Subgroups — C₂₃×#25095 #19172721

Order 5888.

1048 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×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#451	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#419	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2434	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2362	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2056	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2016	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1981	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1927	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1873	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic
16	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂₃×#470	—	2	4	no	—
28	C₂₃×#462	—	2	4	no	—
29	C₂₃×#451	—	2	4	no	—
30	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂₃×#419	—	2	4	no	—
34	C₂×C₂₃×#103	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×#103	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₂₃×#395	—	2	1	no	—
46	C₂₃×#395	—	2	1	no	—
47	C₂₃×#391	C₂²	1	1	yes	characteristic
48	C₂₃×#391	C₂²	1	1	yes	characteristic
49	C₂₃×#391	C₂²	1	1	yes	characteristic
50	C₂₃×#389	C₂²	1	1	yes	characteristic
51	C₂₃×#388	—	2	1	no	—
52	C₂₃×#388	—	2	1	no	—
53	C₂₃×#387	—	2	1	no	—
54	C₂₃×#385	—	2	1	no	—
55	C₂₃×#385	—	2	1	no	—
56	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊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	—	2	1	no	—
64	C₂²×Q₈×C₂₃	—	2	2	no	—
65	C₂²×Q₈×C₂₃	—	2	2	no	—
66	C₂²×Q₈×C₂₃	—	2	2	no	—
67	C₂²×Q₈×C₂₃	—	2	2	no	—
68	C₂²×Q₈×C₂₃	—	4	1	no	—
69	C₂²×Q₈×C₂₃	—	4	1	no	—
70	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₂₃	—	2	1	no	—
75	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₂₃	—	2	2	no	—
77	C₂×Q₁₆×C₂₃	—	2	2	no	—
78	C₂×Q₁₆×C₂₃	—	2	2	no	—
79	C₂×Q₁₆×C₂₃	—	2	2	no	—
80	C₂×Q₁₆×C₂₃	—	2	2	no	—
81	C₂×Q₁₆×C₂₃	—	2	2	no	—
82	C₂×Q₁₆×C₂₃	—	4	1	no	—
83	C₂×Q₁₆×C₂₃	—	4	1	no	—
84	C₂×Q₁₆×C₂₃	—	4	1	no	—
85	C₂×M₁₆×C₂₃	—	2	2	no	—
86	C₂×M₁₆×C₂₃	—	2	2	no	—
87	C₂×M₁₆×C₂₃	—	4	1	no	—
88	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×C₄⋊Q₈	—	2	2	no	—
93	C₂₃×C₄⋊Q₈	—	2	2	no	—
94	C₂₃×C₄⋊Q₈	—	4	1	no	—
95	C₂₃×C₄⋊Q₈	—	4	1	no	—
96	C₂₃×C₂²⋊Q₈	—	2	2	no	—
97	C₂₃×C₂²⋊Q₈	—	2	2	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	D₄	1	4	yes	—
100	C₄×Q₈×C₂₃	—	2	2	no	—
101	C₄×Q₈×C₂₃	—	4	1	no	—
102	C₂₃×#117	—	2	2	no	—
103	C₂₃×#117	—	4	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	—	2	1	no	—
122	C₂×C₄²×C₂₃	—	2	1	no	—
123	C₂×C₄²×C₂₃	—	2	1	no	—
124	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
126	C₂₃×#106	—	2	2	no	—
127	C₂₃×C₄⋊C₈	—	2	2	no	—
128	C₂₃×C₄⋊C₈	—	4	1	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	—	2	2	no	—
134	C₂₃×#103	—	2	2	no	—
135	C₂₃×#103	—	2	2	no	—
136	C₂₃×#103	—	2	2	no	—
137	C₂₃×#103	—	4	1	no	—
138	C₂₃×#103	D₄	1	4	yes	—
139	C₂₃×C₂²⋊C₈	—	2	2	no	—
140	C₂₃×#95	—	2	1	no	—
141	C₂₃×#95	—	2	1	no	—
142	C₂₃×#95	—	2	1	no	—
143	C₂₃×#95	—	2	1	no	—
144	C₂₃×#95	—	2	1	no	—
145	C₂₃×#95	—	2	1	no	—
146	C₂₃×#95	—	2	1	no	—
147	C₂₃×#95	—	2	1	no	—
148	C₂₃×#95	D₄	1	1	yes	characteristic
