Subgroups — C₂₃×#60713 #19208339

Order 5888.

1878 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3510	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3183	C₂	1	4	yes	maximal, complemented
3	C₂₃×#3175	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3173	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3168	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3140	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3058	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#3057	C₂	1	2	yes	maximal, complemented
12	C₂₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#2949	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
21	C₂₃×#559	—	2	4	no	—
22	C₂₃×#556	—	2	4	no	—
23	C₂₃×#551	—	2	8	no	—
24	C₂₃×#542	C₂²	1	8	yes	complemented
25	C₂₃×#536	C₂²	1	2	yes	—
26	C₂₃×#535	C₂²	1	2	yes	—
27	C₂₃×#533	C₂²	1	2	yes	—
28	C₂₃×#529	C₂²	1	8	yes	complemented
29	C₂×C₂₃×#124	C₂²	1	2	yes	—
30	C₂₃×#519	C₂²	1	2	yes	—
31	C₂₃×#518	C₂²	1	2	yes	—
32	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#500	C₂²	1	2	yes	complemented
34	C₂₃×#499	C₂²	1	2	yes	complemented
35	C₂₃×#499	C₂²	1	2	yes	complemented
36	C₂₃×#498	C₂²	1	2	yes	complemented
37	C₂₃×#495	C₂²	1	2	yes	—
38	C₂₃×#495	C₂²	1	2	yes	—
39	C₂₃×#495	C₂²	1	2	yes	—
40	C₂₃×#495	C₂²	1	2	yes	—
41	C₂₃×#489	C₂²	1	8	yes	complemented
42	C₂₃×#488	C₂²	1	8	yes	complemented
43	C₂₃×#470	C₂²	1	4	yes	complemented
44	C₂₃×#469	C₂²	1	4	yes	—
45	C₂₃×#468	C₂²	1	8	yes	complemented
46	C₂₃×#467	C₂²	1	4	yes	complemented
47	C₂₃×#466	C₂²	1	4	yes	—
48	C₂₃×#465	C₂²	1	8	yes	complemented
49	C₂₃×#442	C₂²	1	4	yes	—
50	C₂₃×#441	C₂²	1	4	yes	complemented
51	C₂₃×#440	C₂²	1	8	yes	complemented
52	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
53	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
54	C₄×D₈×C₂₃	C₂²	1	4	yes	—
55	C₂₃×#427	C₂²	1	1	yes	characteristic
56	C₂₃×#427	C₂²	1	2	yes	—
57	C₂₃×#427	C₂²	1	1	yes	characteristic
58	C₂₃×#419	C₂²	1	2	yes	—
59	C₂₃×#419	C₂²	1	2	yes	—
60	C₂₃×#418	C₂²	1	2	yes	—
61	C₂₃×#418	C₂²	1	2	yes	—
62	C₂₃×#405	C₂²	1	1	yes	characteristic
63	C₂₃×#405	C₂²	1	1	yes	characteristic
64	C₂₃×#405	C₂²	1	2	yes	complemented
65	C₂₃×#405	C₂²	1	2	yes	complemented
66	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
67	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
68	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
69	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
70	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
72	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₂₃	—	2	4	no	—
74	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
76	C₂×SD₁₆×C₂₃	—	2	8	no	—
77	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
79	C₂×D₈×C₂₃	—	2	4	no	—
80	C₂×D₈×C₂₃	C₂³	1	2	yes	—
81	C₂×D₈×C₂₃	C₂³	1	2	yes	—
82	C₂₃×#131	—	2	4	no	—
83	C₂₃×#131	—	2	4	no	—
84	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
87	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
88	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
89	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
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	C₂³	1	4	yes	—
94	C₂₃×#127	—	2	2	no	—
95	C₂₃×#126	—	2	8	no	—
96	C₂₃×#126	—	2	8	no	—
