Subgroups — C₂₃×#19613 #19167239

Order 5888.

734 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3167	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3131	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1720	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1719	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1718	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1717	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#533	C₂²	1	2	yes	—
11	C₂₃×#532	C₂²	1	2	yes	—
12	C₂₃×#500	C₂²	1	1	yes	characteristic
13	C₂₃×#500	C₂²	1	1	yes	characteristic
14	C₂₃×#498	C₂²	1	2	yes	—
15	C₂₃×#494	C₂²	1	1	yes	characteristic
16	C₂₃×#493	C₂²	1	1	yes	characteristic
17	C₂₃×#492	C₂²	1	2	yes	—
18	C₂₃×#488	—	2	4	no	—
19	C₂₃×#487	C₂²	1	2	yes	—
20	C₂₃×#487	C₂²	1	2	yes	—
21	C₂₃×#487	C₂²	1	2	yes	—
22	C₂₃×#487	C₂²	1	2	yes	—
23	C₂₃×#486	C₂²	1	2	yes	—
24	C₂₃×#486	C₂²	1	2	yes	—
25	C₂₃×#486	C₂²	1	2	yes	—
26	C₂₃×#486	C₂²	1	2	yes	—
27	C₂₃×#467	—	2	2	no	—
28	C₂₃×#466	—	2	2	no	—
29	C₂₃×#465	—	2	4	no	—
30	C₄×Q₁₆×C₂₃	—	2	2	no	—
31	C₄×SD₁₆×C₂₃	—	2	4	no	—
32	C₄×D₈×C₂₃	—	2	2	no	—
33	C₂₃×#416	—	2	4	no	—
34	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#322	C₂²	1	2	yes	—
38	C₂₃×#322	C₂²	1	2	yes	—
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×Q₁₆×C₂₃	—	2	2	no	—
41	C₂×SD₁₆×C₂₃	—	2	4	no	—
42	C₂×D₈×C₂₃	—	2	2	no	—
43	C₂²×C₈×C₂₃	D₄	1	2	yes	—
44	C₂²×C₈×C₂₃	D₄	1	2	yes	—
45	C₂²×C₈×C₂₃	D₄	1	2	yes	—
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₃×#127	C₂³	1	2	yes	—
50	C₂₃×#125	—	2	2	no	—
51	C₂₃×#124	—	2	2	no	—
52	C₂₃×C₂²⋊Q₈	—	2	2	no	—
53	C₂₃×C₂²⋊Q₈	—	2	2	no	—
54	C₂₃×#121	—	2	2	no	—
55	C₂₃×#121	—	2	2	no	—
56	C₄×Q₈×C₂₃	—	2	2	no	—
57	C₄×Q₈×C₂₃	—	2	2	no	—
58	C₄×D₄×C₂₃	—	2	2	no	—
59	C₄×D₄×C₂₃	—	2	2	no	—
60	C₂₃×#117	—	2	2	no	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×#107	—	2	2	no	—
63	C₂₃×#107	—	2	2	no	—
64	C₂₃×#106	—	2	4	no	—
65	C₂₃×#103	—	2	4	no	—
66	C₂₃×#103	—	2	4	no	—
67	C₂₃×#103	—	2	4	no	—
68	C₂₃×#103	—	2	4	no	—
69	C₂₃×#102	—	2	4	no	—
70	C₂₃×#102	—	2	4	no	—
71	C₂₃×#102	—	2	4	no	—
72	C₂₃×#102	—	2	4	no	—
73	C₄×C₈×C₂₃	D₄	1	2	yes	—
74	C₄×C₈×C₂₃	D₄	1	2	yes	—
75	C₄×C₈×C₂₃	D₄	1	2	yes	—
76	C₄×C₈×C₂₃	C₂³	1	2	yes	—
77	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₂₃	C₂³	1	2	yes	—
79	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	#41×C₂₃	—	4	8	no	—
81	C₂×Q₈×C₂₃	—	2	2	no	—
82	C₂×Q₈×C₂₃	—	2	2	no	—
83	C₂×Q₈×C₂₃	—	2	2	no	—
84	C₂×D₄×C₂₃	—	2	2	no	—
85	C₂×D₄×C₂₃	—	2	2	no	—
86	C₂×D₄×C₂₃	—	2	2	no	—
87	C₂×D₄×C₂₃	—	4	2	no	—
88	C₂²×C₄×C₂₃	—	4	2	no	—
89	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₂₃	—	4	4	no	—
93	SD₁₆×C₂₃	—	4	8	no	—
94	D₈×C₂₃	—	4	4	no	—
95	C₂×C₈×C₂₃	—	2	4	no	—
96	C₂×C₈×C₂₃	—	4	2	no	—
