Subgroups — AGL(1,ℤ/8)×Dic₂₆₉ #46819695

Order 34432.

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×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770717	C₂	1	2	yes	maximal, complemented
4	#31770706	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₂₆₉	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₂₆₉	C₂	1	2	yes	maximal, complemented
7	#31770647	C₂	1	1	yes	maximal, characteristic, complemented
8	#31770610	C₂	1	1	yes	maximal, characteristic, complemented
9	#31770606	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×AGL(1,ℤ/8)×C₂₆₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₂₆₉	—	2	1	no	—
13	C₂×#16864386	C₂²	1	1	yes	characteristic
14	Q₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
15	Q₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
16	D₄×Dic₂₆₉	—	2	1	no	—
17	D₄×Dic₂₆₉	—	2	1	no	—
18	D₄×Dic₂₆₉	—	2	1	no	—
19	D₄×Dic₂₆₉	—	2	1	no	—
20	D₄×Dic₂₆₉	C₂²	1	2	yes	—
21	D₄×Dic₂₆₉	C₂²	1	2	yes	—
22	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
23	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
24	#22575412	—	2	1	no	—
25	#22575412	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₂₆₉	—	2	1	no	—
28	C₂×C₄×Dic₂₆₉	—	2	1	no	—
29	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
30	#22575387	C₂²	1	2	yes	complemented
31	#22575385	C₂²	1	2	yes	complemented
32	#22575377	C₂²	1	2	yes	complemented
33	#22575374	C₂²	1	2	yes	—
34	#22575374	C₂²	1	2	yes	—
35	#22575374	C₂²	1	2	yes	complemented
36	#22575363	C₂²	1	2	yes	complemented
37	C₈×Dic₂₆₉	C₂²	1	2	yes	complemented
38	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
39	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
40	AGL(1,ℤ/8)×C₂₆₉	C₄	1	8	yes	complemented
41	C₂×SD₁₆×C₂₆₉	C₂²	1	2	yes	—
42	C₂×D₈×C₂₆₉	C₂²	1	2	yes	—
43	C₂×M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
44	C₂²×Dic₂₆₉	—	2	1	no	—
45	C₂²×Dic₂₆₉	—	2	1	no	—
46	C₂²×Dic₂₆₉	—	2	2	no	—
47	C₂²×Dic₂₆₉	—	2	2	no	—
48	C₂²×Dic₂₆₉	—	2	1	no	—
49	C₂²×Dic₂₆₉	—	4	4	no	—
50	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
51	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
52	#16864386	—	2	2	no	—
53	C₂×#14524130	C₂³	1	2	yes	—
54	C₄⋊Dic₂₆₉	—	2	1	no	—
55	C₄⋊Dic₂₆₉	—	2	1	no	—
56	C₄⋊Dic₂₆₉	—	2	1	no	—
57	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
60	C₂²⋊Dic₂₆₉	—	2	2	no	—
61	C₂²⋊Dic₂₆₉	—	2	2	no	—
62	C₂²⋊Dic₂₆₉	—	2	1	no	—
63	C₂²⋊Dic₂₆₉	—	2	1	no	—
64	C₂²⋊Dic₂₆₉	—	2	1	no	—
65	C₄×Dic₂₆₉	—	2	1	no	—
66	C₄×Dic₂₆₉	—	2	1	no	—
67	C₄×Dic₂₆₉	—	2	1	no	—
68	C₄×Dic₂₆₉	D₄	1	1	yes	characteristic
69	C₄×Dic₂₆₉	D₄	1	1	yes	characteristic
70	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
71	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
72	C₂⁴×C₂₆₉	—	2	1	no	—
73	#41×C₂₆₉	—	2	2	no	—
