Finite Group Databasebeta

← C2×AGL(1,ℤ/16)×C131 #45719351

Subgroups — C2×AGL(1,ℤ/16)×C131 #45719351

Order 33536.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C131C211yesmaximal, characteristic
2C2×#444×C131C211yesmaximal, characteristic, complemented
3C2×#442×C131C211yesmaximal, characteristic, complemented
4AGL(1,ℤ/16)×C131C218yesmaximal, direct product factor
5C2×#365×C131C211yesmaximal, characteristic, complemented
6C2×#361×C131C211yesmaximal, characteristic, complemented
7C2×#360×C131C211yesmaximal, characteristic, complemented
8C2×#347×C131C211yesmaximal, characteristic, complemented
9C2×#135×C131C2211yescharacteristic
10C22×D8×C131C2211yescharacteristic
11C2×#131×C131—21no—
12C2×C4×D4×C131—21no—
13#509×C131C414yescomplemented
14#509×C131C2214yes—
15C2×SD32×C131C412yescomplemented
16C2×D16×C131C412yescomplemented
17C2×M32×C131C2211yescharacteristic
18C2×M32×C131C2211yescharacteristic
19#444×C131—22no—
20#444×C131C2214yescomplemented
21#442×C131—22no—
22#442×C131C2214yescomplemented
23C2×#108×C131C2211yescharacteristic
24C2×#106×C131C2211yescharacteristic
25C2×#104×C131—21no—
26C2×#102×C131—21no—
27C2×#97×C131C2211yescharacteristic
28#365×C131C2214yescomplemented
29#361×C131C2214yescomplemented
30#360×C131C2214yescomplemented
31#347×C131C2214yescomplemented
32C2×#41×C131—21no—
33C22×D4×C131—21no—
34C23×C4×C131—41no—
35#135×C131—21no—
36#135×C131—21no—
37#135×C131C2×C412yescomplemented
38#135×C131C2312yes—
39C2×Q16×C131C2×C411yescharacteristic, complemented
40C2×SD16×C131—21no—
41C2×D8×C131—21no—
42C2×D8×C131—21no—
43C2×D8×C131—21no—
44C2×D8×C131C2×C412yes—
45C2×D8×C131C2×C412yes—
46C2×D8×C131C2×C411yescharacteristic, complemented
47C2×D8×C131C2312yes—
48#131×C131—24no—
49#131×C131—42no—
50C2×M16×C131—21no—
51C2×M16×C131—41no—
52C22×C8×C131—41no—
53C22×C8×C131D411yescharacteristic
54C22×C8×C131C2311yescharacteristic
55C4×D4×C131—24no—
56C4×D4×C131—42no—
57C2×C4⋊C4×C131—21no—
58C2×C22⋊C4×C131—41no—
59C2×C42×C131D411yescharacteristic
60SD32×C131C2×C418yescomplemented
61D16×C131C2×C418yescomplemented
62M32×C131—21no—
63M32×C131C2×C412yescomplemented
64M32×C131C2312yes—
65M32×C131C2312yes—
66C2×C16×C131—21no—
67C2×C16×C131C2×C412yescomplemented
68#108×C131—21no—
69#108×C131C2312yes—
70#106×C131—21no—
71#106×C131C2312yes—
72#104×C131—24no—
73#102×C131—24no—
74#97×C131D412yes—
75#97×C131C2312yes—
76C24×C131—41no—
77#41×C131—22no—
78#41×C131—22no—
79#41×C131—41no—
80#41×C131—41no—
81C2×Q8×C131—21no—
82C2×D4×C131—22no—
83C2×D4×C131—22no—
84C2×D4×C131—21no—
85C2×D4×C131—22no—
86C2×D4×C131—41no—
87C2×D4×C131—41no—
88C2×D4×C131—41no—
89C2×D4×C131—41no—
90C22×C4×C131—21no—
91C22×C4×C131—44no—
92C22×C4×C131—41no—
93C22×C4×C131—41no—
94C22×C4×C131—84no—
95C22×C4×C131C2×D411yescharacteristic
96Q16×C131—21no—
97Q16×C131C22×C412yescomplemented
98SD16×C131—22no—
99SD16×C131—22no—
100D8×C131—22no—
101D8×C131—22no—
