Subgroups — C₂₃×#19187 #19166813

Order 5888.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3205	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1687	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1667	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈×C₂₃	—	2	1	no	—
9	C₂²×D₈×C₂₃	—	2	1	no	—
10	C₂×C₂₃×#131	—	2	1	no	—
11	C₂₃×#532	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#127	C₂²	1	2	yes	—
13	C₂₃×#493	—	2	1	no	—
14	C₂₃×#493	—	2	1	no	—
15	C₂₃×#493	—	2	2	no	—
16	C₂₃×#493	C₂²	1	2	yes	—
17	C₂₃×#493	C₂²	1	2	yes	—
18	C₂₃×#486	—	2	2	no	—
19	C₂₃×#459	C₂²	1	8	yes	complemented
20	C₂₃×#447	—	2	4	no	—
21	C₂₃×#436	—	2	1	no	—
22	C₂₃×#436	C₂²	1	2	yes	complemented
23	C₈×D₄×C₂₃	—	2	1	no	—
24	C₈×D₄×C₂₃	C₂²	1	2	yes	complemented
25	C₂₃×#431	C₂²	1	2	yes	—
26	C₂₃×#423	C₂²	1	1	yes	characteristic
27	C₂₃×#408	—	2	1	no	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#335	C₂²	1	2	yes	complemented
30	C₂₃×#334	C₂²	1	2	yes	complemented
31	C₂₃×#331	C₂²	1	8	yes	complemented
32	C₂×#41×C₂₃	—	2	1	no	—
33	C₂²×D₄×C₂₃	—	2	2	no	—
34	C₂²×D₄×C₂₃	—	2	2	no	—
35	C₂²×D₄×C₂₃	—	2	2	no	—
36	C₂×D₈×C₂₃	—	2	8	no	—
37	C₂×D₈×C₂₃	—	2	2	no	—
38	C₂×D₈×C₂₃	—	2	2	no	—
39	C₂×D₈×C₂₃	—	2	2	no	—
40	C₂×D₈×C₂₃	—	2	2	no	—
41	C₂×D₈×C₂₃	—	4	4	no	—
42	C₂×D₈×C₂₃	—	4	4	no	—
43	C₂₃×#131	—	4	4	no	—
44	C₂×M₁₆×C₂₃	—	2	1	no	—
45	C₂×M₁₆×C₂₃	—	2	2	no	—
46	C₂²×C₈×C₂₃	—	2	2	no	—
47	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
50	C₂₃×#127	—	2	2	no	—
51	C₂₃×#127	—	2	2	no	—
52	C₂₃×#127	—	2	2	no	—
53	C₂₃×#127	D₄	1	1	yes	characteristic
54	C₂₃×#127	C₂³	1	4	yes	—
55	C₂₃×#124	—	2	1	no	—
56	C₂₃×#121	—	2	1	no	—
57	C₂₃×#121	D₄	1	2	yes	—
58	C₄×D₄×C₂₃	—	2	1	no	—
59	C₄×D₄×C₂₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
63	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
64	C₂₃×#102	—	2	8	no	—
65	C₂₃×C₂²⋊C₈	—	2	2	no	—
66	C₂₃×C₂²⋊C₈	—	2	2	no	—
67	C₂₃×C₂²⋊C₈	—	2	2	no	—
68	C₄×C₈×C₂₃	—	2	2	no	—
69	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂⁴×C₂₃	—	4	2	no	—
74	#41×C₂₃	—	4	2	no	—
75	#41×C₂₃	—	4	2	no	—
76	C₂×Q₈×C₂₃	—	2	1	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	2	1	no	—
79	C₂×D₄×C₂₃	—	2	1	no	—
80	C₂×D₄×C₂₃	—	2	4	no	—
81	C₂×D₄×C₂₃	—	2	4	no	—
82	C₂×D₄×C₂₃	—	2	4	no	—
83	C₂×D₄×C₂₃	—	4	2	no	—
84	C₂×D₄×C₂₃	—	4	2	no	—
85	C₂×D₄×C₂₃	—	4	2	no	—
86	C₂×D₄×C₂₃	—	4	4	no	—
87	C₂×D₄×C₂₃	—	4	4	no	—
88	C₂×D₄×C₂₃	—	8	4	no	—
89	C₂×D₄×C₂₃	—	8	4	no	—
90	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₂₃	—	2	2	no	—
92	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
95	D₈×C₂₃	—	4	4	no	—
96	D₈×C₂₃	—	4	4	no	—
97	D₈×C₂₃	—	4	4	no	—
98	D₈×C₂₃	—	4	4	no	—
99	D₈×C₂₃	—	8	8	no	—
100	M₁₆×C₂₃	—	4	2	no	—
101	M₁₆×C₂₃	—	4	4	no	—
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	4	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	—	4	4	no	—
106	C₂×C₈×C₂₃	D₈	1	4	yes	complemented
107	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₂₃	—	2	2	no	—
114	C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₃	#41	1	1	yes	characteristic
118	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
119	#19187	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	2	2	no	—
121	C₂³×C₂₃	—	4	4	no	—
