Finite Group Databasebeta

← C151×#19051 #50323976

Subgroups — C151×#19051 #50323976

Order 38656.

950 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3545×C151C211yesmaximal, characteristic
2#3238×C151C212yesmaximal, complemented
3C2×#497×C151C211yesmaximal, characteristic, complemented
4#1661×C151C214yesmaximal, complemented
5#1660×C151C212yesmaximal, complemented
6#1659×C151C212yesmaximal, complemented
7#1533×C151C211yesmaximal, characteristic, complemented
8#1532×C151C212yesmaximal, complemented
9C22×Q16×C151—21no—
10C22×SD16×C151—21no—
11#563×C151C2211yescharacteristic
12#562×C151C2212yes—
13H(3,𝔽4)×C151C2211yescharacteristic
14#555×C151—21no—
15#551×C151—21no—
16#551×C151—22no—
17#546×C151—22no—
18#542×C151—22no—
19#538×C151—22no—
20#536×C151—21no—
21#534×C151—21no—
22C2×C4⋊Q8×C151C2211yescharacteristic, complemented
23#529×C151C2212yes—
24C2×#124×C151C2211yescharacteristic
25#501×C151C2214yescomplemented
26#497×C151—21no—
27#497×C151—21no—
28#497×C151C2212yes—
29#497×C151C2212yescomplemented
30#489×C151C2214yescomplemented
31#483×C151—22no—
32#480×C151—22no—
33#477×C151—22no—
34#476×C151—22no—
35#432×C151C2212yescomplemented
36C2×#97×C151C2211yescharacteristic, complemented
37#330×C151C2214yescomplemented
38#329×C151C2212yes—
39#329×C151C2214yescomplemented
40#329×C151C2212yescomplemented
41C2×#41×C151—21no—
42C22×Q8×C151—21no—
43C22×Q8×C151D411yescharacteristic
44C22×D4×C151—21no—
45C22×D4×C151D411yescharacteristic
46C2×Q16×C151—22no—
47C2×Q16×C151—22no—
48C2×Q16×C151—44no—
49C2×SD16×C151—22no—
50C2×SD16×C151—22no—
51C2×SD16×C151—44no—
52C22×C8×C151—21no—
53C4⋊Q8×C151—21no—
54C4⋊Q8×C151—21no—
55C4⋊Q8×C151—21no—
56C4⋊Q8×C151—22no—
57C4⋊Q8×C151C2312yes—
58#126×C151—22no—
59#126×C151—22no—
60#126×C151—22no—
61#125×C151C2312yes—
62#124×C151—21no—
63#124×C151—22no—
64#124×C151—22no—
65#124×C151—21no—
66#124×C151—22no—
67#124×C151D411yescharacteristic
68#124×C151D411yescharacteristic
69#124×C151C2312yes—
70#124×C151C2311yescharacteristic
71#124×C151C2311yescharacteristic
72#123×C151—21no—
73#123×C151—22no—
74#123×C151—22no—
75#123×C151—21no—
76#123×C151—22no—
77#123×C151—22no—
78#123×C151—22no—
79C22⋊Q8×C151—22no—
80C22⋊Q8×C151—22no—
81C22⋊Q8×C151—22no—
82C22⋊Q8×C151—22no—
83C22⋊Q8×C151D412yes—
84#121×C151—22no—
85#121×C151—22no—
86#121×C151—22no—
87#121×C151—22no—
88#121×C151D412yes—
89#120×C151—21no—
90#120×C151—22no—
91#120×C151—21no—
92#120×C151—22no—
93C4×Q8×C151D412yes—
94C4×D4×C151—22no—
95C4×D4×C151—22no—
96C4×D4×C151—22no—
97C4×D4×C151D412yes—
98#117×C151—21no—
99#117×C151—22no—
100#117×C151—22no—
101C2×C4⋊C4×C151—22no—
102C2×C22⋊C4×C151—21no—
103C2×C22⋊C4×C151—22no—
104C2×C22⋊C4×C151—22no—
105C2×C42×C151C2311yescharacteristic
106#107×C151—24no—
107#106×C151—24no—
108C4⋊C8×C151—22no—
109#103×C151—24no—
110#102×C151—24no—
111C22⋊C8×C151—22no—
112#97×C151—21no—
113#97×C151C2312yes—
114#97×C151C2314yes—
115C24×C151—21no—
116C24×C151—21no—
117#41×C151—82no—
118C2×Q8×C151—21no—
119C2×Q8×C151—21no—
120C2×Q8×C151—22no—
121C2×Q8×C151—42no—
122C2×Q8×C151—42no—
123C2×Q8×C151—42no—
124C2×Q8×C151—41no—
125C2×Q8×C151C2×D412yes—
126C2×Q8×C151C2×D411yescharacteristic
127C2×Q8×C151C2×D411yescharacteristic
128C2×D4×C151—22no—
129C2×D4×C151—22no—
130C2×D4×C151—22no—
131C2×D4×C151—21no—
