Subgroups — C₂₃×#19349 #19166975

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3254	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1696	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1688	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1678	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1560	C₂	1	2	yes	maximal, complemented
9	C₂₃×#533	C₂²	1	1	yes	characteristic
10	C₂₃×#532	C₂²	1	2	yes	—
11	C₂₃×#500	C₂²	1	1	yes	characteristic
12	C₂₃×#493	C₂²	1	1	yes	characteristic
13	C₂₃×#486	C₂²	1	2	yes	—
14	C₂₃×#477	C₂²	1	4	yes	—
15	C₂₃×#474	—	2	2	no	—
16	C₂₃×#468	—	2	4	no	—
17	C₂₃×#466	—	2	1	no	—
18	C₂₃×#466	—	2	1	no	—
19	C₂₃×#461	—	2	2	no	—
20	C₂₃×#460	C₂²	1	4	yes	—
21	C₂₃×#446	—	2	1	no	—
22	C₂₃×#446	C₂²	1	2	yes	—
23	C₈×Q₈×C₂₃	—	2	1	no	—
24	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₄×D₈×C₂₃	—	2	1	no	—
26	C₄×D₈×C₂₃	—	2	1	no	—
27	C₂₃×#424	—	2	1	no	—
28	C₂₃×#423	C₂²	1	1	yes	characteristic
29	C₂₃×#423	C₂²	1	2	yes	—
30	C₂₃×#405	—	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×#331	C₂²	1	4	yes	—
36	C₂×#41×C₂₃	—	2	2	no	—
37	C₂×SD₁₆×C₂₃	—	2	4	no	—
38	C₂×D₈×C₂₃	—	2	1	no	—
39	C₂×D₈×C₂₃	—	2	1	no	—
40	C₂×M₁₆×C₂₃	—	2	1	no	—
41	C₂×M₁₆×C₂₃	—	2	2	no	—
42	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×#127	C₂³	1	2	yes	—
48	C₂₃×#125	—	2	1	no	—
49	C₂₃×#124	—	2	2	no	—
50	C₂₃×C₂²⋊Q₈	—	2	1	no	—
51	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
52	C₂₃×#121	—	2	2	no	—
53	C₂₃×#121	—	2	2	no	—
54	C₄×Q₈×C₂₃	—	2	1	no	—
55	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₄×D₄×C₂₃	—	2	2	no	—
57	C₄×D₄×C₂₃	—	2	2	no	—
58	C₂₃×#117	—	2	1	no	—
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#107	—	2	1	no	—
61	C₂₃×#107	—	2	1	no	—
62	C₂₃×#107	—	2	4	no	—
63	C₂₃×C₄⋊C₈	—	2	2	no	—
64	C₂₃×C₄⋊C₈	—	2	2	no	—
65	C₂₃×C₄⋊C₈	—	2	1	no	—
66	C₂₃×C₄⋊C₈	—	2	2	no	—
67	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
69	C₂₃×#103	—	2	4	no	—
70	C₂₃×#102	—	2	2	no	—
71	C₂₃×#102	—	2	2	no	—
72	C₂₃×#102	—	2	4	no	—
73	C₂₃×#97	—	2	2	no	—
74	C₄×C₈×C₂₃	—	2	2	no	—
75	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	#41×C₂₃	—	8	4	no	—
80	C₂×Q₈×C₂₃	—	2	1	no	—
81	C₂×Q₈×C₂₃	—	2	2	no	—
82	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
83	C₂×D₄×C₂₃	—	2	2	no	—
84	C₂×D₄×C₂₃	—	2	2	no	—
85	C₂×D₄×C₂₃	—	2	2	no	—
86	C₂×D₄×C₂₃	—	4	2	no	—
87	C₂²×C₄×C₂₃	—	4	2	no	—
88	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
91	SD₁₆×C₂₃	—	8	4	no	—
92	D₈×C₂₃	—	4	2	no	—
93	D₈×C₂₃	—	4	2	no	—
94	M₁₆×C₂₃	—	4	2	no	—
95	M₁₆×C₂₃	—	8	2	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	1	no	—
100	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₄⋊C₄×C₂₃	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₂₃	—	2	2	no	—
113	C₂²⋊C₄×C₂₃	—	4	2	no	—
114	C₂²⋊C₄×C₂₃	—	4	2	no	—
115	C₄²×C₂₃	—	2	2	no	—
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₂₃	#41	1	1	yes	characteristic
119	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
120	#19349	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₃	—	4	2	no	—
122	C₂³×C₂₃	#121	1	1	yes	characteristic
123	Q₈×C₂₃	—	2	4	no	—
124	Q₈×C₂₃	—	4	2	no	—
125	Q₈×C₂₃	—	8	2	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	2	no	—
128	D₄×C₂₃	—	4	2	no	—
129	D₄×C₂₃	—	4	2	no	—
130	D₄×C₂₃	—	8	2	no	—
131	D₄×C₂₃	—	8	4	no	—
132	C₂×C₄×C₂₃	—	2	1	no	—
133	C₂×C₄×C₂₃	—	2	1	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	8	4	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	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₂₃	—	2	1	no	—
147	C₈×C₂₃	—	2	1	no	—
148	C₈×C₂₃	—	2	1	no	—
149	C₈×C₂₃	—	4	2	no	—
150	C₈×C₂₃	—	4	4	no	—
151	C₈×C₂₃	#137	1	2	yes	—
152	#3254	C₂×C₂₃	1	2	yes	complemented
153	#3169	C₂×C₂₃	1	1	yes	characteristic
154	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#1696	C₂×C₂₃	1	2	yes	complemented
156	#1688	C₂×C₂₃	1	2	yes	complemented
157	#1678	C₂×C₂₃	1	4	yes	complemented
158	#1566	C₂×C₂₃	1	1	yes	characteristic, complemented
159	#1560	C₂×C₂₃	1	2	yes	complemented
160	C₂²×C₂₃	—	2	1	no	—
161	C₂²×C₂₃	—	2	1	no	—
162	C₂²×C₂₃	—	2	1	no	—
163	C₂²×C₂₃	—	8	2	no	—
164	C₂²×C₂₃	—	8	2	no	—
165	C₂²×C₂₃	—	8	2	no	—
166	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
