Subgroups — D₂₃×#466 #15682961

Order 2944.

616 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	C₂×D₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×D₂₃	C₂	1	2	yes	maximal, complemented
4	#13092654	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092653	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092652	C₂	1	2	yes	maximal, complemented
7	#13092651	C₂	1	2	yes	maximal, complemented
8	#13092650	C₂	1	1	yes	maximal, characteristic, complemented
9	#107×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₃	—	2	2	no	—
13	C₂×D₄×D₂₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#121	C₂²	1	2	yes	complemented
18	#10625110	—	2	2	no	—
19	#10625109	C₂²	1	2	yes	complemented
20	C₂×#67272	C₂²	1	2	yes	complemented
21	#10625101	C₂²	1	2	yes	complemented
22	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
23	D₈×D₂₃	—	2	4	no	—
24	C₂×D₁₈₄	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₈×D₂₃	—	2	4	no	—
26	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic, complemented
30	#10625066	C₂²	1	2	yes	complemented
31	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
32	#10625055	—	2	2	no	—
33	C₂²⋊C₄×D₂₃	—	2	2	no	—
34	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#102	C₂²	1	2	yes	complemented
36	#10625014	C₂²	1	2	yes	complemented
37	#10625003	C₂²	1	2	yes	complemented
38	#10625000	C₂²	1	1	yes	characteristic, complemented
39	#10624991	C₂²	1	2	yes	complemented
40	#10624989	C₂²	1	2	yes	complemented
41	C₂³×D₂₃	—	2	2	no	—
42	C₂³×D₂₃	D₄	1	1	yes	characteristic
43	C₂×D₄×C₂₃	—	2	2	no	—
44	C₂×D₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₃	—	2	2	no	—
47	C₂×C₂²⋊D₂₃	—	2	2	no	—
48	C₂×C₂²⋊D₂₃	—	2	2	no	—
49	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
50	D₄×D₂₃	—	2	4	no	—
51	D₄×D₂₃	—	2	4	no	—
52	D₄×D₂₃	—	4	8	no	—
53	C₂×D₉₂	—	2	2	no	—
54	C₂×D₉₂	C₂³	1	2	yes	—
55	C₂×C₄×D₂₃	—	2	2	no	—
56	C₂×C₄×D₂₃	—	2	2	no	—
57	C₂×C₄×D₂₃	—	2	2	no	—
58	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
59	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
60	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
62	D₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊Dic₂₃	—	2	2	no	—
69	#67272	—	2	4	no	—
70	(C₂×D₂₃)⋊C₄	—	2	2	no	—
71	(C₂×D₂₃)⋊C₄	—	2	2	no	—
72	C₄⋊Dic₂₃	C₂³	1	2	yes	—
73	Dic₂₃⋊C₄	—	2	2	no	—
74	C₂×C₂₃⋊C₈	—	2	2	no	—
75	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
77	D₁₈₄	—	2	2	no	—
78	C₈×D₂₃	—	2	2	no	—
79	C₈×D₂₃	—	2	2	no	—
80	C₈×D₂₃	—	2	2	no	—
81	C₈×D₂₃	D₄	1	4	yes	complemented
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₂₃	—	2	2	no	—
85	C₂²×D₂₃	—	2	2	no	—
86	C₂²×D₂₃	—	2	2	no	—
87	C₂²×D₂₃	—	4	4	no	—
88	C₂²×D₂₃	—	4	4	no	—
89	C₂²×D₂₃	—	4	4	no	—
90	C₂²×D₂₃	D₈	1	4	yes	complemented
91	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
93	D₄×C₂₃	—	2	2	no	—
94	D₄×C₂₃	—	2	2	no	—
95	D₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₂₃	—	4	4	no	—
102	C₂²⋊D₂₃	—	4	4	no	—
103	C₂²⋊D₂₃	—	4	4	no	—
104	C₂×Dic₂₃	—	2	2	no	—
105	C₂×Dic₂₃	—	2	2	no	—
106	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
108	D₉₂	—	2	2	no	—
109	D₉₂	—	2	2	no	—
110	D₉₂	—	4	4	no	—
111	C₄×D₂₃	—	2	2	no	—
112	C₄×D₂₃	—	2	1	no	—
113	C₄×D₂₃	—	2	1	no	—
114	C₄×D₂₃	—	2	2	no	—
115	C₄×D₂₃	—	4	4	no	—
116	C₄×D₂₃	C₂×D₄	1	2	yes	—
117	C₄×D₂₃	#41	1	2	yes	—
