Subgroups — C₂₃×#18733 #19166359

Order 5888.

870 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	C₂₃×#3552	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3248	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1652	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1651	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1650	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1649	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1648	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1647	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1483	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#563	—	2	2	no	—
17	C₂₃×#557	C₂²	1	1	yes	characteristic
18	C₂₃×#556	C₂²	1	1	yes	characteristic
19	C₂₃×#554	—	2	1	no	—
20	C₂₃×#553	—	2	1	no	—
21	D₄×Q₈×C₂₃	—	2	1	no	—
22	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#548	—	2	1	no	—
24	C₂₃×#547	C₂²	1	1	yes	characteristic
25	C₂₃×#544	—	2	1	no	—
26	C₂₃×#542	—	2	1	no	—
27	C₂₃×#541	—	2	1	no	—
28	C₂₃×#539	—	2	2	no	—
29	C₂₃×#536	—	2	1	no	—
30	C₂₃×#533	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂₃×#529	C₂²	1	1	yes	characteristic
33	C₂₃×#529	C₂²	1	1	yes	characteristic
34	C₂₃×#501	—	2	2	no	—
35	C₂₃×#500	C₂²	1	1	yes	characteristic
36	C₂₃×#500	C₂²	1	1	yes	characteristic
37	C₂₃×#498	C₂²	1	1	yes	characteristic
38	C₂₃×#498	C₂²	1	1	yes	characteristic
39	C₂₃×#479	—	2	1	no	—
40	C₂₃×#479	—	2	1	no	—
41	C₂₃×#475	C₂²	1	1	yes	characteristic
42	C₂₃×#475	C₂²	1	1	yes	characteristic
43	C₂₃×#474	C₂²	1	1	yes	characteristic
44	C₂₃×#474	C₂²	1	1	yes	characteristic
45	C₂₃×#467	—	2	1	no	—
46	C₂₃×#467	—	2	1	no	—
47	C₂₃×#465	—	2	1	no	—
48	C₂₃×#465	—	2	1	no	—
49	C₂₃×#464	—	2	1	no	—
50	C₂₃×#464	—	2	1	no	—
51	C₂₃×#462	C₂²	1	1	yes	characteristic
52	C₂₃×#462	C₂²	1	1	yes	characteristic
53	C₂₃×#461	C₂²	1	1	yes	characteristic
54	C₂₃×#461	C₂²	1	1	yes	characteristic
55	C₂₃×#428	—	2	2	no	—
56	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂²×Q₈×C₂₃	—	2	1	no	—
73	C₂²×Q₈×C₂₃	—	2	1	no	—
74	C₂×Q₁₆×C₂₃	—	2	1	no	—
75	C₂×Q₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×M₁₆×C₂₃	—	2	1	no	—
79	C₂²×C₈×C₂₃	—	2	1	no	—
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	2	no	—
82	C₂₃×C₄⋊Q₈	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×#126	—	2	2	no	—
92	C₂₃×#126	—	2	2	no	—
93	C₂₃×#126	—	2	2	no	—
94	C₂₃×#126	—	2	1	no	—
95	C₂₃×#126	—	2	1	no	—
96	C₂₃×#125	—	2	2	no	—
97	C₂₃×#125	—	2	2	no	—
98	C₂₃×#125	—	2	1	no	—
99	C₂₃×#125	—	2	2	no	—
100	C₂₃×#125	—	2	1	no	—
101	C₂₃×#125	D₄	1	1	yes	characteristic
102	C₂₃×#125	D₄	1	1	yes	characteristic
103	C₂₃×#124	—	2	2	no	—
104	C₂₃×#124	—	2	1	no	—
105	C₂₃×#124	D₄	1	1	yes	characteristic
106	C₂₃×#124	D₄	1	1	yes	characteristic
107	C₂₃×#123	—	2	2	no	—
108	C₂₃×#123	—	2	2	no	—
109	C₂₃×#123	—	2	1	no	—
110	C₂₃×#123	—	2	1	no	—
111	C₂₃×#123	—	2	2	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	2	no	—
114	C₂₃×C₂²⋊Q₈	—	2	2	no	—
