Subgroups — C₂₃×#20746 #19168372

Order 5888.

1298 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#520	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2314	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1824	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1823	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1788	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1788	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1763	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#520	—	2	4	no	—
18	C₂₃×#520	C₄	1	8	yes	complemented
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	—	2	1	no	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#117	—	2	1	no	—
26	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#433	—	2	4	no	—
30	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	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×#103	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#103	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×#102	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#388	—	2	1	no	—
45	C₂₃×#388	—	2	1	no	—
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#387	—	2	1	no	—
48	C₂₃×#384	C₂²	1	1	yes	characteristic
49	C₂₃×#382	C₂²	1	1	yes	characteristic
50	C₂₃×#380	C₂²	1	1	yes	characteristic
51	C₂₃×#380	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₄×C₄⋊C₄×C₂₃	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×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#340	C₂²	1	1	yes	characteristic
61	C₂₃×#340	C₂²	1	1	yes	characteristic
62	C₂×#41×C₂₃	—	2	2	no	—
63	C₂×#41×C₂₃	—	2	2	no	—
64	C₂×#41×C₂₃	—	2	2	no	—
65	C₂×#41×C₂₃	—	2	2	no	—
66	C₂×#41×C₂₃	—	2	2	no	—
67	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
68	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₂₃	—	2	1	no	—
70	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂³×C₄×C₂₃	—	2	1	no	—
73	C₂³×C₄×C₂₃	—	2	1	no	—
74	C₂³×C₄×C₂₃	—	2	1	no	—
75	C₂³×C₄×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	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
83	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
84	C₄×D₄×C₂₃	—	2	8	no	—
85	C₄×D₄×C₂₃	—	2	8	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
89	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
90	C₂₃×#117	—	2	8	no	—
91	C₂₃×#117	C₂×C₄	1	4	yes	—
92	C₂₃×#117	D₄	1	4	yes	—
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	—	2	2	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
101	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	C₂³	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×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	—	2	1	no	—
116	C₂×C₄²×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	—	2	1	no	—
118	C₂×C₄²×C₂₃	—	2	1	no	—
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	—	2	2	no	—
122	C₂₃×C₄⋊C₈	—	2	2	no	—
123	C₂₃×#103	—	2	2	no	—
124	C₂₃×#103	—	2	2	no	—
125	C₂₃×#103	—	2	2	no	—
126	C₂₃×#103	—	2	2	no	—
127	C₂₃×#102	—	2	2	no	—
128	C₂₃×#102	—	2	2	no	—
129	C₂₃×#102	—	2	2	no	—
130	C₂₃×#102	—	2	2	no	—
131	C₂₃×C₂²⋊C₈	—	2	2	no	—
132	C₂₃×C₂²⋊C₈	—	2	2	no	—
133	C₂₃×#97	—	2	2	no	—
134	C₄×C₈×C₂₃	—	2	2	no	—
135	C₂₃×#95	—	2	1	no	—
136	C₂₃×#95	—	2	1	no	—
137	C₂₃×#95	—	2	1	no	—
138	C₂₃×#95	—	2	1	no	—
139	C₂₃×#95	—	2	1	no	—
140	C₂₃×#95	—	2	1	no	—
141	C₂⁴×C₂₃	—	2	1	no	—
142	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
143	#41×C₂₃	—	2	8	no	—
144	#41×C₂₃	—	2	8	no	—
145	#41×C₂₃	—	2	8	no	—
146	#41×C₂₃	—	2	8	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	C₂²×C₄	1	2	yes	—
151	C₂×Q₈×C₂₃	#41	1	2	yes	—
152	C₂×Q₈×C₂₃	#41	1	2	yes	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	2	4	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
160	C₂×D₄×C₂₃	#41	1	2	yes	—
161	C₂×D₄×C₂₃	#41	1	2	yes	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	4	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	2	1	no	—
