Subgroups — C₂₃×#19746 #19167372

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3182	C₂	1	2	yes	maximal
3	C₂₃×#3165	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1723	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1722	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1716	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1714	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1705	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#137	—	2	2	no	—
13	C₂₃×#533	C₂²	1	2	yes	—
14	C₂₃×#532	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×#501	—	2	4	no	—
17	C₂₃×#500	C₂²	1	2	yes	—
18	C₂₃×#500	C₂²	1	2	yes	—
19	C₂₃×#500	C₂²	1	2	yes	—
20	C₂₃×#498	C₂²	1	2	yes	—
21	C₂₃×#498	C₂²	1	2	yes	—
22	C₂₃×#498	C₂²	1	1	yes	characteristic
23	C₂₃×#497	—	2	2	no	—
24	C₂₃×#494	C₂²	1	1	yes	characteristic
25	C₂₃×#494	C₂²	1	1	yes	characteristic
26	C₂₃×#494	C₂²	1	1	yes	characteristic
27	C₂₃×#492	C₂²	1	2	yes	—
28	C₂₃×#492	C₂²	1	1	yes	characteristic
29	C₂₃×#492	C₂²	1	1	yes	characteristic
30	C₂₃×#487	C₂²	1	2	yes	—
31	C₂₃×#487	C₂²	1	2	yes	—
32	C₂₃×#486	C₂²	1	2	yes	—
33	C₂₃×#470	—	2	2	no	—
34	C₂₃×#468	—	2	4	no	—
35	C₂₃×#441	—	2	2	no	—
36	C₂₃×#440	—	2	4	no	—
37	C₂₃×#428	—	2	4	no	—
38	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
39	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#322	C₂²	1	2	yes	—
41	C₂₃×#322	C₂²	1	1	yes	characteristic
42	C₈²×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	—	2	1	no	—
45	C₂₃×#137	—	4	8	no	—
46	C₂×Q₁₆×C₂₃	—	2	2	no	—
47	C₂×Q₁₆×C₂₃	—	2	2	no	—
48	C₂×Q₁₆×C₂₃	—	2	2	no	—
49	C₂×SD₁₆×C₂₃	—	2	2	no	—
50	C₂×SD₁₆×C₂₃	—	2	2	no	—
51	C₂×SD₁₆×C₂₃	—	2	4	no	—
52	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
53	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
54	C₂₃×C₄⋊Q₈	—	2	2	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₂₃×#127	C₂³	1	1	yes	characteristic
62	C₂₃×#125	—	2	2	no	—
63	C₂₃×#124	—	2	1	no	—
64	C₂₃×C₂²⋊Q₈	—	2	2	no	—
65	C₂₃×C₂²⋊Q₈	—	2	2	no	—
66	C₂₃×#121	—	2	2	no	—
67	C₄×Q₈×C₂₃	—	2	2	no	—
68	C₄×Q₈×C₂₃	—	2	2	no	—
69	C₄×D₄×C₂₃	—	2	2	no	—
70	C₂₃×#117	—	2	2	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₂₃×#107	—	2	4	no	—
74	C₂₃×#107	—	2	4	no	—
75	C₂₃×#107	—	2	4	no	—
76	C₂₃×#106	—	2	4	no	—
77	C₂₃×#106	—	2	4	no	—
78	C₂₃×#106	—	2	2	no	—
79	C₂₃×#103	—	2	4	no	—
80	C₂₃×#103	—	2	4	no	—
81	C₂₃×#102	—	2	4	no	—
82	C₂₃×#97	D₄	1	4	yes	—
83	C₂₃×#97	D₄	1	2	yes	—
84	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₂₃	C₂³	1	2	yes	—
86	C₄×C₈×C₂₃	C₂³	1	2	yes	—
87	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
88	#41×C₂₃	—	4	4	no	—
89	C₂×Q₈×C₂₃	—	2	2	no	—
90	C₂×Q₈×C₂₃	—	2	2	no	—
91	C₂×Q₈×C₂₃	—	2	1	no	—
92	C₂×Q₈×C₂₃	—	2	1	no	—
93	C₂×Q₈×C₂₃	—	2	1	no	—
94	C₂×Q₈×C₂₃	—	4	4	no	—
95	C₂×Q₈×C₂₃	—	4	1	no	—
96	C₂×D₄×C₂₃	—	2	2	no	—
97	C₂×D₄×C₂₃	—	2	1	no	—
98	C₂×D₄×C₂₃	—	4	1	no	—
99	C₂²×C₄×C₂₃	—	4	1	no	—
100	C₂²×C₄×C₂₃	—	4	1	no	—
101	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₂₃	—	4	4	no	—
104	Q₁₆×C₂₃	—	4	4	no	—
105	Q₁₆×C₂₃	—	4	4	no	—
106	SD₁₆×C₂₃	—	4	4	no	—
107	SD₁₆×C₂₃	—	4	4	no	—
108	SD₁₆×C₂₃	—	4	8	no	—
109	M₁₆×C₂₃	—	2	4	no	—
110	M₁₆×C₂₃	—	2	8	no	—
111	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₄⋊C₄×C₂₃	—	2	2	no	—
119	C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₄⋊C₄×C₂₃	—	2	2	no	—
122	C₄⋊C₄×C₂₃	—	4	2	no	—
123	C₄⋊C₄×C₂₃	—	4	2	no	—