115	C₂₃×C₂²⋊Q₈	—	2	2	no	—
116	C₂₃×C₂²⋊Q₈	—	2	2	no	—
117	C₂₃×C₂²⋊Q₈	—	2	1	no	—
118	C₂₃×C₂²⋊Q₈	—	2	1	no	—
119	C₂₃×C₂²⋊Q₈	—	2	1	no	—
120	C₂₃×C₂²⋊Q₈	—	2	2	no	—
121	C₂₃×C₂²⋊Q₈	—	2	1	no	—
122	C₂₃×C₂²⋊Q₈	—	2	1	no	—
123	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
124	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
125	C₂₃×#121	—	2	2	no	—
126	C₂₃×#121	—	2	1	no	—
127	C₂₃×#121	D₄	1	1	yes	characteristic
128	C₂₃×#121	D₄	1	1	yes	characteristic
129	C₄×Q₈×C₂₃	—	2	1	no	—
130	C₄×Q₈×C₂₃	—	2	1	no	—
131	C₄×Q₈×C₂₃	—	2	1	no	—
132	C₄×Q₈×C₂₃	—	2	1	no	—
133	C₄×Q₈×C₂₃	—	2	1	no	—
134	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
135	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
136	C₄×D₄×C₂₃	—	2	2	no	—
137	C₄×D₄×C₂₃	—	2	2	no	—
138	C₄×D₄×C₂₃	—	2	1	no	—
139	C₄×D₄×C₂₃	—	2	1	no	—
140	C₄×D₄×C₂₃	—	2	1	no	—
141	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
143	C₂₃×#117	—	2	1	no	—
144	C₂₃×#117	—	2	1	no	—
145	C₂₃×#117	—	2	1	no	—
146	C₂₃×#117	—	2	1	no	—
147	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
150	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
151	C₂₃×#107	—	2	2	no	—
152	C₂₃×#107	—	2	2	no	—
153	C₂₃×#107	—	2	1	no	—
154	C₂₃×#107	—	2	1	no	—
155	C₂₃×#106	—	2	2	no	—
156	C₂₃×#106	—	2	2	no	—
157	C₂₃×#106	—	2	1	no	—
158	C₂₃×#106	—	2	1	no	—
159	C₂₃×C₄⋊C₈	—	2	1	no	—
160	C₂₃×C₄⋊C₈	—	2	1	no	—
161	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
165	C₂₃×#103	—	2	1	no	—
166	C₂₃×#103	—	2	1	no	—
167	C₂₃×#103	—	2	1	no	—
168	C₂₃×#103	—	2	1	no	—
169	C₂₃×#103	—	2	1	no	—
170	C₂₃×#103	—	2	1	no	—
171	C₂₃×#102	—	2	1	no	—
172	C₂₃×#102	—	2	1	no	—
173	C₂₃×#97	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	#41×C₂₃	—	8	1	no	—
177	#41×C₂₃	—	8	1	no	—
178	C₂×Q₈×C₂₃	—	2	2	no	—
179	C₂×Q₈×C₂₃	—	2	2	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
182	C₂×Q₈×C₂₃	—	2	1	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	8	1	no	—
185	C₂×Q₈×C₂₃	—	8	1	no	—
186	C₂×Q₈×C₂₃	—	8	1	no	—
187	C₂×Q₈×C₂₃	—	8	1	no	—
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×D₄×C₂₃	—	2	1	no	—
191	C₂×D₄×C₂₃	—	2	1	no	—
192	C₂×D₄×C₂₃	—	2	2	no	—
193	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₂₃	—	2	2	no	—
196	C₂²×C₄×C₂₃	—	2	2	no	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	1	no	—
199	C₂²×C₄×C₂₃	—	2	1	no	—
200	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
203	Q₁₆×C₂₃	—	8	1	no	—
204	Q₁₆×C₂₃	—	8	1	no	—
205	SD₁₆×C₂₃	—	8	1	no	—
206	SD₁₆×C₂₃	—	8	1	no	—
207	M₁₆×C₂₃	—	2	4	no	—
208	C₂×C₈×C₂₃	—	2	1	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	—	8	1	no	—
215	C₂×C₈×C₂₃	—	8	1	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	2	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	2	no	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	2	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	—	2	1	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	1	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	2	1	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	4	1	no	—
