Subgroups — D₂₃×#461 #15682811

Order 2944.

544 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×SD₁₆×D₂₃	C₂	1	2	yes	maximal
2	C₄⋊Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092623	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092622	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092621	C₂	1	2	yes	maximal, complemented
7	#13092620	C₂	1	2	yes	maximal, complemented
8	#13092619	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₃	C₂²	1	2	yes	—
13	C₂×D₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	—	2	1	no	—
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	#10625115	C₂²	1	2	yes	complemented
19	C₂×#67274	C₂²	1	2	yes	—
20	D₄×Dic₂₃	C₂²	1	1	yes	characteristic
21	C₂×#67273	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₂₃	—	2	1	no	—
23	SD₁₆×D₂₃	—	2	8	no	—
24	C₂×#67263	C₂²	1	2	yes	—
25	C₂×C₈×D₂₃	C₂²	1	2	yes	—
26	#10625064	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₂₃	—	2	1	no	—
28	C₄⋊C₄×D₂₃	—	2	1	no	—
29	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
30	#10625059	—	2	1	no	—
31	#10625059	—	2	1	no	—
32	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
33	C₂²⋊C₄×D₂₃	—	2	1	no	—
34	C₄×D₉₂	C₂²	1	1	yes	characteristic
35	C₄²×D₂₃	C₂²	1	1	yes	characteristic
36	#10625040	C₂²	1	1	yes	characteristic, complemented
37	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#103	C₂²	1	2	yes	complemented
39	#10625017	C₂²	1	2	yes	complemented
40	#10624998	C₂²	1	2	yes	complemented
41	Dic₂₃⋊C₈	C₂²	1	2	yes	complemented
42	#10624992	C₂²	1	2	yes	complemented
43	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
44	C₂³×D₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂×C₂²⋊D₂₃	—	2	1	no	—
49	C₂²×Dic₂₃	—	2	1	no	—
50	Q₈×D₂₃	—	2	4	no	—
51	Q₈×D₂₃	—	2	4	no	—
52	D₄×D₂₃	—	2	2	no	—
53	D₄×D₂₃	D₄	1	4	yes	—
54	C₂×D₉₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₂₃	—	2	1	no	—
56	C₂×C₄×D₂₃	—	2	2	no	—
57	C₂×C₄×D₂₃	—	2	1	no	—
58	C₂×C₄×D₂₃	—	2	1	no	—
59	C₂×C₄×D₂₃	—	4	4	no	—
60	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
61	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
62	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
63	C₂×Dic₄₆	—	2	2	no	—
64	C₂×Dic₄₆	C₂³	1	2	yes	—
65	SD₁₆×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²⋊Dic₂₃	—	2	1	no	—
73	#67274	—	2	4	no	—
74	#67273	—	2	4	no	—
75	(C₂×D₂₃)⋊C₄	—	2	1	no	—
76	(C₂×D₂₃)⋊C₄	—	2	1	no	—
77	C₄⋊Dic₂₃	—	2	1	no	—
78	C₄⋊Dic₂₃	—	2	1	no	—
79	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
80	Dic₂₃⋊C₄	—	2	1	no	—
81	Dic₂₃⋊C₄	—	2	2	no	—
82	Dic₂₃⋊C₄	—	2	2	no	—
83	C₄×Dic₂₃	D₄	1	1	yes	characteristic
84	C₄×Dic₂₃	D₄	1	1	yes	characteristic
85	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
87	#67263	—	2	4	no	—
88	C₈×D₂₃	—	2	4	no	—
89	C₂³×C₂₃	—	2	1	no	—
90	C₂²×D₂₃	—	2	1	no	—
91	C₂²×D₂₃	—	2	4	no	—
92	C₂²×D₂₃	—	4	2	no	—
93	C₂²×D₂₃	—	4	2	no	—
94	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₂₃	—	2	2	no	—
96	Q₈×C₂₃	—	2	2	no	—
97	D₄×C₂₃	—	2	1	no	—
98	D₄×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₂₃	—	2	4	no	—
106	C₂×Dic₂₃	—	2	2	no	—
107	C₂×Dic₂₃	—	2	1	no	—
108	C₂×Dic₂₃	—	4	2	no	—
109	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
