Subgroups — C₂₃×#19654 #19167280

Order 5888.

798 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3227	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3206	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1716	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1715	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1708	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1705	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#135	—	2	2	no	—
11	C₂₃×#532	C₂²	1	1	yes	characteristic
12	C₂₃×#532	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂₃×#498	—	2	2	no	—
15	C₂₃×#498	C₂²	1	4	yes	—
16	C₂₃×#498	C₂²	1	2	yes	—
17	C₂₃×#498	C₂²	1	2	yes	—
18	C₂₃×#495	—	2	2	no	—
19	C₂₃×#494	C₂²	1	1	yes	characteristic
20	C₂₃×#493	C₂²	1	1	yes	characteristic
21	C₂₃×#492	C₂²	1	4	yes	—
22	C₂₃×#492	C₂²	1	1	yes	characteristic
23	C₂₃×#492	C₂²	1	1	yes	characteristic
24	C₂₃×#487	C₂²	1	4	yes	—
25	C₂₃×#486	C₂²	1	4	yes	—
26	C₂₃×#481	—	2	4	no	—
27	C₂₃×#476	—	2	4	no	—
28	C₂₃×#461	—	2	4	no	—
29	C₂₃×#448	—	2	4	no	—
30	C₂₃×#431	C₂²	1	2	yes	—
31	C₂×C₂₃×#106	—	2	2	no	—
32	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#322	C₂²	1	2	yes	—
34	C₈²×C₂₃	C₂²	1	2	yes	—
35	C₂×#41×C₂₃	—	2	1	no	—
36	C₂×#41×C₂₃	—	2	1	no	—
37	C₂²×Q₈×C₂₃	—	2	2	no	—
38	C₂₃×#135	—	4	8	no	—
39	C₂×Q₁₆×C₂₃	—	2	2	no	—
40	C₂×SD₁₆×C₂₃	—	2	2	no	—
41	C₂×SD₁₆×C₂₃	—	2	2	no	—
42	C₂×SD₁₆×C₂₃	—	2	8	no	—
43	C₂×D₈×C₂₃	—	2	2	no	—
44	C₂²×C₈×C₂₃	D₄	1	2	yes	—
45	C₂₃×C₄⋊Q₈	—	2	2	no	—
46	C₂₃×C₄⋊Q₈	—	2	2	no	—
47	C₂₃×C₄⋊Q₈	—	2	2	no	—
48	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
49	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	C₂₃×#127	C₂³	1	1	yes	characteristic
52	C₂₃×#127	C₂³	1	1	yes	characteristic
53	C₂₃×#124	—	2	1	no	—
54	C₂₃×#124	—	2	1	no	—
55	C₂₃×#121	—	2	2	no	—
56	C₂₃×#121	—	2	2	no	—
57	C₄×D₄×C₂₃	—	2	2	no	—
58	C₄×D₄×C₂₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×#106	—	2	4	no	—
63	C₂₃×#106	—	2	4	no	—
64	C₂₃×#106	—	2	8	no	—
65	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
66	C₂₃×#103	—	2	8	no	—
67	C₂₃×#102	—	2	8	no	—
68	C₂₃×C₂²⋊C₈	D₄	1	4	yes	—
69	C₄×C₈×C₂₃	D₄	1	2	yes	—
70	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂³	1	4	yes	—
72	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	#41×C₂₃	—	4	4	no	—
74	#41×C₂₃	—	4	4	no	—
75	C₂×Q₈×C₂₃	—	2	4	no	—
76	C₂×Q₈×C₂₃	—	2	1	no	—
77	C₂×Q₈×C₂₃	—	2	1	no	—
78	C₂×Q₈×C₂₃	—	4	2	no	—
79	C₂×Q₈×C₂₃	—	8	4	no	—
80	C₂×D₄×C₂₃	—	2	2	no	—
81	C₂×D₄×C₂₃	—	2	2	no	—
82	C₂×D₄×C₂₃	—	2	1	no	—
83	C₂×D₄×C₂₃	—	2	1	no	—
84	C₂×D₄×C₂₃	—	4	1	no	—
85	C₂×D₄×C₂₃	—	4	1	no	—
86	C₂²×C₄×C₂₃	—	4	2	no	—
87	C₂²×C₄×C₂₃	—	4	1	no	—
88	C₂²×C₄×C₂₃	—	4	1	no	—
89	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₂₃	—	4	4	no	—
92	SD₁₆×C₂₃	—	4	4	no	—
93	SD₁₆×C₂₃	—	4	4	no	—
94	SD₁₆×C₂₃	—	8	8	no	—
95	D₈×C₂₃	—	4	4	no	—
96	C₂×C₈×C₂₃	SD₁₆	1	8	yes	complemented
97	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
98	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₂₃	—	2	4	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₂²⋊C₄×C₂₃	—	4	2	no	—
