Subgroups — D₂₃×#472 #15683100

Order 2944.

488 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,ℤ/8)×D₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092688	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092687	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092686	C₂	1	2	yes	maximal, complemented
7	#13092685	C₂	1	2	yes	maximal, complemented
8	#13092684	C₂	1	1	yes	maximal, characteristic, complemented
9	#108×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#100×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₂₃	C₂²	1	2	yes	—
14	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
15	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂₃×#131	C₂²	1	1	yes	characteristic, complemented
17	#10625120	C₂²	1	2	yes	complemented
18	#10625119	C₂²	1	1	yes	characteristic, complemented
19	#10625103	C₂²	1	2	yes	complemented
20	C₂×#67272	C₂²	1	2	yes	complemented
21	#10625085	—	2	2	no	—
22	SD₁₆×D₂₃	—	2	2	no	—
23	#10625082	—	2	2	no	—
24	D₈×D₂₃	—	2	2	no	—
25	D₈×D₂₃	—	2	1	no	—
26	D₈×D₂₃	—	2	1	no	—
27	D₈×D₂₃	—	2	1	no	—
28	D₈×D₂₃	—	2	1	no	—
29	#10625079	C₂²	1	2	yes	complemented
30	#10625078	—	2	1	no	—
31	#10625078	C₂²	1	1	yes	characteristic, complemented
32	M₁₆×D₂₃	—	2	1	no	—
33	M₁₆×D₂₃	C₂²	1	1	yes	characteristic
34	M₁₆×D₂₃	C₂²	1	2	yes	—
35	C₂×D₁₈₄	C₂²	1	1	yes	characteristic, complemented
36	#10625072	—	2	1	no	—
37	#10625072	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₈×D₂₃	—	2	1	no	—
39	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#100	C₂²	1	2	yes	complemented
42	#10625015	C₂²	1	2	yes	complemented
43	#10625005	C₂²	1	2	yes	complemented
44	#10625005	C₂²	1	2	yes	complemented
45	#10625004	C₂²	1	2	yes	complemented
46	#10625001	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₂₃	—	2	2	no	—
48	#41×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₂₃	—	2	2	no	—
51	#67292	D₄	1	1	yes	characteristic
52	Q₈×D₂₃	D₄	1	1	yes	characteristic
53	#67290	—	2	1	no	—
54	#67290	D₄	1	1	yes	characteristic
55	D₄×D₂₃	—	2	1	no	—
56	D₄×D₂₃	—	2	2	no	—
57	D₄×D₂₃	—	2	2	no	—
58	D₄×D₂₃	—	2	2	no	—
59	D₄×D₂₃	—	2	2	no	—
60	D₄×D₂₃	D₄	1	1	yes	characteristic
61	#67288	—	2	1	no	—
62	#67288	C₂³	1	1	yes	characteristic
63	C₂×D₉₂	C₂³	1	2	yes	—
64	C₂×C₄×D₂₃	—	2	1	no	—
65	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
66	SD₁₆×C₂₃	—	2	2	no	—
67	D₈×C₂₃	—	2	2	no	—
68	D₈×C₂₃	—	2	1	no	—
69	D₈×C₂₃	—	2	1	no	—
70	M₁₆×C₂₃	—	2	1	no	—
71	M₁₆×C₂₃	C₂³	1	1	yes	characteristic
72	M₁₆×C₂₃	C₂³	1	2	yes	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	#67274	—	2	2	no	—
76	#67273	—	2	2	no	—
77	#67272	—	2	2	no	—
78	#67272	—	2	2	no	—
79	#67272	—	2	2	no	—
80	#67272	—	2	2	no	—
81	#67267	—	2	1	no	—
82	#67267	C₂³	1	1	yes	characteristic
83	#67267	C₂³	1	2	yes	—
84	C₂×C₂₃⋊C₈	—	2	1	no	—
85	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
86	D₁₈₄	—	2	2	no	—
87	D₁₈₄	—	2	1	no	—
88	D₁₈₄	—	2	1	no	—
89	#67263	—	2	2	no	—
90	#67262	—	2	1	no	—
91	#67262	—	2	2	no	—
92	#67262	—	2	1	no	—
93	#67262	—	2	1	no	—
94	C₈×D₂₃	—	2	1	no	—
95	C₈×D₂₃	—	2	2	no	—
96	C₈×D₂₃	—	2	1	no	—
97	C₈×D₂₃	—	2	1	no	—
98	C₈×D₂₃	D₄	1	1	yes	characteristic
99	C₈×D₂₃	D₄	1	1	yes	characteristic
100	C₈×D₂₃	D₄	1	1	yes	characteristic
101	C₈×D₂₃	D₄	1	1	yes	characteristic