97	C₂×C₈×C₂₃	—	4	2	no	—
98	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₄⋊C₄×C₂₃	—	2	2	no	—
108	C₄⋊C₄×C₂₃	—	4	2	no	—
109	C₄⋊C₄×C₂₃	—	4	2	no	—
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₂²⋊C₄×C₂₃	—	4	2	no	—
112	C₂²⋊C₄×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	—	4	2	no	—
114	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
118	#19613	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₂₃	—	4	2	no	—
120	C₂³×C₂₃	#127	1	1	yes	characteristic
121	Q₈×C₂₃	—	4	4	no	—
122	Q₈×C₂₃	—	4	4	no	—
123	Q₈×C₂₃	—	4	4	no	—
124	D₄×C₂₃	—	4	4	no	—
125	D₄×C₂₃	—	4	4	no	—
126	D₄×C₂₃	—	4	4	no	—
127	D₄×C₂₃	—	8	4	no	—
128	C₂×C₄×C₂₃	—	2	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	8	4	no	—
134	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
135	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
137	C₂×C₄×C₂₃	#135	1	4	yes	—
138	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₂₃	—	2	4	no	—
142	C₈×C₂₃	—	2	2	no	—
143	C₈×C₂₃	—	2	4	no	—
144	C₈×C₂₃	—	2	2	no	—
145	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#3169	C₂×C₂₃	1	1	yes	characteristic
147	#3167	C₂×C₂₃	1	2	yes	complemented
148	#3131	C₂×C₂₃	1	2	yes	complemented
149	#1720	C₂×C₂₃	1	2	yes	complemented
150	#1719	C₂×C₂₃	1	2	yes	complemented
151	#1718	C₂×C₂₃	1	2	yes	complemented
152	#1717	C₂×C₂₃	1	2	yes	complemented
153	C₂×#322	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	2	1	no	—
156	C₂²×C₂₃	—	8	2	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	—	8	2	no	—
159	C₂²×C₂₃	#495	1	2	yes	—
160	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	8	2	no	—
164	C₄×C₂₃	—	8	2	no	—
165	C₄×C₂₃	—	8	2	no	—
166	C₄×C₂₃	#495	1	2	yes	—
167	C₄×C₂₃	C₂×#135	1	2	yes	—
168	C₄×C₂₃	#575	1	2	yes	—
169	C₄×C₂₃	#575	1	2	yes	—
170	#533	C₂²×C₂₃	1	2	yes	—
171	#532	C₂²×C₂₃	1	2	yes	—
172	#500	C₂²×C₂₃	1	1	yes	characteristic
173	#500	C₂²×C₂₃	1	1	yes	characteristic
174	#498	C₂²×C₂₃	1	2	yes	—
175	#494	C₂²×C₂₃	1	1	yes	characteristic
176	#493	C₂²×C₂₃	1	1	yes	characteristic
177	#492	C₂²×C₂₃	1	2	yes	—
178	#488	—	2	4	no	—
179	#487	C₂²×C₂₃	1	2	yes	—
180	#487	C₂²×C₂₃	1	2	yes	—
181	#487	C₂²×C₂₃	1	2	yes	—
182	#487	C₂²×C₂₃	1	2	yes	—
183	#486	C₂²×C₂₃	1	2	yes	—
184	#486	C₂²×C₂₃	1	2	yes	—
185	#486	C₂²×C₂₃	1	2	yes	—
186	#486	C₂²×C₂₃	1	2	yes	—
187	#467	—	2	2	no	—
188	#466	—	2	2	no	—
189	#465	—	2	4	no	—
190	C₄×Q₁₆	—	2	2	no	—
191	C₄×SD₁₆	—	2	4	no	—
192	C₄×D₈	—	2	2	no	—
193	#416	—	2	4	no	—
194	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
197	#322	C₂²×C₂₃	1	2	yes	—
198	#322	C₂²×C₂₃	1	2	yes	—
199	C₂×C₂₃	—	2	2	no	—
200	C₂×C₂₃	—	16	2	no	—
201	C₂×C₂₃	C₂×#495	1	1	yes	characteristic
202	C₂×C₂₃	#3150	1	1	yes	characteristic
203	C₂×C₂₃	#3150	1	1	yes	characteristic