97	C₂₃×#125	C₂³	1	4	yes	—
98	C₂₃×#124	—	2	2	no	—
99	C₂₃×#124	—	2	2	no	—
100	C₂₃×#124	—	2	8	no	—
101	C₂₃×#124	—	2	4	no	—
102	C₂₃×#124	C₂³	1	4	yes	—
103	C₂₃×#124	C₂³	1	4	yes	—
104	C₂₃×#123	—	2	4	no	—
105	C₂₃×#123	—	2	8	no	—
106	C₂₃×#123	—	2	8	no	—
107	C₂₃×C₂²⋊Q₈	—	2	4	no	—
108	C₂₃×C₂²⋊Q₈	—	2	8	no	—
109	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
110	C₂₃×#121	C₂³	1	8	yes	—
111	C₂₃×#120	—	2	4	no	—
112	C₄×Q₈×C₂₃	—	2	4	no	—
113	C₄×Q₈×C₂₃	—	2	4	no	—
114	C₄×Q₈×C₂₃	D₄	1	2	yes	—
115	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
116	C₄×D₄×C₂₃	—	2	8	no	—
117	C₄×D₄×C₂₃	—	2	4	no	—
118	C₄×D₄×C₂₃	D₄	1	4	yes	—
119	C₄×D₄×C₂₃	D₄	1	2	yes	—
120	C₄×D₄×C₂₃	C₂³	1	8	yes	—
121	C₂₃×#117	—	2	4	no	—
122	C₂₃×#117	—	2	8	no	—
123	C₂₃×#117	C₂³	1	2	yes	—
124	C₂₃×#117	C₂³	1	2	yes	—
125	C₂₃×#117	C₂³	1	1	yes	characteristic
126	C₂₃×#117	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
128	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
129	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
131	C₂₃×#107	C₂³	1	2	yes	—
132	C₂₃×#107	C₂³	1	2	yes	—
133	C₂₃×#107	C₂³	1	4	yes	—
134	C₂₃×#106	C₂³	1	4	yes	—
135	C₂₃×#106	C₂³	1	2	yes	—
136	C₂₃×#106	C₂³	1	2	yes	—
137	C₂₃×#103	C₂³	1	8	yes	—
138	C₂₃×#103	C₂³	1	8	yes	—
139	C₂₃×#102	C₂³	1	8	yes	—
140	C₂₃×#102	C₂³	1	8	yes	—
141	C₂₃×#97	C₂³	1	2	yes	—
142	C₂₃×#97	C₂³	1	2	yes	—
143	C₂₃×#97	C₂³	1	2	yes	—
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₂₃	C₂³	1	2	yes	—
149	C₄×C₈×C₂₃	C₂³	1	2	yes	—
150	C₄×C₈×C₂₃	C₂³	1	2	yes	—
151	C₂⁴×C₂₃	—	2	2	no	—
152	#41×C₂₃	—	2	2	no	—
153	#41×C₂₃	—	2	2	no	—
154	C₂×Q₈×C₂₃	—	2	2	no	—
155	C₂×Q₈×C₂₃	—	2	4	no	—
156	C₂×Q₈×C₂₃	—	2	4	no	—
157	C₂×Q₈×C₂₃	—	2	4	no	—
158	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
160	C₂×D₄×C₂₃	—	2	8	no	—
161	C₂×D₄×C₂₃	—	2	2	no	—
162	C₂×D₄×C₂₃	—	2	4	no	—
163	C₂×D₄×C₂₃	—	2	4	no	—
164	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₂₃	—	2	2	no	—
168	C₂²×C₄×C₂₃	—	2	8	no	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	Q₁₆×C₂₃	—	2	4	no	—
174	Q₁₆×C₂₃	—	2	4	no	—
175	SD₁₆×C₂₃	—	2	8	no	—
176	SD₁₆×C₂₃	—	2	8	no	—
177	D₈×C₂₃	—	2	4	no	—
178	D₈×C₂₃	—	2	4	no	—
179	M₁₆×C₂₃	—	2	2	no	—
180	M₁₆×C₂₃	—	2	2	no	—
181	M₁₆×C₂₃	—	2	4	no	—
182	M₁₆×C₂₃	—	2	4	no	—
183	C₂×C₈×C₂₃	—	2	4	no	—
184	C₂×C₈×C₂₃	—	2	4	no	—
185	C₂×C₈×C₂₃	#41	1	4	yes	—
186	C₂×C₈×C₂₃	#41	1	4	yes	—
187	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
188	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
189	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
190	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
191	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
192	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	8	no	—
195	C₄⋊C₄×C₂₃	—	2	8	no	—
196	C₄⋊C₄×C₂₃	—	2	4	no	—
197	C₄⋊C₄×C₂₃	—	2	4	no	—
198	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
200	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	2	no	—
204	C₂²⋊C₄×C₂₃	—	2	8	no	—
205	C₂²⋊C₄×C₂₃	—	2	8	no	—
206	C₂²⋊C₄×C₂₃	—	2	8	no	—
207	C₂²⋊C₄×C₂₃	—	2	8	no	—
208	C₂²⋊C₄×C₂₃	—	2	8	no	—
209	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
210	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
211	C₄²×C₂₃	—	2	8	no	—
212	C₄²×C₂₃	—	2	2	no	—