239	C₄⋊C₄×C₂₃	—	4	1	no	—
240	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
241	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
244	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
245	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
246	C₂²⋊C₄×C₂₃	—	2	2	no	—
247	C₂²⋊C₄×C₂₃	—	2	2	no	—
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	2	no	—
250	C₂²⋊C₄×C₂₃	—	2	1	no	—
251	C₂²⋊C₄×C₂₃	—	2	1	no	—
252	C₂²⋊C₄×C₂₃	—	2	1	no	—
253	C₂²⋊C₄×C₂₃	—	2	1	no	—
254	C₂²⋊C₄×C₂₃	—	2	1	no	—
255	C₂²⋊C₄×C₂₃	—	2	1	no	—
256	C₂²⋊C₄×C₂₃	—	2	1	no	—
257	C₂²⋊C₄×C₂₃	—	2	1	no	—
258	C₂²⋊C₄×C₂₃	—	2	1	no	—
259	C₂²⋊C₄×C₂₃	—	2	1	no	—
260	C₂²⋊C₄×C₂₃	—	4	1	no	—
261	C₄²×C₂₃	—	2	2	no	—
262	C₄²×C₂₃	—	2	2	no	—
263	C₄²×C₂₃	—	2	1	no	—
264	C₄²×C₂₃	—	2	1	no	—
265	C₄²×C₂₃	—	4	1	no	—
266	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
268	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
269	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
270	#18733	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂³×C₂₃	—	2	1	no	—
272	C₂³×C₂₃	—	2	1	no	—
273	C₂³×C₂₃	#120	1	1	yes	characteristic
274	Q₈×C₂₃	—	2	1	no	—
275	Q₈×C₂₃	—	2	1	no	—
276	Q₈×C₂₃	—	2	1	no	—
277	Q₈×C₂₃	—	2	1	no	—
278	Q₈×C₂₃	—	4	1	no	—
279	Q₈×C₂₃	—	4	1	no	—
280	Q₈×C₂₃	—	4	1	no	—
281	Q₈×C₂₃	—	4	1	no	—
282	Q₈×C₂₃	—	4	1	no	—
283	Q₈×C₂₃	—	4	1	no	—
284	Q₈×C₂₃	—	4	1	no	—
285	Q₈×C₂₃	—	4	1	no	—
286	Q₈×C₂₃	—	8	2	no	—
287	Q₈×C₂₃	—	8	2	no	—
288	D₄×C₂₃	—	2	1	no	—
289	D₄×C₂₃	—	2	1	no	—
290	D₄×C₂₃	—	2	1	no	—
291	D₄×C₂₃	—	2	1	no	—
292	D₄×C₂₃	—	4	2	no	—
293	D₄×C₂₃	—	4	2	no	—
294	D₄×C₂₃	—	8	2	no	—
295	C₂×C₄×C₂₃	—	2	2	no	—
296	C₂×C₄×C₂₃	—	2	1	no	—
297	C₂×C₄×C₂₃	—	2	1	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	1	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	2	1	no	—
302	C₂×C₄×C₂₃	—	2	1	no	—
303	C₂×C₄×C₂₃	—	2	1	no	—
304	C₂×C₄×C₂₃	—	2	1	no	—
305	C₂×C₄×C₂₃	—	2	2	no	—
306	C₂×C₄×C₂₃	—	2	2	no	—
307	C₂×C₄×C₂₃	—	2	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	—	8	2	no	—
315	C₂×C₄×C₂₃	—	8	2	no	—
316	C₂×C₄×C₂₃	—	8	2	no	—
317	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
318	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
319	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
320	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
323	C₈×C₂₃	—	2	2	no	—
324	C₈×C₂₃	—	2	2	no	—
325	C₈×C₂₃	—	4	1	no	—
326	C₈×C₂₃	—	4	1	no	—
327	C₈×C₂₃	—	4	1	no	—
328	C₈×C₂₃	—	4	1	no	—
329	#3552	C₂×C₂₃	1	1	yes	characteristic
330	#3248	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#3248	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#3165	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1654	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1653	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1652	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1651	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1650	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1649	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1648	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#1647	C₂×C₂₃	1	1	yes	characteristic, complemented
