Subgroups — C₂₃×#10149 #19157775

Order 5888.

528 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3138	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#533	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂₃×#500	—	2	1	no	—
12	C₂₃×#490	—	2	2	no	—
13	C₂₃×#487	—	2	1	no	—
14	C₂₃×#486	—	2	2	no	—
15	C₂₃×#486	—	2	1	no	—
16	C₂₃×#486	—	2	1	no	—
17	C₂₃×#486	—	2	2	no	—
18	C₂₃×#484	—	2	2	no	—
19	C₂₃×#477	—	2	2	no	—
20	C₂₃×#431	C₂²	1	1	yes	characteristic
21	C₂₃×#423	C₂²	1	1	yes	characteristic
22	C₂₃×#418	—	2	2	no	—
23	C₂×C₂₃×#102	—	2	2	no	—
24	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#343	—	2	2	no	—
27	C₂₃×#331	—	2	2	no	—
28	C₂₃×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₂₃	—	2	1	no	—
30	C₂²×D₄×C₂₃	—	2	1	no	—
31	C₂²×D₄×C₂₃	—	2	1	no	—
32	C₂²×D₄×C₂₃	—	2	1	no	—
33	C₂×M₁₆×C₂₃	—	2	1	no	—
34	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
35	C₂²×C₈×C₂₃	—	2	1	no	—
36	C₂₃×C₄⋊Q₈	—	2	1	no	—
37	C₂₃×C₄⋊Q₈	—	2	1	no	—
38	C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	—	2	1	no	—
40	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₂₃×#127	—	2	1	no	—
43	C₂₃×#127	—	2	1	no	—
44	C₂₃×#127	—	2	1	no	—
45	C₂₃×#127	D₄	1	2	yes	—
46	C₂₃×#125	—	2	1	no	—
47	C₂₃×C₂²⋊Q₈	—	2	1	no	—
48	C₂₃×C₂²⋊Q₈	—	2	1	no	—
49	C₄×Q₈×C₂₃	—	2	1	no	—
50	C₄×Q₈×C₂₃	—	2	1	no	—
51	C₂₃×#117	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×#107	—	2	4	no	—
56	C₂₃×C₄⋊C₈	—	2	1	no	—
57	C₂₃×C₄⋊C₈	—	2	1	no	—
58	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
59	C₂₃×#103	—	2	4	no	—
60	C₂₃×#102	—	2	8	no	—
61	C₂₃×#102	—	2	4	no	—
62	C₂₃×#102	—	2	4	no	—
63	C₂₃×C₂²⋊C₈	—	2	1	no	—
64	C₂₃×#97	—	2	1	no	—
65	C₄×C₈×C₂₃	—	2	1	no	—
66	C₄×C₈×C₂₃	—	2	1	no	—
67	C₄×C₈×C₂₃	—	2	1	no	—
68	C₄×C₈×C₂₃	D₄	1	2	yes	—
69	C₂⁴×C₂₃	—	4	1	no	—
70	C₂×Q₈×C₂₃	—	2	1	no	—
71	C₂×Q₈×C₂₃	—	2	1	no	—
72	C₂×Q₈×C₂₃	—	4	1	no	—
73	C₂×Q₈×C₂₃	—	4	1	no	—
74	C₂×Q₈×C₂₃	—	8	2	no	—
75	C₂×D₄×C₂₃	—	2	2	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	4	4	no	—
79	C₂×D₄×C₂₃	—	4	1	no	—
80	C₂×D₄×C₂₃	—	4	1	no	—
81	C₂×D₄×C₂₃	—	4	1	no	—
82	C₂×D₄×C₂₃	—	4	2	no	—
83	C₂×D₄×C₂₃	—	4	2	no	—
84	C₂×D₄×C₂₃	—	8	2	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
89	M₁₆×C₂₃	—	2	2	no	—
90	M₁₆×C₂₃	—	2	2	no	—
91	M₁₆×C₂₃	—	4	2	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	—	2	2	no	—
95	C₂×C₈×C₂₃	—	4	2	no	—
96	C₂×C₈×C₂₃	—	4	1	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₄⋊C₄×C₂₃	—	4	1	no	—
104	C₄⋊C₄×C₂₃	—	4	1	no	—
105	C₄⋊C₄×C₂₃	—	4	1	no	—
106	C₄⋊C₄×C₂₃	—	4	1	no	—
107	C₂²⋊C₄×C₂₃	—	4	1	no	—
108	C₄²×C₂₃	—	4	1	no	—
109	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
113	#10149	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₃	—	4	2	no	—
115	C₂³×C₂₃	—	8	2	no	—
116	C₂³×C₂₃	—	8	2	no	—
117	C₂³×C₂₃	—	8	2	no	—
118	C₂³×C₂₃	#121	1	1	yes	characteristic
119	Q₈×C₂₃	—	4	2	no	—
120	Q₈×C₂₃	—	4	2	no	—
121	Q₈×C₂₃	—	8	2	no	—
122	Q₈×C₂₃	—	8	2	no	—
123	D₄×C₂₃	—	4	4	no	—
124	D₄×C₂₃	—	4	4	no	—
125	D₄×C₂₃	—	8	2	no	—
126	D₄×C₂₃	—	8	2	no	—
127	D₄×C₂₃	—	8	2	no	—
128	D₄×C₂₃	—	8	1	no	—
129	D₄×C₂₃	—	8	1	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	4	1	no	—
133	C₂×C₄×C₂₃	—	4	1	no	—
134	C₂×C₄×C₂₃	—	4	2	no	—
135	C₂×C₄×C₂₃	—	8	2	no	—
136	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
142	C₂×C₄×C₂₃	#135	1	2	yes	—
143	C₂×C₄×C₂₃	#137	1	2	yes	—
144	C₈×C₂₃	—	4	2	no	—
145	C₈×C₂₃	—	4	2	no	—
146	C₈×C₂₃	—	4	2	no	—
147	C₈×C₂₃	—	8	1	no	—
148	#3230	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#3138	C₂×C₂₃	1	1	yes	characteristic
150	C₂×#486	C₂×C₂₃	1	1	yes	characteristic
151	#1997	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#1545	C₂×C₂₃	1	1	yes	characteristic