122	C₂³×C₂₃	—	8	4	no	—
123	C₂³×C₂₃	—	8	4	no	—
124	C₂³×C₂₃	—	8	4	no	—
125	C₂³×C₂₃	#121	1	1	yes	characteristic
126	Q₈×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	2	4	no	—
128	D₄×C₂₃	—	4	4	no	—
129	D₄×C₂₃	—	4	4	no	—
130	D₄×C₂₃	—	4	4	no	—
131	D₄×C₂₃	—	4	8	no	—
132	D₄×C₂₃	—	4	2	no	—
133	D₄×C₂₃	—	4	2	no	—
134	D₄×C₂₃	—	8	4	no	—
135	D₄×C₂₃	—	8	4	no	—
136	D₄×C₂₃	—	8	4	no	—
137	D₄×C₂₃	—	8	4	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	4	2	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₂₃	—	2	1	no	—
152	C₈×C₂₃	—	2	1	no	—
153	C₈×C₂₃	—	4	2	no	—
154	C₈×C₂₃	—	4	4	no	—
155	C₈×C₂₃	C₂×D₈	1	2	yes	—
156	C₈×C₂₃	C₂×D₈	1	2	yes	—
157	#3205	C₂×C₂₃	1	2	yes	complemented
158	C₂×#493	C₂×C₂₃	1	1	yes	characteristic, complemented
159	#2931	C₂×C₂₃	1	1	yes	characteristic, complemented
160	#1687	C₂×C₂₃	1	4	yes	complemented
161	#1667	C₂×C₂₃	1	4	yes	complemented
162	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
163	#1545	C₂×C₂₃	1	2	yes	complemented
164	C₂²×C₂₃	—	2	1	no	—
165	C₂²×C₂₃	—	2	1	no	—
166	C₂²×C₂₃	—	4	2	no	—
167	C₂²×C₂₃	—	4	2	no	—
168	C₂²×C₂₃	—	4	2	no	—
169	C₂²×C₂₃	—	8	4	no	—
170	C₂²×C₂₃	—	8	4	no	—
171	C₂²×C₂₃	—	8	4	no	—
172	C₂²×C₂₃	—	16	8	no	—
173	C₂²×C₂₃	#459	1	2	yes	—
174	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
175	C₄×C₂₃	—	2	1	no	—
176	C₄×C₂₃	—	2	1	no	—
177	C₄×C₂₃	—	4	2	no	—
178	C₄×C₂₃	#459	1	2	yes	—
179	C₄×C₂₃	#472	1	2	yes	—
180	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
181	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₂₃	#576	1	2	yes	—
183	C₂²×D₈	—	2	1	no	—
184	C₂²×D₈	—	2	1	no	—
185	C₂×#131	—	2	1	no	—
186	#532	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×#127	C₂²×C₂₃	1	2	yes	—
188	#493	—	2	2	no	—
189	#493	—	2	1	no	—
190	#493	—	2	1	no	—
191	#493	C₂²×C₂₃	1	2	yes	—
192	#493	C₂²×C₂₃	1	2	yes	—
193	#486	—	2	2	no	—
194	#459	C₂²×C₂₃	1	8	yes	complemented
195	#447	—	2	4	no	—
196	#436	—	2	1	no	—
197	#436	C₂²×C₂₃	1	2	yes	complemented
198	C₈×D₄	—	2	1	no	—
199	C₈×D₄	C₂²×C₂₃	1	2	yes	complemented
200	#431	C₂²×C₂₃	1	2	yes	—
201	#423	C₂²×C₂₃	1	1	yes	characteristic
202	#408	—	2	1	no	—
203	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
204	#335	C₂²×C₂₃	1	2	yes	complemented
205	#334	C₂²×C₂₃	1	2	yes	complemented
206	#331	C₂²×C₂₃	1	8	yes	complemented
207	C₂×C₂₃	—	2	2	no	—
208	C₂×C₂₃	—	8	2	no	—
209	C₂×C₂₃	—	16	4	no	—
210	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
211	C₂×C₂₃	#3058	1	1	yes	characteristic
212	C₂×C₂₃	C₂×#472	1	1	yes	characteristic
213	C₂×#41	—	2	1	no	—
214	C₂²×D₄	—	2	2	no	—
215	C₂²×D₄	—	2	2	no	—
216	C₂²×D₄	—	2	2	no	—
217	C₂×D₈	—	2	8	no	—
218	C₂×D₈	—	2	2	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	4	4	no	—
223	C₂×D₈	—	4	4	no	—
224	#131	—	4	4	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
230	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
231	#127	—	2	2	no	—
232	#127	—	2	2	no	—
233	#127	—	2	2	no	—
234	#127	D₄×C₂₃	1	1	yes	characteristic
235	#127	C₂³×C₂₃	1	4	yes	—
236	#124	—	2	1	no	—
237	#121	—	2	1	no	—
238	#121	D₄×C₂₃	1	2	yes	—
239	C₄×D₄	—	2	1	no	—
240	C₄×D₄	C₂³×C₂₃	1	2	yes	—
241	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