341	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1483	C₂×C₂₃	1	1	yes	characteristic, complemented
344	C₂²×C₂₃	—	2	1	no	—
345	C₂²×C₂₃	—	2	1	no	—
346	C₂²×C₂₃	—	4	1	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	—	4	1	no	—
352	C₂²×C₂₃	#456	1	2	yes	—
353	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
354	C₄×C₂₃	—	2	1	no	—
355	C₄×C₂₃	—	2	1	no	—
356	C₄×C₂₃	—	2	1	no	—
357	C₄×C₂₃	—	2	1	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	4	2	no	—
360	C₄×C₂₃	—	4	1	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	1	no	—
366	C₄×C₂₃	—	8	1	no	—
367	C₄×C₂₃	—	8	1	no	—
368	C₄×C₂₃	#576	1	1	yes	characteristic
369	C₄×C₂₃	#577	1	1	yes	characteristic
370	C₄×C₂₃	#577	1	1	yes	characteristic
371	C₄×C₂₃	#578	1	1	yes	characteristic
372	#563	—	2	2	no	—
373	#557	C₂²×C₂₃	1	1	yes	characteristic
374	#556	C₂²×C₂₃	1	1	yes	characteristic
375	#554	—	2	1	no	—
376	#553	—	2	1	no	—
377	D₄×Q₈	—	2	1	no	—
378	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
379	#548	—	2	1	no	—
380	#547	C₂²×C₂₃	1	1	yes	characteristic
381	#544	—	2	1	no	—
382	#542	—	2	1	no	—
383	#541	—	2	1	no	—
384	#539	—	2	2	no	—
385	#536	—	2	1	no	—
386	#533	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
388	#529	C₂²×C₂₃	1	1	yes	characteristic
389	#529	C₂²×C₂₃	1	1	yes	characteristic
390	#501	—	2	2	no	—
391	#500	C₂²×C₂₃	1	1	yes	characteristic
392	#500	C₂²×C₂₃	1	1	yes	characteristic
393	#498	C₂²×C₂₃	1	1	yes	characteristic
394	#498	C₂²×C₂₃	1	1	yes	characteristic
395	#479	—	2	1	no	—
396	#479	—	2	1	no	—
397	#475	C₂²×C₂₃	1	1	yes	characteristic
398	#475	C₂²×C₂₃	1	1	yes	characteristic
399	#474	C₂²×C₂₃	1	1	yes	characteristic
400	#474	C₂²×C₂₃	1	1	yes	characteristic
401	#467	—	2	1	no	—
402	#467	—	2	1	no	—
403	#465	—	2	1	no	—
404	#465	—	2	1	no	—
405	#464	—	2	1	no	—
406	#464	—	2	1	no	—
407	#462	C₂²×C₂₃	1	1	yes	characteristic
408	#462	C₂²×C₂₃	1	1	yes	characteristic
409	#461	C₂²×C₂₃	1	1	yes	characteristic
410	#461	C₂²×C₂₃	1	1	yes	characteristic
411	#428	—	2	2	no	—
412	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
413	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
414	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
415	#332	C₂²×C₂₃	1	1	yes	characteristic
416	#332	C₂²×C₂₃	1	1	yes	characteristic
417	#332	C₂²×C₂₃	1	1	yes	characteristic
418	#332	C₂²×C₂₃	1	1	yes	characteristic
419	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
420	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
421	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
422	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
424	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
