Subgroups — AGL(1,ℤ/8)×Dic₁₀₁ #27179942

Order 12928.

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×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265289	C₂	1	2	yes	maximal, complemented
4	#20265278	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
7	#20265219	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265182	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265178	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×AGL(1,ℤ/8)×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₀₁	—	2	1	no	—
13	C₂×#13108058	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
15	Q₈×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
16	D₄×Dic₁₀₁	—	2	1	no	—
17	D₄×Dic₁₀₁	—	2	1	no	—
18	D₄×Dic₁₀₁	—	2	1	no	—
19	D₄×Dic₁₀₁	—	2	1	no	—
20	D₄×Dic₁₀₁	C₂²	1	2	yes	—
21	D₄×Dic₁₀₁	C₂²	1	2	yes	—
22	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
23	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
24	#15692214	—	2	1	no	—
25	#15692214	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
28	C₂×C₄×Dic₁₀₁	—	2	1	no	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	#15692189	C₂²	1	2	yes	complemented
31	#15692187	C₂²	1	2	yes	complemented
32	#15692179	C₂²	1	2	yes	complemented
33	#15692176	C₂²	1	2	yes	—
34	#15692176	C₂²	1	2	yes	—
35	#15692176	C₂²	1	2	yes	complemented
36	#15692165	C₂²	1	2	yes	complemented
37	C₈×Dic₁₀₁	C₂²	1	2	yes	complemented
38	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
40	AGL(1,ℤ/8)×C₁₀₁	C₄	1	8	yes	complemented
41	C₂×SD₁₆×C₁₀₁	C₂²	1	2	yes	—
42	C₂×D₈×C₁₀₁	C₂²	1	2	yes	—
43	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	—	2	1	no	—
45	C₂²×Dic₁₀₁	—	2	2	no	—
46	C₂²×Dic₁₀₁	—	2	2	no	—
47	C₂²×Dic₁₀₁	—	2	1	no	—
48	C₂²×Dic₁₀₁	—	2	1	no	—
49	C₂²×Dic₁₀₁	—	4	4	no	—
50	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
51	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
52	#13108058	—	2	2	no	—
53	C₂×#11717579	C₂³	1	2	yes	—
54	C₄⋊Dic₁₀₁	—	2	1	no	—
55	C₄⋊Dic₁₀₁	—	2	1	no	—
56	C₄⋊Dic₁₀₁	—	2	1	no	—
57	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
60	C₂²⋊Dic₁₀₁	—	2	1	no	—
61	C₂²⋊Dic₁₀₁	—	2	2	no	—
62	C₂²⋊Dic₁₀₁	—	2	2	no	—
63	C₂²⋊Dic₁₀₁	—	2	1	no	—
64	C₂²⋊Dic₁₀₁	—	2	1	no	—
65	C₄×Dic₁₀₁	—	2	1	no	—
66	C₄×Dic₁₀₁	—	2	1	no	—
67	C₄×Dic₁₀₁	—	2	1	no	—
68	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
69	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
70	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
71	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
72	C₂⁴×C₁₀₁	—	2	1	no	—
73	#41×C₁₀₁	—	2	2	no	—
74	#41×C₁₀₁	C₂×C₄	1	4	yes	complemented
75	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₀₁	—	2	1	no	—
77	C₂×D₄×C₁₀₁	—	2	1	no	—
78	C₂×D₄×C₁₀₁	—	2	2	no	—
79	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
80	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
81	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	—	2	1	no	—
83	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
84	SD₁₆×C₁₀₁	C₂×C₄	1	8	yes	complemented
85	D₈×C₁₀₁	C₂×C₄	1	8	yes	complemented
86	M₁₆×C₁₀₁	C₂×C₄	1	4	yes	complemented
87	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
88	C₂³×C₁₀₁	—	2	4	no	—
89	C₂³×C₁₀₁	—	2	1	no	—
90	C₂³×C₁₀₁	—	2	2	no	—
91	C₂³×C₁₀₁	—	4	2	no	—
92	C₂³×C₁₀₁	—	4	2	no	—
93	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₁₀₁	—	2	1	no	—
95	Q₈×C₁₀₁	C₂²×C₄	1	2	yes	complemented
96	D₄×C₁₀₁	—	2	4	no	—
97	D₄×C₁₀₁	—	2	4	no	—
98	D₄×C₁₀₁	—	2	2	no	—
99	D₄×C₁₀₁	—	2	1	no	—
100	D₄×C₁₀₁	C₂²×C₄	1	4	yes	—
101	D₄×C₁₀₁	C₂²×C₄	1	2	yes	complemented