149	C₂₃×#95	D₄	1	1	yes	characteristic
150	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
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	8	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	—	2	1	no	—
159	C₂×Q₈×C₂₃	—	2	1	no	—
160	C₂×Q₈×C₂₃	—	2	1	no	—
161	C₂×Q₈×C₂₃	—	2	1	no	—
162	C₂×Q₈×C₂₃	—	2	1	no	—
163	C₂×Q₈×C₂₃	—	2	1	no	—
164	C₂×Q₈×C₂₃	—	2	1	no	—
165	C₂×Q₈×C₂₃	—	2	1	no	—
166	C₂×Q₈×C₂₃	—	2	1	no	—
167	C₂×Q₈×C₂₃	—	2	1	no	—
168	C₂×Q₈×C₂₃	—	2	1	no	—
169	C₂×Q₈×C₂₃	—	2	1	no	—
170	C₂×Q₈×C₂₃	—	2	1	no	—
171	C₂×Q₈×C₂₃	—	2	1	no	—
172	C₂×Q₈×C₂₃	—	2	1	no	—
173	C₂×Q₈×C₂₃	—	4	2	no	—
174	C₂×Q₈×C₂₃	—	4	2	no	—
175	C₂×Q₈×C₂₃	—	4	4	no	—
176	C₂×Q₈×C₂₃	—	4	4	no	—
177	C₂×Q₈×C₂₃	—	4	4	no	—
178	C₂×Q₈×C₂₃	—	4	2	no	—
179	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
180	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
181	C₂×Q₈×C₂₃	#41	1	2	yes	—
182	C₂²×C₄×C₂₃	—	2	1	no	—
183	C₂²×C₄×C₂₃	—	2	1	no	—
184	C₂²×C₄×C₂₃	—	2	1	no	—
185	C₂²×C₄×C₂₃	—	2	1	no	—
186	C₂²×C₄×C₂₃	—	2	1	no	—
187	C₂²×C₄×C₂₃	—	2	1	no	—
188	C₂²×C₄×C₂₃	—	2	1	no	—
189	C₂²×C₄×C₂₃	—	2	1	no	—
190	C₂²×C₄×C₂₃	—	2	1	no	—
191	C₂²×C₄×C₂₃	—	2	1	no	—
192	C₂²×C₄×C₂₃	—	2	1	no	—
193	C₂²×C₄×C₂₃	—	2	1	no	—
194	C₂²×C₄×C₂₃	—	4	2	no	—
195	C₂²×C₄×C₂₃	—	4	2	no	—
196	C₂²×C₄×C₂₃	—	4	2	no	—
197	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₂₃	#41	1	2	yes	—
204	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
205	Q₁₆×C₂₃	—	4	4	no	—
206	Q₁₆×C₂₃	—	4	4	no	—
207	Q₁₆×C₂₃	—	4	4	no	—
208	M₁₆×C₂₃	—	4	4	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	2	1	no	—
211	C₂×C₈×C₂₃	—	2	1	no	—
212	C₂×C₈×C₂₃	—	2	1	no	—
213	C₂×C₈×C₂₃	—	2	1	no	—
214	C₂×C₈×C₂₃	—	2	1	no	—
215	C₂×C₈×C₂₃	—	2	1	no	—
216	C₂×C₈×C₂₃	—	2	1	no	—
217	C₂×C₈×C₂₃	—	2	1	no	—
218	C₂×C₈×C₂₃	—	2	1	no	—
219	C₂×C₈×C₂₃	—	2	1	no	—
220	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
221	C₄⋊C₄×C₂₃	—	2	4	no	—
222	C₄⋊C₄×C₂₃	—	2	1	no	—
223	C₄⋊C₄×C₂₃	—	2	1	no	—
224	C₄⋊C₄×C₂₃	—	2	1	no	—
225	C₄⋊C₄×C₂₃	—	2	1	no	—
226	C₄⋊C₄×C₂₃	—	2	1	no	—
227	C₄⋊C₄×C₂₃	—	2	1	no	—
228	C₄⋊C₄×C₂₃	—	4	2	no	—
229	C₄⋊C₄×C₂₃	—	4	2	no	—
230	C₄⋊C₄×C₂₃	—	4	2	no	—
231	C₄⋊C₄×C₂₃	—	4	2	no	—
232	C₄⋊C₄×C₂₃	—	4	2	no	—
233	C₄⋊C₄×C₂₃	—	4	2	no	—
234	C₄⋊C₄×C₂₃	—	4	2	no	—
235	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
237	C₂²⋊C₄×C₂₃	—	2	4	no	—
238	C₂²⋊C₄×C₂₃	—	2	2	no	—
239	C₂²⋊C₄×C₂₃	—	2	2	no	—
240	C₂²⋊C₄×C₂₃	—	4	2	no	—
241	C₂²⋊C₄×C₂₃	—	4	2	no	—
242	C₄²×C₂₃	—	2	1	no	—
243	C₄²×C₂₃	—	2	1	no	—
244	C₄²×C₂₃	—	2	1	no	—
245	C₄²×C₂₃	—	2	1	no	—
246	C₄²×C₂₃	—	4	2	no	—
247	C₄²×C₂₃	—	4	2	no	—
248	C₄²×C₂₃	—	4	2	no	—
249	#25095	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂³×C₂₃	—	2	1	no	—
251	C₂³×C₂₃	—	2	1	no	—
252	C₂³×C₂₃	—	2	1	no	—
253	C₂³×C₂₃	—	2	1	no	—
254	C₂³×C₂₃	—	2	1	no	—
255	C₂³×C₂₃	—	2	1	no	—
256	C₂³×C₂₃	#121	1	2	yes	—
257	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
258	Q₈×C₂₃	—	2	2	no	—
259	Q₈×C₂₃	—	2	2	no	—
260	Q₈×C₂₃	—	2	2	no	—
261	Q₈×C₂₃	—	2	2	no	—
262	Q₈×C₂₃	—	2	2	no	—
263	Q₈×C₂₃	—	2	2	no	—
264	Q₈×C₂₃	—	4	8	no	—
265	Q₈×C₂₃	—	4	1	no	—
266	Q₈×C₂₃	—	4	1	no	—