425	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
426	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
427	C₂×C₂₃	—	2	2	no	—
428	C₂×C₂₃	—	8	1	no	—
429	C₂×C₂₃	—	8	1	no	—
430	C₂×C₂₃	#3012	1	1	yes	characteristic
431	C₂×C₂₃	#3013	1	1	yes	characteristic
432	C₂×C₂₃	C₂×#456	1	1	yes	characteristic
433	C₂×#41	—	2	1	no	—
434	C₂²×Q₈	—	2	1	no	—
435	C₂²×Q₈	—	2	1	no	—
436	C₂×Q₁₆	—	2	1	no	—
437	C₂×Q₁₆	—	2	1	no	—
438	C₂×SD₁₆	—	2	1	no	—
439	C₂×SD₁₆	—	2	1	no	—
440	C₂×M₁₆	—	2	1	no	—
441	C₂²×C₈	—	2	1	no	—
442	C₂²×C₈	—	2	1	no	—
443	C₄⋊Q₈	—	2	2	no	—
444	C₄⋊Q₈	—	2	1	no	—
445	C₄⋊Q₈	—	2	1	no	—
446	C₄⋊Q₈	—	2	1	no	—
447	C₄⋊Q₈	—	2	1	no	—
448	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
449	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
451	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
452	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
453	#126	—	2	2	no	—
454	#126	—	2	2	no	—
455	#126	—	2	2	no	—
456	#126	—	2	1	no	—
457	#126	—	2	1	no	—
458	#125	—	2	2	no	—
459	#125	—	2	2	no	—
460	#125	—	2	2	no	—
461	#125	—	2	1	no	—
462	#125	—	2	1	no	—
463	#125	D₄×C₂₃	1	1	yes	characteristic
464	#125	D₄×C₂₃	1	1	yes	characteristic
465	#124	—	2	2	no	—
466	#124	—	2	1	no	—
467	#124	D₄×C₂₃	1	1	yes	characteristic
468	#124	D₄×C₂₃	1	1	yes	characteristic
469	#123	—	2	2	no	—
470	#123	—	2	2	no	—
471	#123	—	2	1	no	—
472	#123	—	2	1	no	—
473	#123	—	2	2	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	2	no	—
476	C₂²⋊Q₈	—	2	2	no	—
477	C₂²⋊Q₈	—	2	2	no	—
478	C₂²⋊Q₈	—	2	2	no	—
479	C₂²⋊Q₈	—	2	1	no	—
480	C₂²⋊Q₈	—	2	2	no	—
481	C₂²⋊Q₈	—	2	1	no	—
482	C₂²⋊Q₈	—	2	1	no	—
483	C₂²⋊Q₈	—	2	1	no	—
484	C₂²⋊Q₈	—	2	1	no	—
485	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
486	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
487	#121	—	2	2	no	—
488	#121	—	2	1	no	—
489	#121	D₄×C₂₃	1	1	yes	characteristic
490	#121	D₄×C₂₃	1	1	yes	characteristic
491	C₄×Q₈	—	2	1	no	—
492	C₄×Q₈	—	2	1	no	—
493	C₄×Q₈	—	2	1	no	—
494	C₄×Q₈	—	2	1	no	—
495	C₄×Q₈	—	2	1	no	—
496	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
497	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
498	C₄×D₄	—	2	2	no	—
499	C₄×D₄	—	2	2	no	—
500	C₄×D₄	—	2	1	no	—
501	C₄×D₄	—	2	1	no	—
502	C₄×D₄	—	2	1	no	—
503	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
504	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
505	#117	—	2	1	no	—
506	#117	—	2	1	no	—
507	#117	—	2	1	no	—
508	#117	—	2	1	no	—
509	C₂×C₄⋊C₄	—	2	1	no	—
510	C₂×C₄⋊C₄	—	2	1	no	—
511	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
512	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
513	#107	—	2	2	no	—
514	#107	—	2	2	no	—
515	#107	—	2	1	no	—
516	#107	—	2	1	no	—
517	#106	—	2	2	no	—
518	#106	—	2	2	no	—
519	#106	—	2	1	no	—
520	#106	—	2	1	no	—
521	C₄⋊C₈	—	2	1	no	—
522	C₄⋊C₈	—	2	1	no	—
523	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
524	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
