Subgroups — AGL(1,ℤ/8)×Dic₂₃ #15682207

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092529	C₂	1	2	yes	maximal, complemented
4	#13092518	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₂₃	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
7	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092422	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092418	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₂₃	—	2	1	no	—
15	C₂₃×AGL(1,ℤ/8)	C₄	1	8	yes	complemented
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
20	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
21	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
22	D₄×Dic₂₃	—	2	1	no	—
23	D₄×Dic₂₃	—	2	1	no	—
24	D₄×Dic₂₃	—	2	1	no	—
25	D₄×Dic₂₃	—	2	1	no	—
26	D₄×Dic₂₃	C₂²	1	2	yes	—
27	D₄×Dic₂₃	C₂²	1	2	yes	—
28	D₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
29	D₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
30	#10625097	—	2	1	no	—
31	#10625097	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₂₃	—	2	1	no	—
34	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂×#67267	C₂²	1	1	yes	characteristic
36	#10625017	C₂²	1	2	yes	complemented
37	#10625014	C₂²	1	2	yes	—
38	#10625014	C₂²	1	2	yes	—
39	#10625014	C₂²	1	2	yes	complemented
40	#10625000	C₂²	1	2	yes	complemented
41	#10624999	C₂²	1	2	yes	complemented
42	#10624997	C₂²	1	2	yes	complemented
43	C₈×Dic₂₃	C₂²	1	2	yes	complemented
44	C₂⁴×C₂₃	—	2	1	no	—
45	#41×C₂₃	—	2	2	no	—
46	#41×C₂₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
52	C₂×D₄×C₂₃	C₂³	1	2	yes	—
53	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₂₃	—	2	1	no	—
57	C₂²×Dic₂₃	—	2	1	no	—
58	C₂²×Dic₂₃	—	2	1	no	—
59	C₂²×Dic₂₃	—	2	2	no	—
60	C₂²×Dic₂₃	—	2	2	no	—
61	C₂²×Dic₂₃	—	4	4	no	—
62	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
63	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
64	SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
65	D₈×C₂₃	C₂×C₄	1	8	yes	complemented
66	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
67	C₂×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂²⋊Dic₂₃	—	2	1	no	—
69	C₂²⋊Dic₂₃	—	2	1	no	—
70	C₂²⋊Dic₂₃	—	2	1	no	—
71	C₂²⋊Dic₂₃	—	2	2	no	—
72	C₂²⋊Dic₂₃	—	2	2	no	—
73	C₄⋊Dic₂₃	—	2	1	no	—
74	C₄⋊Dic₂₃	—	2	1	no	—
75	C₄⋊Dic₂₃	—	2	1	no	—
76	C₄⋊Dic₂₃	C₂³	1	2	yes	—
77	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₂₃	—	2	1	no	—
80	C₄×Dic₂₃	—	2	1	no	—
81	C₄×Dic₂₃	—	2	1	no	—
82	C₄×Dic₂₃	D₄	1	1	yes	characteristic
83	C₄×Dic₂₃	D₄	1	1	yes	characteristic
84	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
85	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
86	#67267	—	2	2	no	—
87	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
88	C₂³×C₂₃	—	2	4	no	—
89	C₂³×C₂₃	—	2	1	no	—
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	4	2	no	—
92	C₂³×C₂₃	—	4	2	no	—
93	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₂₃	—	2	1	no	—
95	Q₈×C₂₃	C₂²×C₄	1	2	yes	complemented
96	D₄×C₂₃	—	2	4	no	—
97	D₄×C₂₃	—	2	4	no	—
98	D₄×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	2	1	no	—
100	D₄×C₂₃	C₂²×C₄	1	4	yes	—
101	D₄×C₂₃	C₂²×C₄	1	2	yes	complemented
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#41	1	2	yes	—
