Subgroups — D₂₃×#460 #15682825

Order 2944.

584 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
2	#127×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092618	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092617	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092616	C₂	1	2	yes	maximal, complemented
7	#13092615	C₂	1	2	yes	maximal, complemented
8	#13092614	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₂₃	—	2	1	no	—
14	C₂×D₄×D₂₃	—	2	1	no	—
15	C₂×D₄×D₂₃	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂×#67274	C₂²	1	2	yes	complemented
21	#10625111	—	2	1	no	—
22	#10625111	—	2	1	no	—
23	#10625111	C₂²	1	2	yes	—
24	C₂×#67273	C₂²	1	2	yes	complemented
25	SD₁₆×D₂₃	—	2	8	no	—
26	C₂×#67263	C₂²	1	2	yes	complemented
27	C₂×C₈×D₂₃	C₂²	1	2	yes	—
28	C₄⋊C₄×D₂₃	—	2	1	no	—
29	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
30	#10625058	—	2	1	no	—
31	#10625058	C₂²	1	1	yes	characteristic, complemented
32	#10625045	C₂²	1	1	yes	characteristic
33	C₄²×D₂₃	—	2	1	no	—
34	C₄²×D₂₃	C₂²	1	1	yes	characteristic
35	C₄×Dic₄₆	—	2	1	no	—
36	C₄×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
37	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#102	C₂²	1	2	yes	complemented
39	#10625014	C₂²	1	2	yes	complemented
40	#10625003	C₂²	1	2	yes	complemented
41	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
42	#10624991	C₂²	1	2	yes	complemented
43	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	C₂³×D₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	C₂³	1	2	yes	—
49	C₂×C₂²⋊D₂₃	—	2	2	no	—
50	C₂×C₂²⋊D₂₃	—	2	2	no	—
51	C₂×C₂²⋊D₂₃	—	2	2	no	—
52	Q₈×D₂₃	—	2	2	no	—
53	Q₈×D₂₃	D₄	1	4	yes	complemented
54	D₄×D₂₃	—	2	4	no	—
55	D₄×D₂₃	—	2	4	no	—
56	D₄×D₂₃	—	4	4	no	—
57	D₄×D₂₃	—	4	4	no	—
58	C₂×D₉₂	—	2	1	no	—
59	C₂×D₉₂	—	2	1	no	—
60	C₂×D₉₂	C₂³	1	2	yes	—
61	C₂×C₄×D₂₃	—	2	1	no	—
62	C₂×C₄×D₂₃	—	2	1	no	—
63	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
64	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
66	C₂×Dic₄₆	—	2	1	no	—
67	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
68	SD₁₆×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₄²×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	#67274	—	2	4	no	—
75	#67273	—	2	4	no	—
76	C₄⋊Dic₂₃	—	2	1	no	—
77	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
78	Dic₂₃⋊C₄	—	2	1	no	—
79	Dic₂₃⋊C₄	—	2	1	no	—
80	Dic₂₃⋊C₄	—	2	1	no	—
81	C₄×Dic₂₃	—	2	1	no	—
82	C₄×Dic₂₃	—	2	1	no	—
83	C₄×Dic₂₃	—	2	1	no	—
84	C₄×Dic₂₃	D₄	1	1	yes	characteristic
85	C₄×Dic₂₃	D₄	1	1	yes	characteristic
86	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
88	#67263	—	2	4	no	—
89	C₈×D₂₃	—	2	4	no	—
90	C₂³×C₂₃	—	2	2	no	—
91	C₂²×D₂₃	—	2	2	no	—
92	C₂²×D₂₃	—	4	4	no	—
93	C₂²×D₂₃	—	4	4	no	—
94	C₂²×D₂₃	—	4	4	no	—
95	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	1	no	—
97	Q₈×C₂₃	C₂×D₄	1	2	yes	complemented
98	D₄×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	2	2	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₂₃	—	4	4	no	—
108	C₂²⋊D₂₃	—	4	4	no	—
109	C₂²⋊D₂₃	—	4	4	no	—
110	C₂×Dic₂₃	—	2	1	no	—
111	C₂×Dic₂₃	—	2	1	no	—
112	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
115	D₉₂	—	2	2	no	—
116	D₉₂	—	2	2	no	—
117	D₉₂	—	4	2	no	—
118	D₉₂	—	4	2	no	—
119	C₄×D₂₃	—	2	4	no	—
120	C₄×D₂₃	—	2	2	no	—
121	C₄×D₂₃	—	4	2	no	—