107	C₂²⋊C₄×C₂₃	—	4	2	no	—
108	C₄²×C₂₃	C₂×D₄	1	2	yes	—
109	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
111	#19654	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₂₃	—	4	1	no	—
113	C₂³×C₂₃	—	4	1	no	—
114	C₂³×C₂₃	#127	1	1	yes	characteristic
115	Q₈×C₂₃	—	4	2	no	—
116	Q₈×C₂₃	—	4	2	no	—
117	Q₈×C₂₃	—	4	8	no	—
118	Q₈×C₂₃	—	8	4	no	—
119	D₄×C₂₃	—	4	2	no	—
120	D₄×C₂₃	—	4	2	no	—
121	D₄×C₂₃	—	4	4	no	—
122	D₄×C₂₃	—	4	4	no	—
123	D₄×C₂₃	—	8	2	no	—
124	D₄×C₂₃	—	8	2	no	—
125	C₂×C₄×C₂₃	—	2	4	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	—	8	4	no	—
130	C₂×C₄×C₂₃	—	8	2	no	—
131	C₂×C₄×C₂₃	—	8	2	no	—
132	C₂×C₄×C₂₃	#127	1	2	yes	—
133	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
135	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
137	C₈×C₂₃	—	2	8	no	—
138	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
139	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
140	#3227	C₂×C₂₃	1	2	yes	complemented
141	#3206	C₂×C₂₃	1	2	yes	complemented
142	C₂×#498	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#3147	C₂×C₂₃	1	1	yes	characteristic
144	#1716	C₂×C₂₃	1	2	yes	complemented
145	#1715	C₂×C₂₃	1	2	yes	complemented
146	#1708	C₂×C₂₃	1	2	yes	complemented
147	#1705	C₂×C₂₃	1	2	yes	complemented
148	#1450	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂²×C₂₃	—	2	2	no	—
150	C₂²×C₂₃	—	8	1	no	—
151	C₂²×C₂₃	—	8	1	no	—
152	C₂²×C₂₃	—	8	2	no	—
153	C₂²×C₂₃	—	8	2	no	—
154	C₂²×C₂₃	#492	1	2	yes	—
155	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	8	1	no	—
158	C₄×C₂₃	—	8	1	no	—
159	C₄×C₂₃	—	8	4	no	—
160	C₄×C₂₃	#492	1	2	yes	—
161	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
162	C₄×C₂₃	#577	1	4	yes	—
163	C₂×#135	—	2	2	no	—
164	#532	C₂²×C₂₃	1	1	yes	characteristic
165	#532	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
167	#498	—	2	2	no	—
168	#498	C₂²×C₂₃	1	2	yes	—
169	#498	C₂²×C₂₃	1	2	yes	—
170	#498	C₂²×C₂₃	1	4	yes	—
171	#495	—	2	2	no	—
172	#494	C₂²×C₂₃	1	1	yes	characteristic
173	#493	C₂²×C₂₃	1	1	yes	characteristic
174	#492	C₂²×C₂₃	1	1	yes	characteristic
175	#492	C₂²×C₂₃	1	1	yes	characteristic
176	#492	C₂²×C₂₃	1	4	yes	—
177	#487	C₂²×C₂₃	1	4	yes	—
178	#486	C₂²×C₂₃	1	4	yes	—
179	#481	—	2	4	no	—
180	#476	—	2	4	no	—
181	#461	—	2	4	no	—
182	#448	—	2	4	no	—
183	#431	C₂²×C₂₃	1	2	yes	—
184	C₂×#106	—	2	2	no	—
185	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
186	#322	C₂²×C₂₃	1	2	yes	—
187	C₈²	C₂²×C₂₃	1	2	yes	—
188	C₂×C₂₃	—	2	2	no	—
189	C₂×C₂₃	—	16	1	no	—
190	C₂×C₂₃	—	16	1	no	—
191	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
192	C₂×C₂₃	#3154	1	2	yes	—
193	C₂×#41	—	2	1	no	—
194	C₂×#41	—	2	1	no	—
195	C₂²×Q₈	—	2	2	no	—
196	#135	—	4	8	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×SD₁₆	—	2	2	no	—
199	C₂×SD₁₆	—	2	2	no	—
200	C₂×SD₁₆	—	2	8	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂²×C₈	D₄×C₂₃	1	2	yes	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