167	C₄×C₂₃	—	2	1	no	—
168	C₄×C₂₃	—	2	1	no	—
169	C₄×C₂₃	—	2	1	no	—
170	C₄×C₂₃	—	4	2	no	—
171	C₄×C₂₃	—	4	2	no	—
172	C₄×C₂₃	—	8	2	no	—
173	C₄×C₂₃	#472	1	2	yes	—
174	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
175	C₄×C₂₃	#575	1	1	yes	characteristic
176	C₄×C₂₃	#577	1	2	yes	—
177	#533	C₂²×C₂₃	1	1	yes	characteristic
178	#532	C₂²×C₂₃	1	2	yes	—
179	#500	C₂²×C₂₃	1	1	yes	characteristic
180	#493	C₂²×C₂₃	1	1	yes	characteristic
181	#486	C₂²×C₂₃	1	2	yes	—
182	#477	C₂²×C₂₃	1	4	yes	—
183	#474	—	2	2	no	—
184	#468	—	2	4	no	—
185	#466	—	2	1	no	—
186	#466	—	2	1	no	—
187	#461	—	2	2	no	—
188	#460	C₂²×C₂₃	1	4	yes	—
189	#446	—	2	1	no	—
190	#446	C₂²×C₂₃	1	2	yes	—
191	C₈×Q₈	—	2	1	no	—
192	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
193	C₄×D₈	—	2	1	no	—
194	C₄×D₈	—	2	1	no	—
195	#424	—	2	1	no	—
196	#423	C₂²×C₂₃	1	1	yes	characteristic
197	#423	C₂²×C₂₃	1	2	yes	—
198	#405	—	2	1	no	—
199	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
200	#334	C₂²×C₂₃	1	2	yes	—
201	#334	C₂²×C₂₃	1	2	yes	—
202	#333	C₂²×C₂₃	1	4	yes	—
203	#331	C₂²×C₂₃	1	4	yes	—
204	C₂×C₂₃	—	4	1	no	—
205	C₂×C₂₃	—	16	2	no	—
206	C₂×C₂₃	#3037	1	1	yes	characteristic
207	C₂×C₂₃	#3060	1	1	yes	characteristic
208	C₂×C₂₃	C₂×#472	1	1	yes	characteristic
209	C₂×#41	—	2	2	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
217	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
218	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
219	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
220	#127	C₂³×C₂₃	1	2	yes	—
221	#125	—	2	1	no	—
222	#124	—	2	2	no	—
223	C₂²⋊Q₈	—	2	1	no	—
224	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
225	#121	—	2	2	no	—
226	#121	—	2	2	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	#117	—	2	1	no	—
232	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
233	#107	—	2	1	no	—
234	#107	—	2	1	no	—
235	#107	—	2	4	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
241	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	4	no	—
246	#97	—	2	2	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
249	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
252	C₂₃	#19349	1	1	yes	characteristic, direct product factor, Sylow
253	#41	—	8	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
265	SD₁₆	—	8	4	no	—
266	D₈	—	4	2	no	—
267	D₈	—	4	2	no	—
268	M₁₆	—	4	2	no	—
269	M₁₆	—	8	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
292	C₄²	#41×C₂₃	1	1	yes	characteristic
293	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
294	C₂³	—	4	2	no	—
295	C₂³	C₂₃×#121	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	4	2	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	4	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
314	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
315	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
316	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
318	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	4	2	no	—
323	C₈	—	4	4	no	—
324	C₈	C₂₃×#137	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₂₃×#472	1	2	yes	—
339	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
340	C₄	C₂₃×#575	1	1	yes	characteristic
341	C₄	C₂₃×#577	1	2	yes	—
342	C₂	—	4	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₂₃×#3037	1	1	yes	characteristic
345	C₂	C₂₃×#3060	1	1	yes	characteristic
346	C₂	C₂×C₂₃×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3169	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×#3254	C₂	semidirect
3. C₂₃×#2960	C₂	semidirect
4. C₂₃×#1696	C₂	semidirect
5. C₂₃×#1688	C₂	semidirect
6. C₂₃×#1678	C₂	semidirect
7. C₂₃×#1566	C₂	semidirect
8. C₂₃×#1560	C₂	semidirect
120. #19349	C₂₃	direct (N × H)
152. #3254	C₂×C₂₃	semidirect
154. #2960	C₂×C₂₃	semidirect
155. #1696	C₂×C₂₃	semidirect
156. #1688	C₂×C₂₃	semidirect
157. #1678	C₂×C₂₃	semidirect
158. #1566	C₂×C₂₃	semidirect
159. #1560	C₂×C₂₃	semidirect
252. C₂₃	#19349	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×#3254	2
C₂₃×#3169	2
C₂₃×#2960	2
C₂₃×#1696	2
C₂₃×#1688	2
C₂₃×#1678	2
C₂₃×#1566	2
C₂₃×#1560	2
#19349	23

Subgroups — C23×#19349 #19166975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19349 #19166975