213	C₄²×C₂₃	—	2	2	no	—
214	C₄²×C₂₃	C₂×D₄	1	4	yes	—
215	C₄²×C₂₃	C₂×D₄	1	2	yes	—
216	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
217	C₄²×C₂₃	C₂⁴	1	2	yes	—
218	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
219	#60713	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₂₃	—	2	4	no	—
221	C₂³×C₂₃	—	4	4	no	—
222	C₂³×C₂₃	—	8	4	no	—
223	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
224	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₂₃	—	2	2	no	—
226	Q₈×C₂₃	—	2	4	no	—
227	Q₈×C₂₃	—	2	4	no	—
228	Q₈×C₂₃	—	4	4	no	—
229	Q₈×C₂₃	—	8	4	no	—
230	D₄×C₂₃	—	2	4	no	—
231	D₄×C₂₃	—	2	2	no	—
232	D₄×C₂₃	—	2	4	no	—
233	D₄×C₂₃	—	2	4	no	—
234	D₄×C₂₃	—	4	4	no	—
235	D₄×C₂₃	—	8	8	no	—
236	C₂×C₄×C₂₃	—	2	2	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	4	no	—
239	C₂×C₄×C₂₃	—	2	4	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	4	4	no	—
242	C₂×C₄×C₂₃	—	4	2	no	—
243	C₂×C₄×C₂₃	—	4	4	no	—
244	C₂×C₄×C₂₃	—	8	8	no	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
246	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
251	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
252	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
253	C₈×C₂₃	—	2	2	no	—
254	C₈×C₂₃	—	2	2	no	—
255	C₈×C₂₃	C₂×#41	1	2	yes	—
256	C₈×C₂₃	C₂×#41	1	2	yes	—
257	C₈×C₂₃	C₂×#41	1	2	yes	—
258	C₈×C₂₃	C₂×#41	1	2	yes	—
259	#3510	C₂×C₂₃	1	2	yes	complemented
260	#3183	C₂×C₂₃	1	4	yes	complemented
261	#3175	C₂×C₂₃	1	4	yes	complemented
262	#3173	C₂×C₂₃	1	4	yes	complemented
263	#3168	C₂×C₂₃	1	4	yes	complemented
264	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
265	C₂×#495	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#3140	C₂×C₂₃	1	2	yes	complemented
267	#3059	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#3058	C₂×C₂₃	1	1	yes	characteristic
269	#3057	C₂×C₂₃	1	2	yes	complemented
270	#2950	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#2949	C₂×C₂₃	1	2	yes	complemented
272	#2948	C₂×C₂₃	1	1	yes	characteristic, complemented
273	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
274	C₂²×C₂₃	—	2	2	no	—
275	C₂²×C₂₃	—	2	4	no	—
276	C₂²×C₂₃	—	2	2	no	—
277	C₂²×C₂₃	—	4	8	no	—
278	C₂²×C₂₃	—	4	4	no	—
279	C₂²×C₂₃	—	16	4	no	—
280	C₂²×C₂₃	#532	1	2	yes	—
281	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	2	4	no	—
284	C₄×C₂₃	—	2	2	no	—
285	C₄×C₂₃	—	2	2	no	—
286	C₄×C₂₃	—	4	4	no	—
287	C₄×C₂₃	—	4	4	no	—
288	C₄×C₂₃	—	4	4	no	—
289	C₄×C₂₃	#532	1	2	yes	—
290	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
291	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
292	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
293	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
294	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
297	#559	—	2	4	no	—
298	#556	—	2	4	no	—
299	#551	—	2	8	no	—
300	#542	C₂²×C₂₃	1	8	yes	complemented
301	#536	C₂²×C₂₃	1	2	yes	—
302	#535	C₂²×C₂₃	1	2	yes	—
303	#533	C₂²×C₂₃	1	2	yes	—
304	#529	C₂²×C₂₃	1	8	yes	complemented
305	C₂×#124	C₂²×C₂₃	1	2	yes	—
306	#519	C₂²×C₂₃	1	2	yes	—
307	#518	C₂²×C₂₃	1	2	yes	—
308	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
309	#500	C₂²×C₂₃	1	2	yes	complemented
310	#499	C₂²×C₂₃	1	2	yes	complemented