107	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₂₃	—	2	2	no	—
109	C₂×Dic₂₃	—	2	1	no	—
110	C₂×Dic₂₃	—	2	1	no	—
111	C₂×Dic₂₃	—	2	2	no	—
112	C₂×Dic₂₃	—	4	1	no	—
113	C₂×Dic₂₃	—	4	2	no	—
114	C₂×Dic₂₃	—	4	4	no	—
115	C₂×Dic₂₃	—	4	2	no	—
116	C₂×Dic₂₃	—	4	2	no	—
117	C₂×Dic₂₃	—	4	4	no	—
118	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
122	C₈×C₂₃	C₂²×C₄	1	4	yes	complemented
123	C₂₃⋊C₈	—	2	2	no	—
124	C₄×AGL(1,ℤ/8)	—	23	1	no	maximal, Sylow
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	4	no	—
129	C₂²×C₂₃	—	4	8	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	1	no	—
133	C₂²×C₂₃	—	4	2	no	—
134	C₂²×C₂₃	C₄×D₄	1	2	yes	—
135	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	C₄×D₄	1	2	yes	—
138	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
140	Dic₂₃	—	2	1	no	—
141	Dic₂₃	—	2	1	no	—
142	Dic₂₃	—	2	1	no	—
143	Dic₂₃	—	4	1	no	—
144	Dic₂₃	—	4	2	no	—
145	Dic₂₃	—	4	1	no	—
146	Dic₂₃	AGL(1,ℤ/8)	1	2	yes	direct product factor
147	C₂×AGL(1,ℤ/8)	D₂₃	1	1	yes	characteristic
148	C₄×#41	—	23	1	no	—
149	C₂×C₄×D₄	—	23	1	no	—
150	#442	—	23	2	no	—
151	#440	—	23	2	no	—
152	C₄×SD₁₆	—	23	2	no	—
153	C₄×D₈	—	23	2	no	—
154	#428	—	23	1	no	—
155	#418	—	23	1	no	—
156	#417	—	23	1	no	—
157	C₄×M₁₆	—	23	1	no	—
158	C₂×C₂₃	—	2	2	no	—
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	—	4	4	no	—
161	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₂₃	#575	1	1	yes	characteristic
164	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
165	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
166	C₂³×C₄	—	46	1	no	—
167	AGL(1,ℤ/8)	Dic₂₃	1	8	yes	direct product factor
168	C₂×SD₁₆	C₂×D₂₃	1	2	yes	—
169	C₂×D₈	C₂×D₂₃	1	2	yes	—
170	C₂×M₁₆	—	23	1	no	—
171	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
172	C₄×Q₈	—	23	1	no	—
173	C₄×Q₈	—	23	1	no	—
174	C₄×D₄	—	23	2	no	—
175	C₄×D₄	—	23	2	no	—
176	C₄×D₄	—	23	1	no	—
177	C₄×D₄	—	23	1	no	—
178	C₄×D₄	—	46	1	no	—
179	C₄×D₄	—	46	1	no	—
180	C₄×D₄	—	46	1	no	—
181	C₄×D₄	—	46	1	no	—
182	#117	—	23	1	no	—
183	#117	—	46	1	no	—
184	C₂×C₄⋊C₄	—	23	1	no	—
185	C₂×C₂²⋊C₄	—	46	1	no	—
186	C₂×C₄²	—	23	1	no	—
187	C₂×C₄²	—	46	1	no	—
188	#107	—	23	2	no	—
189	#106	—	23	2	no	—
190	#103	—	23	2	no	—
191	#102	—	23	2	no	—
192	#102	—	23	2	no	—
193	#102	—	23	2	no	—
194	#97	—	23	2	no	—
195	C₄×C₈	—	23	2	no	—
196	C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	—	2	1	no	—
198	#41	—	2	2	no	—
199	#41	C₂×Dic₂₃	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₂×Dic₂₃	1	4	yes	complemented
205	C₂×D₄	C₂²×D₂₃	1	2	yes	—
206	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	23	1	no	—
209	C₂²×C₄	—	23	1	no	—
210	C₂²×C₄	—	46	1	no	—
211	C₂²×C₄	—	46	1	no	—
212	C₂²×C₄	—	46	2	no	—
213	C₂²×C₄	—	46	2	no	—
214	C₂²×C₄	—	46	1	no	—
215	C₂²×C₄	—	92	4	no	—
216	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
217	SD₁₆	C₂×Dic₂₃	1	8	yes	complemented
218	D₈	C₂×Dic₂₃	1	8	yes	complemented
219	M₁₆	—	46	2	no	—
220	M₁₆	C₂×Dic₂₃	1	4	yes	complemented
221	C₂×C₈	—	23	2	no	—
222	C₂×C₈	C₂²×D₂₃	1	2	yes	—
223	C₄⋊C₄	—	23	1	no	—
224	C₄⋊C₄	—	23	1	no	—
225	C₄⋊C₄	—	23	2	no	—
226	C₄⋊C₄	—	46	1	no	—
227	C₄⋊C₄	—	46	1	no	—
228	C₄⋊C₄	—	46	1	no	—
229	C₂²⋊C₄	—	46	2	no	—
230	C₂²⋊C₄	—	46	2	no	—
231	C₂²⋊C₄	—	46	1	no	—