118	C₈×C₂₃	—	2	1	no	—
119	C₈×C₂₃	C₂×D₄	1	2	yes	complemented
120	C₂₃⋊C₈	—	2	1	no	—
121	C₂₃⋊C₈	C₂×D₄	1	2	yes	complemented
122	C₂×#466	—	23	1	no	maximal, Sylow
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	—	4	2	no	—
127	C₂²×C₂₃	—	4	2	no	—
128	C₂²×C₂₃	C₂×D₈	1	2	yes	complemented
129	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×D₂₃	—	2	1	no	—
131	C₂×D₂₃	—	2	1	no	—
132	C₂×D₂₃	—	2	8	no	—
133	C₂×D₂₃	—	4	2	no	—
134	C₂×D₂₃	—	4	2	no	—
135	C₂×D₂₃	—	4	2	no	—
136	C₂×D₂₃	—	8	8	no	—
137	C₂×D₂₃	#121	1	2	yes	—
138	C₂×D₂₃	C₂×D₈	1	2	yes	—
139	C₂×D₂₃	C₂×D₈	1	2	yes	complemented
140	C₂×D₂₃	#135	1	2	yes	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	4	2	no	—
143	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
145	Dic₂₃	—	2	1	no	—
146	Dic₂₃	—	4	2	no	—
147	Dic₂₃	#121	1	2	yes	—
148	C₂²×D₈	—	23	1	no	—
149	C₂³×C₈	—	23	1	no	—
150	C₂×#121	—	23	2	no	—
151	#466	—	23	1	no	—
152	#466	—	23	1	no	—
153	#466	—	23	2	no	—
154	#466	—	23	1	no	—
155	#466	—	23	2	no	—
156	#466	D₂₃	1	1	yes	characteristic, direct product factor
157	C₂×#107	—	23	1	no	—
158	C₂×#102	—	23	2	no	—
159	C₂×C₂₃	—	2	2	no	—
160	C₂×C₂₃	—	8	2	no	—
161	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
162	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
164	D₂₃	—	2	2	no	—
165	D₂₃	—	8	2	no	—
166	D₂₃	#466	1	4	yes	direct product factor
167	C₂²×D₄	—	23	2	no	—
168	C₂²×D₄	—	46	2	no	—
169	C₂³×C₄	—	23	1	no	—
170	C₂×D₈	—	23	2	no	—
171	C₂×D₈	—	23	1	no	—
172	C₂×D₈	—	46	4	no	—
173	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic, complemented
174	C₂²×C₈	—	23	1	no	—
175	C₂²×C₈	—	23	1	no	—
176	C₂²×C₈	—	23	1	no	—
177	C₂²×C₈	—	23	1	no	—
178	C₂²×C₈	—	23	1	no	—
179	C₂²×C₈	—	46	4	no	—
180	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
181	#121	—	23	2	no	—
182	#121	—	23	2	no	—
183	#121	—	23	2	no	—
184	#121	—	46	2	no	—
185	#121	—	46	2	no	—
186	#121	C₂×D₂₃	1	2	yes	complemented
187	C₂×C₄⋊C₄	—	23	2	no	—
188	C₂×C₂²⋊C₄	—	46	2	no	—
189	#107	—	23	1	no	—
190	#107	—	23	2	no	—
191	#107	C₂×D₂₃	1	1	yes	characteristic, complemented
192	#102	—	23	2	no	—
193	#102	—	23	2	no	—
194	#102	—	23	2	no	—
195	#102	C₂×D₂₃	1	2	yes	complemented
196	C₂₃	C₂×#466	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	—	23	1	no	—
198	C₂⁴	—	46	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	23	2	no	—
201	C₂×D₄	—	46	2	no	—
202	C₂×D₄	—	46	4	no	—
203	C₂×D₄	—	46	4	no	—
204	C₂×D₄	—	46	2	no	—
205	C₂×D₄	—	46	2	no	—
206	C₂×D₄	—	46	2	no	—
207	C₂×D₄	—	92	8	no	—
208	C₂×D₄	C₂²×D₂₃	1	2	yes	—
209	C₂²×C₄	—	23	1	no	—
210	C₂²×C₄	—	23	1	no	—
211	C₂²×C₄	—	23	1	no	—
212	C₂²×C₄	—	23	1	no	—
213	C₂²×C₄	—	23	1	no	—
214	C₂²×C₄	—	46	2	no	—
215	C₂²×C₄	—	46	2	no	—
216	C₂²×C₄	—	46	2	no	—
217	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
218	D₈	—	2	2	no	—
219	D₈	—	46	4	no	—
220	D₈	—	46	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	23	1	no	—
223	C₂×C₈	—	23	4	no	—
224	C₂×C₈	—	23	1	no	—
225	C₂×C₈	—	46	2	no	—
226	C₂×C₈	—	46	2	no	—
227	C₂×C₈	—	46	2	no	—
228	C₂×C₈	—	46	2	no	—
229	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
231	C₄⋊C₄	—	23	2	no	—
232	C₄⋊C₄	—	46	2	no	—
233	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	46	2	no	—
236	C₂²⋊C₄	—	46	2	no	—
237	C₂²⋊C₄	—	46	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	23	1	no	—
