Subgroups — D₂₃×#467 #15682967

Order 2944.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×D₂₃	C₂	1	2	yes	maximal
4	#13092659	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092658	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092657	C₂	1	2	yes	maximal, complemented
7	#13092656	C₂	1	2	yes	maximal, complemented
8	#13092655	C₂	1	1	yes	maximal, characteristic, complemented
9	#107×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
17	#10625117	C₂²	1	2	yes	—
18	C₂×#67275	C₂²	1	2	yes	complemented
19	#10625095	C₂²	1	2	yes	—
20	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
21	Q₁₆×D₂₃	—	2	4	no	—
22	C₂×Dic₉₂	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₈×D₂₃	—	2	4	no	—
24	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
28	#10625068	C₂²	1	2	yes	—
29	#10625067	—	2	2	no	—
30	C₄⋊C₄×D₂₃	—	2	2	no	—
31	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
32	C₂²⋊C₄×D₂₃	—	2	2	no	—
33	#10625049	—	2	2	no	—
34	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#103	C₂²	1	2	yes	complemented
36	#10625017	C₂²	1	2	yes	complemented
37	#10625000	C₂²	1	1	yes	characteristic, complemented
38	#10624998	C₂²	1	2	yes	complemented
39	#10624992	C₂²	1	2	yes	complemented
40	#10624989	C₂²	1	2	yes	complemented
41	C₂³×D₂₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
45	Q₈×D₂₃	—	2	4	no	—
46	Q₈×D₂₃	—	2	4	no	—
47	C₂×C₄×D₂₃	—	2	2	no	—
48	C₂×C₄×D₂₃	—	2	2	no	—
49	C₂×C₄×D₂₃	—	2	2	no	—
50	C₂×C₄×D₂₃	—	2	2	no	—
51	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₄₆	—	2	2	no	—
56	C₂×Dic₄₆	C₂³	1	2	yes	—
57	Q₁₆×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊Dic₂₃	—	2	2	no	—
65	#67275	—	2	4	no	—
66	(C₂×D₂₃)⋊C₄	—	2	2	no	—
67	(C₂×D₂₃)⋊C₄	—	2	2	no	—
68	C₄⋊Dic₂₃	—	2	2	no	—
69	C₄⋊Dic₂₃	C₂³	1	2	yes	—
70	Dic₂₃⋊C₄	—	2	2	no	—
71	Dic₂₃⋊C₄	—	2	2	no	—
72	Dic₂₃⋊C₄	—	2	2	no	—
73	C₂×C₂₃⋊C₈	—	2	2	no	—
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
76	Dic₉₂	—	2	2	no	—
77	C₈×D₂₃	—	2	2	no	—
78	C₈×D₂₃	—	2	2	no	—
79	C₈×D₂₃	—	2	2	no	—
80	C₈×D₂₃	D₄	1	4	yes	—
81	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₂₃	—	2	2	no	—
83	C₂²×D₂₃	—	2	2	no	—
84	C₂²×D₂₃	Q₁₆	1	4	yes	complemented
85	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₂₃	—	2	2	no	—
88	Q₈×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₂₃	—	2	2	no	—
96	C₂×Dic₂₃	—	2	2	no	—
97	C₂×Dic₂₃	—	2	2	no	—
98	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
100	C₄×D₂₃	—	2	2	no	—
101	C₄×D₂₃	—	2	1	no	—
102	C₄×D₂₃	—	2	1	no	—
103	C₄×D₂₃	—	2	2	no	—
104	C₄×D₂₃	—	4	4	no	—
105	C₄×D₂₃	—	4	4	no	—
106	C₄×D₂₃	C₂×D₄	1	2	yes	—
107	C₄×D₂₃	#41	1	2	yes	—
108	Dic₄₆	—	2	2	no	—
109	Dic₄₆	—	2	2	no	—
110	Dic₄₆	—	4	4	no	—
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	C₂×D₄	1	2	yes	—
113	C₂₃⋊C₈	—	2	1	no	—
