Subgroups — C₂₃×#19635 #19167261

Order 5888.

878 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3208	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3138	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1717	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1709	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₂₃	—	2	2	no	—
10	C₂₃×#533	C₂²	1	2	yes	—
11	C₂×C₂₃×#127	C₂²	1	2	yes	—
12	C₂₃×#500	C₂²	1	1	yes	characteristic, complemented
13	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
14	C₂₃×#493	—	2	2	no	—
15	C₂₃×#493	C₂²	1	2	yes	—
16	C₂₃×#493	C₂²	1	2	yes	—
17	C₂₃×#492	C₂²	1	4	yes	complemented
18	C₂₃×#490	—	2	4	no	—
19	C₂₃×#487	C₂²	1	2	yes	complemented
20	C₂₃×#486	C₂²	1	4	yes	complemented
21	C₂₃×#486	C₂²	1	4	yes	complemented
22	C₂₃×#486	C₂²	1	4	yes	complemented
23	C₂₃×#467	—	2	2	no	—
24	C₂₃×#460	—	2	4	no	—
25	C₂₃×#450	—	2	4	no	—
26	C₄×Q₁₆×C₂₃	—	2	2	no	—
27	C₂₃×#431	C₂²	1	1	yes	characteristic
28	C₂₃×#418	—	2	4	no	—
29	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#322	C₂²	1	2	yes	complemented
32	C₈²×C₂₃	C₂²	1	2	yes	complemented
33	C₂²×D₄×C₂₃	—	2	2	no	—
34	C₂²×D₄×C₂₃	—	2	2	no	—
35	C₂²×D₄×C₂₃	—	2	2	no	—
36	C₂×Q₁₆×C₂₃	—	2	2	no	—
37	C₂×SD₁₆×C₂₃	—	2	8	no	—
38	C₂×D₈×C₂₃	—	2	4	no	—
39	C₂×D₈×C₂₃	—	2	4	no	—
40	C₂×D₈×C₂₃	—	4	8	no	—
41	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
42	C₂²×C₈×C₂₃	D₄	1	2	yes	—
43	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₂₃×#127	—	2	2	no	—
46	C₂₃×#127	—	2	2	no	—
47	C₂₃×#127	—	2	2	no	—
48	C₂₃×#127	C₂³	1	4	yes	—
49	C₂₃×#125	—	2	2	no	—
50	C₂₃×C₂²⋊Q₈	—	2	2	no	—
51	C₂₃×C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈×C₂₃	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	2	2	no	—
54	C₂₃×#117	—	2	2	no	—
55	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×#107	—	2	2	no	—
57	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
58	C₂₃×#103	—	2	4	no	—
59	C₂₃×#102	—	2	8	no	—
60	C₂₃×#102	—	2	8	no	—
61	C₂₃×#102	—	2	8	no	—
62	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
63	C₂₃×#97	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₂₃	C₂³	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	2	yes	—
68	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂⁴×C₂₃	—	4	2	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	2	2	no	—
72	C₂×D₄×C₂₃	—	2	4	no	—
73	C₂×D₄×C₂₃	—	2	4	no	—
74	C₂×D₄×C₂₃	—	2	4	no	—
75	C₂×D₄×C₂₃	—	4	8	no	—
76	C₂×D₄×C₂₃	—	4	2	no	—
77	C₂×D₄×C₂₃	—	4	2	no	—
78	C₂×D₄×C₂₃	—	4	2	no	—
79	C₂×D₄×C₂₃	—	4	8	no	—
80	C₂×D₄×C₂₃	—	8	4	no	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₂₃	—	4	4	no	—
85	SD₁₆×C₂₃	—	8	8	no	—
86	D₈×C₂₃	—	4	8	no	—
87	D₈×C₂₃	—	4	8	no	—
88	M₁₆×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	2	4	no	—
90	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	4	2	no	—
99	C₄⋊C₄×C₂₃	—	4	2	no	—
100	C₂²⋊C₄×C₂₃	—	4	2	no	—
101	C₄²×C₂₃	—	4	2	no	—
102	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
106	#19635	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₃	—	4	4	no	—
108	C₂³×C₂₃	—	8	4	no	—
109	C₂³×C₂₃	—	8	4	no	—
110	C₂³×C₂₃	—	8	4	no	—
111	C₂³×C₂₃	#127	1	1	yes	characteristic
112	Q₈×C₂₃	—	4	4	no	—
113	Q₈×C₂₃	—	4	4	no	—
114	D₄×C₂₃	—	4	8	no	—
115	D₄×C₂₃	—	4	8	no	—
116	D₄×C₂₃	—	4	8	no	—
117	D₄×C₂₃	—	8	4	no	—
118	D₄×C₂₃	—	8	4	no	—
119	D₄×C₂₃	—	8	4	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#135	1	4	yes	—
128	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₂₃	—	2	4	no	—
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	2	4	no	—
134	C₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
135	#3208	C₂×C₂₃	1	2	yes	complemented
136	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
137	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
138	#3138	C₂×C₂₃	1	2	yes	complemented
139	#1717	C₂×C₂₃	1	4	yes	complemented
140	#1709	C₂×C₂₃	1	2	yes	complemented
141	#1706	C₂×C₂₃	1	2	yes	complemented
142	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	8	4	no	—
146	C₂²×C₂₃	—	8	4	no	—
147	C₂²×C₂₃	—	8	4	no	—
148	C₂²×C₂₃	—	16	8	no	—
149	C₂²×C₂₃	#495	1	2	yes	—
150	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	8	2	no	—
154	C₄×C₂₃	—	8	2	no	—
155	C₄×C₂₃	#495	1	2	yes	—
156	C₄×C₂₃	C₂×#135	1	2	yes	—
157	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
158	C₄×C₂₃	#576	1	2	yes	—
159	C₂²×D₈	—	2	2	no	—
160	#533	C₂²×C₂₃	1	2	yes	—
161	C₂×#127	C₂²×C₂₃	1	2	yes	—