172	C₂²×C₄×C₂₃	—	2	4	no	—
173	C₂²×C₄×C₂₃	—	2	4	no	—
174	C₂²×C₄×C₂₃	—	2	4	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	—	2	1	no	—
179	C₂²×C₄×C₂₃	—	8	1	no	—
180	C₂²×C₄×C₂₃	—	8	1	no	—
181	C₂²×C₄×C₂₃	—	8	1	no	—
182	C₂²×C₄×C₂₃	—	8	1	no	—
183	C₂²×C₄×C₂₃	—	8	1	no	—
184	C₂²×C₄×C₂₃	—	8	1	no	—
185	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
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	2	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	—	2	1	no	—
195	C₂×C₈×C₂₃	—	2	1	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	C₄⋊C₄×C₂₃	—	2	4	no	—
206	C₄⋊C₄×C₂₃	—	2	4	no	—
207	C₄⋊C₄×C₂₃	—	2	4	no	—
208	C₄⋊C₄×C₂₃	—	2	1	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	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	8	1	no	—
219	C₄⋊C₄×C₂₃	—	8	1	no	—
220	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
221	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
222	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
223	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
224	C₂²⋊C₄×C₂₃	—	2	4	no	—
225	C₂²⋊C₄×C₂₃	—	2	4	no	—
226	C₂²⋊C₄×C₂₃	—	8	1	no	—
227	C₂²⋊C₄×C₂₃	—	8	1	no	—
228	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
229	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
230	C₄²×C₂₃	—	2	4	no	—
231	C₄²×C₂₃	—	2	4	no	—
232	C₄²×C₂₃	—	8	1	no	—
233	C₄²×C₂₃	—	8	1	no	—
234	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
235	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
236	C₄²×C₂₃	C₂×D₄	1	2	yes	—
237	C₄²×C₂₃	C₂×D₄	1	2	yes	—
238	#20746	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂³×C₂₃	—	2	2	no	—
240	C₂³×C₂₃	—	2	1	no	—
241	C₂³×C₂₃	—	2	1	no	—
242	C₂³×C₂₃	—	2	2	no	—
243	C₂³×C₂₃	—	2	2	no	—
244	C₂³×C₂₃	—	2	1	no	—
245	C₂³×C₂₃	—	2	1	no	—
246	C₂³×C₂₃	—	2	1	no	—
247	C₂³×C₂₃	—	2	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	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
254	Q₈×C₂₃	—	2	2	no	—
255	Q₈×C₂₃	—	2	2	no	—
256	Q₈×C₂₃	—	2	2	no	—
257	Q₈×C₂₃	#117	1	4	yes	—
258	D₄×C₂₃	—	2	16	no	—
259	D₄×C₂₃	—	2	2	no	—
260	D₄×C₂₃	—	2	2	no	—
261	D₄×C₂₃	—	2	2	no	—
262	D₄×C₂₃	#117	1	4	yes	—
263	C₂×C₄×C₂₃	—	2	16	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	2	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	1	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	—	2	1	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	2	2	no	—
281	C₂×C₄×C₂₃	—	2	2	no	—
282	C₂×C₄×C₂₃	—	2	2	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	—	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	—	8	4	no	—
300	C₂×C₄×C₂₃	—	8	4	no	—
301	C₂×C₄×C₂₃	—	8	4	no	—
302	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
303	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
304	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
305	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
306	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
307	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
308	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
310	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
311	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
312	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
313	C₈×C₂₃	—	4	1	no	—
314	C₈×C₂₃	—	4	1	no	—
315	C₈×C₂₃	—	4	1	no	—
316	C₈×C₂₃	—	4	1	no	—
317	C₂×#520	C₂×C₂₃	1	1	yes	characteristic
318	C₂×#433	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#2314	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1824	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1823	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1805	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1804	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1788	C₂×C₂₃	1	1	yes	characteristic
327	#1788	C₂×C₂₃	1	1	yes	characteristic
328	#1763	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#1762	C₂×C₂₃	1	1	yes	characteristic, complemented