74	#41×C₂₆₉	C₂×C₄	1	4	yes	complemented
75	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₂₆₉	—	2	1	no	—
77	C₂×D₄×C₂₆₉	—	2	1	no	—
78	C₂×D₄×C₂₆₉	—	2	2	no	—
79	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	complemented
80	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
81	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₂₆₉	—	2	1	no	—
83	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
84	SD₁₆×C₂₆₉	C₂×C₄	1	8	yes	complemented
85	D₈×C₂₆₉	C₂×C₄	1	8	yes	complemented
86	M₁₆×C₂₆₉	C₂×C₄	1	4	yes	complemented
87	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
88	C₂×Dic₂₆₉	—	2	2	no	—
89	C₂×Dic₂₆₉	—	2	1	no	—
90	C₂×Dic₂₆₉	—	2	1	no	—
91	C₂×Dic₂₆₉	—	2	2	no	—
92	C₂×Dic₂₆₉	—	4	1	no	—
93	C₂×Dic₂₆₉	—	4	2	no	—
94	C₂×Dic₂₆₉	—	4	4	no	—
95	C₂×Dic₂₆₉	—	4	2	no	—
96	C₂×Dic₂₆₉	—	4	2	no	—
97	C₂×Dic₂₆₉	—	4	4	no	—
98	C₂×Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
101	C₂×Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
102	#14524130	—	2	2	no	—
103	C₂³×C₂₆₉	—	2	4	no	—
104	C₂³×C₂₆₉	—	2	1	no	—
105	C₂³×C₂₆₉	—	2	2	no	—
106	C₂³×C₂₆₉	—	4	2	no	—
107	C₂³×C₂₆₉	—	4	2	no	—
108	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₂₆₉	—	2	1	no	—
110	Q₈×C₂₆₉	C₂²×C₄	1	2	yes	complemented
111	D₄×C₂₆₉	—	2	4	no	—
112	D₄×C₂₆₉	—	2	4	no	—
113	D₄×C₂₆₉	—	2	2	no	—
114	D₄×C₂₆₉	—	2	1	no	—
115	D₄×C₂₆₉	C₂²×C₄	1	4	yes	—
116	D₄×C₂₆₉	C₂²×C₄	1	2	yes	complemented
117	C₂×C₄×C₂₆₉	—	2	1	no	—
118	C₂×C₄×C₂₆₉	—	2	4	no	—
119	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₆₉	#41	1	2	yes	—
122	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
123	C₈×C₂₆₉	C₂²×C₄	1	4	yes	complemented
124	C₂²×C₂₆₉	—	2	1	no	—
125	C₂²×C₂₆₉	—	2	1	no	—
126	C₂²×C₂₆₉	—	2	2	no	—
127	C₂²×C₂₆₉	—	2	4	no	—
128	C₂²×C₂₆₉	—	4	8	no	—
129	C₂²×C₂₆₉	—	4	2	no	—
130	C₂²×C₂₆₉	—	4	1	no	—
131	C₂²×C₂₆₉	—	4	2	no	—
132	C₂²×C₂₆₉	—	4	1	no	—
133	C₂²×C₂₆₉	C₄×D₄	1	2	yes	—
134	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₂₆₉	—	2	2	no	—
136	C₄×C₂₆₉	C₄×D₄	1	2	yes	—
137	C₄×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
138	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
139	Dic₂₆₉	—	2	1	no	—
140	Dic₂₆₉	—	2	1	no	—
141	Dic₂₆₉	—	2	1	no	—
142	Dic₂₆₉	—	4	1	no	—
143	Dic₂₆₉	—	4	2	no	—
144	Dic₂₆₉	—	4	1	no	—
145	Dic₂₆₉	AGL(1,ℤ/8)	1	2	yes	direct product factor
146	C₂×C₂₆₉	—	2	2	no	—
147	C₂×C₂₆₉	—	4	2	no	—
148	C₂×C₂₆₉	—	4	4	no	—
149	C₂×C₂₆₉	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×C₂₆₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₂₆₉	#575	1	1	yes	characteristic
152	C₂₆₉	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	269	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	D₂₆₉	1	1	yes	characteristic
155	C₄×#41	—	269	1	no	—
156	C₂×C₄×D₄	—	269	1	no	—
157	#442	—	269	2	no	—
158	#440	—	269	2	no	—
159	C₄×SD₁₆	—	269	2	no	—
