Subgroups — C₂₃×#19648 #19167274

Order 5888.

664 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3231	C₂	1	2	yes	maximal
2	C₂₃×#3185	C₂	1	2	yes	maximal
3	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1718	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1716	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#533	C₂²	1	2	yes	—
13	C₂₃×#533	C₂²	1	2	yes	—
14	C₂₃×#500	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	2	yes	—
16	C₂₃×#500	C₂²	1	1	yes	characteristic
17	C₂₃×#500	C₂²	1	1	yes	characteristic
18	C₂₃×#499	—	2	2	no	—
19	C₂₃×#498	C₂²	1	2	yes	—
20	C₂₃×#498	C₂²	1	2	yes	—
21	C₂₃×#498	C₂²	1	2	yes	—
22	C₂₃×#494	C₂²	1	1	yes	characteristic
23	C₂₃×#494	C₂²	1	2	yes	—
24	C₂₃×#487	C₂²	1	2	yes	—
25	C₂₃×#487	C₂²	1	2	yes	—
26	C₂₃×#487	C₂²	1	2	yes	—
27	C₂₃×#487	C₂²	1	2	yes	—
28	C₂₃×#487	C₂²	1	2	yes	—
29	C₂₃×#485	—	2	4	no	—
30	C₂₃×#479	—	2	4	no	—
31	C₂₃×#470	—	2	4	no	—
32	C₂₃×#467	—	2	2	no	—
33	C₂₃×#441	—	2	4	no	—
34	C₄×Q₁₆×C₂₃	—	2	2	no	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#427	—	2	2	no	—
37	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#322	C₂²	1	1	yes	characteristic
40	C₂₃×#322	C₂²	1	1	yes	characteristic
41	C₈²×C₂₃	C₂²	1	1	yes	characteristic
42	C₈²×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆×C₂₃	—	2	2	no	—
44	C₂×Q₁₆×C₂₃	—	2	4	no	—
45	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
46	C₂²×C₈×C₂₃	D₄	1	2	yes	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₂₃×#125	—	2	2	no	—
52	C₂₃×#125	—	2	2	no	—
53	C₂₃×C₂²⋊Q₈	—	2	2	no	—
54	C₂₃×C₂²⋊Q₈	—	2	2	no	—
55	C₂₃×C₂²⋊Q₈	—	2	2	no	—
56	C₂₃×C₂²⋊Q₈	—	2	2	no	—
57	C₄×Q₈×C₂₃	—	2	2	no	—
58	C₄×Q₈×C₂₃	—	2	2	no	—
59	C₄×Q₈×C₂₃	—	2	2	no	—
60	C₄×Q₈×C₂₃	—	2	2	no	—
61	C₂₃×#117	—	2	2	no	—
62	C₂₃×#117	—	2	2	no	—
63	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#107	—	2	2	no	—
65	C₂₃×#107	—	2	2	no	—
66	C₂₃×#107	—	2	2	no	—
67	C₂₃×#107	—	2	4	no	—
68	C₂₃×#106	—	2	4	no	—
69	C₂₃×#106	—	2	4	no	—
70	C₂₃×#106	—	2	4	no	—
71	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
72	C₂₃×#103	—	2	4	no	—
73	C₂₃×#103	—	2	4	no	—
74	C₂₃×#103	—	2	4	no	—
75	C₂₃×#103	—	2	4	no	—
76	C₂₃×#103	—	2	4	no	—
77	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₂₃×#97	D₄	1	2	yes	—
79	C₄×C₈×C₂₃	D₄	1	2	yes	—
80	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₂₃	—	2	2	no	—
87	C₂×Q₈×C₂₃	—	2	2	no	—
88	C₂×Q₈×C₂₃	—	2	2	no	—
89	C₂×Q₈×C₂₃	—	2	2	no	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₂₃	—	4	8	no	—
94	Q₁₆×C₂₃	—	4	4	no	—
95	M₁₆×C₂₃	—	2	4	no	—
96	C₂×C₈×C₂₃	—	2	4	no	—
97	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₄⋊C₄×C₂₃	—	2	2	no	—
108	C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₄⋊C₄×C₂₃	—	2	2	no	—
110	C₄⋊C₄×C₂₃	—	2	2	no	—
111	C₄⋊C₄×C₂₃	—	4	2	no	—
112	C₄⋊C₄×C₂₃	—	4	2	no	—
113	C₄⋊C₄×C₂₃	—	4	2	no	—
114	C₄⋊C₄×C₂₃	—	4	2	no	—
115	C₂²⋊C₄×C₂₃	—	4	2	no	—
116	C₂²⋊C₄×C₂₃	—	4	2	no	—
117	C₄²×C₂₃	—	4	2	no	—
118	C₄²×C₂₃	—	4	2	no	—
119	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
123	#19648	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₂₃	#127	1	1	yes	characteristic
125	Q₈×C₂₃	—	4	4	no	—
126	Q₈×C₂₃	—	4	4	no	—
127	Q₈×C₂₃	—	4	4	no	—
128	Q₈×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	4	2	no	—
135	C₂×C₄×C₂₃	—	4	2	no	—
136	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
137	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₂₃	—	2	2	no	—
143	C₈×C₂₃	—	2	2	no	—
144	C₈×C₂₃	—	2	2	no	—
145	C₈×C₂₃	—	2	2	no	—
146	C₈×C₂₃	#135	1	4	yes	—
147	C₈×C₂₃	#137	1	4	yes	—
148	#3231	C₂×C₂₃	1	2	yes	—
149	#3185	C₂×C₂₃	1	2	yes	—
150	#3170	C₂×C₂₃	1	1	yes	characteristic
151	#3160	C₂×C₂₃	1	1	yes	characteristic
152	#1718	C₂×C₂₃	1	2	yes	complemented
153	#1716	C₂×C₂₃	1	2	yes	complemented
154	#1713	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#1710	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
157	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
158	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