102	C₂³×C₂₃	—	2	2	no	—
103	C₂²×D₂₃	—	2	1	no	—
104	C₂²×D₂₃	—	2	2	no	—
105	C₂²×D₂₃	—	4	2	no	—
106	C₂²×D₂₃	—	4	2	no	—
107	C₂²×D₂₃	—	8	4	no	—
108	C₂²×D₂₃	#41	1	1	yes	characteristic
109	Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
110	D₄×C₂₃	—	2	1	no	—
111	D₄×C₂₃	—	2	2	no	—
112	D₄×C₂₃	—	2	2	no	—
113	D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₂₃	—	2	1	no	—
117	C₂²⋊D₂₃	—	2	1	no	—
118	C₂²⋊D₂₃	—	2	1	no	—
119	C₂²⋊D₂₃	—	4	2	no	—
120	C₂²⋊D₂₃	—	4	2	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	#41	1	1	yes	characteristic
123	D₉₂	—	2	1	no	—
124	D₉₂	—	2	2	no	—
125	D₉₂	—	2	2	no	—
126	D₉₂	C₂×D₄	1	1	yes	characteristic
127	C₄×D₂₃	—	2	1	no	—
128	C₄×D₂₃	—	2	1	no	—
129	C₄×D₂₃	—	2	1	no	—
130	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
131	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
132	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
134	Dic₄₆	—	2	1	no	—
135	Dic₄₆	C₂×D₄	1	1	yes	characteristic
136	C₈×C₂₃	—	2	1	no	—
137	C₈×C₂₃	—	2	2	no	—
138	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂₃⋊C₈	—	2	1	no	—
141	C₂₃⋊C₈	—	2	2	no	—
142	C₂₃⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂₃⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×#472	—	23	1	no	maximal, Sylow
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	4	2	no	—
147	C₂²×C₂₃	—	8	2	no	—
148	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
149	C₂×D₂₃	—	2	2	no	—
150	C₂×D₂₃	—	2	1	no	—
151	C₂×D₂₃	—	4	2	no	—
152	C₂×D₂₃	—	4	2	no	—
153	C₂×D₂₃	—	8	4	no	—
154	C₂×D₂₃	—	8	4	no	—
155	C₂×D₂₃	—	8	2	no	—
156	C₂×D₂₃	#121	1	1	yes	characteristic
157	C₂×D₂₃	#121	1	1	yes	characteristic
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
161	Dic₂₃	—	2	1	no	—
162	Dic₂₃	#121	1	1	yes	characteristic
163	Dic₂₃	#121	1	1	yes	characteristic
164	C₂×AGL(1,ℤ/8)	—	23	2	no	—
165	C₂²×D₈	—	23	1	no	—
166	C₂×#131	—	23	1	no	—
167	#472	—	23	2	no	—
168	#472	—	23	1	no	—
169	#472	—	23	1	no	—
170	#472	—	23	1	no	—
171	#472	—	23	2	no	—
172	#472	D₂₃	1	1	yes	characteristic, direct product factor
173	C₂×#108	—	23	1	no	—
174	C₂×#100	—	23	2	no	—
175	C₂×C₂₃	—	2	1	no	—
176	C₂×C₂₃	—	4	1	no	—
177	C₂×C₂₃	—	8	2	no	—
178	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
179	D₂₃	—	2	1	no	—
180	D₂₃	—	4	1	no	—
181	D₂₃	—	8	2	no	—
182	D₂₃	#472	1	2	yes	direct product factor
183	C₂×#41	—	23	1	no	—
184	C₂²×D₄	—	23	2	no	—
185	AGL(1,ℤ/8)	—	23	2	no	—
186	AGL(1,ℤ/8)	—	23	2	no	—
187	AGL(1,ℤ/8)	—	23	2	no	—
188	AGL(1,ℤ/8)	—	46	2	no	—
189	AGL(1,ℤ/8)	—	46	2	no	—
190	AGL(1,ℤ/8)	C₂×D₂₃	1	2	yes	complemented
191	C₂×SD₁₆	—	46	2	no	—
192	C₂×D₈	—	23	2	no	—
193	C₂×D₈	—	23	1	no	—
194	C₂×D₈	—	46	1	no	—
195	C₂×D₈	—	46	1	no	—
196	C₂×D₈	—	46	1	no	—
197	C₂×D₈	—	46	1	no	—
198	C₂×D₈	—	46	2	no	—
199	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic, complemented
200	#131	—	23	1	no	—
201	#131	—	23	1	no	—
202	#131	—	23	1	no	—
203	#131	—	46	1	no	—
204	#131	—	46	1	no	—
205	#131	C₂×D₂₃	1	1	yes	characteristic, complemented
206	C₂×M₁₆	—	23	1	no	—
207	C₂×M₁₆	—	23	2	no	—
208	C₂×M₁₆	—	46	1	no	—
209	C₂²×C₈	—	23	1	no	—
210	C₂²×C₈	—	46	1	no	—
211	#108	—	23	2	no	—
212	#108	—	23	1	no	—
213	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
214	#100	—	23	2	no	—
