95	—	2	1	no	—
486	#95	—	2	1	no	—
487	#95	—	2	1	no	—
488	#95	—	2	1	no	—
489	#95	D₄×C₂₃	1	1	yes	characteristic
490	#95	D₄×C₂₃	1	1	yes	characteristic
491	C₂₃	#23007	1	1	yes	characteristic, direct product factor, Sylow
492	C₂⁴	—	2	1	no	—
493	C₂⁴	—	2	1	no	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
509	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
510	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
511	C₂×D₄	#41×C₂₃	1	2	yes	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	—	4	2	no	—
521	C₂²×C₄	—	4	2	no	—
522	C₂²×C₄	—	4	2	no	—
523	C₂²×C₄	—	4	2	no	—
524	C₂²×C₄	—	4	2	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	—	4	2	no	—
527	C₂²×C₄	—	8	1	no	—
528	C₂²×C₄	—	8	1	no	—
529	C₂²×C₄	—	8	1	no	—
530	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
537	SD₁₆	—	2	8	no	—
538	D₈	—	2	8	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
549	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
550	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	4	1	no	—
561	C₄⋊C₄	—	4	2	no	—
562	C₄⋊C₄	—	4	2	no	—
563	C₄⋊C₄	—	4	2	no	—
564	C₄⋊C₄	—	4	2	no	—
565	C₄⋊C₄	—	8	1	no	—
566	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
567	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
568	C₂²⋊C₄	—	2	4	no	—
569	C₂²⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	4	2	no	—
572	C₂²⋊C₄	—	4	2	no	—
573	C₂²⋊C₄	—	4	2	no	—
574	C₂²⋊C₄	—	4	2	no	—
575	C₂²⋊C₄	—	8	1	no	—
576	C₄²	—	2	1	no	—
577	C₄²	—	2	1	no	—
578	C₄²	—	2	1	no	—
579	C₄²	—	2	1	no	—
580	C₄²	—	4	2	no	—
581	C₄²	—	8	1	no	—
582	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
583	C₄²	#41×C₂₃	1	2	yes	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	2	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	4	1	no	—
590	C₂³	—	4	1	no	—
591	C₂³	—	4	1	no	—
592	C₂³	—	4	1	no	—
593	C₂³	—	4	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
599	Q₈	—	2	2	no	—
600	Q₈	—	2	2	no	—
601	Q₈	—	4	1	no	—
602	Q₈	—	4	1	no	—
603	D₄	—	2	2	no	—
604	D₄	—	2	2	no	—
605	D₄	—	2	2	no	—
606	D₄	—	2	2	no	—
607	D₄	—	2	2	no	—
608	D₄	—	4	1	no	—
609	D₄	—	4	1	no	—
610	D₄	C₄×D₄×C₂₃	1	4	yes	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	4	8	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	—	8	4	no	—
647	C₂×C₄	—	8	4	no	—
648	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
649	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
650	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
651	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
652	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
653	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
654	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
655	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
656	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
657	C₈	—	2	2	no	—
658	C₈	—	2	2	no	—
659	C₈	—	2	2	no	—
660	C₈	—	4	1	no	—
661	C₂²	—	2	4	no	—
662	C₂²	—	4	2	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	4	2	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	8	1	no	—
672	C₂²	—	8	1	no	—
673	C₂²	—	8	1	no	—
674	C₂²	—	8	1	no	—
675	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
676	C₂²	D₄²×C₂₃	1	1	yes	characteristic
677	C₂²	C₂₃×#548	1	1	yes	characteristic
678	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
679	C₂²	C₂₃×#575	1	1	yes	characteristic
680	C₂²	C₂₃×#577	1	1	yes	characteristic
681	C₂²	C₂₃×#577	1	1	yes	characteristic
682	C₄	—	2	2	no	—
683	C₄	—	2	2	no	—
684	C₄	—	2	2	no	—
685	C₄	—	4	2	no	—
686	C₄	—	4	1	no	—
687	C₄	—	4	1	no	—
688	C₄	—	4	1	no	—
689	C₄	—	4	2	no	—
690	C₄	—	4	2	no	—
691	C₄	—	8	1	no	—
692	C₄	—	8	1	no	—
693	C₄	—	8	1	no	—
694	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
695	C₄	C₂₃×#520	1	1	yes	characteristic
696	C₄	C₂₃×#547	1	1	yes	characteristic
697	C₄	C₂₃×#553	1	1	yes	characteristic
698	C₂	—	4	4	no	—
699	C₂	—	8	2	no	—
700	C₂	C₂₃×#2361	1	1	yes	characteristic
701	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
702	C₂	C₂₃×#2952	1	1	yes	characteristic
703	C₂	C₂₃×#3265	1	1	yes	characteristic
704	C₂	C₂₃×#3267	1	1	yes	characteristic
705	C₂	C₂₃×#3373	1	1	yes	characteristic
706	C₂	C₂₃×#3376	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	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×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×C₄×SD₁₆×C₂₃	C₂	semidirect
4. C₂₃×#2367	C₂	semidirect
5. D₄×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1959	C₂	semidirect
7. C₂₃×#1948	C₂	semidirect
8. C₂₃×#1935	C₂	semidirect
9. C₂₃×#1934	C₂	semidirect
11. C₂₃×#1877	C₂	semidirect
12. C₂₃×#1865	C₂	semidirect
13. C₂₃×#1864	C₂	semidirect
14. C₂₃×#1771	C₂	semidirect
15. C₂₃×#1762	C₂	semidirect
25. C₂₃×#463	C₄	semidirect
71. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
80. C₂₃×#124	C₂×C₄	semidirect
90. C₄×D₄×C₂₃	C₂×C₄	semidirect
119. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
122. C₂₃×#103	C₂×C₄	semidirect
126. C₂₃×#102	C₂×C₄	semidirect
230. #23007	C₂₃	direct (N × H)
310. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
311. #2367	C₂×C₂₃	semidirect
312. D₄×C₄⋊C₄	C₂×C₂₃	semidirect
313. #1959	C₂×C₂₃	semidirect
314. #1948	C₂×C₂₃	semidirect
315. #1935	C₂×C₂₃	semidirect
316. #1934	C₂×C₂₃	semidirect
318. #1877	C₂×C₂₃	semidirect
319. #1865	C₂×C₂₃	semidirect
320. #1864	C₂×C₂₃	semidirect
321. #1771	C₂×C₂₃	semidirect
322. #1762	C₂×C₂₃	semidirect
369. #463	C₄×C₂₃	semidirect
424. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
433. #124	C₂×C₄×C₂₃	semidirect
443. C₄×D₄	C₂×C₄×C₂₃	semidirect
472. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
475. #103	C₂×C₄×C₂₃	semidirect
479. #102	C₂×C₄×C₂₃	semidirect
491. C₂₃	#23007	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×#442	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#2367	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1959	2
C₂₃×#1948	2
C₂₃×#1935	2
C₂₃×#1934	2
C₂₃×#1879	2
C₂₃×#1877	2
C₂₃×#1865	2
C₂₃×#1864	2
C₂₃×#1771	2
C₂₃×#1762	2
#23007	23

Subgroups — C23×#23007 #19170633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23007 #19170633