102D8×C131—21no—
103D8×C131—22no—
104D8×C131C22×C414yes—
105D8×C131C22×C412yescomplemented
106M16×C131—22no—
107M16×C131—44no—
108M16×C131—41no—
109C2×C8×C131—21no—
110C2×C8×C131—21no—
111C2×C8×C131—44no—
112C2×C8×C131—41no—
113C2×C8×C131C22×C411yescharacteristic
114C2×C8×C131C22×C411yescharacteristic
115C2×C8×C131C2×D412yes—
116C2×C8×C131C2×D411yescharacteristic
117C2×C8×C131#4111yescharacteristic
118C2×C8×C131#4111yescharacteristic
119C2×C8×C131C2411yescharacteristic
120C4⋊C4×C131—22no—
121C4⋊C4×C131—41no—
122C22⋊C4×C131—44no—
123C42×C131—21no—
124C42×C131C2×D412yes—
125C16×C131—22no—
126C16×C131C22×C414yescomplemented
127C23×C131—42no—
128C23×C131—42no—
129C23×C131—41no—
130C23×C131—42no—
131C23×C131—82no—
132C23×C131—82no—
133C23×C131AGL(1,ℤ/8)11yescharacteristic
134Q8×C131—22no—
135Q8×C131—41no—
136D4×C131—24no—
137D4×C131—22no—
138D4×C131—42no—
139D4×C131—42no—
140D4×C131—42no—
141D4×C131—42no—
142D4×C131—42no—
143D4×C131—41no—
144C2×C4×C131—22no—
145C2×C4×C131—42no—
146C2×C4×C131—41no—
147C2×C4×C131—42no—
148C2×C4×C131—42no—
149C2×C4×C131—42no—
150C2×C4×C131—88no—
151C2×C4×C131—82no—
152C2×C4×C131C4×D411yescharacteristic
153C2×C4×C131C4×D411yescharacteristic
154C2×C4×C131AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C131AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C131AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C131C22×D411yescharacteristic
158C8×C131—22no—
159C8×C131—42no—
160C8×C131C4×D411yescharacteristic
161C8×C131C4×D411yescharacteristic
162C8×C131C23×C411yescharacteristic
163C8×C131C2×#4111yescharacteristic
164C22×C131—22no—
165C22×C131—42no—
166C22×C131—44no—
167C22×C131—82no—
168C22×C131—82no—
169C22×C131—84no—
170C22×C131—82no—
171C22×C131—84no—
172C22×C131#44211yescharacteristic
173C22×C131#44211yescharacteristic
174C22×C131C2×AGL(1,ℤ/8)11yescharacteristic
175C4×C131—42no—
176C4×C131—42no—
177C4×C131—82no—
178C4×C131#44211yescharacteristic
179C4×C131#44211yescharacteristic
180C4×C131C2×C4×D411yescharacteristic
181C4×C131C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C131—21no—
183C2×C131—21no—
184C2×C131—82no—
185C2×C131—84no—
186C2×C131AGL(1,ℤ/16)12yesdirect product factor
187C2×C131C2×#44211yescharacteristic
188C2×AGL(1,ℤ/16)C13111yesmaximal, characteristic, direct product factor, Sylow
189C131C2×AGL(1,ℤ/16)11yescharacteristic, direct product factor, Sylow
190C2×#509C2×C13111yescharacteristic
191C2×#444C2×C13111yescharacteristic, complemented
192C2×#442C2×C13111yescharacteristic, complemented
193AGL(1,ℤ/16)C2×C13118yesdirect product factor
194C2×#365C2×C13111yescharacteristic, complemented
195C2×#361C2×C13111yescharacteristic, complemented
196C2×#360C2×C13111yescharacteristic, complemented
197C2×#347C2×C13111yescharacteristic, complemented
198C2×#135C22×C13111yescharacteristic
199C22×D8C22×C13111yescharacteristic
200C2×#131—21no—
201C2×C4×D4—21no—
202#509C4×C13114yescomplemented