160	C₄×D₈	—	269	2	no	—
161	#428	—	269	1	no	—
162	#418	—	269	1	no	—
163	#417	—	269	1	no	—
164	C₄×M₁₆	—	269	1	no	—
165	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
166	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
167	C₂³×C₄	—	538	1	no	—
168	AGL(1,ℤ/8)	Dic₂₆₉	1	8	yes	direct product factor
169	C₂×SD₁₆	C₂×D₂₆₉	1	2	yes	—
170	C₂×D₈	C₂×D₂₆₉	1	2	yes	—
171	C₂×M₁₆	—	269	1	no	—
172	C₂×M₁₆	C₂×D₂₆₉	1	1	yes	characteristic
173	C₄×Q₈	—	269	1	no	—
174	C₄×Q₈	—	269	1	no	—
175	C₄×D₄	—	269	2	no	—
176	C₄×D₄	—	269	2	no	—
177	C₄×D₄	—	269	1	no	—
178	C₄×D₄	—	269	1	no	—
179	C₄×D₄	—	538	1	no	—
180	C₄×D₄	—	538	1	no	—
181	C₄×D₄	—	538	1	no	—
182	C₄×D₄	—	538	1	no	—
183	#117	—	269	1	no	—
184	#117	—	538	1	no	—
185	C₂×C₄⋊C₄	—	269	1	no	—
186	C₂×C₂²⋊C₄	—	538	1	no	—
187	C₂×C₄²	—	269	1	no	—
188	C₂×C₄²	—	538	1	no	—
189	#107	—	269	2	no	—
190	#106	—	269	2	no	—
191	#103	—	269	2	no	—
192	#102	—	269	2	no	—
193	#102	—	269	2	no	—
194	#102	—	269	2	no	—
195	#97	—	269	2	no	—
196	C₄×C₈	—	269	2	no	—
197	C₂⁴	—	2	1	no	—
198	#41	—	2	2	no	—
199	#41	C₂×Dic₂₆₉	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₂₆₉	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×Dic₂₆₉	1	4	yes	complemented
205	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
206	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	269	1	no	—
209	C₂²×C₄	—	269	1	no	—
210	C₂²×C₄	—	538	1	no	—
211	C₂²×C₄	—	538	1	no	—
212	C₂²×C₄	—	538	2	no	—
213	C₂²×C₄	—	538	2	no	—
214	C₂²×C₄	—	538	1	no	—
215	C₂²×C₄	—	1076	4	no	—
216	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
217	SD₁₆	C₂×Dic₂₆₉	1	8	yes	complemented
218	D₈	C₂×Dic₂₆₉	1	8	yes	complemented
219	M₁₆	—	538	2	no	—
220	M₁₆	C₂×Dic₂₆₉	1	4	yes	complemented
221	C₂×C₈	—	269	2	no	—
222	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
223	C₄⋊C₄	—	269	1	no	—
224	C₄⋊C₄	—	269	1	no	—
225	C₄⋊C₄	—	269	2	no	—
226	C₄⋊C₄	—	538	1	no	—
227	C₄⋊C₄	—	538	1	no	—
228	C₄⋊C₄	—	538	1	no	—
229	C₂²⋊C₄	—	538	2	no	—
230	C₂²⋊C₄	—	538	2	no	—
231	C₂²⋊C₄	—	538	1	no	—
232	C₂²⋊C₄	—	538	1	no	—
233	C₂²⋊C₄	—	538	1	no	—
234	C₄²	—	269	1	no	—
235	C₄²	—	269	1	no	—
236	C₄²	—	269	1	no	—
237	C₄²	—	269	1	no	—
238	C₄²	—	538	1	no	—
239	C₄²	—	538	1	no	—
240	C₄²	—	538	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×D₂₆₉	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂²×Dic₂₆₉	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×Dic₂₆₉	1	4	yes	—
254	D₄	C₂²×Dic₂₆₉	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	269	1	no	—
258	C₂×C₄	—	269	1	no	—
259	C₂×C₄	—	269	1	no	—
260	C₂×C₄	—	269	1	no	—
261	C₂×C₄	—	538	2	no	—
262	C₂×C₄	—	538	1	no	—
263	C₂×C₄	—	538	1	no	—
264	C₂×C₄	—	538	2	no	—
265	C₂×C₄	—	1076	4	no	—
266	C₂×C₄	—	1076	2	no	—
267	C₂×C₄	—	1076	1	no	—
268	C₂×C₄	—	1076	4	no	—
269	C₂×C₄	—	1076	2	no	—