204	C₂×#41	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂²×C₈	D₄×C₂₃	1	2	yes	—
209	C₂²×C₈	D₄×C₂₃	1	2	yes	—
210	C₂²×C₈	D₄×C₂₃	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
214	#127	C₂³×C₂₃	1	2	yes	—
215	#125	—	2	2	no	—
216	#124	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	#117	—	2	2	no	—
226	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	2	no	—
229	#106	—	2	4	no	—
230	#103	—	2	4	no	—
231	#103	—	2	4	no	—
232	#103	—	2	4	no	—
233	#103	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	C₄×C₈	D₄×C₂₃	1	2	yes	—
239	C₄×C₈	D₄×C₂₃	1	2	yes	—
240	C₄×C₈	D₄×C₂₃	1	2	yes	—
241	C₄×C₈	C₂³×C₂₃	1	2	yes	—
242	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₂₃	1	2	yes	—
244	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
245	C₂₃	#19613	1	1	yes	characteristic, direct product factor, Sylow
246	#41	—	4	8	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	SD₁₆	—	4	8	no	—
260	D₈	—	4	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	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	C₂⁴×C₂₃	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	C₂₃×#127	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
300	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
301	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
302	C₂×C₄	C₂₃×#135	1	4	yes	—
303	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	4	no	—
309	C₈	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂₃×#495	1	2	yes	—
316	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₂₃×#495	1	2	yes	—
323	C₄	C₂×C₂₃×#135	1	2	yes	—
324	C₄	C₂₃×#575	1	2	yes	—
325	C₄	C₂₃×#575	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	16	2	no	—
328	C₂	C₂×C₂₃×#495	1	1	yes	characteristic
329	C₂	C₂₃×#3150	1	1	yes	characteristic
330	C₂	C₂₃×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#127
prime radical	C₂₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3170	C₂	semidirect
3. C₂₃×#3167	C₂	semidirect
4. C₂₃×#3131	C₂	semidirect
5. C₂₃×#1720	C₂	semidirect
6. C₂₃×#1719	C₂	semidirect
7. C₂₃×#1718	C₂	semidirect
8. C₂₃×#1717	C₂	semidirect
9. C₂×C₂₃×#322	C₂	semidirect
118. #19613	C₂₃	direct (N × H)
145. #3170	C₂×C₂₃	semidirect
147. #3167	C₂×C₂₃	semidirect
148. #3131	C₂×C₂₃	semidirect
149. #1720	C₂×C₂₃	semidirect
150. #1719	C₂×C₂₃	semidirect
151. #1718	C₂×C₂₃	semidirect
152. #1717	C₂×C₂₃	semidirect
153. C₂×#322	C₂×C₂₃	semidirect
245. C₂₃	#19613	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×#3170	2
C₂₃×#3169	2
C₂₃×#3167	2
C₂₃×#3131	2
C₂₃×#1720	2
C₂₃×#1719	2
C₂₃×#1718	2
C₂₃×#1717	2
C₂×C₂₃×#322	2
#19613	23

Subgroups — C23×#19613 #19167239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19613 #19167239