203#509C22×C13114yes—
204C2×SD32C4×C13112yescomplemented
205C2×D16C4×C13112yescomplemented
206C2×M32C22×C13111yescharacteristic
207C2×M32C22×C13111yescharacteristic
208#444—22no—
209#444C22×C13114yescomplemented
210#442—22no—
211#442C22×C13114yescomplemented
212C2×#108C22×C13111yescharacteristic
213C2×#106C22×C13111yescharacteristic
214C2×#104—21no—
215C2×#102—21no—
216C2×#97C22×C13111yescharacteristic
217#365C22×C13114yescomplemented
218#361C22×C13114yescomplemented
219#360C22×C13114yescomplemented
220#347C22×C13114yescomplemented
221C2×#41—21no—
222C22×D4—21no—
223C23×C4—41no—
224#135—21no—
225#135—21no—
226#135C2×C4×C13112yescomplemented
227#135C23×C13112yes—
228C2×Q16C2×C4×C13111yescharacteristic, complemented
229C2×SD16—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8C2×C4×C13112yes—
234C2×D8C2×C4×C13112yes—
235C2×D8C2×C4×C13111yescharacteristic, complemented
236C2×D8C23×C13112yes—
237#131—24no—
238#131—42no—
239C2×M16—21no—
240C2×M16—41no—
241C22×C8—41no—
242C22×C8D4×C13111yescharacteristic
243C22×C8C23×C13111yescharacteristic
244C4×D4—24no—
245C4×D4—42no—
246C2×C4⋊C4—21no—
247C2×C22⋊C4—41no—
248C2×C42D4×C13111yescharacteristic
249SD32C2×C4×C13118yescomplemented
250D16C2×C4×C13118yescomplemented
251M32—21no—
252M32C2×C4×C13112yescomplemented
253M32C23×C13112yes—
254M32C23×C13112yes—
255C2×C16—21no—
256C2×C16C2×C4×C13112yescomplemented
257#108—21no—
258#108C23×C13112yes—
259#106—21no—
260#106C23×C13112yes—
261#104—24no—
262#102—24no—
263#97D4×C13112yes—
264#97C23×C13112yes—
265C24—41no—
266#41—22no—
267#41—22no—
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—21no—
280C22×C4—44no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—84no—
284C22×C4C2×D4×C13111yescharacteristic
285Q16—21no—
286Q16C22×C4×C13112yescomplemented
287SD16—22no—
288SD16—22no—
289D8—22no—
290D8—22no—
291D8—21no—
292D8—22no—
293D8C22×C4×C13114yes—
294D8C22×C4×C13112yescomplemented
295M16—22no—
296M16—44no—
297M16—41no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—44no—
301C2×C8—41no—
302C2×C8C22×C4×C13111yescharacteristic
303C2×C8C22×C4×C13111yescharacteristic
304C2×C8C2×D4×C13112yes—
305C2×C8C2×D4×C13111yescharacteristic
306C2×C8#41×C13111yescharacteristic
307C2×C8#41×C13111yescharacteristic
308C2×C8C24×C13111yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C22⋊C4—44no—
312C42—21no—
313C42C2×D4×C13112yes—
314C16—22no—
315C16C22×C4×C13114yescomplemented
316C23—41no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—82no—
321C23—82no—
322C23AGL(1,ℤ/8)×C13111yescharacteristic
323Q8—22no—
324Q8—41no—
325D4—24no—
326D4—22no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—41no—
332D4—42no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—88no—
340C2×C4—82no—
341C2×C4C4×D4×C13111yescharacteristic
342C2×C4C4×D4×C13111yescharacteristic
343C2×C4AGL(1,ℤ/8)×C13111yescharacteristic
344C2×C4AGL(1,ℤ/8)×C13111yescharacteristic
