Subgroups — C₂₃×#19194 #19166820

Order 5888.

638 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3246	C₂	1	2	yes	maximal
2	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1695	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1693	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1671	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1550	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#131	—	2	1	no	—
10	C₂₃×#533	C₂²	1	2	yes	—
11	C₂₃×#532	C₂²	1	1	yes	characteristic
12	C₂₃×#500	C₂²	1	1	yes	characteristic
13	C₂₃×#500	C₂²	1	1	yes	characteristic
14	C₂₃×#499	—	2	1	no	—
15	C₂₃×#499	—	2	1	no	—
16	C₂₃×#498	C₂²	1	2	yes	—
17	C₂₃×#476	C₂²	1	4	yes	—
18	C₂₃×#475	—	2	2	no	—
19	C₂₃×#465	—	2	4	no	—
20	C₂₃×#461	C₂²	1	4	yes	—
21	C₂₃×#460	—	2	2	no	—
22	C₂₃×#436	—	2	1	no	—
23	C₂₃×#436	C₂²	1	2	yes	—
24	C₈×D₄×C₂₃	—	2	1	no	—
25	C₈×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#427	—	2	1	no	—
27	C₂₃×#427	—	2	1	no	—
28	C₂₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#334	C₂²	1	2	yes	—
33	C₂₃×#334	C₂²	1	2	yes	—
34	C₂₃×#333	C₂²	1	4	yes	—
35	C₂₃×#332	C₂²	1	4	yes	—
36	C₂×#41×C₂₃	—	2	1	no	—
37	C₂×SD₁₆×C₂₃	—	2	4	no	—
38	C₂₃×#131	—	4	4	no	—
39	C₂×M₁₆×C₂₃	—	2	1	no	—
40	C₂×M₁₆×C₂₃	—	2	2	no	—
41	C₂²×C₈×C₂₃	—	2	2	no	—
42	C₂²×C₈×C₂₃	—	2	2	no	—
43	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×#127	D₄	1	1	yes	characteristic
49	C₂₃×#125	—	2	2	no	—
50	C₂₃×#124	—	2	1	no	—
51	C₂₃×C₂²⋊Q₈	—	2	2	no	—
52	C₂₃×C₂²⋊Q₈	—	2	2	no	—
53	C₂₃×#121	—	2	1	no	—
54	C₂₃×#121	D₄	1	2	yes	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₄×Q₈×C₂₃	—	2	2	no	—
57	C₄×D₄×C₂₃	—	2	1	no	—
58	C₄×D₄×C₂₃	C₂³	1	2	yes	—
59	C₂₃×#117	—	2	2	no	—
60	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×#107	—	2	1	no	—
62	C₂₃×#107	—	2	1	no	—
63	C₂₃×#107	—	2	1	no	—
64	C₂₃×#107	—	2	1	no	—
65	C₂₃×#106	—	2	2	no	—
66	C₂₃×#106	—	2	2	no	—
67	C₂₃×#106	—	2	4	no	—
68	C₂₃×C₄⋊C₈	—	2	2	no	—
69	C₂₃×C₄⋊C₈	—	2	1	no	—
70	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
71	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
72	C₂₃×#103	—	2	4	no	—
73	C₂₃×#102	—	2	4	no	—
74	C₂₃×C₂²⋊C₈	—	2	2	no	—
75	C₂₃×C₂²⋊C₈	—	2	2	no	—
76	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	#41×C₂₃	—	4	2	no	—
81	#41×C₂₃	—	4	2	no	—
82	C₂×Q₈×C₂₃	—	2	1	no	—
83	C₂×Q₈×C₂₃	—	2	2	no	—
84	C₂×Q₈×C₂₃	—	2	2	no	—
85	C₂×D₄×C₂₃	—	2	2	no	—
86	C₂×D₄×C₂₃	—	2	1	no	—
87	C₂×D₄×C₂₃	—	2	1	no	—
88	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×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	#41	1	1	yes	characteristic
93	SD₁₆×C₂₃	—	8	4	no	—
94	M₁₆×C₂₃	—	4	2	no	—
95	M₁₆×C₂₃	—	4	4	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	2	4	no	—
98	C₂×C₈×C₂₃	—	2	2	no	—
99	C₂×C₈×C₂₃	—	4	4	no	—
100	C₂×C₈×C₂₃	—	8	2	no	—
101	C₂×C₈×C₂₃	SD₁₆	1	4	yes	complemented
102	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₂₃	—	2	1	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	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂²⋊C₄×C₂₃	—	2	2	no	—
116	C₂²⋊C₄×C₂₃	—	4	2	no	—
117	C₄²×C₂₃	—	4	2	no	—
118	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₂₃	#41	1	1	yes	characteristic
121	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
122	#19194	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₂₃	—	2	2	no	—
124	C₂³×C₂₃	#121	1	1	yes	characteristic
125	Q₈×C₂₃	—	4	2	no	—
126	Q₈×C₂₃	—	4	2	no	—
127	Q₈×C₂₃	—	4	2	no	—
128	Q₈×C₂₃	—	8	2	no	—
129	D₄×C₂₃	—	2	4	no	—
130	D₄×C₂₃	—	4	2	no	—
131	D₄×C₂₃	—	4	4	no	—
132	D₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	2	2	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	4	1	no	—
136	C₂×C₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	4	2	no	—
140	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₂₃	#137	1	2	yes	—
145	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₂₃	—	2	1	no	—
149	C₈×C₂₃	—	2	1	no	—
150	C₈×C₂₃	—	4	2	no	—
151	C₈×C₂₃	—	4	4	no	—
152	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
153	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
154	#3246	C₂×C₂₃	1	2	yes	—
155	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#2931	C₂×C₂₃	1	1	yes	characteristic
157	#1695	C₂×C₂₃	1	2	yes	complemented
158	#1693	C₂×C₂₃	1	2	yes	complemented
159	#1671	C₂×C₂₃	1	4	yes	complemented
160	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#1550	C₂×C₂₃	1	2	yes	complemented
162	C₂²×C₂₃	—	2	1	no	—
163	C₂²×C₂₃	—	2	1	no	—
164	C₂²×C₂₃	—	4	2	no	—
165	C₂²×C₂₃	—	4	2	no	—
166	C₂²×C₂₃	—	4	2	no	—
167	C₂²×C₂₃	#461	1	2	yes	—
168	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
169	C₄×C₂₃	—	2	1	no	—
