Finite Group Databasebeta

← C151×#17686 #50322611

Subgroups — C151×#17686 #50322611

Order 38656.

902 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3510×C151C211yesmaximal, characteristic
2#2970×C151C212yesmaximal, complemented
3#1535×C151C212yesmaximal, complemented
4#1533×C151C211yesmaximal, characteristic, complemented
5#1532×C151C212yesmaximal, complemented
6#1531×C151C212yesmaximal, complemented
7#1529×C151C212yesmaximal, complemented
8#1524×C151C212yesmaximal, complemented
9C2×#329×C151C211yesmaximal, characteristic, complemented
10#559×C151—22no—
11#556×C151—22no—
12#551×C151—22no—
13#551×C151—22no—
14#542×C151C412yes—
15#542×C151C412yes—
16#536×C151C411yescharacteristic
17#535×C151C411yescharacteristic
18#533×C151C2211yescharacteristic
19#529×C151C2212yes—
20#529×C151C2212yes—
21C2×#124×C151C2211yescharacteristic
22#519×C151C411yescharacteristic
23#518×C151C411yescharacteristic
24C4×#41×C151C2211yescharacteristic
25#446×C151—22no—
26#435×C151—22no—
27#435×C151—22no—
28C8×D4×C151—22no—
29#432×C151C2212yes—
30#432×C151C2212yescomplemented
31#432×C151C2212yescomplemented
32C4×M16×C151C2212yescomplemented
33C2×#97×C151C2212yes—
34C2×#97×C151C2212yescomplemented
35#330×C151C2212yescomplemented
36#330×C151C2212yescomplemented
37#330×C151C2212yescomplemented
38#330×C151C2212yescomplemented
39#329×C151C2211yescharacteristic
40#329×C151C2212yes—
41#329×C151C2211yescharacteristic
42#329×C151C2212yes—
43#329×C151C2211yescharacteristic
44#329×C151C2211yescharacteristic
45C2×#41×C151C2×C411yescharacteristic
46C22×Q8×C151C2×C411yescharacteristic
47C22×D4×C151C2×C411yescharacteristic
48C2×M16×C151—22no—
49C22×C8×C151—22no—
50C22×C8×C151—22no—
51C4⋊Q8×C151—21no—
52C4⋊Q8×C151C2×C412yes—
53#127×C151—21no—
54#126×C151—22no—
55#126×C151—22no—
56#126×C151—22no—
57#126×C151—22no—
58#125×C151C2312yes—
59#124×C151—21no—
60#124×C151—22no—
61#124×C151—22no—
62#124×C151—22no—
63#124×C151—21no—
64#124×C151C2×C411yescharacteristic
65#124×C151C2×C411yescharacteristic
66#124×C151C2311yescharacteristic
67#124×C151C2311yescharacteristic
68#123×C151—22no—
69#123×C151—22no—
70#123×C151—22no—
71#123×C151—22no—
72#123×C151—22no—
73C22⋊Q8×C151—22no—
74C22⋊Q8×C151—22no—
75C22⋊Q8×C151—22no—
76C22⋊Q8×C151C2×C412yes—
77C22⋊Q8×C151C2×C412yes—
78#121×C151C2×C412yes—
79#121×C151C2×C412yes—
80#120×C151—22no—
81C4×Q8×C151—22no—
82C4×Q8×C151—22no—
83C4×Q8×C151C2×C412yes—
84C4×Q8×C151C2×C412yes—
85C4×Q8×C151D412yes—
86C4×D4×C151—22no—
87C4×D4×C151—22no—
88C4×D4×C151—22no—
89C4×D4×C151C2×C412yes—
90C4×D4×C151C2×C412yes—
91C4×D4×C151D412yes—
92C4×D4×C151D412yes—
93C4×D4×C151D412yes—
94#117×C151—21no—
95#117×C151—21no—
96#117×C151—22no—
97#117×C151—22no—
98#117×C151C2×C411yescharacteristic
99#117×C151C2×C411yescharacteristic
100#117×C151C2×C411yescharacteristic
101#117×C151C2×C412yes—
102C2×C22⋊C4×C151—22no—
103C2×C22⋊C4×C151—21no—
104C2×C22⋊C4×C151—21no—
105C2×C42×C151C2311yescharacteristic
106C2×C42×C151C2312yes—
107C4⋊C8×C151—22no—
108C4⋊C8×C151—22no—
109C4⋊C8×C151—22no—
110C22⋊C8×C151—22no—
111C22⋊C8×C151—22no—
112C22⋊C8×C151—22no—
113#97×C151—22no—
114#97×C151—22no—
115#97×C151C2312yes—
116#97×C151C2312yes—