232	C₂²⋊C₄	—	46	1	no	—
233	C₂²⋊C₄	—	46	1	no	—
234	C₄²	—	23	1	no	—
235	C₄²	—	23	1	no	—
236	C₄²	—	23	1	no	—
237	C₄²	—	23	1	no	—
238	C₄²	—	46	1	no	—
239	C₄²	—	46	1	no	—
240	C₄²	—	46	1	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	D₄×D₂₃	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂²×Dic₂₃	1	2	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	C₂²×Dic₂₃	1	4	yes	—
254	D₄	C₂²×Dic₂₃	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	23	1	no	—
258	C₂×C₄	—	23	1	no	—
259	C₂×C₄	—	23	1	no	—
260	C₂×C₄	—	23	1	no	—
261	C₂×C₄	—	46	2	no	—
262	C₂×C₄	—	46	1	no	—
263	C₂×C₄	—	46	1	no	—
264	C₂×C₄	—	46	2	no	—
265	C₂×C₄	—	92	4	no	—
266	C₂×C₄	—	92	2	no	—
267	C₂×C₄	—	92	1	no	—
268	C₂×C₄	—	92	4	no	—
269	C₂×C₄	—	92	2	no	—
270	C₂×C₄	—	92	2	no	—
271	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
272	C₂×C₄	#67290	1	2	yes	—
273	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
274	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
275	C₈	—	46	2	no	—
276	C₈	C₂²×Dic₂₃	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	D₄×Dic₂₃	1	2	yes	—
287	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	23	2	no	—
290	C₄	—	46	1	no	—
291	C₄	—	46	1	no	—
292	C₄	—	46	1	no	—
293	C₄	—	92	2	no	—
294	C₄	—	92	1	no	—
295	C₄	—	92	1	no	—
296	C₄	D₄×Dic₂₃	1	2	yes	—
297	C₄	C₂×#67290	1	1	yes	characteristic
298	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	4	2	no	—
302	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
304	C₂	#13093213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₂
socle	C₂²×C₂₃	C₂²×D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₂₃
hypocenter	C₂₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #41×Dic₂₃	C₂	semidirect
2. C₂×D₄×Dic₂₃	C₂	semidirect
3. #13092529	C₂	semidirect
4. #13092518	C₂	semidirect
5. SD₁₆×Dic₂₃	C₂	semidirect
6. D₈×Dic₂₃	C₂	semidirect
7. #13092459	C₂	semidirect
8. #13092422	C₂	semidirect
9. #13092418	C₂	semidirect
10. M₁₆×Dic₂₃	C₂	semidirect
15. C₂₃×AGL(1,ℤ/8)	C₄	semidirect
20. Q₈×Dic₂₃	C₂²	semidirect
21. Q₈×Dic₂₃	C₂²	semidirect
28. D₄×Dic₂₃	C₂²	semidirect
29. D₄×Dic₂₃	C₂²	semidirect
36. #10625017	C₂²	semidirect
39. #10625014	C₂²	semidirect
40. #10625000	C₂²	semidirect
41. #10624999	C₂²	semidirect
42. #10624997	C₂²	semidirect
43. C₈×Dic₂₃	C₂²	semidirect
46. #41×C₂₃	C₂×C₄	semidirect
51. C₂×D₄×C₂₃	C₂×C₄	semidirect
64. SD₁₆×C₂₃	C₂×C₄	semidirect
65. D₈×C₂₃	C₂×C₄	semidirect
66. M₁₆×C₂₃	C₂×C₄	semidirect
95. Q₈×C₂₃	C₂²×C₄	semidirect
101. D₄×C₂₃	C₂²×C₄	semidirect
122. C₈×C₂₃	C₂²×C₄	semidirect
146. Dic₂₃	AGL(1,ℤ/8)	direct (N × H)
167. AGL(1,ℤ/8)	Dic₂₃	direct (N × H)
196. C₂₃	C₄×AGL(1,ℤ/8)	semidirect
199. #41	C₂×Dic₂₃	semidirect
204. C₂×D₄	C₂×Dic₂₃	semidirect
217. SD₁₆	C₂×Dic₂₃	semidirect
218. D₈	C₂×Dic₂₃	semidirect
220. M₁₆	C₂×Dic₂₃	semidirect
248. Q₈	C₂²×Dic₂₃	semidirect
254. D₄	C₂²×Dic₂₃	semidirect
276. C₈	C₂²×Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₂₃	2
C₂×D₄×Dic₂₃	2
#13092529	2
#13092518	2
SD₁₆×Dic₂₃	2
D₈×Dic₂₃	2
#13092459	2
#13092422	2
#13092418	2
M₁₆×Dic₂₃	2
C₂×C₂₃×AGL(1,ℤ/8)	2
C₄×AGL(1,ℤ/8)	23

Subgroups — AGL(1,ℤ/8)×Dic23 #15682207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,ℤ/8)×Dic₂₃ #15682207