110	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
112	D₉₂	—	2	1	no	—
113	D₉₂	C₂×D₄	1	2	yes	—
114	C₄×D₂₃	—	2	2	no	—
115	C₄×D₂₃	—	4	4	no	—
116	C₄×D₂₃	—	4	2	no	—
117	C₄×D₂₃	—	4	2	no	—
118	C₄×D₂₃	—	4	2	no	—
119	C₄×D₂₃	SD₁₆	1	4	yes	complemented
120	C₄×D₂₃	C₂×D₄	1	2	yes	—
121	C₄×D₂₃	#41	1	2	yes	—
122	Dic₄₆	—	2	2	no	—
123	Dic₄₆	—	2	2	no	—
124	Dic₄₆	—	4	4	no	—
125	C₈×C₂₃	—	2	2	no	—
126	C₂₃⋊C₈	—	2	2	no	—
127	C₂×#461	—	23	1	no	maximal, Sylow
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	4	1	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₂₃	—	2	2	no	—
133	C₂×D₂₃	—	4	1	no	—
134	C₂×D₂₃	—	4	1	no	—
135	C₂×D₂₃	—	4	8	no	—
136	C₂×D₂₃	#121	1	2	yes	—
137	C₂×D₂₃	C₂×SD₁₆	1	2	yes	—
138	C₂×D₂₃	#137	1	2	yes	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	4	2	no	—
141	C₄×C₂₃	—	4	1	no	—
142	C₄×C₂₃	C₂×SD₁₆	1	2	yes	complemented
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	—	4	1	no	—
148	Dic₂₃	#121	1	2	yes	—
149	Dic₂₃	C₂×SD₁₆	1	2	yes	complemented
150	C₂²×SD₁₆	—	23	2	no	—
151	C₂×C₄⋊Q₈	—	23	1	no	—
152	C₂×C₄×D₄	—	23	1	no	—
153	#461	—	23	2	no	—
154	#461	—	23	1	no	—
155	#461	—	23	1	no	—
156	#461	—	23	1	no	—
157	#461	—	23	2	no	—
158	#461	D₂₃	1	1	yes	characteristic, direct product factor
159	C₂×C₄⋊C₈	—	23	1	no	—
160	C₂×#103	—	23	2	no	—
161	C₂×C₂₃	—	4	2	no	—
162	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
163	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
164	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
165	D₂₃	—	4	2	no	—
166	D₂₃	#461	1	4	yes	direct product factor
167	C₂²×Q₈	—	23	2	no	—
168	C₂²×D₄	—	23	1	no	—
169	C₂³×C₄	—	46	1	no	—
170	C₂×SD₁₆	—	23	2	no	—
171	C₂×SD₁₆	—	23	2	no	—
172	C₂×SD₁₆	—	23	2	no	—
173	C₂×SD₁₆	—	46	8	no	—
174	C₂×SD₁₆	C₂×D₂₃	1	2	yes	—
175	C₂²×C₈	—	23	2	no	—
176	C₄⋊Q₈	—	23	2	no	—
177	C₄⋊Q₈	—	23	1	no	—
178	C₄⋊Q₈	—	46	1	no	—
179	C₄⋊Q₈	—	46	1	no	—
180	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
181	C₄×D₄	—	23	1	no	—
182	C₄×D₄	—	23	1	no	—
183	C₄×D₄	—	23	1	no	—
184	C₄×D₄	—	46	1	no	—
185	C₄×D₄	—	46	1	no	—
186	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
187	C₂×C₄⋊C₄	—	23	1	no	—
188	C₂×C₄⋊C₄	—	46	1	no	—
189	C₂×C₄⋊C₄	—	46	1	no	—
190	C₂×C₂²⋊C₄	—	46	1	no	—
191	C₂×C₄²	—	23	1	no	—
192	C₄⋊C₈	—	23	2	no	—
193	C₄⋊C₈	—	23	1	no	—
194	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
195	#103	—	23	2	no	—
196	#103	—	23	2	no	—
197	#103	—	23	2	no	—
198	#103	C₂×D₂₃	1	2	yes	complemented
199	C₂₃	C₂×#461	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	—	46	1	no	—
201	C₂×Q₈	—	23	2	no	—
202	C₂×Q₈	—	46	2	no	—
203	C₂×Q₈	—	46	4	no	—
204	C₂×Q₈	—	46	4	no	—
205	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
206	C₂×D₄	—	23	4	no	—
207	C₂×D₄	—	23	1	no	—
208	C₂×D₄	—	46	2	no	—
209	C₂×D₄	—	46	1	no	—
210	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	23	1	no	—
213	C₂²×C₄	—	23	1	no	—
214	C₂²×C₄	—	23	1	no	—
215	C₂²×C₄	—	46	1	no	—
216	C₂²×C₄	—	46	1	no	—
217	C₂²×C₄	—	46	2	no	—
218	C₂²×C₄	—	46	1	no	—
219	C₂²×C₄	—	46	1	no	—
220	C₂²×C₄	—	92	4	no	—
221	SD₁₆	—	2	4	no	—
222	SD₁₆	—	46	4	no	—
223	SD₁₆	—	46	4	no	—
224	SD₁₆	—	46	4	no	—
225	C₂×C₈	—	23	2	no	—
226	C₂×C₈	—	46	4	no	—