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	2	4	no	—
104	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₁	#41	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	1	no	—
110	C₂×Dic₁₀₁	—	2	1	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	4	1	no	—
113	C₂×Dic₁₀₁	—	4	2	no	—
114	C₂×Dic₁₀₁	—	4	4	no	—
115	C₂×Dic₁₀₁	—	4	2	no	—
116	C₂×Dic₁₀₁	—	4	2	no	—
117	C₂×Dic₁₀₁	—	4	4	no	—
118	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
122	C₈×C₁₀₁	C₂²×C₄	1	4	yes	complemented
123	#11717579	—	2	2	no	—
124	C₂²×C₁₀₁	—	2	1	no	—
125	C₂²×C₁₀₁	—	2	1	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	4	no	—
128	C₂²×C₁₀₁	—	4	8	no	—
129	C₂²×C₁₀₁	—	4	2	no	—
130	C₂²×C₁₀₁	—	4	1	no	—
131	C₂²×C₁₀₁	—	4	1	no	—
132	C₂²×C₁₀₁	—	4	2	no	—
133	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
134	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₁₀₁	—	2	2	no	—
136	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
137	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
138	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
139	Dic₁₀₁	—	2	1	no	—
140	Dic₁₀₁	—	2	1	no	—
141	Dic₁₀₁	—	2	1	no	—
142	Dic₁₀₁	—	4	1	no	—
143	Dic₁₀₁	—	4	2	no	—
144	Dic₁₀₁	—	4	1	no	—
145	Dic₁₀₁	AGL(1,ℤ/8)	1	2	yes	direct product factor
146	C₂×C₁₀₁	—	2	2	no	—
147	C₂×C₁₀₁	—	4	2	no	—
148	C₂×C₁₀₁	—	4	4	no	—
149	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×C₁₀₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₁₀₁	#575	1	1	yes	characteristic
152	C₄×AGL(1,ℤ/8)	—	101	1	no	maximal, Sylow
153	C₁₀₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
154	C₂×AGL(1,ℤ/8)	D₁₀₁	1	1	yes	characteristic
155	C₄×#41	—	101	1	no	—
156	C₂×C₄×D₄	—	101	1	no	—
157	#442	—	101	2	no	—
158	#440	—	101	2	no	—
159	C₄×SD₁₆	—	101	2	no	—
160	C₄×D₈	—	101	2	no	—
161	#428	—	101	1	no	—
162	#418	—	101	1	no	—
163	#417	—	101	1	no	—
164	C₄×M₁₆	—	101	1	no	—
165	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
167	C₂³×C₄	—	202	1	no	—
168	AGL(1,ℤ/8)	Dic₁₀₁	1	8	yes	direct product factor
169	C₂×SD₁₆	C₂×D₁₀₁	1	2	yes	—
170	C₂×D₈	C₂×D₁₀₁	1	2	yes	—
171	C₂×M₁₆	—	101	1	no	—
172	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
173	C₄×Q₈	—	101	1	no	—
174	C₄×Q₈	—	101	1	no	—
175	C₄×D₄	—	101	2	no	—
176	C₄×D₄	—	101	2	no	—
177	C₄×D₄	—	101	1	no	—
178	C₄×D₄	—	101	1	no	—
179	C₄×D₄	—	202	1	no	—
180	C₄×D₄	—	202	1	no	—
181	C₄×D₄	—	202	1	no	—
182	C₄×D₄	—	202	1	no	—
183	#117	—	101	1	no	—
184	#117	—	202	1	no	—
185	C₂×C₄⋊C₄	—	101	1	no	—
186	C₂×C₂²⋊C₄	—	202	1	no	—
187	C₂×C₄²	—	101	1	no	—
188	C₂×C₄²	—	202	1	no	—
189	#107	—	101	2	no	—
190	#106	—	101	2	no	—
191	#103	—	101	2	no	—
192	#102	—	101	2	no	—
193	#102	—	101	2	no	—
194	#102	—	101	2	no	—
195	#97	—	101	2	no	—
196	C₄×C₈	—	101	2	no	—
197	C₂⁴	—	2	1	no	—
198	#41	—	2	2	no	—
199	#41	C₂×Dic₁₀₁	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₁₀₁	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×Dic₁₀₁	1	4	yes	complemented
205	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
206	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	101	1	no	—
210	C₂²×C₄	—	404	4	no	—
211	C₂²×C₄	—	202	1	no	—
212	C₂²×C₄	—	202	1	no	—
213	C₂²×C₄	—	202	2	no	—
214	C₂²×C₄	—	202	2	no	—
215	C₂²×C₄	—	202	1	no	—
216	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
217	SD₁₆	C₂×Dic₁₀₁	1	8	yes	complemented
218	D₈	C₂×Dic₁₀₁	1	8	yes	complemented
219	M₁₆	—	202	2	no	—
220	M₁₆	C₂×Dic₁₀₁	1	4	yes	complemented
221	C₂×C₈	—	101	2	no	—
222	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
223	C₄⋊C₄	—	101	1	no	—
224	C₄⋊C₄	—	101	1	no	—
225	C₄⋊C₄	—	101	2	no	—