162	#500	C₂²×C₂₃	1	1	yes	characteristic, complemented
163	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
164	#493	—	2	2	no	—
165	#493	C₂²×C₂₃	1	2	yes	—
166	#493	C₂²×C₂₃	1	2	yes	—
167	#492	C₂²×C₂₃	1	4	yes	complemented
168	#490	—	2	4	no	—
169	#487	C₂²×C₂₃	1	2	yes	complemented
170	#486	C₂²×C₂₃	1	4	yes	complemented
171	#486	C₂²×C₂₃	1	4	yes	complemented
172	#486	C₂²×C₂₃	1	4	yes	complemented
173	#467	—	2	2	no	—
174	#460	—	2	4	no	—
175	#450	—	2	4	no	—
176	C₄×Q₁₆	—	2	2	no	—
177	#431	C₂²×C₂₃	1	1	yes	characteristic
178	#418	—	2	4	no	—
179	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
181	#322	C₂²×C₂₃	1	2	yes	complemented
182	C₈²	C₂²×C₂₃	1	2	yes	complemented
183	C₂×C₂₃	—	2	2	no	—
184	C₂×C₂₃	—	16	4	no	—
185	C₂×C₂₃	C₂×#495	1	1	yes	characteristic
186	C₂×C₂₃	#3151	1	1	yes	characteristic
187	C₂×C₂₃	#3155	1	1	yes	characteristic
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×SD₁₆	—	2	8	no	—
193	C₂×D₈	—	2	4	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	4	8	no	—
196	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
197	C₂²×C₈	D₄×C₂₃	1	2	yes	—
198	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
200	#127	—	2	2	no	—
201	#127	—	2	2	no	—
202	#127	—	2	2	no	—
203	#127	C₂³×C₂₃	1	4	yes	—
204	#125	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	#117	—	2	2	no	—
210	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
211	#107	—	2	2	no	—
212	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
213	#103	—	2	4	no	—
214	#102	—	2	8	no	—
215	#102	—	2	8	no	—
216	#102	—	2	8	no	—
217	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
218	#97	D₄×C₂₃	1	2	yes	—
219	C₄×C₈	D₄×C₂₃	1	2	yes	—
220	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₂₃	1	2	yes	—
222	C₄×C₈	C₂³×C₂₃	1	2	yes	—
223	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
224	C₂₃	#19635	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	8	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
240	Q₁₆	—	4	4	no	—
241	SD₁₆	—	8	8	no	—
242	D₈	—	4	8	no	—
243	D₈	—	4	8	no	—
244	M₁₆	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
248	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
262	C₂³	—	4	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	8	4	no	—
265	C₂³	—	8	4	no	—
266	C₂³	C₂₃×#127	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	8	no	—
271	D₄	—	4	8	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
280	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
281	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
282	C₂×C₄	C₂₃×#135	1	4	yes	—
283	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	4	no	—
289	C₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	16	8	no	—
296	C₂²	C₂₃×#495	1	2	yes	—
297	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₂₃×#495	1	2	yes	—
303	C₄	C₂×C₂₃×#135	1	2	yes	—
304	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
305	C₄	C₂₃×#576	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	C₂×C₂₃×#495	1	1	yes	characteristic
309	C₂	C₂₃×#3151	1	1	yes	characteristic
310	C₂	C₂₃×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#493	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×#3208	C₂	semidirect
2. C₂₃×#3170	C₂	semidirect
4. C₂₃×#3138	C₂	semidirect
5. C₂₃×#1717	C₂	semidirect
6. C₂₃×#1709	C₂	semidirect
7. C₂₃×#1706	C₂	semidirect
8. C₈×M₁₆×C₂₃	C₂	semidirect
12. C₂₃×#500	C₂²	semidirect
13. C₂₃×#494	C₂²	semidirect
17. C₂₃×#492	C₂²	semidirect
19. C₂₃×#487	C₂²	semidirect
20. C₂₃×#486	C₂²	semidirect
21. C₂₃×#486	C₂²	semidirect
22. C₂₃×#486	C₂²	semidirect
31. C₂₃×#322	C₂²	semidirect
32. C₈²×C₂₃	C₂²	semidirect
106. #19635	C₂₃	direct (N × H)
135. #3208	C₂×C₂₃	semidirect
136. #3170	C₂×C₂₃	semidirect
138. #3138	C₂×C₂₃	semidirect
139. #1717	C₂×C₂₃	semidirect
140. #1709	C₂×C₂₃	semidirect
141. #1706	C₂×C₂₃	semidirect
142. C₈×M₁₆	C₂×C₂₃	semidirect
162. #500	C₂²×C₂₃	semidirect
163. #494	C₂²×C₂₃	semidirect
167. #492	C₂²×C₂₃	semidirect
169. #487	C₂²×C₂₃	semidirect
170. #486	C₂²×C₂₃	semidirect
171. #486	C₂²×C₂₃	semidirect
172. #486	C₂²×C₂₃	semidirect
181. #322	C₂²×C₂₃	semidirect
182. C₈²	C₂²×C₂₃	semidirect
224. C₂₃	#19635	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×#3208	2
C₂₃×#3170	2
C₂×C₂₃×#493	2
C₂₃×#3138	2
C₂₃×#1717	2
C₂₃×#1709	2
C₂₃×#1706	2
C₈×M₁₆×C₂₃	2
#19635	23

Subgroups — C23×#19635 #19167261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19635 #19167261