330	C₄×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
331	C₄×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
332	C₂²×C₂₃	—	2	2	no	—
333	C₂²×C₂₃	—	2	2	no	—
334	C₂²×C₂₃	—	2	2	no	—
335	C₂²×C₂₃	—	2	4	no	—
336	C₂²×C₂₃	—	4	1	no	—
337	C₂²×C₂₃	—	4	2	no	—
338	C₂²×C₂₃	—	4	1	no	—
339	C₂²×C₂₃	—	4	2	no	—
340	C₂²×C₂₃	—	4	1	no	—
341	C₂²×C₂₃	—	4	2	no	—
342	C₂²×C₂₃	—	4	1	no	—
343	C₂²×C₂₃	—	4	2	no	—
344	C₂²×C₂₃	—	4	2	no	—
345	C₂²×C₂₃	—	4	2	no	—
346	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
347	C₂²×C₂₃	#525	1	1	yes	characteristic
348	C₂²×C₂₃	#525	1	1	yes	characteristic
349	C₂²×C₂₃	#576	1	1	yes	characteristic
350	C₂²×C₂₃	#577	1	1	yes	characteristic
351	C₂²×C₂₃	#577	1	1	yes	characteristic
352	C₂²×C₂₃	#578	1	1	yes	characteristic
353	C₄×C₂₃	—	2	2	no	—
354	C₄×C₂₃	—	2	2	no	—
355	C₄×C₂₃	—	2	2	no	—
356	C₄×C₂₃	—	2	2	no	—
357	C₄×C₂₃	—	2	2	no	—
358	C₄×C₂₃	—	2	4	no	—
359	C₄×C₂₃	—	4	2	no	—
360	C₄×C₂₃	—	4	2	no	—
361	C₄×C₂₃	—	8	1	no	—
362	C₄×C₂₃	—	8	1	no	—
363	C₄×C₂₃	—	8	1	no	—
364	C₄×C₂₃	—	8	1	no	—
365	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
366	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
367	C₄×C₂₃	#525	1	1	yes	characteristic
368	C₄×C₂₃	#525	1	1	yes	characteristic
369	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
370	#520	—	2	4	no	—
371	#520	C₄×C₂₃	1	8	yes	complemented
372	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₄×D₄	—	2	1	no	—
375	C₂×C₄×D₄	—	2	1	no	—
376	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#117	—	2	1	no	—
379	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
380	C₂²×C₄⋊C₄	—	2	1	no	—
381	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
382	#433	—	2	4	no	—
383	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
389	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
392	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
394	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
397	#388	—	2	1	no	—
398	#388	—	2	1	no	—
399	#387	—	2	1	no	—
400	#387	—	2	1	no	—
401	#384	C₂²×C₂₃	1	1	yes	characteristic
402	#382	C₂²×C₂₃	1	1	yes	characteristic
403	#380	C₂²×C₂₃	1	1	yes	characteristic
404	#380	C₂²×C₂₃	1	1	yes	characteristic
405	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
406	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
407	#340	C₂²×C₂₃	1	1	yes	characteristic
408	#340	C₂²×C₂₃	1	1	yes	characteristic
409	#340	C₂²×C₂₃	1	1	yes	characteristic
410	#340	C₂²×C₂₃	1	1	yes	characteristic
411	#340	C₂²×C₂₃	1	1	yes	characteristic
412	#340	C₂²×C₂₃	1	1	yes	characteristic
413	#340	C₂²×C₂₃	1	1	yes	characteristic
414	#340	C₂²×C₂₃	1	1	yes	characteristic
415	C₂×C₂₃	—	4	2	no	—
416	C₂×C₂₃	—	4	4	no	—
417	C₂×C₂₃	#2280	1	1	yes	characteristic
418	C₂×C₂₃	#2897	1	1	yes	characteristic
419	C₂×C₂₃	#2898	1	1	yes	characteristic
420	C₂×C₂₃	#3119	1	1	yes	characteristic
421	C₂×C₂₃	#3120	1	1	yes	characteristic
422	C₂×C₂₃	#3124	1	1	yes	characteristic
423	C₂×C₂₃	#3125	1	1	yes	characteristic
424	C₂×#41	—	2	2	no	—
425	C₂×#41	—	2	2	no	—
426	C₂×#41	—	2	2	no	—
427	C₂×#41	—	2	2	no	—
428	C₂×#41	—	2	2	no	—
429	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
430	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₂²×D₄	—	2	1	no	—
432	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
433	C₂³×C₄	—	2	1	no	—
434	C₂³×C₄	—	2	1	no	—
435	C₂³×C₄	—	2	1	no	—
436	C₂³×C₄	—	2	1	no	—
437	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
438	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
439	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
440	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
441	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
442	C₄×Q₈	—	2	2	no	—
443	C₄×Q₈	—	2	2	no	—
444	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
445	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
446	C₄×D₄	—	2	8	no	—
447	C₄×D₄	—	2	8	no	—
448	C₄×D₄	—	2	2	no	—
449	C₄×D₄	—	2	2	no	—