226	C₄⋊C₄	—	202	1	no	—
227	C₄⋊C₄	—	202	1	no	—
228	C₄⋊C₄	—	202	1	no	—
229	C₂²⋊C₄	—	202	2	no	—
230	C₂²⋊C₄	—	202	2	no	—
231	C₂²⋊C₄	—	202	1	no	—
232	C₂²⋊C₄	—	202	1	no	—
233	C₂²⋊C₄	—	202	1	no	—
234	C₄²	—	101	1	no	—
235	C₄²	—	101	1	no	—
236	C₄²	—	101	1	no	—
237	C₄²	—	101	1	no	—
238	C₄²	—	202	1	no	—
239	C₄²	—	202	1	no	—
240	C₄²	—	202	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂²×Dic₁₀₁	1	2	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	2	no	—
251	D₄	—	2	1	no	—
252	D₄	—	2	4	no	—
253	D₄	C₂²×Dic₁₀₁	1	4	yes	—
254	D₄	C₂²×Dic₁₀₁	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	101	1	no	—
258	C₂×C₄	—	101	1	no	—
259	C₂×C₄	—	101	1	no	—
260	C₂×C₄	—	101	1	no	—
261	C₂×C₄	—	404	4	no	—
262	C₂×C₄	—	404	2	no	—
263	C₂×C₄	—	404	2	no	—
264	C₂×C₄	—	404	4	no	—
265	C₂×C₄	—	404	2	no	—
266	C₂×C₄	—	404	1	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	1	no	—
269	C₂×C₄	—	202	1	no	—
270	C₂×C₄	—	202	2	no	—
271	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
272	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
273	C₂×C₄	#13108073	1	2	yes	—
274	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
275	C₈	—	202	2	no	—
276	C₈	C₂²×Dic₁₀₁	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×Dic₁₀₁	1	2	yes	—
287	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	101	2	no	—
290	C₄	—	404	2	no	—
291	C₄	—	404	1	no	—
292	C₄	—	404	1	no	—
293	C₄	—	202	1	no	—
294	C₄	—	202	1	no	—
295	C₄	—	202	1	no	—
296	C₄	D₄×Dic₁₀₁	1	2	yes	—
297	C₄	C₂³×Dic₁₀₁	1	1	yes	characteristic
298	C₄	C₂×#13108073	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	4	2	no	—
302	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₁₀₁	1	1	yes	characteristic
304	C₂	#20265973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	C₂×AGL(1,ℤ/8)×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₁₀₁
hypocenter	C₁₀₁	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
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×Dic₁₀₁	C₂	semidirect
2. C₂×D₄×Dic₁₀₁	C₂	semidirect
3. #20265289	C₂	semidirect
4. #20265278	C₂	semidirect
5. SD₁₆×Dic₁₀₁	C₂	semidirect
6. D₈×Dic₁₀₁	C₂	semidirect
7. #20265219	C₂	semidirect
8. #20265182	C₂	semidirect
9. #20265178	C₂	semidirect
10. M₁₆×Dic₁₀₁	C₂	semidirect
14. Q₈×Dic₁₀₁	C₂²	semidirect
15. Q₈×Dic₁₀₁	C₂²	semidirect
22. D₄×Dic₁₀₁	C₂²	semidirect
23. D₄×Dic₁₀₁	C₂²	semidirect
30. #15692189	C₂²	semidirect
31. #15692187	C₂²	semidirect
32. #15692179	C₂²	semidirect
35. #15692176	C₂²	semidirect
36. #15692165	C₂²	semidirect
37. C₈×Dic₁₀₁	C₂²	semidirect
40. AGL(1,ℤ/8)×C₁₀₁	C₄	semidirect
74. #41×C₁₀₁	C₂×C₄	semidirect
79. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
84. SD₁₆×C₁₀₁	C₂×C₄	semidirect
85. D₈×C₁₀₁	C₂×C₄	semidirect
86. M₁₆×C₁₀₁	C₂×C₄	semidirect
95. Q₈×C₁₀₁	C₂²×C₄	semidirect
101. D₄×C₁₀₁	C₂²×C₄	semidirect
122. C₈×C₁₀₁	C₂²×C₄	semidirect
145. Dic₁₀₁	AGL(1,ℤ/8)	direct (N × H)
153. C₁₀₁	C₄×AGL(1,ℤ/8)	semidirect
168. AGL(1,ℤ/8)	Dic₁₀₁	direct (N × H)
199. #41	C₂×Dic₁₀₁	semidirect
204. C₂×D₄	C₂×Dic₁₀₁	semidirect
217. SD₁₆	C₂×Dic₁₀₁	semidirect
218. D₈	C₂×Dic₁₀₁	semidirect
220. M₁₆	C₂×Dic₁₀₁	semidirect
248. Q₈	C₂²×Dic₁₀₁	semidirect
254. D₄	C₂²×Dic₁₀₁	semidirect
276. C₈	C₂²×Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₁₀₁	2
C₂×D₄×Dic₁₀₁	2
#20265289	2
#20265278	2
SD₁₆×Dic₁₀₁	2
D₈×Dic₁₀₁	2
#20265219	2
#20265182	2
#20265178	2
M₁₆×Dic₁₀₁	2
C₂×AGL(1,ℤ/8)×C₁₀₁	2
C₄×AGL(1,ℤ/8)	101

Subgroups — AGL(1,ℤ/8)×Dic101 #27179942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,ℤ/8)×Dic₁₀₁ #27179942