240	C₂³	—	23	4	no	—
241	C₂³	—	23	1	no	—
242	C₂³	—	46	2	no	—
243	C₂³	—	46	2	no	—
244	C₂³	—	46	2	no	—
245	C₂³	—	92	4	no	—
246	C₂³	—	92	4	no	—
247	C₂³	—	92	4	no	—
248	C₂³	D₄×D₂₃	1	1	yes	characteristic
249	D₄	—	2	2	no	—
250	D₄	—	2	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	46	2	no	—
253	D₄	—	46	2	no	—
254	D₄	—	92	4	no	—
255	D₄	—	92	4	no	—
256	D₄	—	92	4	no	—
257	D₄	—	92	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	23	1	no	—
262	C₂×C₄	—	23	2	no	—
263	C₂×C₄	—	23	2	no	—
264	C₂×C₄	—	23	1	no	—
265	C₂×C₄	—	46	2	no	—
266	C₂×C₄	—	46	2	no	—
267	C₂×C₄	—	46	2	no	—
268	C₂×C₄	—	46	1	no	—
269	C₂×C₄	—	46	1	no	—
270	C₂×C₄	—	46	2	no	—
271	C₂×C₄	—	92	4	no	—
272	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	23	2	no	—
276	C₈	—	46	1	no	—
277	C₈	D₄×D₂₃	1	2	yes	complemented
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	23	2	no	—
284	C₂²	—	23	2	no	—
285	C₂²	—	23	2	no	—
286	C₂²	—	23	2	no	—
287	C₂²	—	46	8	no	—
288	C₂²	—	46	1	no	—
289	C₂²	—	46	1	no	—
290	C₂²	—	92	2	no	—
291	C₂²	—	92	2	no	—
292	C₂²	—	92	2	no	—
293	C₂²	—	184	8	no	—
294	C₂²	D₈×D₂₃	1	2	yes	complemented
295	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	23	2	no	—
299	C₄	—	46	1	no	—
300	C₄	—	92	2	no	—
301	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
302	C₄	#41×D₂₃	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	23	4	no	—
306	C₂	—	46	2	no	—
307	C₂	—	184	2	no	—
308	C₂	#121×D₂₃	1	1	yes	characteristic
309	C₂	C₂×D₈×D₂₃	1	1	yes	characteristic
310	C₂	#135×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#466	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#466	D₂₃
hypocenter	C₂₃	C₂×#466
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×D₂₃	C₂	semidirect
2. C₂²×C₈×D₂₃	C₂	semidirect
3. #121×D₂₃	C₂	semidirect
4. #13092654	C₂	semidirect
5. #13092653	C₂	semidirect
6. #13092652	C₂	semidirect
7. #13092651	C₂	semidirect
8. #13092650	C₂	semidirect
9. #107×D₂₃	C₂	semidirect
10. #102×D₂₃	C₂	semidirect
11. C₂₃×#466	C₂	semidirect
15. C₂×D₈×C₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₃	C₂²	semidirect
17. C₂₃×#121	C₂²	semidirect
19. #10625109	C₂²	semidirect
20. C₂×#67272	C₂²	semidirect
21. #10625101	C₂²	semidirect
22. C₂²×C₂₃⋊C₈	C₂²	semidirect
24. C₂×D₁₈₄	C₂²	semidirect
28. C₂×C₈×D₂₃	C₂²	semidirect
29. C₂×C₈×D₂₃	C₂²	semidirect
30. #10625066	C₂²	semidirect
34. C₂₃×#107	C₂²	semidirect
35. C₂₃×#102	C₂²	semidirect
36. #10625014	C₂²	semidirect
37. #10625003	C₂²	semidirect
38. #10625000	C₂²	semidirect
39. #10624991	C₂²	semidirect
40. #10624989	C₂²	semidirect
81. C₈×D₂₃	D₄	semidirect
90. C₂²×D₂₃	D₈	semidirect
119. C₈×C₂₃	C₂×D₄	semidirect
121. C₂₃⋊C₈	C₂×D₄	semidirect
128. C₂²×C₂₃	C₂×D₈	semidirect
139. C₂×D₂₃	C₂×D₈	semidirect
156. #466	D₂₃	direct (N × H)
166. D₂₃	#466	direct (N × H)
173. C₂×D₈	C₂×D₂₃	semidirect
180. C₂²×C₈	C₂×D₂₃	semidirect
186. #121	C₂×D₂₃	semidirect
191. #107	C₂×D₂₃	semidirect
195. #102	C₂×D₂₃	semidirect
196. C₂₃	C₂×#466	semidirect
277. C₈	D₄×D₂₃	semidirect
294. C₂²	D₈×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×D₂₃	2
C₂²×C₈×D₂₃	2
#121×D₂₃	2
#13092654	2
#13092653	2
#13092652	2
#13092651	2
#13092650	2
#107×D₂₃	2
#102×D₂₃	2
C₂₃×#466	2
C₂×#466	23

Subgroups — D23×#466 #15682961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#466 #15682961