Subgroups — D₂₃×#473 #15683106

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#137×D₂₃	C₂	1	2	yes	maximal
2	C₂×Q₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092693	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092692	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092691	C₂	1	2	yes	maximal, complemented
7	#13092690	C₂	1	2	yes	maximal, complemented
8	#13092689	C₂	1	1	yes	maximal, characteristic, complemented
9	#108×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#101×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₂₃	C₂²	1	2	yes	—
14	C₂₃×#137	C₂²	1	2	yes	—
15	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂₃×#131	C₂²	1	1	yes	characteristic
17	#10625122	C₂²	1	2	yes	—
18	#10625119	C₂²	1	1	yes	characteristic
19	C₂×#67275	C₂²	1	2	yes	complemented
20	#10625113	C₂²	1	2	yes	—
21	#10625089	—	2	2	no	—
22	Q₁₆×D₂₃	—	2	2	no	—
23	Q₁₆×D₂₃	—	2	1	no	—
24	Q₁₆×D₂₃	—	2	1	no	—
25	Q₁₆×D₂₃	—	2	1	no	—
26	Q₁₆×D₂₃	—	2	1	no	—
27	#10625086	—	2	2	no	—
28	SD₁₆×D₂₃	—	2	2	no	—
29	#10625080	C₂²	1	2	yes	—
30	#10625078	—	2	1	no	—
31	#10625078	C₂²	1	1	yes	characteristic
32	M₁₆×D₂₃	—	2	1	no	—
33	M₁₆×D₂₃	C₂²	1	1	yes	characteristic
34	M₁₆×D₂₃	C₂²	1	2	yes	—
35	C₂×Dic₉₂	C₂²	1	1	yes	characteristic, complemented
36	#10625072	—	2	1	no	—
37	#10625072	C₂²	1	1	yes	characteristic
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×#101	C₂²	1	2	yes	complemented
42	#10625018	C₂²	1	2	yes	complemented
43	#10625006	C₂²	1	2	yes	complemented
44	#10625006	C₂²	1	2	yes	complemented
45	#10625004	C₂²	1	2	yes	complemented
46	#10625001	C₂²	1	1	yes	characteristic, complemented
47	#41×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
49	#67292	D₄	1	1	yes	characteristic
50	Q₈×D₂₃	—	2	2	no	—
51	Q₈×D₂₃	—	2	2	no	—
52	Q₈×D₂₃	—	2	2	no	—
53	Q₈×D₂₃	—	2	2	no	—
54	Q₈×D₂₃	D₄	1	1	yes	characteristic
55	#67290	—	2	1	no	—
56	#67290	D₄	1	1	yes	characteristic
57	D₄×D₂₃	—	2	1	no	—
58	D₄×D₂₃	D₄	1	1	yes	characteristic
59	#67288	—	2	1	no	—
60	#67288	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₂₃	—	2	1	no	—
62	C₂×C₄×D₂₃	—	2	2	no	—
63	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
64	C₂×Dic₄₆	—	2	2	no	—
65	C₂×Dic₄₆	C₂³	1	2	yes	—
66	Q₁₆×C₂₃	—	2	2	no	—
67	Q₁₆×C₂₃	—	2	1	no	—
68	Q₁₆×C₂₃	—	2	1	no	—
69	SD₁₆×C₂₃	—	2	2	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	#67275	—	2	2	no	—
76	#67275	—	2	2	no	—
77	#67275	—	2	2	no	—
78	#67275	—	2	2	no	—
79	#67274	—	2	2	no	—
80	#67273	—	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	Dic₉₂	—	2	2	no	—
87	Dic₉₂	—	2	1	no	—
88	Dic₉₂	—	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	C₂²×D₂₃	—	2	1	no	—
103	C₂²×D₂₃	#41	1	1	yes	characteristic
104	Q₈×C₂₃	—	2	2	no	—
105	Q₈×C₂₃	—	2	2	no	—
106	Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
107	D₄×C₂₃	—	2	1	no	—
108	D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₂₃	—	2	1	no	—
113	C₂²⋊D₂₃	—	2	1	no	—
114	C₂²⋊D₂₃	—	2	1	no	—
115	C₂×Dic₂₃	—	2	1	no	—
116	C₂×Dic₂₃	—	2	2	no	—
117	C₂×Dic₂₃	#41	1	1	yes	characteristic
118	D₉₂	—	2	1	no	—
119	D₉₂	C₂×D₄	1	1	yes	characteristic
120	C₄×D₂₃	—	2	1	no	—
121	C₄×D₂₃	—	2	1	no	—
122	C₄×D₂₃	—	2	1	no	—
123	C₄×D₂₃	—	4	2	no	—
124	C₄×D₂₃	—	4	2	no	—
125	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
126	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
127	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
128	C₄×D₂₃	C₂×D₄	1	1	yes	characteristic
129	Dic₄₆	—	2	1	no	—
130	Dic₄₆	—	2	2	no	—
131	Dic₄₆	—	2	2	no	—
132	Dic₄₆	—	4	2	no	—
133	Dic₄₆	—	4	2	no	—
134	Dic₄₆	C₂×D₄	1	1	yes	characteristic
135	C₈×C₂₃	—	2	1	no	—
136	C₈×C₂₃	—	2	2	no	—
137	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂₃⋊C₈	—	2	1	no	—
140	C₂₃⋊C₈	—	2	2	no	—
141	C₂₃⋊C₈	C₂×D₄	1	1	yes	characteristic
142	C₂₃⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×#473	—	23	1	no	maximal, Sylow
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
146	C₂×D₂₃	—	2	2	no	—
147	C₂×D₂₃	—	2	1	no	—
148	C₂×D₂₃	—	4	2	no	—
149	C₂×D₂₃	#121	1	1	yes	characteristic
150	C₂×D₂₃	#121	1	1	yes	characteristic
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
154	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	Dic₂₃	—	2	1	no	—
156	Dic₂₃	—	4	2	no	—
157	Dic₂₃	#121	1	1	yes	characteristic
158	Dic₂₃	#121	1	1	yes	characteristic
159	C₂×#137	—	23	2	no	—
160	C₂²×Q₁₆	—	23	1	no	—
161	C₂×#131	—	23	1	no	—