153	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#1278	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	2	1	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	—	8	2	no	—
159	C₂²×C₂₃	—	8	2	no	—
160	C₂²×C₂₃	—	16	4	no	—
161	C₂²×C₂₃	#392	1	1	yes	characteristic
162	C₂²×C₂₃	#464	1	2	yes	—
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	2	1	no	—
165	C₄×C₂₃	—	8	1	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	—	8	1	no	—
168	C₄×C₂₃	#452	1	2	yes	—
169	C₄×C₂₃	#454	1	2	yes	—
170	C₄×C₂₃	#472	1	2	yes	—
171	C₄×C₂₃	#495	1	1	yes	characteristic
172	C₄×C₂₃	#497	1	1	yes	characteristic
173	#533	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
176	#500	—	2	1	no	—
177	#490	—	2	2	no	—
178	#487	—	2	1	no	—
179	#486	—	2	1	no	—
180	#486	—	2	1	no	—
181	#486	—	2	2	no	—
182	#486	—	2	2	no	—
183	#484	—	2	2	no	—
184	#477	—	2	2	no	—
185	#431	C₂²×C₂₃	1	1	yes	characteristic
186	#423	C₂²×C₂₃	1	1	yes	characteristic
187	#418	—	2	2	no	—
188	C₂×#102	—	2	2	no	—
189	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
191	#343	—	2	2	no	—
192	#331	—	2	2	no	—
193	Q₈⋊C₈	—	2	2	no	—
194	C₂×C₂₃	—	2	2	no	—
195	C₂×C₂₃	—	16	2	no	—
196	C₂×C₂₃	#2003	1	1	yes	characteristic
197	C₂×C₂₃	#2015	1	1	yes	characteristic
198	C₂×C₂₃	#2018	1	1	yes	characteristic
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
205	C₂²×C₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
211	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
212	#127	—	2	1	no	—
213	#127	—	2	1	no	—
214	#127	—	2	1	no	—
215	#127	D₄×C₂₃	1	2	yes	—
216	#125	—	2	1	no	—
217	C₂²⋊Q₈	—	2	1	no	—
218	C₂²⋊Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	#117	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
224	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
225	#107	—	2	4	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
229	#103	—	2	4	no	—
230	#102	—	2	8	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	D₄×C₂₃	1	2	yes	—
239	C₂₃	#10149	1	1	yes	characteristic, direct product factor, Sylow
240	C₂⁴	—	4	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	8	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	8	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	—	8	2	no	—
286	C₂³	—	8	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	C₂₃×#121	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
307	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
308	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
309	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
310	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
312	C₂×C₄	C₂₃×#135	1	2	yes	—
313	C₂×C₄	C₂₃×#137	1	2	yes	—
314	C₈	—	4	2	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	8	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	4	no	—
324	C₂²	C₂₃×#392	1	1	yes	characteristic
325	C₂²	C₂₃×#464	1	2	yes	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₂₃×#452	1	2	yes	—
332	C₄	C₂₃×#454	1	2	yes	—
333	C₄	C₂₃×#472	1	2	yes	—
334	C₄	C₂₃×#495	1	1	yes	characteristic
335	C₄	C₂₃×#497	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₂₃×#2003	1	1	yes	characteristic
339	C₂	C₂₃×#2015	1	1	yes	characteristic
340	C₂	C₂₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×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×#3230	C₂	semidirect
4. C₂₃×#1997	C₂	semidirect
6. C₂₃×#1493	C₂	semidirect
7. C₂₃×#1278	C₂	semidirect
113. #10149	C₂₃	direct (N × H)
148. #3230	C₂×C₂₃	semidirect
151. #1997	C₂×C₂₃	semidirect
153. #1493	C₂×C₂₃	semidirect
154. #1278	C₂×C₂₃	semidirect
239. C₂₃	#10149	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×#3230	2
C₂₃×#3138	2
C₂×C₂₃×#486	2
C₂₃×#1997	2
C₂₃×#1545	2
C₂₃×#1493	2
C₂₃×#1278	2
#10149	23

Subgroups — C23×#10149 #19157775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10149 #19157775