345C2×C4AGL(1,ℤ/8)×C13111yescharacteristic
346C2×C4C22×D4×C13111yescharacteristic
347C8—22no—
348C8—42no—
349C8C4×D4×C13111yescharacteristic
350C8C4×D4×C13111yescharacteristic
351C8C23×C4×C13111yescharacteristic
352C8C2×#41×C13111yescharacteristic
353C22—22no—
354C22—42no—
355C22—44no—
356C22—82no—
357C22—82no—
358C22—84no—
359C22—84no—
360C22—82no—
361C22#442×C13111yescharacteristic
362C22#442×C13111yescharacteristic
363C22C2×AGL(1,ℤ/8)×C13111yescharacteristic
364C4—42no—
365C4—42no—
366C4—82no—
367C4#442×C13111yescharacteristic
368C4#442×C13111yescharacteristic
369C4C2×C4×D4×C13111yescharacteristic
370C4C2×AGL(1,ℤ/8)×C13111yescharacteristic
371C2—21no—
372C2—21no—
373C2—84no—
374C2—82no—
375C2AGL(1,ℤ/16)×C13112yesdirect product factor
376C2C2×#442×C13111yescharacteristic
subgroupG/H
center, socleC22×C131#442
commutatorC8C23×C4×C131
Frattini, cosocleC2×C8C24×C131
prime radicalC2×#509×C131C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C2×#97×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#444×C131C2semidirect
3. C2×#442×C131C2semidirect
4. AGL(1,ℤ/16)×C131C2direct (N × H)
5. C2×#365×C131C2semidirect
6. C2×#361×C131C2semidirect
7. C2×#360×C131C2semidirect
8. C2×#347×C131C2semidirect
13. #509×C131C4semidirect
15. C2×SD32×C131C4semidirect
16. C2×D16×C131C4semidirect
20. #444×C131C22semidirect
22. #442×C131C22semidirect
28. #365×C131C22semidirect
29. #361×C131C22semidirect
30. #360×C131C22semidirect
31. #347×C131C22semidirect
37. #135×C131C2×C4semidirect
39. C2×Q16×C131C2×C4semidirect
46. C2×D8×C131C2×C4semidirect
60. SD32×C131C2×C4semidirect
61. D16×C131C2×C4semidirect
63. M32×C131C2×C4semidirect
67. C2×C16×C131C2×C4semidirect
97. Q16×C131C22×C4semidirect
105. D8×C131C22×C4semidirect
126. C16×C131C22×C4semidirect
186. C2×C131AGL(1,ℤ/16)direct (N × H)
188. C2×AGL(1,ℤ/16)C131direct (N × H)
189. C131C2×AGL(1,ℤ/16)direct (N × H)
191. C2×#444C2×C131semidirect
192. C2×#442C2×C131semidirect
193. AGL(1,ℤ/16)C2×C131direct (N × H)
194. C2×#365C2×C131semidirect
195. C2×#361C2×C131semidirect
196. C2×#360C2×C131semidirect
197. C2×#347C2×C131semidirect
202. #509C4×C131semidirect
204. C2×SD32C4×C131semidirect
205. C2×D16C4×C131semidirect
209. #444C22×C131semidirect
211. #442C22×C131semidirect
217. #365C22×C131semidirect
218. #361C22×C131semidirect
219. #360C22×C131semidirect
220. #347C22×C131semidirect
226. #135C2×C4×C131semidirect
228. C2×Q16C2×C4×C131semidirect
235. C2×D8C2×C4×C131semidirect
249. SD32C2×C4×C131semidirect
250. D16C2×C4×C131semidirect
252. M32C2×C4×C131semidirect
256. C2×C16C2×C4×C131semidirect
286. Q16C22×C4×C131semidirect
294. D8C22×C4×C131semidirect
315. C16C22×C4×C131semidirect
375. C2AGL(1,ℤ/16)×C131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#509×C1312
C2×#444×C1312
C2×#442×C1312
AGL(1,ℤ/16)×C1312
C2×#365×C1312
C2×#361×C1312
C2×#360×C1312
C2×#347×C1312
C2×AGL(1,ℤ/16)131