267	Q₈×C₂₃	—	4	1	no	—
268	Q₈×C₂₃	—	4	1	no	—
269	Q₈×C₂₃	—	4	1	no	—
270	Q₈×C₂₃	—	4	1	no	—
271	Q₈×C₂₃	—	4	1	no	—
272	Q₈×C₂₃	—	4	1	no	—
273	Q₈×C₂₃	—	4	1	no	—
274	Q₈×C₂₃	—	4	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	1	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	2	no	—
282	C₂×C₄×C₂₃	—	2	2	no	—
283	C₂×C₄×C₂₃	—	2	2	no	—
284	C₂×C₄×C₂₃	—	2	2	no	—
285	C₂×C₄×C₂₃	—	2	2	no	—
286	C₂×C₄×C₂₃	—	2	2	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	2	no	—
289	C₂×C₄×C₂₃	—	2	2	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	2	2	no	—
292	C₂×C₄×C₂₃	—	2	1	no	—
293	C₂×C₄×C₂₃	—	2	1	no	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	1	no	—
296	C₂×C₄×C₂₃	—	4	8	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	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	4	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	4	1	no	—
312	C₂×C₄×C₂₃	—	4	1	no	—
313	C₂×C₄×C₂₃	—	4	1	no	—
314	C₂×C₄×C₂₃	—	4	1	no	—
315	C₂×C₄×C₂₃	—	4	1	no	—
316	C₂×C₄×C₂₃	#121	1	2	yes	—
317	C₂×C₄×C₂₃	#137	1	2	yes	—
318	C₂×C₄×C₂₃	#137	1	2	yes	—
319	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
322	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
323	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
324	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
325	C₈×C₂₃	—	2	2	no	—
326	C₈×C₂₃	—	4	1	no	—
327	C₈×C₂₃	—	4	1	no	—
328	C₈×C₂₃	—	4	1	no	—
329	C₂×#470	C₂×C₂₃	1	1	yes	characteristic
330	C₂×#462	C₂×C₂₃	1	1	yes	characteristic, complemented
331	C₂×#451	C₂×C₂₃	1	1	yes	characteristic
332	C₂×#419	C₂×C₂₃	1	1	yes	characteristic
333	#2434	C₂×C₂₃	1	1	yes	characteristic
334	#2362	C₂×C₂₃	1	1	yes	characteristic
335	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#2056	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#2016	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1981	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#1927	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1873	C₂×C₂₃	1	1	yes	characteristic
343	#1851	C₂×C₂₃	1	1	yes	characteristic
344	C₂²×C₂₃	—	2	2	no	—
345	C₂²×C₂₃	—	2	2	no	—
346	C₂²×C₂₃	—	2	2	no	—
347	C₂²×C₂₃	—	4	1	no	—
348	C₂²×C₂₃	—	4	1	no	—
349	C₂²×C₂₃	—	4	1	no	—
350	C₂²×C₂₃	—	4	1	no	—
351	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
352	C₂²×C₂₃	D₄²	1	1	yes	characteristic
353	C₂²×C₂₃	#546	1	1	yes	characteristic
354	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
355	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
356	C₂²×C₂₃	#578	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	1	no	—
362	C₄×C₂₃	—	4	2	no	—
363	C₄×C₂₃	—	4	2	no	—
364	C₄×C₂₃	—	4	2	no	—
365	C₄×C₂₃	—	4	1	no	—
366	C₄×C₂₃	—	4	2	no	—
367	C₄×C₂₃	—	4	2	no	—
368	C₄×C₂₃	—	8	1	no	—
369	C₄×C₂₃	—	8	1	no	—
370	C₄×C₂₃	—	8	1	no	—
371	C₄×C₂₃	#525	1	1	yes	characteristic
372	C₄×C₂₃	#538	1	1	yes	characteristic
373	C₄×C₂₃	D₄²	1	1	yes	characteristic
374	C₄×C₂₃	#546	1	1	yes	characteristic
375	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
376	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
377	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