525	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
526	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
527	#103	—	2	1	no	—
528	#103	—	2	1	no	—
529	#103	—	2	1	no	—
530	#103	—	2	1	no	—
531	#103	—	2	1	no	—
532	#103	—	2	1	no	—
533	#102	—	2	1	no	—
534	#102	—	2	1	no	—
535	#97	—	2	1	no	—
536	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
537	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
538	C₂₃	#18733	1	1	yes	characteristic, direct product factor, Sylow
539	#41	—	8	1	no	—
540	#41	—	8	1	no	—
541	C₂×Q₈	—	2	2	no	—
542	C₂×Q₈	—	2	2	no	—
543	C₂×Q₈	—	2	1	no	—
544	C₂×Q₈	—	2	1	no	—
545	C₂×Q₈	—	2	1	no	—
546	C₂×Q₈	—	2	1	no	—
547	C₂×Q₈	—	8	1	no	—
548	C₂×Q₈	—	8	1	no	—
549	C₂×Q₈	—	8	1	no	—
550	C₂×Q₈	—	8	1	no	—
551	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
552	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	2	no	—
556	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	—	2	2	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	2	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
564	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
565	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
566	Q₁₆	—	8	1	no	—
567	Q₁₆	—	8	1	no	—
568	SD₁₆	—	8	1	no	—
569	SD₁₆	—	8	1	no	—
570	M₁₆	—	2	4	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	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	8	1	no	—
578	C₂×C₈	—	8	1	no	—
579	C₄⋊C₄	—	2	2	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	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	—	2	1	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	4	1	no	—
602	C₄⋊C₄	—	4	1	no	—
603	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
604	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
605	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
606	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
607	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
608	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	2	no	—
612	C₂²⋊C₄	—	2	2	no	—
613	C₂²⋊C₄	—	2	1	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	2	1	no	—
616	C₂²⋊C₄	—	2	1	no	—
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	2	1	no	—
621	C₂²⋊C₄	—	2	1	no	—
622	C₂²⋊C₄	—	2	1	no	—
623	C₂²⋊C₄	—	4	1	no	—
624	C₄²	—	2	2	no	—
625	C₄²	—	2	2	no	—
626	C₄²	—	2	1	no	—
627	C₄²	—	2	1	no	—
628	C₄²	—	4	1	no	—
629	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
630	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
631	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
632	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
633	C₂³	—	2	1	no	—
634	C₂³	—	2	1	no	—
635	C₂³	C₂₃×#120	1	1	yes	characteristic
636	Q₈	—	2	1	no	—
637	Q₈	—	2	1	no	—
638	Q₈	—	2	1	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	4	1	no	—
641	Q₈	—	4	1	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	—	4	1	no	—
645	Q₈	—	4	1	no	—
646	Q₈	—	4	1	no	—
647	Q₈	—	4	1	no	—
648	Q₈	—	8	2	no	—