170	C₄×C₂₃	—	2	1	no	—
171	C₄×C₂₃	—	2	1	no	—
172	C₄×C₂₃	—	4	2	no	—
173	C₄×C₂₃	—	8	2	no	—
174	C₄×C₂₃	—	8	2	no	—
175	C₄×C₂₃	#461	1	2	yes	—
176	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
177	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
178	C₄×C₂₃	#577	1	2	yes	—
179	C₂×#131	—	2	1	no	—
180	#533	C₂²×C₂₃	1	2	yes	—
181	#532	C₂²×C₂₃	1	1	yes	characteristic
182	#500	C₂²×C₂₃	1	1	yes	characteristic
183	#500	C₂²×C₂₃	1	1	yes	characteristic
184	#499	—	2	1	no	—
185	#499	—	2	1	no	—
186	#498	C₂²×C₂₃	1	2	yes	—
187	#476	C₂²×C₂₃	1	4	yes	—
188	#475	—	2	2	no	—
189	#465	—	2	4	no	—
190	#461	C₂²×C₂₃	1	4	yes	—
191	#460	—	2	2	no	—
192	#436	—	2	1	no	—
193	#436	C₂²×C₂₃	1	2	yes	—
194	C₈×D₄	—	2	1	no	—
195	C₈×D₄	C₂²×C₂₃	1	2	yes	—
196	#427	—	2	1	no	—
197	#427	—	2	1	no	—
198	#423	C₂²×C₂₃	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
200	#408	—	2	1	no	—
201	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
202	#334	C₂²×C₂₃	1	2	yes	—
203	#334	C₂²×C₂₃	1	2	yes	—
204	#333	C₂²×C₂₃	1	4	yes	—
205	#332	C₂²×C₂₃	1	4	yes	—
206	C₂×C₂₃	—	2	2	no	—
207	C₂×C₂₃	—	8	2	no	—
208	C₂×C₂₃	C₂×#461	1	1	yes	characteristic
209	C₂×C₂₃	#3060	1	1	yes	characteristic
210	C₂×C₂₃	#3067	1	1	yes	characteristic
211	C₂×#41	—	2	1	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	#131	—	4	4	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
220	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
221	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
222	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
223	#127	D₄×C₂₃	1	1	yes	characteristic
224	#125	—	2	2	no	—
225	#124	—	2	1	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	#121	—	2	1	no	—
229	#121	D₄×C₂₃	1	2	yes	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	C₂³×C₂₃	1	2	yes	—
234	#117	—	2	2	no	—
235	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
236	#107	—	2	1	no	—
237	#107	—	2	1	no	—
238	#107	—	2	1	no	—
239	#107	—	2	1	no	—
240	#106	—	2	2	no	—
241	#106	—	2	2	no	—
242	#106	—	2	4	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
246	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
247	#103	—	2	4	no	—
248	#102	—	2	4	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	—	2	2	no	—
251	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
252	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
255	C₂₃	#19194	1	1	yes	characteristic, direct product factor, Sylow
256	#41	—	4	2	no	—
257	#41	—	4	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
269	SD₁₆	—	8	4	no	—
270	M₁₆	—	4	2	no	—
271	M₁₆	—	4	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	4	no	—
276	C₂×C₈	—	8	2	no	—
277	C₂×C₈	SD₁₆×C₂₃	1	4	yes	complemented
278	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
296	C₄²	#41×C₂₃	1	1	yes	characteristic
297	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
298	C₂³	—	2	2	no	—
299	C₂³	C₂₃×#121	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
316	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
317	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
318	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
319	C₂×C₄	C₂₃×#137	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
322	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	4	no	—
327	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
328	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₂₃×#461	1	2	yes	—
335	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₂₃×#461	1	2	yes	—
343	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
344	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
345	C₄	C₂₃×#577	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	8	2	no	—
348	C₂	C₂×C₂₃×#461	1	1	yes	characteristic
349	C₂	C₂₃×#3060	1	1	yes	characteristic
350	C₂	C₂₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
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×#3160	C₂	semidirect
4. C₂₃×#1695	C₂	semidirect
5. C₂₃×#1693	C₂	semidirect
6. C₂₃×#1671	C₂	semidirect
7. C₂×C₂₃×#334	C₂	semidirect
8. C₂₃×#1550	C₂	semidirect
101. C₂×C₈×C₂₃	SD₁₆	semidirect
122. #19194	C₂₃	direct (N × H)
155. #3160	C₂×C₂₃	semidirect
157. #1695	C₂×C₂₃	semidirect
158. #1693	C₂×C₂₃	semidirect
159. #1671	C₂×C₂₃	semidirect
160. C₂×#334	C₂×C₂₃	semidirect
161. #1550	C₂×C₂₃	semidirect
255. C₂₃	#19194	direct (N × H)
277. 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×#3246	2
C₂₃×#3160	2
C₂₃×#2931	2
C₂₃×#1695	2
C₂₃×#1693	2
C₂₃×#1671	2
C₂×C₂₃×#334	2
C₂₃×#1550	2
#19194	23

Subgroups — C23×#19194 #19166820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19194 #19166820