244	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
245	#102	—	2	8	no	—
246	C₂²⋊C₈	—	2	2	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
251	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
254	C₂₃	#19187	1	1	yes	characteristic, direct product factor, Sylow
255	C₂⁴	—	4	2	no	—
256	#41	—	4	2	no	—
257	#41	—	4	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	4	no	—
270	C₂×D₄	—	8	4	no	—
271	C₂×D₄	—	8	4	no	—
272	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
277	D₈	—	4	4	no	—
278	D₈	—	4	4	no	—
279	D₈	—	4	4	no	—
280	D₈	—	4	4	no	—
281	D₈	—	8	8	no	—
282	M₁₆	—	4	2	no	—
283	M₁₆	—	4	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	4	no	—
288	C₂×C₈	D₈×C₂₃	1	4	yes	complemented
289	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
299	C₄²	#41×C₂₃	1	1	yes	characteristic
300	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	—	4	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	8	4	no	—
306	C₂³	C₂₃×#121	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	8	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	4	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
325	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
326	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
328	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
329	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	4	no	—
336	C₈	C₂×D₈×C₂₃	1	2	yes	—
337	C₈	C₂×D₈×C₂₃	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	8	4	no	—
345	C₂²	—	8	4	no	—
346	C₂²	—	16	8	no	—
347	C₂²	C₂₃×#459	1	2	yes	—
348	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	C₂₃×#459	1	2	yes	—
353	C₄	C₂₃×#472	1	2	yes	—
354	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
355	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
356	C₄	C₂₃×#576	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	—	16	4	no	—
360	C₂	C₂×C₂₃×#459	1	1	yes	characteristic
361	C₂	C₂₃×#3058	1	1	yes	characteristic
362	C₂	C₂×C₂₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
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×#3205	C₂	semidirect
2. C₂×C₂₃×#493	C₂	semidirect
3. C₂₃×#2931	C₂	semidirect
4. C₂₃×#1687	C₂	semidirect
5. C₂₃×#1667	C₂	semidirect
6. C₂₃×#1564	C₂	semidirect
7. C₂₃×#1545	C₂	semidirect
19. C₂₃×#459	C₂²	semidirect
22. C₂₃×#436	C₂²	semidirect
24. C₈×D₄×C₂₃	C₂²	semidirect
29. C₂₃×#335	C₂²	semidirect
30. C₂₃×#334	C₂²	semidirect
31. C₂₃×#331	C₂²	semidirect
106. C₂×C₈×C₂₃	D₈	semidirect
119. #19187	C₂₃	direct (N × H)
157. #3205	C₂×C₂₃	semidirect
158. C₂×#493	C₂×C₂₃	semidirect
159. #2931	C₂×C₂₃	semidirect
160. #1687	C₂×C₂₃	semidirect
161. #1667	C₂×C₂₃	semidirect
162. #1564	C₂×C₂₃	semidirect
163. #1545	C₂×C₂₃	semidirect
194. #459	C₂²×C₂₃	semidirect
197. #436	C₂²×C₂₃	semidirect
199. C₈×D₄	C₂²×C₂₃	semidirect
204. #335	C₂²×C₂₃	semidirect
205. #334	C₂²×C₂₃	semidirect
206. #331	C₂²×C₂₃	semidirect
254. C₂₃	#19187	direct (N × H)
288. C₂×C₈	D₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3205	2
C₂×C₂₃×#493	2
C₂₃×#2931	2
C₂₃×#1687	2
C₂₃×#1667	2
C₂₃×#1564	2
C₂₃×#1545	2
#19187	23

Subgroups — C23×#19187 #19166813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19187 #19166813