114	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
115	C₂×#467	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	complemented
119	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₂₃	—	2	1	no	—
121	C₂×D₂₃	—	2	1	no	—
122	C₂×D₂₃	—	2	8	no	—
123	C₂×D₂₃	#121	1	2	yes	—
124	C₂×D₂₃	C₂×Q₁₆	1	2	yes	—
125	C₂×D₂₃	C₂×Q₁₆	1	2	yes	complemented
126	C₂×D₂₃	#135	1	2	yes	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	4	2	no	—
130	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	Dic₂₃	—	2	1	no	—
133	Dic₂₃	—	4	2	no	—
134	Dic₂₃	—	4	2	no	—
135	Dic₂₃	#121	1	2	yes	—
136	C₂²×Q₁₆	—	23	1	no	—
137	C₂³×C₈	—	23	1	no	—
138	C₂×C₂²⋊Q₈	—	23	2	no	—
139	#467	—	23	1	no	—
140	#467	—	23	1	no	—
141	#467	—	23	2	no	—
142	#467	—	23	1	no	—
143	#467	—	23	2	no	—
144	#467	D₂₃	1	1	yes	characteristic, direct product factor
145	C₂×#107	—	23	1	no	—
146	C₂×#103	—	23	2	no	—
147	C₂×C₂₃	—	2	2	no	—
148	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
149	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
150	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
151	D₂₃	—	2	2	no	—
152	D₂₃	#467	1	4	yes	direct product factor
153	C₂²×Q₈	—	23	2	no	—
154	C₂³×C₄	—	23	1	no	—
155	C₂×Q₁₆	—	23	2	no	—
156	C₂×Q₁₆	—	23	1	no	—
157	C₂×Q₁₆	—	46	4	no	—
158	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
159	C₂²×C₈	—	23	1	no	—
160	C₂²×C₈	—	23	1	no	—
161	C₂²×C₈	—	23	1	no	—
162	C₂²×C₈	—	23	1	no	—
163	C₂²×C₈	—	23	1	no	—
164	C₂²×C₈	—	46	4	no	—
165	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
166	C₂²⋊Q₈	—	23	2	no	—
167	C₂²⋊Q₈	—	23	2	no	—
168	C₂²⋊Q₈	—	23	2	no	—
169	C₂²⋊Q₈	—	46	2	no	—
170	C₂²⋊Q₈	—	46	2	no	—
171	C₂²⋊Q₈	C₂×D₂₃	1	2	yes	—
172	C₂×C₄⋊C₄	—	23	2	no	—
173	C₂×C₄⋊C₄	—	46	2	no	—
174	C₂×C₂²⋊C₄	—	46	2	no	—
175	#107	—	23	1	no	—
176	#107	—	23	2	no	—
177	#107	C₂×D₂₃	1	1	yes	characteristic, complemented
178	#103	—	23	2	no	—
179	#103	—	23	2	no	—
180	#103	—	23	2	no	—
181	#103	C₂×D₂₃	1	2	yes	complemented
182	C₂₃	C₂×#467	1	1	yes	characteristic, complemented, Sylow
183	C₂⁴	—	23	1	no	—
184	C₂×Q₈	—	23	2	no	—
185	C₂×Q₈	—	46	2	no	—
186	C₂×Q₈	—	46	4	no	—
187	C₂×Q₈	—	46	4	no	—
188	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
189	C₂²×C₄	—	23	1	no	—
190	C₂²×C₄	—	23	1	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	23	1	no	—
193	C₂²×C₄	—	23	1	no	—
194	C₂²×C₄	—	46	2	no	—
195	C₂²×C₄	—	46	2	no	—
196	C₂²×C₄	—	46	2	no	—
197	C₂²×C₄	—	46	2	no	—
198	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	46	4	no	—
201	Q₁₆	—	46	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	23	1	no	—
204	C₂×C₈	—	23	4	no	—
205	C₂×C₈	—	23	1	no	—
206	C₂×C₈	—	46	2	no	—
207	C₂×C₈	—	46	2	no	—
208	C₂×C₈	—	46	2	no	—
209	C₂×C₈	—	46	2	no	—
210	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	23	2	no	—
214	C₄⋊C₄	—	46	2	no	—
215	C₄⋊C₄	—	46	2	no	—