162	#473	—	23	2	no	—
163	#473	—	23	1	no	—
164	#473	—	23	1	no	—
165	#473	—	23	1	no	—
166	#473	—	23	2	no	—
167	#473	D₂₃	1	1	yes	characteristic, direct product factor
168	C₂×#108	—	23	1	no	—
169	C₂×#101	—	23	2	no	—
170	C₂×C₂₃	—	2	1	no	—
171	C₂×C₂₃	—	4	1	no	—
172	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
173	D₂₃	—	2	1	no	—
174	D₂₃	—	4	1	no	—
175	D₂₃	#473	1	2	yes	direct product factor
176	C₂×#41	—	23	1	no	—
177	C₂²×Q₈	—	23	2	no	—
178	#137	—	23	2	no	—
179	#137	—	23	2	no	—
180	#137	—	23	2	no	—
181	#137	—	46	2	no	—
182	#137	—	46	2	no	—
183	#137	C₂×D₂₃	1	2	yes	—
184	C₂×Q₁₆	—	23	2	no	—
185	C₂×Q₁₆	—	23	1	no	—
186	C₂×Q₁₆	—	46	1	no	—
187	C₂×Q₁₆	—	46	1	no	—
188	C₂×Q₁₆	—	46	1	no	—
189	C₂×Q₁₆	—	46	1	no	—
190	C₂×Q₁₆	—	46	2	no	—
191	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
192	C₂×SD₁₆	—	46	2	no	—
193	#131	—	23	1	no	—
194	#131	—	23	1	no	—
195	#131	—	23	1	no	—
196	#131	—	46	1	no	—
197	#131	—	46	1	no	—
198	#131	C₂×D₂₃	1	1	yes	characteristic
199	C₂×M₁₆	—	23	1	no	—
200	C₂×M₁₆	—	23	2	no	—
201	C₂×M₁₆	—	46	1	no	—
202	C₂²×C₈	—	23	1	no	—
203	C₂²×C₈	—	46	1	no	—
204	#108	—	23	2	no	—
205	#108	—	23	1	no	—
206	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
207	#101	—	23	2	no	—
208	#101	—	23	2	no	—
209	#101	—	23	2	no	—
210	#101	C₂×D₂₃	1	2	yes	complemented
211	C₂₃	C₂×#473	1	1	yes	characteristic, complemented, Sylow
212	#41	—	23	1	no	—
213	#41	—	23	1	no	—
214	#41	—	23	1	no	—
215	#41	—	46	1	no	—
216	#41	—	46	1	no	—
217	#41	C₂²×D₂₃	1	1	yes	characteristic
218	C₂×Q₈	—	23	1	no	—
219	C₂×Q₈	—	23	2	no	—
220	C₂×Q₈	—	46	2	no	—
221	C₂×Q₈	—	46	2	no	—
222	C₂×Q₈	—	46	2	no	—
223	C₂×Q₈	—	46	2	no	—
224	C₂×Q₈	—	46	2	no	—
225	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
226	C₂×D₄	—	23	1	no	—
227	C₂×D₄	—	46	1	no	—
228	C₂²×C₄	—	23	1	no	—
229	C₂²×C₄	—	46	1	no	—
230	C₂²×C₄	—	46	2	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	1	no	—
233	Q₁₆	—	2	1	no	—
234	Q₁₆	—	46	2	no	—
235	Q₁₆	—	46	2	no	—
236	Q₁₆	—	46	2	no	—
237	Q₁₆	—	46	2	no	—
238	Q₁₆	—	46	2	no	—
239	Q₁₆	—	46	1	no	—
240	Q₁₆	—	46	1	no	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	46	2	no	—
243	SD₁₆	—	46	2	no	—
244	SD₁₆	—	46	2	no	—
245	M₁₆	—	2	1	no	—
246	M₁₆	—	23	1	no	—
247	M₁₆	—	23	2	no	—
248	M₁₆	—	46	1	no	—
249	M₁₆	—	46	1	no	—
250	M₁₆	—	46	2	no	—
251	M₁₆	—	46	1	no	—
252	M₁₆	—	46	1	no	—
253	M₁₆	C₂²×D₂₃	1	1	yes	characteristic
254	M₁₆	C₂²×D₂₃	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	23	1	no	—
257	C₂×C₈	—	23	1	no	—
258	C₂×C₈	—	23	1	no	—
259	C₂×C₈	—	23	1	no	—
260	C₂×C₈	—	23	1	no	—
261	C₂×C₈	—	46	1	no	—
262	C₂×C₈	—	46	1	no	—
263	C₂×C₈	—	46	2	no	—
264	C₂×C₈	—	46	1	no	—
265	C₂×C₈	—	46	1	no	—
266	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
267	C₂³	—	23	1	no	—
268	C₂³	—	46	1	no	—
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	23	1	no	—
272	Q₈	—	46	1	no	—
273	Q₈	—	46	2	no	—
274	Q₈	—	46	2	no	—
275	Q₈	—	92	2	no	—
276	Q₈	—	92	2	no	—
277	Q₈	D₄×D₂₃	1	1	yes	characteristic
278	D₄	—	2	1	no	—
279	D₄	—	23	1	no	—
280	D₄	—	46	1	no	—
281	D₄	—	46	1	no	—
282	D₄	—	46	1	no	—
283	D₄	—	46	1	no	—
284	D₄	D₄×D₂₃	1	1	yes	characteristic
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	23	1	no	—
288	C₂×C₄	—	23	1	no	—
289	C₂×C₄	—	23	1	no	—
290	C₂×C₄	—	23	1	no	—
291	C₂×C₄	—	23	1	no	—
292	C₂×C₄	—	46	1	no	—
293	C₂×C₄	—	46	1	no	—
294	C₂×C₄	—	46	1	no	—
295	C₂×C₄	—	46	1	no	—
296	C₂×C₄	—	46	2	no	—
297	C₂×C₄	—	92	2	no	—
298	C₂×C₄	—	92	2	no	—
299	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	2	no	—
302	C₈	—	23	1	no	—
303	C₈	—	23	1	no	—
304	C₈	—	46	1	no	—
305	C₈	—	46	2	no	—
306	C₈	D₄×D₂₃	1	1	yes	characteristic
307	C₈	D₄×D₂₃	1	1	yes	characteristic
308	C₂²	—	2	1	no	—
309	C₂²	—	23	1	no	—
310	C₂²	—	23	1	no	—
311	C₂²	—	46	2	no	—
312	C₂²	—	46	1	no	—
313	C₂²	—	92	2	no	—
314	C₂²	#41×D₂₃	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	23	1	no	—
318	C₄	—	23	1	no	—
319	C₄	—	46	1	no	—
320	C₄	—	92	2	no	—
321	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
322	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	4	1	no	—
325	C₂	—	23	2	no	—
326	C₂	—	46	1	no	—
327	C₂	—	92	1	no	—
328	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×#473	C₂
socle	C₂×C₂₃	C₂×#121
prime radical	#41×D₂₃	C₂²
hypercenter	#473	D₂₃
hypocenter	C₂₃	C₂×#473
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #473