159	C₂²×C₂₃	—	2	1	no	—
160	C₂²×C₂₃	—	2	1	no	—
161	C₂²×C₂₃	—	2	1	no	—
162	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	2	1	no	—
165	C₄×C₂₃	—	2	1	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	—	8	2	no	—
168	C₄×C₂₃	—	8	2	no	—
169	C₄×C₂₃	—	8	2	no	—
170	C₄×C₂₃	C₂×#135	1	2	yes	—
171	C₄×C₂₃	C₂×#137	1	2	yes	—
172	C₄×C₂₃	#578	1	2	yes	—
173	#533	C₂²×C₂₃	1	2	yes	—
174	#533	C₂²×C₂₃	1	2	yes	—
175	#500	C₂²×C₂₃	1	1	yes	characteristic
176	#500	C₂²×C₂₃	1	1	yes	characteristic
177	#500	C₂²×C₂₃	1	1	yes	characteristic
178	#500	C₂²×C₂₃	1	2	yes	—
179	#499	—	2	2	no	—
180	#498	C₂²×C₂₃	1	2	yes	—
181	#498	C₂²×C₂₃	1	2	yes	—
182	#498	C₂²×C₂₃	1	2	yes	—
183	#494	C₂²×C₂₃	1	1	yes	characteristic
184	#494	C₂²×C₂₃	1	2	yes	—
185	#487	C₂²×C₂₃	1	2	yes	—
186	#487	C₂²×C₂₃	1	2	yes	—
187	#487	C₂²×C₂₃	1	2	yes	—
188	#487	C₂²×C₂₃	1	2	yes	—
189	#487	C₂²×C₂₃	1	2	yes	—
190	#485	—	2	4	no	—
191	#479	—	2	4	no	—
192	#470	—	2	4	no	—
193	#467	—	2	2	no	—
194	#441	—	2	4	no	—
195	C₄×Q₁₆	—	2	2	no	—
196	#431	C₂²×C₂₃	1	1	yes	characteristic
197	#427	—	2	2	no	—
198	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
200	#322	C₂²×C₂₃	1	1	yes	characteristic
201	#322	C₂²×C₂₃	1	1	yes	characteristic
202	C₈²	C₂²×C₂₃	1	1	yes	characteristic
203	C₈²	C₂²×C₂₃	1	1	yes	characteristic
204	C₂×C₂₃	—	4	1	no	—
205	C₂×C₂₃	#3147	1	1	yes	characteristic
206	C₂×C₂₃	#3152	1	1	yes	characteristic
207	C₂×C₂₃	#3156	1	1	yes	characteristic
208	C₂×Q₁₆	—	2	2	no	—
209	C₂×Q₁₆	—	2	4	no	—
210	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
211	C₂²×C₈	D₄×C₂₃	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
216	#125	—	2	2	no	—
217	#125	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	2	2	no	—
228	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
229	#107	—	2	2	no	—
230	#107	—	2	2	no	—
231	#107	—	2	2	no	—
232	#107	—	2	4	no	—
233	#106	—	2	4	no	—
234	#106	—	2	4	no	—
235	#106	—	2	4	no	—
236	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
237	#103	—	2	4	no	—
238	#103	—	2	4	no	—
239	#103	—	2	4	no	—
240	#103	—	2	4	no	—
241	#103	—	2	4	no	—
242	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
243	#97	D₄×C₂₃	1	2	yes	—
244	C₄×C₈	D₄×C₂₃	1	2	yes	—
245	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
251	C₂₃	#19648	1	1	yes	characteristic, direct product factor, Sylow
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
259	Q₁₆	—	4	8	no	—
260	Q₁₆	—	4	4	no	—
261	M₁₆	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
267	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
288	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
289	C₂³	C₂₃×#127	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	4	no	—
293	Q₈	—	4	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
302	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
303	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	2	no	—
311	C₈	C₂₃×#135	1	4	yes	—
312	C₈	C₂₃×#137	1	4	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×C₂₃×#135	1	2	yes	—
325	C₄	C₂×C₂₃×#137	1	2	yes	—
326	C₄	C₂₃×#578	1	2	yes	—
327	C₂	—	4	1	no	—
328	C₂	C₂₃×#3147	1	1	yes	characteristic
329	C₂	C₂₃×#3152	1	1	yes	characteristic
330	C₂	C₂₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	#127
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
5. C₂₃×#1718	C₂	semidirect
6. C₂₃×#1716	C₂	semidirect
7. C₂₃×#1713	C₂	semidirect
8. C₂₃×#1710	C₂	semidirect
9. C₂₃×#1707	C₂	semidirect
10. C₂₃×#1707	C₂	semidirect
123. #19648	C₂₃	direct (N × H)
152. #1718	C₂×C₂₃	semidirect
153. #1716	C₂×C₂₃	semidirect
154. #1713	C₂×C₂₃	semidirect
155. #1710	C₂×C₂₃	semidirect
156. #1707	C₂×C₂₃	semidirect
157. #1707	C₂×C₂₃	semidirect
251. C₂₃	#19648	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×#3231	2
C₂₃×#3185	2
C₂₃×#3170	2
C₂₃×#3160	2
C₂₃×#1718	2
C₂₃×#1716	2
C₂₃×#1713	2
C₂₃×#1710	2
C₂₃×#1707	2
C₈×M₁₆×C₂₃	2
#19648	23

Subgroups — C23×#19648 #19167274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19648 #19167274