311	#499	C₂²×C₂₃	1	2	yes	complemented
312	#498	C₂²×C₂₃	1	2	yes	complemented
313	#495	C₂²×C₂₃	1	2	yes	—
314	#495	C₂²×C₂₃	1	2	yes	—
315	#495	C₂²×C₂₃	1	2	yes	—
316	#495	C₂²×C₂₃	1	2	yes	—
317	#489	C₂²×C₂₃	1	8	yes	complemented
318	#488	C₂²×C₂₃	1	8	yes	complemented
319	#470	C₂²×C₂₃	1	4	yes	complemented
320	#469	C₂²×C₂₃	1	4	yes	—
321	#468	C₂²×C₂₃	1	8	yes	complemented
322	#467	C₂²×C₂₃	1	4	yes	complemented
323	#466	C₂²×C₂₃	1	4	yes	—
324	#465	C₂²×C₂₃	1	8	yes	complemented
325	#442	C₂²×C₂₃	1	4	yes	—
326	#441	C₂²×C₂₃	1	4	yes	complemented
327	#440	C₂²×C₂₃	1	8	yes	complemented
328	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
329	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
330	C₄×D₈	C₂²×C₂₃	1	4	yes	—
331	#427	C₂²×C₂₃	1	1	yes	characteristic
332	#427	C₂²×C₂₃	1	2	yes	—
333	#427	C₂²×C₂₃	1	1	yes	characteristic
334	#419	C₂²×C₂₃	1	2	yes	—
335	#419	C₂²×C₂₃	1	2	yes	—
336	#418	C₂²×C₂₃	1	2	yes	—
337	#418	C₂²×C₂₃	1	2	yes	—
338	#405	C₂²×C₂₃	1	1	yes	characteristic
339	#405	C₂²×C₂₃	1	1	yes	characteristic
340	#405	C₂²×C₂₃	1	2	yes	complemented
341	#405	C₂²×C₂₃	1	2	yes	complemented
342	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
343	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
344	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
346	C₂×C₂₃	—	2	2	no	—
347	C₂×C₂₃	—	4	2	no	—
348	C₂×C₂₃	—	8	4	no	—
349	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
350	C₂×C₂₃	#3584	1	2	yes	—
351	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
352	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
353	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
354	C₂×Q₁₆	—	2	4	no	—
355	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
356	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
357	C₂×SD₁₆	—	2	8	no	—
358	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
359	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
360	C₂×D₈	—	2	4	no	—
361	C₂×D₈	C₂³×C₂₃	1	2	yes	—
362	C₂×D₈	C₂³×C₂₃	1	2	yes	—
363	#131	—	2	4	no	—
364	#131	—	2	4	no	—
365	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
367	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
368	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
369	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
370	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
371	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
373	C₄⋊Q₈	—	2	2	no	—
374	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
375	#127	—	2	2	no	—
376	#126	—	2	8	no	—
377	#126	—	2	8	no	—
378	#125	C₂³×C₂₃	1	4	yes	—
379	#124	—	2	2	no	—
380	#124	—	2	2	no	—
381	#124	—	2	8	no	—
382	#124	—	2	4	no	—
383	#124	C₂³×C₂₃	1	4	yes	—
384	#124	C₂³×C₂₃	1	4	yes	—
385	#123	—	2	4	no	—
386	#123	—	2	8	no	—
387	#123	—	2	8	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	C₂²⋊Q₈	—	2	8	no	—
390	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
391	#121	C₂³×C₂₃	1	8	yes	—
392	#120	—	2	4	no	—
393	C₄×Q₈	—	2	4	no	—
394	C₄×Q₈	—	2	4	no	—
395	C₄×Q₈	D₄×C₂₃	1	2	yes	—
396	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
397	C₄×D₄	—	2	8	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	D₄×C₂₃	1	4	yes	—
400	C₄×D₄	D₄×C₂₃	1	2	yes	—
401	C₄×D₄	C₂³×C₂₃	1	8	yes	—
402	#117	—	2	4	no	—