207	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
209	#127	C₂³×C₂₃	1	1	yes	characteristic
210	#127	C₂³×C₂₃	1	1	yes	characteristic
211	#124	—	2	1	no	—
212	#124	—	2	1	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
220	#106	—	2	4	no	—
221	#106	—	2	4	no	—
222	#106	—	2	8	no	—
223	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
224	#103	—	2	8	no	—
225	#102	—	2	8	no	—
226	C₂²⋊C₈	D₄×C₂₃	1	4	yes	—
227	C₄×C₈	D₄×C₂₃	1	2	yes	—
228	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₂₃	1	4	yes	—
230	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
231	C₂₃	#19654	1	1	yes	characteristic, direct product factor, Sylow
232	#41	—	4	4	no	—
233	#41	—	4	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	8	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
250	Q₁₆	—	4	4	no	—
251	SD₁₆	—	4	4	no	—
252	SD₁₆	—	4	4	no	—
253	SD₁₆	—	8	8	no	—
254	D₈	—	4	4	no	—
255	C₂×C₈	SD₁₆×C₂₃	1	8	yes	complemented
256	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
257	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₂₃	1	2	yes	—
269	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
270	C₂³	—	4	1	no	—
271	C₂³	—	4	1	no	—
272	C₂³	C₂₃×#127	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	8	no	—
276	Q₈	—	8	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	4	no	—
281	D₄	—	8	2	no	—
282	D₄	—	8	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	C₂₃×#127	1	2	yes	—
291	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
292	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
293	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
295	C₈	—	2	8	no	—
296	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
297	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₂₃×#492	1	2	yes	—
304	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	4	no	—
309	C₄	C₂₃×#492	1	2	yes	—
310	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
311	C₄	C₂₃×#577	1	4	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	—	16	1	no	—
315	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
316	C₂	C₂₃×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3147	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
1. C₂₃×#3227	C₂	semidirect
2. C₂₃×#3206	C₂	semidirect
3. C₂×C₂₃×#498	C₂	semidirect
5. C₂₃×#1716	C₂	semidirect
6. C₂₃×#1715	C₂	semidirect
7. C₂₃×#1708	C₂	semidirect
8. C₂₃×#1705	C₂	semidirect
9. C₂₃×#1450	C₂	semidirect
96. C₂×C₈×C₂₃	SD₁₆	semidirect
111. #19654	C₂₃	direct (N × H)
140. #3227	C₂×C₂₃	semidirect
141. #3206	C₂×C₂₃	semidirect
142. C₂×#498	C₂×C₂₃	semidirect
144. #1716	C₂×C₂₃	semidirect
145. #1715	C₂×C₂₃	semidirect
146. #1708	C₂×C₂₃	semidirect
147. #1705	C₂×C₂₃	semidirect
148. #1450	C₂×C₂₃	semidirect
231. C₂₃	#19654	direct (N × H)
255. C₂×C₈	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3227	2
C₂₃×#3206	2
C₂×C₂₃×#498	2
C₂₃×#3147	2
C₂₃×#1716	2
C₂₃×#1715	2
C₂₃×#1708	2
C₂₃×#1705	2
C₂₃×#1450	2
#19654	23

Subgroups — C23×#19654 #19167280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19654 #19167280