124	C₄⋊C₄×C₂₃	—	4	2	no	—
125	C₂²⋊C₄×C₂₃	—	4	2	no	—
126	C₂²⋊C₄×C₂₃	—	4	2	no	—
127	C₄²×C₂₃	—	4	2	no	—
128	C₄²×C₂₃	C₂×D₄	1	2	yes	—
129	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
131	#19746	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₂₃	—	4	1	no	—
133	C₂³×C₂₃	#127	1	1	yes	characteristic
134	Q₈×C₂₃	—	4	2	no	—
135	Q₈×C₂₃	—	4	2	no	—
136	Q₈×C₂₃	—	4	2	no	—
137	Q₈×C₂₃	—	4	4	no	—
138	Q₈×C₂₃	—	4	4	no	—
139	Q₈×C₂₃	—	8	2	no	—
140	D₄×C₂₃	—	4	2	no	—
141	D₄×C₂₃	—	4	4	no	—
142	D₄×C₂₃	—	8	2	no	—
143	C₂×C₄×C₂₃	—	2	2	no	—
144	C₂×C₄×C₂₃	—	2	4	no	—
145	C₂×C₄×C₂₃	—	4	2	no	—
146	C₂×C₄×C₂₃	—	4	2	no	—
147	C₂×C₄×C₂₃	—	4	1	no	—
148	C₂×C₄×C₂₃	—	4	1	no	—
149	C₂×C₄×C₂₃	—	4	1	no	—
150	C₂×C₄×C₂₃	—	8	2	no	—
151	C₂×C₄×C₂₃	—	8	2	no	—
152	C₂×C₄×C₂₃	#127	1	2	yes	—
153	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
154	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
156	C₈×C₂₃	—	2	2	no	—
157	C₈×C₂₃	—	2	4	no	—
158	C₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
159	C₈×C₂₃	#137	1	8	yes	complemented
160	#3185	C₂×C₂₃	1	1	yes	characteristic, complemented
161	#3182	C₂×C₂₃	1	2	yes	—
162	#3165	C₂×C₂₃	1	2	yes	complemented
163	#3152	C₂×C₂₃	1	1	yes	characteristic
164	#1723	C₂×C₂₃	1	2	yes	complemented
165	#1722	C₂×C₂₃	1	1	yes	characteristic, complemented
166	#1716	C₂×C₂₃	1	1	yes	characteristic, complemented
167	#1714	C₂×C₂₃	1	2	yes	complemented
168	#1713	C₂×C₂₃	1	1	yes	characteristic, complemented
169	#1705	C₂×C₂₃	1	1	yes	characteristic, complemented
170	#1455	C₂×C₂₃	1	1	yes	characteristic, complemented
171	C₂²×C₂₃	—	2	1	no	—
172	C₂²×C₂₃	—	2	2	no	—
173	C₂²×C₂₃	—	8	1	no	—
174	C₂²×C₂₃	—	8	2	no	—
175	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
176	C₄×C₂₃	—	2	1	no	—
177	C₄×C₂₃	—	2	2	no	—
178	C₄×C₂₃	—	8	1	no	—
179	C₄×C₂₃	—	8	1	no	—
180	C₄×C₂₃	—	8	1	no	—
181	C₄×C₂₃	—	8	2	no	—
182	C₄×C₂₃	—	8	2	no	—
183	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
184	C₄×C₂₃	C₂×#137	1	4	yes	—
185	C₂×#137	—	2	2	no	—
186	#533	C₂²×C₂₃	1	2	yes	—
187	#532	C₂²×C₂₃	1	1	yes	characteristic
188	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
189	#501	—	2	4	no	—
190	#500	C₂²×C₂₃	1	2	yes	—
191	#500	C₂²×C₂₃	1	2	yes	—
192	#500	C₂²×C₂₃	1	2	yes	—
193	#498	C₂²×C₂₃	1	2	yes	—
194	#498	C₂²×C₂₃	1	2	yes	—
195	#498	C₂²×C₂₃	1	1	yes	characteristic
196	#497	—	2	2	no	—
197	#494	C₂²×C₂₃	1	1	yes	characteristic
198	#494	C₂²×C₂₃	1	1	yes	characteristic
199	#494	C₂²×C₂₃	1	1	yes	characteristic
200	#492	C₂²×C₂₃	1	1	yes	characteristic
201	#492	C₂²×C₂₃	1	1	yes	characteristic
202	#492	C₂²×C₂₃	1	2	yes	—
203	#487	C₂²×C₂₃	1	2	yes	—
204	#487	C₂²×C₂₃	1	2	yes	—
205	#486	C₂²×C₂₃	1	2	yes	—
206	#470	—	2	2	no	—
207	#468	—	2	4	no	—
208	#441	—	2	2	no	—
209	#440	—	2	4	no	—
210	#428	—	2	4	no	—
211	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
212	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
213	#322	C₂²×C₂₃	1	2	yes	—
214	#322	C₂²×C₂₃	1	1	yes	characteristic
215	C₈²	C₂²×C₂₃	1	1	yes	characteristic
216	C₂×C₂₃	—	4	1	no	—
217	C₂×C₂₃	—	16	1	no	—
218	C₂×C₂₃	#3151	1	1	yes	characteristic
219	C₂×C₂₃	#3152	1	2	yes	—
220	C₂×#41	—	2	1	no	—
221	C₂²×Q₈	—	2	1	no	—
222	#137	—	4	8	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×SD₁₆	—	2	4	no	—
229	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
230	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