378	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
379	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
386	#470	—	2	4	no	—
387	#462	—	2	4	no	—
388	#451	—	2	4	no	—
389	C₂×#106	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	#419	—	2	4	no	—
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×#103	C₂²×C₂₃	1	1	yes	characteristic
400	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
401	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
402	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
404	#395	—	2	1	no	—
405	#395	—	2	1	no	—
406	#391	C₂²×C₂₃	1	1	yes	characteristic
407	#391	C₂²×C₂₃	1	1	yes	characteristic
408	#391	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	#387	—	2	1	no	—
413	#385	—	2	1	no	—
414	#385	—	2	1	no	—
415	C₄×C₄⋊C₄	—	2	1	no	—
416	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
417	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
418	#340	C₂²×C₂₃	1	1	yes	characteristic
419	#340	C₂²×C₂₃	1	1	yes	characteristic
420	#336	C₂²×C₂₃	1	1	yes	characteristic
421	#336	C₂²×C₂₃	1	1	yes	characteristic
422	C₂×C₂₃	—	4	2	no	—
423	C₂×C₂₃	#2424	1	1	yes	characteristic
424	C₂×C₂₃	#3039	1	1	yes	characteristic
425	C₂×C₂₃	#3044	1	1	yes	characteristic
426	C₂×C₂₃	#3270	1	1	yes	characteristic
427	C₂×C₂₃	#3276	1	1	yes	characteristic
428	C₂×C₂₃	#3312	1	1	yes	characteristic
429	C₂×C₂₃	#3328	1	1	yes	characteristic
430	C₂²×Q₈	—	2	1	no	—
431	C₂²×Q₈	—	2	2	no	—
432	C₂²×Q₈	—	2	2	no	—
433	C₂²×Q₈	—	2	2	no	—
434	C₂²×Q₈	—	2	2	no	—
435	C₂²×Q₈	—	4	1	no	—
436	C₂²×Q₈	—	4	1	no	—
437	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
438	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
439	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
440	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
441	C₂³×C₄	—	2	1	no	—
442	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
443	C₂×Q₁₆	—	2	2	no	—
444	C₂×Q₁₆	—	2	2	no	—
445	C₂×Q₁₆	—	2	2	no	—
446	C₂×Q₁₆	—	2	2	no	—
447	C₂×Q₁₆	—	2	2	no	—
448	C₂×Q₁₆	—	2	2	no	—
449	C₂×Q₁₆	—	4	1	no	—
450	C₂×Q₁₆	—	4	1	no	—
451	C₂×Q₁₆	—	4	1	no	—
452	C₂×M₁₆	—	2	2	no	—
453	C₂×M₁₆	—	2	2	no	—
454	C₂×M₁₆	—	4	1	no	—
455	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
456	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
457	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
458	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
459	C₄⋊Q₈	—	2	2	no	—
460	C₄⋊Q₈	—	2	2	no	—
461	C₄⋊Q₈	—	4	1	no	—
462	C₄⋊Q₈	—	4	1	no	—
463	C₂²⋊Q₈	—	2	2	no	—
464	C₂²⋊Q₈	—	2	2	no	—
465	C₂²⋊Q₈	—	2	2	no	—
466	C₂²⋊Q₈	D₄×C₂₃	1	4	yes	—
467	C₄×Q₈	—	2	2	no	—
468	C₄×Q₈	—	4	1	no	—
469	#117	—	2	2	no	—
470	#117	—	4	1	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄⋊C₄	—	2	1	no	—
473	C₂×C₄⋊C₄	—	2	1	no	—
474	C₂×C₄⋊C₄	—	2	1	no	—
475	C₂×C₄⋊C₄	—	2	1	no	—
476	C₂×C₄⋊C₄	—	2	1	no	—
477	C₂×C₄⋊C₄	—	2	1	no	—
478	C₂×C₄⋊C₄	—	2	1	no	—
479	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
480	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
481	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
482	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