117#97×C151C2312yes—
118#97×C151C2312yes—
119C4×C8×C151—22no—
120C24×C151—21no—
121#41×C151—42no—
122C2×Q8×C151—21no—
123C2×Q8×C151—22no—
124C2×Q8×C151—42no—
125C2×Q8×C151C22⋊C411yescharacteristic
126C2×Q8×C151C22×C411yescharacteristic
127C2×Q8×C151C22×C411yescharacteristic
128C2×D4×C151—22no—
129C2×D4×C151—22no—
130C2×D4×C151—21no—
131C2×D4×C151—42no—
132C2×D4×C151C22⋊C412yes—
133C2×D4×C151C22⋊C411yescharacteristic
134C2×D4×C151C22×C411yescharacteristic
135C2×D4×C151C22×C411yescharacteristic
136C22×C4×C151—21no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151C22×C411yescharacteristic
140C22×C4×C151C22×C412yes—
141C22×C4×C151C2×D412yes—
142C22×C4×C151C2×D412yes—
143C22×C4×C151C2×D412yes—
144M16×C151—44no—
145C2×C8×C151—22no—
146C2×C8×C151—22no—
147C2×C8×C151—22no—
148C2×C8×C151—22no—
149C2×C8×C151—44no—
150C2×C8×C151—44no—
151C4⋊C4×C151—22no—
152C4⋊C4×C151—22no—
153C4⋊C4×C151—22no—
154C4⋊C4×C151—22no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—22no—
157C4⋊C4×C151—22no—
158C4⋊C4×C151C22⋊C412yes—
159C4⋊C4×C151C22⋊C412yes—
160C4⋊C4×C151C22⋊C412yes—
161C4⋊C4×C151C22×C412yes—
162C4⋊C4×C151C22×C412yes—
163C22⋊C4×C151—21no—
164C22⋊C4×C151—21no—
165C22⋊C4×C151—22no—
166C22⋊C4×C151—22no—
167C22⋊C4×C151—22no—
168C22⋊C4×C151—22no—
169C22⋊C4×C151—22no—
170C22⋊C4×C151—22no—
171C22⋊C4×C151—22no—
172C22⋊C4×C151—22no—
173C22⋊C4×C151—22no—
174C22⋊C4×C151—22no—
175C22⋊C4×C151C22⋊C411yescharacteristic
176C22⋊C4×C151C22⋊C411yescharacteristic
177C22⋊C4×C151C22⋊C412yes—
178C22⋊C4×C151C22⋊C412yes—
179C42×C151—22no—
180C42×C151—22no—
181C42×C151—21no—
182C42×C151—21no—
183C42×C151C22×C412yes—
184C42×C151C22×C411yescharacteristic
185C42×C151C2×D412yes—
186C42×C151C2×D412yes—
187C42×C151C2×D412yes—
188C42×C151C2411yescharacteristic
189C23×C151—21no—
190C23×C151—21no—
191C23×C151—42no—
192C23×C151—82no—
193C23×C151C2×C22⋊C412yes—
194C23×C151C2×C22⋊C411yescharacteristic
195Q8×C151—41no—
196Q8×C151—41no—
197Q8×C151—41no—
198Q8×C151—42no—
199Q8×C151—82no—
200D4×C151—24no—
201D4×C151—41no—
202D4×C151—41no—
203D4×C151—41no—
204D4×C151—42no—
205D4×C151—82no—
206D4×C151—82no—
207C2×C4×C151—22no—
208C2×C4×C151—21no—
209C2×C4×C151—21no—
210C2×C4×C151—22no—
211C2×C4×C151—22no—
212C2×C4×C151—42no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—82no—
218C2×C4×C151—82no—
219C2×C4×C151C2×C22⋊C411yescharacteristic
220C2×C4×C151C2×C22⋊C412yes—
221C2×C4×C151C2×C22⋊C412yes—
222C2×C4×C151C2×C22⋊C412yes—
223C2×C4×C151C2×C22⋊C412yes—
224C2×C4×C151C23×C411yescharacteristic
225C2×C4×C151C22×D412yes—
226C8×C151—24no—
227C8×C151—42no—
228C8×C151—42no—
229C8×C151—42no—
230C22×C151—22no—
231C22×C151—22no—
232C22×C151—22no—
233C22×C151—22no—
234C22×C151—41no—
235C22×C151—41no—
236C22×C151—42no—
237C22×C151—42no—
238C22×C151—162no—
239C22×C151#41312yes—
240C22×C151C22×C22⋊C411yescharacteristic
241C4×C151—21no—
242C4×C151—21no—
243C4×C151—22no—
244C4×C151—22no—
245C4×C151—22no—
246C4×C151—22no—
247C4×C151—22no—
248C4×C151—41no—
249C4×C151—41no—
250C4×C151—42no—
251C4×C151—42no—
252C2×C151—22no—
253C2×C151—42no—
254C2×C151—81no—
255C2×C151—81no—
256C2×C151C2×#41311yescharacteristic
257C2×C151#290112yes—