122	C₄×D₂₃	SD₁₆	1	4	yes	complemented
123	C₄×D₂₃	C₂×D₄	1	2	yes	—
124	C₄×D₂₃	#41	1	2	yes	—
125	Dic₄₆	—	2	1	no	—
126	Dic₄₆	—	2	4	no	—
127	Dic₄₆	C₂×D₄	1	2	yes	complemented
128	C₈×C₂₃	—	2	2	no	—
129	C₂₃⋊C₈	—	2	2	no	—
130	C₂×#460	—	23	1	no	maximal, Sylow
131	C₂²×C₂₃	—	4	2	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	4	2	no	—
134	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₂₃	—	4	2	no	—
136	C₂×D₂₃	—	4	2	no	—
137	C₂×D₂₃	—	4	2	no	—
138	C₂×D₂₃	—	8	8	no	—
139	C₂×D₂₃	#121	1	2	yes	—
140	C₂×D₂₃	C₂×SD₁₆	1	2	yes	—
141	C₂×D₂₃	AGL(1,ℤ/8)	1	2	yes	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	1	no	—
145	C₄×C₂₃	C₂×SD₁₆	1	2	yes	complemented
146	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
148	Dic₂₃	—	2	2	no	—
149	Dic₂₃	—	2	1	no	—
150	Dic₂₃	—	4	1	no	—
151	Dic₂₃	#121	1	2	yes	—
152	Dic₂₃	C₂×SD₁₆	1	2	yes	complemented
153	C₂²×SD₁₆	—	23	2	no	—
154	C₂×#127	—	23	1	no	—
155	C₂×C₄×Q₈	—	23	1	no	—
156	#460	—	23	2	no	—
157	#460	—	23	1	no	—
158	#460	—	23	1	no	—
159	#460	—	23	1	no	—
160	#460	—	23	2	no	—
161	#460	D₂₃	1	1	yes	characteristic, direct product factor
162	C₂×C₄⋊C₈	—	23	1	no	—
163	C₂×#102	—	23	2	no	—
164	C₂×C₂₃	—	8	2	no	—
165	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
166	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
167	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	D₂₃	—	8	2	no	—
169	D₂₃	#460	1	4	yes	direct product factor
170	C₂²×Q₈	—	23	1	no	—
171	C₂²×D₄	—	23	2	no	—
172	C₂²×D₄	—	46	1	no	—
173	C₂²×D₄	—	46	1	no	—
174	C₂×SD₁₆	—	23	2	no	—
175	C₂×SD₁₆	—	23	2	no	—
176	C₂×SD₁₆	—	23	2	no	—
177	C₂×SD₁₆	—	46	8	no	—
178	C₂×SD₁₆	C₂×D₂₃	1	2	yes	complemented
179	C₂²×C₈	—	23	2	no	—
180	#127	—	23	2	no	—
181	#127	—	23	1	no	—
182	#127	—	46	1	no	—
183	#127	—	46	1	no	—
184	#127	C₂×D₂₃	1	1	yes	characteristic
185	C₄×Q₈	—	23	1	no	—
186	C₄×Q₈	—	23	1	no	—
187	C₄×Q₈	—	23	1	no	—
188	C₄×Q₈	—	46	1	no	—
189	C₄×Q₈	—	46	1	no	—
190	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
191	C₂×C₄⋊C₄	—	23	1	no	—
192	C₂×C₄⋊C₄	—	46	1	no	—
193	C₂×C₄²	—	23	1	no	—
194	C₂×C₄²	—	46	1	no	—
195	C₄⋊C₈	—	23	2	no	—
196	C₄⋊C₈	—	23	1	no	—
197	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
198	#102	—	23	2	no	—
199	#102	—	23	2	no	—
200	#102	—	23	2	no	—
201	#102	C₂×D₂₃	1	2	yes	complemented
202	C₂₃	C₂×#460	1	1	yes	characteristic, complemented, Sylow
203	C₂⁴	—	46	2	no	—
204	C₂×Q₈	—	23	4	no	—
205	C₂×Q₈	—	23	1	no	—
206	C₂×Q₈	—	46	1	no	—
207	C₂×Q₈	—	46	2	no	—
208	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	23	2	no	—
212	C₂×D₄	—	46	4	no	—
213	C₂×D₄	—	46	4	no	—
214	C₂×D₄	—	46	2	no	—
215	C₂×D₄	—	46	2	no	—
216	C₂×D₄	—	46	2	no	—
217	C₂×D₄	—	46	1	no	—
218	C₂×D₄	—	46	1	no	—
219	C₂×D₄	—	92	4	no	—
220	C₂×D₄	—	92	4	no	—
221	C₂×D₄	C₂²×D₂₃	1	2	yes	—
222	C₂²×C₄	—	23	1	no	—
223	C₂²×C₄	—	23	1	no	—
224	C₂²×C₄	—	23	1	no	—
225	C₂²×C₄	—	46	1	no	—
226	C₂²×C₄	—	46	1	no	—
227	SD₁₆	—	2	4	no	—
228	SD₁₆	—	46	4	no	—
229	SD₁₆	—	46	4	no	—
230	SD₁₆	—	46	4	no	—
231	C₂×C₈	—	23	2	no	—
232	C₂×C₈	—	46	4	no	—
233	C₂×C₈	C₂²×D₂₃	1	2	yes	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	23	1	no	—
236	C₄⋊C₄	—	46	1	no	—
237	C₄⋊C₄	—	46	1	no	—
238	C₄⋊C₄	—	46	1	no	—
239	C₄⋊C₄	—	46	1	no	—
240	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	—	23	1	no	—
243	C₄²	—	23	1	no	—