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	D₄×C₂₃	1	1	yes	characteristic
487	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
488	C₂×C₄²	—	2	1	no	—
489	C₂×C₄²	—	2	1	no	—
490	C₂×C₄²	—	2	1	no	—
491	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
492	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
493	#106	—	2	2	no	—
494	C₄⋊C₈	—	2	2	no	—
495	C₄⋊C₈	—	4	1	no	—
496	#103	—	2	2	no	—
497	#103	—	2	2	no	—
498	#103	—	2	2	no	—
499	#103	—	2	2	no	—
500	#103	—	2	2	no	—
501	#103	—	2	2	no	—
502	#103	—	2	2	no	—
503	#103	—	2	2	no	—
504	#103	—	4	1	no	—
505	#103	D₄×C₂₃	1	4	yes	—
506	C₂²⋊C₈	—	2	2	no	—
507	#95	—	2	1	no	—
508	#95	—	2	1	no	—
509	#95	—	2	1	no	—
510	#95	—	2	1	no	—
511	#95	—	2	1	no	—
512	#95	—	2	1	no	—
513	#95	—	2	1	no	—
514	#95	—	2	1	no	—
515	#95	D₄×C₂₃	1	1	yes	characteristic
516	#95	D₄×C₂₃	1	1	yes	characteristic
517	C₂₃	#25095	1	1	yes	characteristic, direct product factor, Sylow
518	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	8	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	1	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	1	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	4	2	no	—
542	C₂×Q₈	—	4	2	no	—
543	C₂×Q₈	—	4	4	no	—
544	C₂×Q₈	—	4	4	no	—
545	C₂×Q₈	—	4	4	no	—
546	C₂×Q₈	—	4	2	no	—
547	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
548	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
549	C₂×Q₈	#41×C₂₃	1	2	yes	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	—	2	1	no	—
558	C₂²×C₄	—	2	1	no	—
559	C₂²×C₄	—	2	1	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	4	2	no	—
563	C₂²×C₄	—	4	2	no	—
564	C₂²×C₄	—	4	2	no	—
565	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
566	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
568	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
569	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
570	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
571	C₂²×C₄	#41×C₂₃	1	2	yes	—
572	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
573	Q₁₆	—	4	4	no	—
574	Q₁₆	—	4	4	no	—
575	Q₁₆	—	4	4	no	—
576	M₁₆	—	4	4	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	—	2	1	no	—
581	C₂×C₈	—	2	1	no	—
582	C₂×C₈	—	2	1	no	—
583	C₂×C₈	—	2	1	no	—
584	C₂×C₈	—	2	1	no	—
585	C₂×C₈	—	2	1	no	—
586	C₂×C₈	—	2	1	no	—
587	C₂×C₈	—	2	1	no	—
588	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
589	C₄⋊C₄	—	2	4	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	4	2	no	—
597	C₄⋊C₄	—	4	2	no	—
598	C₄⋊C₄	—	4	2	no	—
599	C₄⋊C₄	—	4	2	no	—
600	C₄⋊C₄	—	4	2	no	—
601	C₄⋊C₄	—	4	2	no	—
602	C₄⋊C₄	—	4	2	no	—
603	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
604	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
605	C₂²⋊C₄	—	2	4	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	2	no	—
608	C₂²⋊C₄	—	4	2	no	—
609	C₂²⋊C₄	—	4	2	no	—
610	C₄²	—	2	1	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	1	no	—
613	C₄²	—	2	1	no	—
614	C₄²	—	4	2	no	—
615	C₄²	—	4	2	no	—
616	C₄²	—	4	2	no	—
617	C₂³	—	2	1	no	—
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	—	2	1	no	—
623	C₂³	C₂₃×#121	1	2	yes	—