270	C₂×C₄	—	1076	2	no	—
271	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
272	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
273	C₂×C₄	#16864401	1	2	yes	—
274	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
275	C₈	—	538	2	no	—
276	C₈	C₂²×Dic₂₆₉	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	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	D₄×Dic₂₆₉	1	2	yes	—
287	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	269	2	no	—
290	C₄	—	538	1	no	—
291	C₄	—	538	1	no	—
292	C₄	—	538	1	no	—
293	C₄	—	1076	1	no	—
294	C₄	—	1076	2	no	—
295	C₄	—	1076	1	no	—
296	C₄	D₄×Dic₂₆₉	1	2	yes	—
297	C₄	C₂³×Dic₂₆₉	1	1	yes	characteristic
298	C₄	C₂×#16864401	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	4	2	no	—
302	C₂	C₂×D₄×Dic₂₆₉	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₂₆₉	1	1	yes	characteristic
304	C₂	#31771401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₆₉
commutator	C₄×C₂₆₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting, prime radical	C₂×AGL(1,ℤ/8)×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₂₆₉
hypocenter	C₂₆₉	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
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×Dic₂₆₉	C₂	semidirect
2. C₂×D₄×Dic₂₆₉	C₂	semidirect
3. #31770717	C₂	semidirect
4. #31770706	C₂	semidirect
5. SD₁₆×Dic₂₆₉	C₂	semidirect
6. D₈×Dic₂₆₉	C₂	semidirect
7. #31770647	C₂	semidirect
8. #31770610	C₂	semidirect
9. #31770606	C₂	semidirect
10. M₁₆×Dic₂₆₉	C₂	semidirect
14. Q₈×Dic₂₆₉	C₂²	semidirect
15. Q₈×Dic₂₆₉	C₂²	semidirect
22. D₄×Dic₂₆₉	C₂²	semidirect
23. D₄×Dic₂₆₉	C₂²	semidirect
30. #22575387	C₂²	semidirect
31. #22575385	C₂²	semidirect
32. #22575377	C₂²	semidirect
35. #22575374	C₂²	semidirect
36. #22575363	C₂²	semidirect
37. C₈×Dic₂₆₉	C₂²	semidirect
40. AGL(1,ℤ/8)×C₂₆₉	C₄	semidirect
74. #41×C₂₆₉	C₂×C₄	semidirect
79. C₂×D₄×C₂₆₉	C₂×C₄	semidirect
84. SD₁₆×C₂₆₉	C₂×C₄	semidirect
85. D₈×C₂₆₉	C₂×C₄	semidirect
86. M₁₆×C₂₆₉	C₂×C₄	semidirect
110. Q₈×C₂₆₉	C₂²×C₄	semidirect
116. D₄×C₂₆₉	C₂²×C₄	semidirect
123. C₈×C₂₆₉	C₂²×C₄	semidirect
145. Dic₂₆₉	AGL(1,ℤ/8)	direct (N × H)
152. C₂₆₉	C₄×AGL(1,ℤ/8)	semidirect
168. AGL(1,ℤ/8)	Dic₂₆₉	direct (N × H)
199. #41	C₂×Dic₂₆₉	semidirect
204. C₂×D₄	C₂×Dic₂₆₉	semidirect
217. SD₁₆	C₂×Dic₂₆₉	semidirect
218. D₈	C₂×Dic₂₆₉	semidirect
220. M₁₆	C₂×Dic₂₆₉	semidirect
248. Q₈	C₂²×Dic₂₆₉	semidirect
254. D₄	C₂²×Dic₂₆₉	semidirect
276. C₈	C₂²×Dic₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₂₆₉	2
C₂×D₄×Dic₂₆₉	2
#31770717	2
#31770706	2
SD₁₆×Dic₂₆₉	2
D₈×Dic₂₆₉	2
#31770647	2
#31770610	2
#31770606	2
M₁₆×Dic₂₆₉	2
C₂×AGL(1,ℤ/8)×C₂₆₉	2
C₄×AGL(1,ℤ/8)	269

Subgroups — AGL(1,ℤ/8)×Dic269 #46819695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,ℤ/8)×Dic₂₆₉ #46819695