403	#117	—	2	8	no	—
404	#117	C₂³×C₂₃	1	2	yes	—
405	#117	C₂³×C₂₃	1	2	yes	—
406	#117	C₂³×C₂₃	1	1	yes	characteristic
407	#117	C₂³×C₂₃	1	2	yes	—
408	C₂×C₂²⋊C₄	—	2	4	no	—
409	C₂×C₂²⋊C₄	—	2	4	no	—
410	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
412	#107	C₂³×C₂₃	1	2	yes	—
413	#107	C₂³×C₂₃	1	2	yes	—
414	#107	C₂³×C₂₃	1	4	yes	—
415	#106	C₂³×C₂₃	1	4	yes	—
416	#106	C₂³×C₂₃	1	2	yes	—
417	#106	C₂³×C₂₃	1	2	yes	—
418	#103	C₂³×C₂₃	1	8	yes	—
419	#103	C₂³×C₂₃	1	8	yes	—
420	#102	C₂³×C₂₃	1	8	yes	—
421	#102	C₂³×C₂₃	1	8	yes	—
422	#97	C₂³×C₂₃	1	2	yes	—
423	#97	C₂³×C₂₃	1	2	yes	—
424	#97	C₂³×C₂₃	1	2	yes	—
425	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
426	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
428	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₂₃	1	2	yes	—
430	C₄×C₈	C₂³×C₂₃	1	2	yes	—
431	C₄×C₈	C₂³×C₂₃	1	2	yes	—
432	C₂₃	#60713	1	1	yes	characteristic, direct product factor, Sylow
433	C₂⁴	—	2	2	no	—
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
447	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
448	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	8	no	—
451	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
453	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
455	Q₁₆	—	2	4	no	—
456	Q₁₆	—	2	4	no	—
457	SD₁₆	—	2	8	no	—
458	SD₁₆	—	2	8	no	—
459	D₈	—	2	4	no	—
460	D₈	—	2	4	no	—
461	M₁₆	—	2	2	no	—
462	M₁₆	—	2	2	no	—
463	M₁₆	—	2	4	no	—
464	M₁₆	—	2	4	no	—
465	C₂×C₈	—	2	4	no	—
466	C₂×C₈	—	2	4	no	—
467	C₂×C₈	#41×C₂₃	1	4	yes	—
468	C₂×C₈	#41×C₂₃	1	4	yes	—
469	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
470	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
471	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
472	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
473	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
474	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	8	no	—
477	C₄⋊C₄	—	2	8	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
481	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
482	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
483	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
492	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
493	C₄²	—	2	8	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	2	no	—
496	C₄²	C₂×D₄×C₂₃	1	4	yes	—
497	C₄²	C₂×D₄×C₂₃	1	2	yes	—
498	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
499	C₄²	C₂⁴×C₂₃	1	2	yes	—
500	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
501	C₂³	—	2	4	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	8	4	no	—
504	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
505	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
506	Q₈	—	2	2	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	2	4	no	—
509	Q₈	—	4	4	no	—
510	Q₈	—	8	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	2	no	—
515	D₄	—	4	4	no	—
516	D₄	—	8	8	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	8	8	no	—
526	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
527	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
530	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
532	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