231	C₄⋊Q₈	—	2	2	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
234	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
235	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
236	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
237	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
238	#127	C₂³×C₂₃	1	1	yes	characteristic
239	#125	—	2	2	no	—
240	#124	—	2	1	no	—
241	C₂²⋊Q₈	—	2	2	no	—
242	C₂²⋊Q₈	—	2	2	no	—
243	#121	—	2	2	no	—
244	C₄×Q₈	—	2	2	no	—
245	C₄×Q₈	—	2	2	no	—
246	C₄×D₄	—	2	2	no	—
247	#117	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
250	#107	—	2	4	no	—
251	#107	—	2	4	no	—
252	#107	—	2	4	no	—
253	#106	—	2	4	no	—
254	#106	—	2	4	no	—
255	#106	—	2	2	no	—
256	#103	—	2	4	no	—
257	#103	—	2	4	no	—
258	#102	—	2	4	no	—
259	#97	D₄×C₂₃	1	4	yes	—
260	#97	D₄×C₂₃	1	2	yes	—
261	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₂₃	1	2	yes	—
263	C₄×C₈	C₂³×C₂₃	1	2	yes	—
264	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
265	C₂₃	#19746	1	1	yes	characteristic, direct product factor, Sylow
266	#41	—	4	4	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
281	Q₁₆	—	4	4	no	—
282	Q₁₆	—	4	4	no	—
283	Q₁₆	—	4	4	no	—
284	SD₁₆	—	4	4	no	—
285	SD₁₆	—	4	4	no	—
286	SD₁₆	—	4	8	no	—
287	M₁₆	—	2	4	no	—
288	M₁₆	—	2	8	no	—
289	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
290	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
291	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₄²	—	4	2	no	—
306	C₄²	C₂×D₄×C₂₃	1	2	yes	—
307	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
308	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
309	C₂³	—	4	1	no	—
310	C₂³	C₂₃×#127	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	4	no	—
315	Q₈	—	4	4	no	—
316	Q₈	—	8	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	4	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	8	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	C₂₃×#127	1	2	yes	—
330	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	2	4	no	—
335	C₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
336	C₈	C₂₃×#137	1	8	yes	complemented
337	C₂²	—	2	1	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
350	C₄	C₂×C₂₃×#137	1	4	yes	—
351	C₂	—	4	1	no	—
352	C₂	—	16	1	no	—
353	C₂	C₂₃×#3151	1	1	yes	characteristic
354	C₂	C₂₃×#3152	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
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
1. C₂₃×#3185	C₂	semidirect
3. C₂₃×#3165	C₂	semidirect
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1722	C₂	semidirect
7. C₂₃×#1716	C₂	semidirect
8. C₂₃×#1714	C₂	semidirect
9. C₂₃×#1713	C₂	semidirect
10. C₂₃×#1705	C₂	semidirect
11. C₂₃×#1455	C₂	semidirect
131. #19746	C₂₃	direct (N × H)
159. C₈×C₂₃	#137	semidirect
160. #3185	C₂×C₂₃	semidirect
162. #3165	C₂×C₂₃	semidirect
164. #1723	C₂×C₂₃	semidirect
165. #1722	C₂×C₂₃	semidirect
166. #1716	C₂×C₂₃	semidirect
167. #1714	C₂×C₂₃	semidirect
168. #1713	C₂×C₂₃	semidirect
169. #1705	C₂×C₂₃	semidirect
170. #1455	C₂×C₂₃	semidirect
265. C₂₃	#19746	direct (N × H)
336. C₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3185	2
C₂₃×#3182	2
C₂₃×#3165	2
C₂₃×#3152	2
C₂₃×#1723	2
C₂₃×#1722	2
C₂₃×#1716	2
C₂₃×#1714	2
C₂₃×#1713	2
C₂₃×#1705	2
C₂₃×#1455	2
#19746	23

Subgroups — C23×#19746 #19167372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19746 #19167372