227	C₂×C₈	C₂²×D₂₃	1	2	yes	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	23	1	no	—
231	C₄⋊C₄	—	46	1	no	—
232	C₄⋊C₄	—	46	2	no	—
233	C₄⋊C₄	—	46	2	no	—
234	C₄⋊C₄	—	46	1	no	—
235	C₄⋊C₄	—	46	1	no	—
236	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	46	1	no	—
239	C₂²⋊C₄	—	46	1	no	—
240	C₂²⋊C₄	—	46	1	no	—
241	C₄²	—	23	1	no	—
242	C₄²	—	23	1	no	—
243	C₄²	—	23	1	no	—
244	C₄²	C₂²×D₂₃	1	1	yes	characteristic
245	C₂³	—	2	1	no	—
246	C₂³	—	23	1	no	—
247	C₂³	—	46	4	no	—
248	C₂³	—	46	1	no	—
249	C₂³	—	92	2	no	—
250	C₂³	—	92	2	no	—
251	Q₈	—	2	2	no	—
252	Q₈	—	2	2	no	—
253	Q₈	—	46	2	no	—
254	Q₈	—	46	2	no	—
255	Q₈	—	92	4	no	—
256	D₄	—	2	1	no	—
257	D₄	—	23	2	no	—
258	D₄	—	46	4	no	—
259	D₄	—	46	1	no	—
260	D₄	D₄×D₂₃	1	2	yes	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	23	1	no	—
265	C₂×C₄	—	23	1	no	—
266	C₂×C₄	—	23	4	no	—
267	C₂×C₄	—	23	2	no	—
268	C₂×C₄	—	23	2	no	—
269	C₂×C₄	—	23	1	no	—
270	C₂×C₄	—	46	2	no	—
271	C₂×C₄	—	46	1	no	—
272	C₂×C₄	—	46	2	no	—
273	C₂×C₄	—	92	2	no	—
274	C₂×C₄	—	92	2	no	—
275	C₂×C₄	—	92	4	no	—
276	C₂×C₄	—	92	2	no	—
277	C₂×C₄	—	92	2	no	—
278	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
279	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	46	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	23	2	no	—
287	C₂²	—	23	2	no	—
288	C₂²	—	23	2	no	—
289	C₂²	—	46	2	no	—
290	C₂²	—	92	8	no	—
291	C₂²	—	92	1	no	—
292	C₂²	—	92	1	no	—
293	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	1	no	—
297	C₄	—	23	2	no	—
298	C₄	—	23	2	no	—
299	C₄	—	46	1	no	—
300	C₄	—	92	2	no	—
301	C₄	—	92	1	no	—
302	C₄	SD₁₆×D₂₃	1	2	yes	complemented
303	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
304	C₄	#41×D₂₃	1	1	yes	characteristic
305	C₂	—	4	2	no	—
306	C₂	—	23	4	no	—
307	C₂	—	92	2	no	—
308	C₂	#121×D₂₃	1	1	yes	characteristic
309	C₂	C₂×SD₁₆×D₂₃	1	1	yes	characteristic
310	C₂	#137×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×#461	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×D₄×D₂₃	C₂²
hypercenter	#461	D₂₃
hypocenter	C₂₃	C₂×#461
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
2. C₄⋊Q₈×D₂₃	C₂	semidirect
4. #13092623	C₂	semidirect
5. #13092622	C₂	semidirect
6. #13092621	C₂	semidirect
7. #13092620	C₂	semidirect
8. #13092619	C₂	semidirect
9. C₄⋊C₈×D₂₃	C₂	semidirect
10. #103×D₂₃	C₂	semidirect
11. C₂₃×#461	C₂	semidirect
16. C₂₃×C₄⋊Q₈	C₂²	semidirect
18. #10625115	C₂²	semidirect
36. #10625040	C₂²	semidirect
37. C₂₃×C₄⋊C₈	C₂²	semidirect
38. C₂₃×#103	C₂²	semidirect
39. #10625017	C₂²	semidirect
40. #10624998	C₂²	semidirect
41. Dic₂₃⋊C₈	C₂²	semidirect
42. #10624992	C₂²	semidirect
43. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
119. C₄×D₂₃	SD₁₆	semidirect
142. C₄×C₂₃	C₂×SD₁₆	semidirect
149. Dic₂₃	C₂×SD₁₆	semidirect
158. #461	D₂₃	direct (N × H)
166. D₂₃	#461	direct (N × H)
180. C₄⋊Q₈	C₂×D₂₃	semidirect
194. C₄⋊C₈	C₂×D₂₃	semidirect
198. #103	C₂×D₂₃	semidirect
199. C₂₃	C₂×#461	semidirect
302. 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
C₄⋊Q₈×D₂₃	2
C₄×D₄×D₂₃	2
#13092623	2
#13092622	2
#13092621	2
#13092620	2
#13092619	2
C₄⋊C₈×D₂₃	2
#103×D₂₃	2
C₂₃×#461	2
C₂×#461	23

Subgroups — D23×#461 #15682811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#461 #15682811