244	C₄²	—	23	1	no	—
245	C₄²	—	46	1	no	—
246	C₄²	—	46	1	no	—
247	C₄²	—	46	1	no	—
248	C₄²	C₂²×D₂₃	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	23	1	no	—
251	C₂³	—	46	2	no	—
252	C₂³	—	92	4	no	—
253	C₂³	—	92	4	no	—
254	C₂³	—	92	4	no	—
255	Q₈	—	2	1	no	—
256	Q₈	—	23	2	no	—
257	Q₈	—	46	4	no	—
258	Q₈	—	46	1	no	—
259	Q₈	D₄×D₂₃	1	2	yes	complemented
260	D₄	—	2	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	46	2	no	—
265	D₄	—	46	2	no	—
266	D₄	—	92	2	no	—
267	D₄	—	92	4	no	—
268	D₄	—	92	4	no	—
269	D₄	—	92	4	no	—
270	D₄	—	92	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	23	1	no	—
274	C₂×C₄	—	23	1	no	—
275	C₂×C₄	—	23	4	no	—
276	C₂×C₄	—	23	2	no	—
277	C₂×C₄	—	23	2	no	—
278	C₂×C₄	—	23	1	no	—
279	C₂×C₄	—	46	2	no	—
280	C₂×C₄	—	46	4	no	—
281	C₂×C₄	—	46	1	no	—
282	C₂×C₄	—	46	1	no	—
283	C₂×C₄	—	92	2	no	—
284	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
285	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	46	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	23	2	no	—
293	C₂²	—	23	2	no	—
294	C₂²	—	23	2	no	—
295	C₂²	—	92	2	no	—
296	C₂²	—	92	2	no	—
297	C₂²	—	92	2	no	—
298	C₂²	—	184	8	no	—
299	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	23	2	no	—
304	C₄	—	23	2	no	—
305	C₄	—	46	2	no	—
306	C₄	—	46	1	no	—
307	C₄	—	92	1	no	—
308	C₄	SD₁₆×D₂₃	1	2	yes	complemented
309	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
310	C₄	#41×D₂₃	1	1	yes	characteristic
311	C₂	—	8	2	no	—
312	C₂	—	23	4	no	—
313	C₂	—	184	2	no	—
314	C₂	#121×D₂₃	1	1	yes	characteristic
315	C₂	C₂×SD₁₆×D₂₃	1	1	yes	characteristic
316	C₂	AGL(1,ℤ/8)×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×#460	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	#127×D₂₃	C₂
hypercenter	#460	D₂₃
hypocenter	C₂₃	C₂×#460
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
1. C₂×SD₁₆×D₂₃	C₂	semidirect
3. C₄×Q₈×D₂₃	C₂	semidirect
4. #13092618	C₂	semidirect
5. #13092617	C₂	semidirect
6. #13092616	C₂	semidirect
7. #13092615	C₂	semidirect
8. #13092614	C₂	semidirect
9. C₄⋊C₈×D₂₃	C₂	semidirect
10. #102×D₂₃	C₂	semidirect
11. C₂₃×#460	C₂	semidirect
16. C₂×SD₁₆×C₂₃	C₂²	semidirect
18. C₄×Q₈×C₂₃	C₂²	semidirect
19. Q₈×Dic₂₃	C₂²	semidirect
20. C₂×#67274	C₂²	semidirect
24. C₂×#67273	C₂²	semidirect
26. C₂×#67263	C₂²	semidirect
31. #10625058	C₂²	semidirect
36. C₄×Dic₄₆	C₂²	semidirect
37. C₂₃×C₄⋊C₈	C₂²	semidirect
38. C₂₃×#102	C₂²	semidirect
39. #10625014	C₂²	semidirect
40. #10625003	C₂²	semidirect
41. Dic₂₃⋊C₈	C₂²	semidirect
42. #10624991	C₂²	semidirect
43. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
53. Q₈×D₂₃	D₄	semidirect
97. Q₈×C₂₃	C₂×D₄	semidirect
122. C₄×D₂₃	SD₁₆	semidirect
127. Dic₄₆	C₂×D₄	semidirect
145. C₄×C₂₃	C₂×SD₁₆	semidirect
152. Dic₂₃	C₂×SD₁₆	semidirect
161. #460	D₂₃	direct (N × H)
169. D₂₃	#460	direct (N × H)
178. C₂×SD₁₆	C₂×D₂₃	semidirect
190. C₄×Q₈	C₂×D₂₃	semidirect
197. C₄⋊C₈	C₂×D₂₃	semidirect
201. #102	C₂×D₂₃	semidirect
202. C₂₃	C₂×#460	semidirect
259. Q₈	D₄×D₂₃	semidirect
308. C₄	SD₁₆×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×D₂₃	2
#127×D₂₃	2
C₄×Q₈×D₂₃	2
#13092618	2
#13092617	2
#13092616	2
#13092615	2
#13092614	2
C₄⋊C₈×D₂₃	2
#102×D₂₃	2
C₂₃×#460	2
C₂×#460	23

Subgroups — D23×#460 #15682825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#460 #15682825