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₂₃	C₂	semidirect
4. #13092693	C₂	semidirect
5. #13092692	C₂	semidirect
6. #13092691	C₂	semidirect
7. #13092690	C₂	semidirect
8. #13092689	C₂	semidirect
9. #108×D₂₃	C₂	semidirect
10. #101×D₂₃	C₂	semidirect
11. C₂₃×#473	C₂	semidirect
15. C₂×Q₁₆×C₂₃	C₂²	semidirect
19. C₂×#67275	C₂²	semidirect
35. C₂×Dic₉₂	C₂²	semidirect
40. C₂₃×#108	C₂²	semidirect
41. C₂₃×#101	C₂²	semidirect
42. #10625018	C₂²	semidirect
43. #10625006	C₂²	semidirect
44. #10625006	C₂²	semidirect
45. #10625004	C₂²	semidirect
46. #10625001	C₂²	semidirect
167. #473	D₂₃	direct (N × H)
175. D₂₃	#473	direct (N × H)
191. C₂×Q₁₆	C₂×D₂₃	semidirect
206. #108	C₂×D₂₃	semidirect
210. #101	C₂×D₂₃	semidirect
211. C₂₃	C₂×#473	semidirect

Maximal subgroups

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

subgroup	index
#137×D₂₃	2
C₂×Q₁₆×D₂₃	2
#131×D₂₃	2
#13092693	2
#13092692	2
#13092691	2
#13092690	2
#13092689	2
#108×D₂₃	2
#101×D₂₃	2
C₂₃×#473	2
C₂×#473	23

Subgroups — D23×#473 #15683106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#473 #15683106