624	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
625	Q₈	—	2	2	no	—
626	Q₈	—	2	2	no	—
627	Q₈	—	2	2	no	—
628	Q₈	—	2	2	no	—
629	Q₈	—	2	2	no	—
630	Q₈	—	2	2	no	—
631	Q₈	—	4	8	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	4	1	no	—
639	Q₈	—	4	1	no	—
640	Q₈	—	4	1	no	—
641	Q₈	—	4	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	2	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	1	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	—	2	2	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	4	8	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	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	C₂₃×#121	1	2	yes	—
684	C₂×C₄	C₂₃×#137	1	2	yes	—
685	C₂×C₄	C₂₃×#137	1	2	yes	—
686	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
687	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
688	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
689	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
690	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
691	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
692	C₈	—	2	2	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₂²	—	2	2	no	—
697	C₂²	—	2	2	no	—
698	C₂²	—	2	2	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
704	C₂²	D₄²×C₂₃	1	1	yes	characteristic
705	C₂²	C₂₃×#546	1	1	yes	characteristic
706	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
707	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
708	C₂²	C₂₃×#578	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	2	no	—
715	C₄	—	4	2	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	—	4	1	no	—
720	C₄	—	8	1	no	—
721	C₄	—	8	1	no	—
722	C₄	—	8	1	no	—
723	C₄	C₂₃×#525	1	1	yes	characteristic
724	C₄	C₂₃×#538	1	1	yes	characteristic
725	C₄	D₄²×C₂₃	1	1	yes	characteristic
726	C₄	C₂₃×#546	1	1	yes	characteristic
727	C₂	—	4	2	no	—
728	C₂	C₂₃×#2424	1	1	yes	characteristic
729	C₂	C₂₃×#3039	1	1	yes	characteristic
730	C₂	C₂₃×#3044	1	1	yes	characteristic
731	C₂	C₂₃×#3270	1	1	yes	characteristic
732	C₂	C₂₃×#3276	1	1	yes	characteristic
733	C₂	C₂₃×#3312	1	1	yes	characteristic
734	C₂	C₂₃×#3328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×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
2. C₂×C₂₃×#462	C₂	semidirect
7. C₂₃×#2066	C₂	semidirect
8. C₂₃×#2056	C₂	semidirect
9. C₂₃×#2043	C₂	semidirect
10. C₂₃×#2016	C₂	semidirect
11. C₂₃×#1981	C₂	semidirect
12. C₂₃×#1927	C₂	semidirect
13. C₂₃×#1878	C₂	semidirect
249. #25095	C₂₃	direct (N × H)
330. C₂×#462	C₂×C₂₃	semidirect
335. #2066	C₂×C₂₃	semidirect
336. #2056	C₂×C₂₃	semidirect
337. #2043	C₂×C₂₃	semidirect
338. #2016	C₂×C₂₃	semidirect
339. #1981	C₂×C₂₃	semidirect
340. #1927	C₂×C₂₃	semidirect
341. #1878	C₂×C₂₃	semidirect
517. C₂₃	#25095	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×#470	2
C₂×C₂₃×#462	2
C₂×C₂₃×#451	2
C₂×C₂₃×#419	2
C₂₃×#2434	2
C₂₃×#2362	2
C₂₃×#2066	2
C₂₃×#2056	2
C₂₃×#2043	2
C₂₃×#2016	2
C₂₃×#1981	2
C₂₃×#1927	2
C₂₃×#1878	2
C₂₃×#1873	2
C₂₃×#1851	2
#25095	23

Subgroups — C23×#25095 #19172721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25095 #19172721