533	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₈	C₂×#41×C₂₃	1	2	yes	—
537	C₈	C₂×#41×C₂₃	1	2	yes	—
538	C₈	C₂×#41×C₂₃	1	2	yes	—
539	C₈	C₂×#41×C₂₃	1	2	yes	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	8	no	—
545	C₂²	—	16	4	no	—
546	C₂²	C₂₃×#532	1	2	yes	—
547	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	4	no	—
551	C₄	—	2	2	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	C₂₃×#532	1	2	yes	—
556	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
557	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
558	C₂	—	2	2	no	—
559	C₂	—	4	2	no	—
560	C₂	—	8	4	no	—
561	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
562	C₂	C₂₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3058	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3510	C₂	semidirect
2. C₂₃×#3183	C₂	semidirect
3. C₂₃×#3175	C₂	semidirect
4. C₂₃×#3173	C₂	semidirect
5. C₂₃×#3168	C₂	semidirect
6. C₂₃×#3160	C₂	semidirect
7. C₂×C₂₃×#495	C₂	semidirect
8. C₂₃×#3140	C₂	semidirect
9. C₂₃×#3059	C₂	semidirect
11. C₂₃×#3057	C₂	semidirect
12. C₂₃×#2950	C₂	semidirect
13. C₂₃×#2949	C₂	semidirect
14. C₂₃×#2948	C₂	semidirect
15. C₄×C₂₃×#131	C₂	semidirect
24. C₂₃×#542	C₂²	semidirect
28. C₂₃×#529	C₂²	semidirect
33. C₂₃×#500	C₂²	semidirect
34. C₂₃×#499	C₂²	semidirect
35. C₂₃×#499	C₂²	semidirect
36. C₂₃×#498	C₂²	semidirect
41. C₂₃×#489	C₂²	semidirect
42. C₂₃×#488	C₂²	semidirect
43. C₂₃×#470	C₂²	semidirect
45. C₂₃×#468	C₂²	semidirect
46. C₂₃×#467	C₂²	semidirect
48. C₂₃×#465	C₂²	semidirect
50. C₂₃×#441	C₂²	semidirect
51. C₂₃×#440	C₂²	semidirect
52. C₄×Q₁₆×C₂₃	C₂²	semidirect
53. C₄×SD₁₆×C₂₃	C₂²	semidirect
64. C₂₃×#405	C₂²	semidirect
65. C₂₃×#405	C₂²	semidirect
67. C₄×M₁₆×C₂₃	C₂²	semidirect
69. C₂×C₄×C₈×C₂₃	C₂²	semidirect
219. #60713	C₂₃	direct (N × H)
259. #3510	C₂×C₂₃	semidirect
260. #3183	C₂×C₂₃	semidirect
261. #3175	C₂×C₂₃	semidirect
262. #3173	C₂×C₂₃	semidirect
263. #3168	C₂×C₂₃	semidirect
264. #3160	C₂×C₂₃	semidirect
265. C₂×#495	C₂×C₂₃	semidirect
266. #3140	C₂×C₂₃	semidirect
267. #3059	C₂×C₂₃	semidirect
269. #3057	C₂×C₂₃	semidirect
270. #2950	C₂×C₂₃	semidirect
271. #2949	C₂×C₂₃	semidirect
272. #2948	C₂×C₂₃	semidirect
273. C₄×#131	C₂×C₂₃	semidirect
300. #542	C₂²×C₂₃	semidirect
304. #529	C₂²×C₂₃	semidirect
309. #500	C₂²×C₂₃	semidirect
310. #499	C₂²×C₂₃	semidirect
311. #499	C₂²×C₂₃	semidirect
312. #498	C₂²×C₂₃	semidirect
317. #489	C₂²×C₂₃	semidirect
318. #488	C₂²×C₂₃	semidirect
319. #470	C₂²×C₂₃	semidirect
321. #468	C₂²×C₂₃	semidirect
322. #467	C₂²×C₂₃	semidirect
324. #465	C₂²×C₂₃	semidirect
326. #441	C₂²×C₂₃	semidirect
327. #440	C₂²×C₂₃	semidirect
328. C₄×Q₁₆	C₂²×C₂₃	semidirect
329. C₄×SD₁₆	C₂²×C₂₃	semidirect
340. #405	C₂²×C₂₃	semidirect
341. #405	C₂²×C₂₃	semidirect
343. C₄×M₁₆	C₂²×C₂₃	semidirect
345. C₂×C₄×C₈	C₂²×C₂₃	semidirect
432. C₂₃	#60713	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3510	2
C₂₃×#3183	2
C₂₃×#3175	2
C₂₃×#3173	2
C₂₃×#3168	2
C₂₃×#3160	2
C₂×C₂₃×#495	2
C₂₃×#3140	2
C₂₃×#3059	2
C₂₃×#3058	2
C₂₃×#3057	2
C₂₃×#2950	2
C₂₃×#2949	2
C₂₃×#2948	2
C₄×C₂₃×#131	2
#60713	23

Subgroups — C23×#60713 #19208339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60713 #19208339