216	C₄⋊C₄	—	46	2	no	—
217	C₄⋊C₄	—	46	2	no	—
218	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	46	2	no	—
221	C₂²⋊C₄	—	46	2	no	—
222	C₂²⋊C₄	—	46	2	no	—
223	C₂³	—	23	1	no	—
224	C₂³	—	23	4	no	—
225	C₂³	—	23	1	no	—
226	C₂³	—	46	2	no	—
227	C₂³	—	46	2	no	—
228	C₂³	D₄×D₂₃	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	Q₈	—	46	2	no	—
232	Q₈	—	46	2	no	—
233	Q₈	—	92	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	23	1	no	—
239	C₂×C₄	—	23	2	no	—
240	C₂×C₄	—	23	2	no	—
241	C₂×C₄	—	23	1	no	—
242	C₂×C₄	—	46	2	no	—
243	C₂×C₄	—	46	2	no	—
244	C₂×C₄	—	46	2	no	—
245	C₂×C₄	—	46	1	no	—
246	C₂×C₄	—	46	1	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	46	2	no	—
249	C₂×C₄	—	92	4	no	—
250	C₂×C₄	—	92	4	no	—
251	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	23	2	no	—
255	C₈	—	46	1	no	—
256	C₈	D₄×D₂₃	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	23	2	no	—
260	C₂²	—	23	2	no	—
261	C₂²	—	23	2	no	—
262	C₂²	—	23	2	no	—
263	C₂²	—	46	8	no	—
264	C₂²	—	46	1	no	—
265	C₂²	—	46	1	no	—
266	C₂²	Q₁₆×D₂₃	1	2	yes	complemented
267	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	23	2	no	—
272	C₄	—	46	1	no	—
273	C₄	—	92	2	no	—
274	C₄	—	92	2	no	—
275	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
276	C₄	#41×D₂₃	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	23	4	no	—
279	C₂	—	46	2	no	—
280	C₂	#121×D₂₃	1	1	yes	characteristic
281	C₂	C₂×Q₁₆×D₂₃	1	1	yes	characteristic
282	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×#467	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×D₂₃	D₄
hypercenter	#467	D₂₃
hypocenter	C₂₃	C₂×#467
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×D₂₃	C₂	semidirect
4. #13092659	C₂	semidirect
5. #13092658	C₂	semidirect
6. #13092657	C₂	semidirect
7. #13092656	C₂	semidirect
8. #13092655	C₂	semidirect
9. #107×D₂₃	C₂	semidirect
10. #103×D₂₃	C₂	semidirect
11. C₂₃×#467	C₂	semidirect
14. C₂×Q₁₆×C₂₃	C₂²	semidirect
18. C₂×#67275	C₂²	semidirect
22. C₂×Dic₉₂	C₂²	semidirect
34. C₂₃×#107	C₂²	semidirect
35. C₂₃×#103	C₂²	semidirect
36. #10625017	C₂²	semidirect
37. #10625000	C₂²	semidirect
38. #10624998	C₂²	semidirect
39. #10624992	C₂²	semidirect
40. #10624989	C₂²	semidirect
84. C₂²×D₂₃	Q₁₆	semidirect
118. C₂²×C₂₃	C₂×Q₁₆	semidirect
125. C₂×D₂₃	C₂×Q₁₆	semidirect
144. #467	D₂₃	direct (N × H)
152. D₂₃	#467	direct (N × H)
158. C₂×Q₁₆	C₂×D₂₃	semidirect
177. #107	C₂×D₂₃	semidirect
181. #103	C₂×D₂₃	semidirect
182. C₂₃	C₂×#467	semidirect
266. C₂²	Q₁₆×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₂₃	2
C₂²×C₈×D₂₃	2
C₂²⋊Q₈×D₂₃	2
#13092659	2
#13092658	2
#13092657	2
#13092656	2
#13092655	2
#107×D₂₃	2
#103×D₂₃	2
C₂₃×#467	2
C₂×#467	23

Subgroups — D23×#467 #15682967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#467 #15682967