450	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
451	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
452	#117	—	2	8	no	—
453	#117	C₂×C₄×C₂₃	1	4	yes	—
454	#117	D₄×C₂₃	1	4	yes	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	—	2	1	no	—
458	C₂×C₄⋊C₄	—	2	2	no	—
459	C₂×C₄⋊C₄	—	2	8	no	—
460	C₂×C₄⋊C₄	—	2	1	no	—
461	C₂×C₄⋊C₄	—	2	1	no	—
462	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
463	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
464	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
466	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
467	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
470	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
471	C₂×C₂²⋊C₄	—	2	1	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	—	2	1	no	—
474	C₂×C₂²⋊C₄	—	2	1	no	—
475	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
476	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
477	C₂×C₄²	—	2	1	no	—
478	C₂×C₄²	—	2	1	no	—
479	C₂×C₄²	—	2	1	no	—
480	C₂×C₄²	—	2	1	no	—
481	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
482	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
483	C₄⋊C₈	—	2	2	no	—
484	C₄⋊C₈	—	2	2	no	—
485	#103	—	2	2	no	—
486	#103	—	2	2	no	—
487	#103	—	2	2	no	—
488	#103	—	2	2	no	—
489	#102	—	2	2	no	—
490	#102	—	2	2	no	—
491	#102	—	2	2	no	—
492	#102	—	2	2	no	—
493	C₂²⋊C₈	—	2	2	no	—
494	C₂²⋊C₈	—	2	2	no	—
495	#97	—	2	2	no	—
496	C₄×C₈	—	2	2	no	—
497	#95	—	2	1	no	—
498	#95	—	2	1	no	—
499	#95	—	2	1	no	—
500	#95	—	2	1	no	—
501	#95	—	2	1	no	—
502	#95	—	2	1	no	—
503	C₂₃	#20746	1	1	yes	characteristic, direct product factor, Sylow
504	C₂⁴	—	2	1	no	—
505	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
506	#41	—	2	8	no	—
507	#41	—	2	8	no	—
508	#41	—	2	8	no	—
509	#41	—	2	8	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
514	C₂×Q₈	#41×C₂₃	1	2	yes	—
515	C₂×Q₈	#41×C₂₃	1	2	yes	—
516	C₂×D₄	—	2	4	no	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	1	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
523	C₂×D₄	#41×C₂₃	1	2	yes	—
524	C₂×D₄	#41×C₂₃	1	2	yes	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	4	no	—
536	C₂²×C₄	—	2	4	no	—
537	C₂²×C₄	—	2	4	no	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	8	1	no	—
543	C₂²×C₄	—	8	1	no	—
544	C₂²×C₄	—	8	1	no	—
545	C₂²×C₄	—	8	1	no	—
546	C₂²×C₄	—	8	1	no	—
547	C₂²×C₄	—	8	1	no	—
548	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
549	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
550	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
551	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
552	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
553	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
554	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
555	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	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	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	4	no	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	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	—	8	1	no	—
582	C₄⋊C₄	—	8	1	no	—
583	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
584	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
585	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
586	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	—	2	4	no	—
589	C₂²⋊C₄	—	8	1	no	—
590	C₂²⋊C₄	—	8	1	no	—
591	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
592	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
593	C₄²	—	2	4	no	—
594	C₄²	—	2	4	no	—
595	C₄²	—	8	1	no	—
596	C₄²	—	8	1	no	—
597	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
598	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
599	C₄²	C₂×D₄×C₂₃	1	2	yes	—
600	C₄²	C₂×D₄×C₂₃	1	2	yes	—
601	C₂³	—	2	1	no	—
602	C₂³	—	2	1	no	—
603	C₂³	—	2	1	no	—
604	C₂³	—	2	1	no	—
605	C₂³	—	2	2	no	—
606	C₂³	—	2	1	no	—
607	C₂³	—	2	1	no	—