258#17686C15111yesmaximal, characteristic, direct product factor, Sylow
259C151#1768611yescharacteristic, direct product factor, Sylow
260#3510C2×C15111yescharacteristic
261#2970C2×C15112yescomplemented
262#1535C2×C15112yescomplemented
263#1533C2×C15111yescharacteristic, complemented
264#1532C2×C15112yescomplemented
265#1531C2×C15112yescomplemented
266#1529C2×C15112yescomplemented
267#1524C2×C15112yescomplemented
268C2×#329C2×C15111yescharacteristic, complemented
269#559—22no—
270#556—22no—
271#551—22no—
272#551—22no—
273#542C4×C15112yes—
274#542C4×C15112yes—
275#536C4×C15111yescharacteristic
276#535C4×C15111yescharacteristic
277#533C22×C15111yescharacteristic
278#529C22×C15112yes—
279#529C22×C15112yes—
280C2×#124C22×C15111yescharacteristic
281#519C4×C15111yescharacteristic
282#518C4×C15111yescharacteristic
283C4×#41C22×C15111yescharacteristic
284#446—22no—
285#435—22no—
286#435—22no—
287C8×D4—22no—
288#432C22×C15112yes—
289#432C22×C15112yescomplemented
290#432C22×C15112yescomplemented
291C4×M16C22×C15112yescomplemented
292C2×#97C22×C15112yes—
293C2×#97C22×C15112yescomplemented
294#330C22×C15112yescomplemented
295#330C22×C15112yescomplemented
296#330C22×C15112yescomplemented
297#330C22×C15112yescomplemented
298#329C22×C15111yescharacteristic
299#329C22×C15112yes—
300#329C22×C15111yescharacteristic
301#329C22×C15112yes—
302#329C22×C15111yescharacteristic
303#329C22×C15111yescharacteristic
304C2×#41C2×C4×C15111yescharacteristic
305C22×Q8C2×C4×C15111yescharacteristic
306C22×D4C2×C4×C15111yescharacteristic
307C2×M16—22no—
308C22×C8—22no—
309C22×C8—22no—
310C4⋊Q8—21no—
311C4⋊Q8C2×C4×C15112yes—
312#127—21no—
313#126—22no—
314#126—22no—
315#126—22no—
316#126—22no—
317#125C23×C15112yes—
318#124—21no—
319#124—21no—
320#124—22no—
321#124—22no—
322#124—22no—
323#124C2×C4×C15111yescharacteristic
324#124C2×C4×C15111yescharacteristic
325#124C23×C15111yescharacteristic
326#124C23×C15111yescharacteristic
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8C2×C4×C15112yes—
336C22⋊Q8C2×C4×C15112yes—
337#121C2×C4×C15112yes—
338#121C2×C4×C15112yes—
339#120—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8C2×C4×C15112yes—
343C4×Q8C2×C4×C15112yes—
344C4×Q8D4×C15112yes—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4C2×C4×C15112yes—
349C4×D4C2×C4×C15112yes—
350C4×D4D4×C15112yes—
351C4×D4D4×C15112yes—
352C4×D4D4×C15112yes—
353#117—21no—
354#117—21no—
355#117—22no—
356#117—22no—
357#117C2×C4×C15111yescharacteristic
358#117C2×C4×C15111yescharacteristic
359#117C2×C4×C15111yescharacteristic
360#117C2×C4×C15112yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42C23×C15111yescharacteristic
365C2×C42C23×C15112yes—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372#97—22no—
373#97—22no—
374#97C23×C15112yes—
375#97C23×C15112yes—
376#97C23×C15112yes—
377#97C23×C15112yes—
378C4×C8—22no—
379C24—21no—
380#41—42no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8C22⋊C4×C15111yescharacteristic
385C2×Q8C22×C4×C15111yescharacteristic
386C2×Q8C22×C4×C15111yescharacteristic
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4C22⋊C4×C15112yes—
392C2×D4C22⋊C4×C15111yescharacteristic
393C2×D4C22×C4×C15111yescharacteristic