649	Q₈	—	8	2	no	—
650	D₄	—	2	1	no	—
651	D₄	—	2	1	no	—
652	D₄	—	2	1	no	—
653	D₄	—	2	1	no	—
654	D₄	—	4	2	no	—
655	D₄	—	4	2	no	—
656	D₄	—	8	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	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	—	8	2	no	—
677	C₂×C₄	—	8	2	no	—
678	C₂×C₄	—	8	2	no	—
679	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
680	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
681	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
682	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
683	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
684	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
685	C₈	—	2	2	no	—
686	C₈	—	2	2	no	—
687	C₈	—	4	1	no	—
688	C₈	—	4	1	no	—
689	C₈	—	4	1	no	—
690	C₈	—	4	1	no	—
691	C₂²	—	2	1	no	—
692	C₂²	—	2	1	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	4	1	no	—
695	C₂²	—	4	1	no	—
696	C₂²	—	4	1	no	—
697	C₂²	—	4	1	no	—
698	C₂²	—	4	1	no	—
699	C₂²	C₂₃×#456	1	2	yes	—
700	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
701	C₄	—	2	1	no	—
702	C₄	—	2	1	no	—
703	C₄	—	2	1	no	—
704	C₄	—	2	1	no	—
705	C₄	—	4	1	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	1	no	—
708	C₄	—	4	2	no	—
709	C₄	—	4	1	no	—
710	C₄	—	4	1	no	—
711	C₄	—	4	1	no	—
712	C₄	—	4	1	no	—
713	C₄	—	8	1	no	—
714	C₄	—	8	1	no	—
715	C₄	C₂₃×#576	1	1	yes	characteristic
716	C₄	C₂₃×#577	1	1	yes	characteristic
717	C₄	C₂₃×#577	1	1	yes	characteristic
718	C₄	C₂₃×#578	1	1	yes	characteristic
719	C₂	—	2	2	no	—
720	C₂	—	8	1	no	—
721	C₂	—	8	1	no	—
722	C₂	C₂₃×#3012	1	1	yes	characteristic
723	C₂	C₂₃×#3013	1	1	yes	characteristic
724	C₂	C₂×C₂₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3552	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3248	C₂	semidirect
3. C₂₃×#3248	C₂	semidirect
4. C₂₃×#3165	C₂	semidirect
5. C₂₃×#1654	C₂	semidirect
6. C₂₃×#1653	C₂	semidirect
7. C₂₃×#1652	C₂	semidirect
8. C₂₃×#1651	C₂	semidirect
9. C₂₃×#1650	C₂	semidirect
10. C₂₃×#1649	C₂	semidirect
11. C₂₃×#1648	C₂	semidirect
12. C₂₃×#1647	C₂	semidirect
13. C₂×C₂₃×#332	C₂	semidirect
14. C₂₃×#1483	C₂	semidirect
15. C₂₃×#1483	C₂	semidirect
270. #18733	C₂₃	direct (N × H)
330. #3248	C₂×C₂₃	semidirect
331. #3248	C₂×C₂₃	semidirect
332. #3165	C₂×C₂₃	semidirect
333. #1654	C₂×C₂₃	semidirect
334. #1653	C₂×C₂₃	semidirect
335. #1652	C₂×C₂₃	semidirect
336. #1651	C₂×C₂₃	semidirect
337. #1650	C₂×C₂₃	semidirect
338. #1649	C₂×C₂₃	semidirect
339. #1648	C₂×C₂₃	semidirect
340. #1647	C₂×C₂₃	semidirect
341. C₂×#332	C₂×C₂₃	semidirect
342. #1483	C₂×C₂₃	semidirect
343. #1483	C₂×C₂₃	semidirect
538. C₂₃	#18733	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×#3552	2
C₂₃×#3248	2
C₂₃×#3165	2
C₂₃×#1654	2
C₂₃×#1653	2
C₂₃×#1652	2
C₂₃×#1651	2
C₂₃×#1650	2
C₂₃×#1649	2
C₂₃×#1648	2
C₂₃×#1647	2
C₂×C₂₃×#332	2
C₂₃×#1483	2
#18733	23

Subgroups — C23×#18733 #19166359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18733 #19166359