215	#100	—	23	2	no	—
216	#100	—	23	2	no	—
217	#100	C₂×D₂₃	1	2	yes	complemented
218	C₂₃	C₂×#472	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	—	46	2	no	—
220	#41	—	23	1	no	—
221	#41	—	23	1	no	—
222	#41	—	23	1	no	—
223	#41	—	46	1	no	—
224	#41	—	46	1	no	—
225	#41	C₂²×D₂₃	1	1	yes	characteristic
226	C₂×Q₈	—	23	1	no	—
227	C₂×D₄	—	23	1	no	—
228	C₂×D₄	—	23	2	no	—
229	C₂×D₄	—	46	2	no	—
230	C₂×D₄	—	46	2	no	—
231	C₂×D₄	—	46	2	no	—
232	C₂×D₄	—	46	2	no	—
233	C₂×D₄	—	46	2	no	—
234	C₂×D₄	—	46	1	no	—
235	C₂×D₄	C₂²×D₂₃	1	2	yes	—
236	C₂²×C₄	—	23	1	no	—
237	C₂²×C₄	—	46	1	no	—
238	SD₁₆	—	2	2	no	—
239	SD₁₆	—	46	2	no	—
240	SD₁₆	—	46	2	no	—
241	SD₁₆	—	46	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	2	1	no	—
244	D₈	—	2	1	no	—
245	D₈	—	46	2	no	—
246	D₈	—	46	2	no	—
247	D₈	—	46	2	no	—
248	D₈	—	46	2	no	—
249	D₈	—	46	2	no	—
250	D₈	—	46	1	no	—
251	D₈	—	46	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	23	1	no	—
254	M₁₆	—	23	2	no	—
255	M₁₆	—	46	1	no	—
256	M₁₆	—	46	1	no	—
257	M₁₆	—	46	2	no	—
258	M₁₆	—	46	1	no	—
259	M₁₆	—	46	1	no	—
260	M₁₆	C₂²×D₂₃	1	1	yes	characteristic
261	M₁₆	C₂²×D₂₃	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	23	1	no	—
264	C₂×C₈	—	23	1	no	—
265	C₂×C₈	—	23	1	no	—
266	C₂×C₈	—	23	1	no	—
267	C₂×C₈	—	23	1	no	—
268	C₂×C₈	—	46	1	no	—
269	C₂×C₈	—	46	1	no	—
270	C₂×C₈	—	46	2	no	—
271	C₂×C₈	—	46	1	no	—
272	C₂×C₈	—	46	1	no	—
273	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
274	C₂³	—	2	2	no	—
275	C₂³	—	23	1	no	—
276	C₂³	—	46	1	no	—
277	C₂³	—	46	2	no	—
278	C₂³	—	92	2	no	—
279	C₂³	—	92	2	no	—
280	C₂³	—	184	4	no	—
281	Q₈	—	23	1	no	—
282	Q₈	—	46	1	no	—
283	Q₈	D₄×D₂₃	1	1	yes	characteristic
284	D₄	—	2	1	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	23	1	no	—
288	D₄	—	46	1	no	—
289	D₄	—	46	1	no	—
290	D₄	—	46	1	no	—
291	D₄	—	46	1	no	—
292	D₄	—	46	2	no	—
293	D₄	—	46	2	no	—
294	D₄	—	92	2	no	—
295	D₄	—	92	2	no	—
296	D₄	D₄×D₂₃	1	1	yes	characteristic
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	23	1	no	—
299	C₂×C₄	—	23	1	no	—
300	C₂×C₄	—	23	1	no	—
301	C₂×C₄	—	23	1	no	—
302	C₂×C₄	—	23	1	no	—
303	C₂×C₄	—	46	1	no	—
304	C₂×C₄	—	46	1	no	—
305	C₂×C₄	—	46	1	no	—
306	C₂×C₄	—	46	1	no	—
307	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	—	23	1	no	—
311	C₈	—	23	1	no	—
312	C₈	—	46	1	no	—
313	C₈	—	46	2	no	—
314	C₈	D₄×D₂₃	1	1	yes	characteristic
315	C₈	D₄×D₂₃	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	23	1	no	—
320	C₂²	—	23	1	no	—
321	C₂²	—	46	2	no	—
322	C₂²	—	46	1	no	—
323	C₂²	—	92	2	no	—
324	C₂²	—	92	2	no	—
325	C₂²	—	184	2	no	—
326	C₂²	—	184	4	no	—
327	C₂²	—	184	4	no	—
328	C₂²	#41×D₂₃	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	23	1	no	—
331	C₄	—	23	1	no	—
332	C₄	—	46	1	no	—
333	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
334	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	4	1	no	—
337	C₂	—	8	2	no	—
338	C₂	—	23	2	no	—
339	C₂	—	46	1	no	—
340	C₂	—	92	1	no	—
341	C₂	—	184	2	no	—
342	C₂	#121×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#121×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#472	C₂
socle	C₂×C₂₃	C₂×#121
hypercenter	#472	D₂₃
hypocenter	C₂₃	C₂×#472
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #472