394C2×D4C22×C4×C15111yescharacteristic
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4C22×C4×C15111yescharacteristic
399C22×C4C22×C4×C15112yes—
400C22×C4C2×D4×C15112yes—
401C22×C4C2×D4×C15112yes—
402C22×C4C2×D4×C15112yes—
403M16—44no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—44no—
409C2×C8—44no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4C22⋊C4×C15112yes—
418C4⋊C4C22⋊C4×C15112yes—
419C4⋊C4C22⋊C4×C15112yes—
420C4⋊C4C22×C4×C15112yes—
421C4⋊C4C22×C4×C15112yes—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4C22⋊C4×C15111yescharacteristic
435C22⋊C4C22⋊C4×C15111yescharacteristic
436C22⋊C4C22⋊C4×C15112yes—
437C22⋊C4C22⋊C4×C15112yes—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42C22×C4×C15112yes—
443C42C22×C4×C15111yescharacteristic
444C42C2×D4×C15112yes—
445C42C2×D4×C15112yes—
446C42C2×D4×C15112yes—
447C42C24×C15111yescharacteristic
448C23—21no—
449C23—21no—
450C23—42no—
451C23—82no—
452C23C2×C22⋊C4×C15112yes—
453C23C2×C22⋊C4×C15111yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—42no—
458Q8—82no—
459D4—24no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—42no—
464D4—82no—
465D4—82no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4C2×C22⋊C4×C15111yescharacteristic
479C2×C4C2×C22⋊C4×C15112yes—
480C2×C4C2×C22⋊C4×C15112yes—
481C2×C4C2×C22⋊C4×C15112yes—
482C2×C4C2×C22⋊C4×C15112yes—
483C2×C4C23×C4×C15111yescharacteristic
484C2×C4C22×D4×C15112yes—
485C8—24no—
486C8—42no—
487C8—42no—
488C8—42no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—162no—
498C22#413×C15112yes—
499C22C22×C22⋊C4×C15111yescharacteristic
500C4—21no—
501C4—21no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—41no—
509C4—42no—
510C4—42no—
511C2—22no—
512C2—42no—
513C2—81no—
514C2—81no—
515C2C2×#413×C15111yescharacteristic
516C2#2901×C15112yes—
subgroupG/H
center, socleC22×C151C22×C22⋊C4
commutatorC2×C4C23×C4×C151
Frattini, cosocleC42C24×C151
prime radical#535×C151C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C4×#41×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2970×C151C2semidirect
3. #1535×C151C2semidirect
4. #1533×C151C2semidirect
5. #1532×C151C2semidirect
6. #1531×C151C2semidirect
7. #1529×C151C2semidirect
8. #1524×C151C2semidirect
9. C2×#329×C151C2semidirect
30. #432×C151C22semidirect
31. #432×C151C22semidirect
32. C4×M16×C151C22semidirect
34. C2×#97×C151C22semidirect
35. #330×C151C22semidirect
36. #330×C151C22semidirect
37. #330×C151C22semidirect
38. #330×C151C22semidirect
258. #17686C151direct (N × H)
259. C151#17686direct (N × H)
261. #2970C2×C151semidirect
262. #1535C2×C151semidirect
263. #1533C2×C151semidirect
264. #1532C2×C151semidirect
265. #1531C2×C151semidirect
266. #1529C2×C151semidirect
267. #1524C2×C151semidirect
268. C2×#329C2×C151semidirect
289. #432C22×C151semidirect
290. #432C22×C151semidirect
291. C4×M16C22×C151semidirect
293. C2×#97C22×C151semidirect
294. #330C22×C151semidirect
295. #330C22×C151semidirect
296. #330C22×C151semidirect
297. #330C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3510×C1512
#2970×C1512
#1535×C1512
#1533×C1512
#1532×C1512
#1531×C1512
#1529×C1512
#1524×C1512
C2×#329×C1512
#17686151