608	C₂³	—	2	2	no	—
609	C₂³	—	2	2	no	—
610	C₂³	—	4	1	no	—
611	C₂³	—	4	1	no	—
612	C₂³	—	4	1	no	—
613	C₂³	—	4	1	no	—
614	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
615	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
616	Q₈	—	2	2	no	—
617	Q₈	—	2	2	no	—
618	Q₈	—	2	2	no	—
619	Q₈	C₂₃×#117	1	4	yes	—
620	D₄	—	2	16	no	—
621	D₄	—	2	2	no	—
622	D₄	—	2	2	no	—
623	D₄	—	2	2	no	—
624	D₄	C₂₃×#117	1	4	yes	—
625	C₂×C₄	—	2	16	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	8	4	no	—
662	C₂×C₄	—	8	4	no	—
663	C₂×C₄	—	8	4	no	—
664	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
665	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
666	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
667	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
668	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
669	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
670	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
671	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
672	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
673	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
674	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
675	C₈	—	4	1	no	—
676	C₈	—	4	1	no	—
677	C₈	—	4	1	no	—
678	C₈	—	4	1	no	—
679	C₂²	—	2	2	no	—
680	C₂²	—	2	4	no	—
681	C₂²	—	2	2	no	—
682	C₂²	—	2	2	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	2	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	4	2	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	1	no	—
690	C₂²	—	4	2	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	2	no	—
693	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
694	C₂²	C₂₃×#525	1	1	yes	characteristic
695	C₂²	C₂₃×#525	1	1	yes	characteristic
696	C₂²	C₂₃×#576	1	1	yes	characteristic
697	C₂²	C₂₃×#577	1	1	yes	characteristic
698	C₂²	C₂₃×#577	1	1	yes	characteristic
699	C₂²	C₂₃×#578	1	1	yes	characteristic
700	C₄	—	2	2	no	—
701	C₄	—	2	4	no	—
702	C₄	—	2	2	no	—
703	C₄	—	2	2	no	—
704	C₄	—	2	2	no	—
705	C₄	—	2	2	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	2	no	—
708	C₄	—	8	1	no	—
709	C₄	—	8	1	no	—
710	C₄	—	8	1	no	—
711	C₄	—	8	1	no	—
712	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
713	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
714	C₄	C₂₃×#525	1	1	yes	characteristic
715	C₄	C₂₃×#525	1	1	yes	characteristic
716	C₂	—	4	2	no	—
717	C₂	—	4	4	no	—
718	C₂	C₂₃×#2280	1	1	yes	characteristic
719	C₂	C₂₃×#2897	1	1	yes	characteristic
720	C₂	C₂₃×#2898	1	1	yes	characteristic
721	C₂	C₂₃×#3119	1	1	yes	characteristic
722	C₂	C₂₃×#3120	1	1	yes	characteristic
723	C₂	C₂₃×#3124	1	1	yes	characteristic
724	C₂	C₂₃×#3125	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×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#433	C₂	semidirect
3. C₂₃×#2314	C₂	semidirect
4. C₂₃×#1824	C₂	semidirect
5. C₂₃×#1823	C₂	semidirect
6. C₂₃×#1805	C₂	semidirect
7. C₂₃×#1804	C₂	semidirect
8. C₂₃×#1803	C₂	semidirect
9. C₂₃×#1802	C₂	semidirect
12. C₂₃×#1763	C₂	semidirect
13. C₂₃×#1762	C₂	semidirect
14. C₄×C₂₃×#103	C₂	semidirect
15. C₄×C₂₃×#102	C₂	semidirect
18. C₂₃×#520	C₄	semidirect
238. #20746	C₂₃	direct (N × H)
318. C₂×#433	C₂×C₂₃	semidirect
319. #2314	C₂×C₂₃	semidirect
320. #1824	C₂×C₂₃	semidirect
321. #1823	C₂×C₂₃	semidirect
322. #1805	C₂×C₂₃	semidirect
323. #1804	C₂×C₂₃	semidirect
324. #1803	C₂×C₂₃	semidirect
325. #1802	C₂×C₂₃	semidirect
328. #1763	C₂×C₂₃	semidirect
329. #1762	C₂×C₂₃	semidirect
330. C₄×#103	C₂×C₂₃	semidirect
331. C₄×#102	C₂×C₂₃	semidirect
371. #520	C₄×C₂₃	semidirect
503. C₂₃	#20746	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×#520	2
C₂×C₂₃×#433	2
C₂₃×#2314	2
C₂₃×#1824	2
C₂₃×#1823	2
C₂₃×#1805	2
C₂₃×#1804	2
C₂₃×#1803	2
C₂₃×#1802	2
C₂₃×#1788	2
C₂₃×#1763	2
C₂₃×#1762	2
C₄×C₂₃×#103	2
C₄×C₂₃×#102	2
#20746	23

Subgroups — C23×#20746 #19168372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20746 #19168372