Complements

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

N	H ≅ G/N	type
1. AGL(1,ℤ/8)×D₂₃	C₂	semidirect
2. C₂×D₈×D₂₃	C₂	semidirect
3. #131×D₂₃	C₂	semidirect
4. #13092688	C₂	semidirect
5. #13092687	C₂	semidirect
6. #13092686	C₂	semidirect
7. #13092685	C₂	semidirect
8. #13092684	C₂	semidirect
9. #108×D₂₃	C₂	semidirect
10. #100×D₂₃	C₂	semidirect
11. C₂₃×#472	C₂	semidirect
14. C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
15. C₂×D₈×C₂₃	C₂²	semidirect
16. C₂₃×#131	C₂²	semidirect
17. #10625120	C₂²	semidirect
18. #10625119	C₂²	semidirect
19. #10625103	C₂²	semidirect
20. C₂×#67272	C₂²	semidirect
29. #10625079	C₂²	semidirect
31. #10625078	C₂²	semidirect
35. C₂×D₁₈₄	C₂²	semidirect
37. #10625072	C₂²	semidirect
40. C₂₃×#108	C₂²	semidirect
41. C₂₃×#100	C₂²	semidirect
42. #10625015	C₂²	semidirect
43. #10625005	C₂²	semidirect
44. #10625005	C₂²	semidirect
45. #10625004	C₂²	semidirect
46. #10625001	C₂²	semidirect
172. #472	D₂₃	direct (N × H)
182. D₂₃	#472	direct (N × H)
190. AGL(1,ℤ/8)	C₂×D₂₃	semidirect
199. C₂×D₈	C₂×D₂₃	semidirect
205. #131	C₂×D₂₃	semidirect
213. #108	C₂×D₂₃	semidirect
217. #100	C₂×D₂₃	semidirect
218. C₂₃	C₂×#472	semidirect

Maximal subgroups

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

subgroup	index
AGL(1,ℤ/8)×D₂₃	2
C₂×D₈×D₂₃	2
#131×D₂₃	2
#13092688	2
#13092687	2
#13092686	2
#13092685	2
#13092684	2
#108×D₂₃	2
#100×D₂₃	2
C₂₃×#472	2
C₂×#472	23

Subgroups — D23×#472 #15683100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#472 #15683100