132C2×D4×C151—22no—
133C2×D4×C151—21no—
134C2×D4×C151—42no—
135C2×D4×C151—82no—
136C2×D4×C151C2×D412yes—
137C2×D4×C151C2×D411yescharacteristic
138C2×D4×C151C2×D411yescharacteristic
139C22×C4×C151—21no—
140C22×C4×C151—22no—
141C22×C4×C151—22no—
142C22×C4×C151—21no—
143C22×C4×C151—22no—
144C22×C4×C151—41no—
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D412yes—
147Q16×C151—44no—
148Q16×C151—44no—
149SD16×C151—44no—
150SD16×C151—44no—
151C2×C8×C151—24no—
152C2×C8×C151—22no—
153C2×C8×C151—24no—
154C4⋊C4×C151—22no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—22no—
157C4⋊C4×C151—22no—
158C4⋊C4×C151—22no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—24no—
162C4⋊C4×C151—42no—
163C4⋊C4×C151C2×D412yes—
164C4⋊C4×C151C2×D412yes—
165C22⋊C4×C151—22no—
166C22⋊C4×C151—21no—
167C22⋊C4×C151—21no—
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×C151—22no—
176C22⋊C4×C151—22no—
177C22⋊C4×C151—22no—
178C22⋊C4×C151—22no—
179C22⋊C4×C151—22no—
180C42×C151—21no—
181C42×C151—21no—
182C42×C151—22no—
183C42×C151C2×D411yescharacteristic
184C42×C151C2×D412yes—
185C42×C151C2411yescharacteristic
186C23×C151—21no—
187C23×C151—22no—
188C23×C151—21no—
189C23×C151—42no—
190C23×C151—82no—
191C23×C151—82no—
192C23×C151—81no—
193C23×C151#12011yescharacteristic
194Q8×C151—41no—
195Q8×C151—41no—
196Q8×C151—42no—
197Q8×C151—42no—
198Q8×C151—42no—
199Q8×C151—81no—
200Q8×C151—82no—
201Q8×C151—81no—
202D4×C151—41no—
203D4×C151—41no—
204D4×C151—42no—
205D4×C151—44no—
206D4×C151—81no—
207D4×C151—81no—
208D4×C151—82no—
209D4×C151—82no—
210D4×C151—82no—
211C2×C4×C151—22no—
212C2×C4×C151—21no—
213C2×C4×C151—21no—
214C2×C4×C151—22no—
215C2×C4×C151—22no—
216C2×C4×C151—22no—
217C2×C4×C151—42no—
218C2×C4×C151—42no—
219C2×C4×C151—42no—
220C2×C4×C151—82no—
221C2×C4×C151—82no—
222C2×C4×C151—82no—
223C2×C4×C151—81no—
224C2×C4×C151—82no—
225C2×C4×C151#12011yescharacteristic
226C2×C4×C151#12012yes—
227C2×C4×C151C22×D411yescharacteristic
228C2×C4×C151C22×D412yes—
229C8×C151—24no—
230C8×C151—44no—
231C22×C151—22no—
232C22×C151—42no—
233C22×C151—41no—
234C22×C151—41no—
235C22×C151—42no—
236C22×C151—42no—
237C22×C151—42no—
238C22×C151—42no—
239C22×C151—162no—
240C22×C151—162no—
241C22×C151#45312yes—
242C22×C151C2×#12011yescharacteristic
243C4×C151—21no—
244C4×C151—21no—
245C4×C151—22no—
246C4×C151—22no—
247C4×C151—41no—
248C4×C151—41no—
249C4×C151—42no—
250C4×C151—42no—
251C4×C151—42no—
252C4×C151—82no—
253C2×C151—22no—
254C2×C151—82no—
255C2×C151—81no—
256C2×C151—81no—
257C2×C151C2×#45311yescharacteristic
258C2×C151#302212yes—
259#19051C15111yesmaximal, characteristic, direct product factor, Sylow
260C151#1905111yescharacteristic, direct product factor, Sylow
261#3545C2×C15111yescharacteristic
262#3238C2×C15112yescomplemented
263C2×#497C2×C15111yescharacteristic, complemented
264#1661C2×C15114yescomplemented
265#1660C2×C15112yescomplemented
266#1659C2×C15112yescomplemented
267#1533C2×C15111yescharacteristic, complemented
268#1532C2×C15112yescomplemented
269C22×Q16—21no—
270C22×SD16—21no—
271#563C22×C15111yescharacteristic
272#562C22×C15112yes—
273H(3,𝔽4)C22×C15111yescharacteristic
274#555—21no—
275#551—21no—
276#551—22no—
277#546—22no—
278#542—22no—
279#538—22no—
280#536—21no—
281#534—21no—
282C2×C4⋊Q8C22×C15111yescharacteristic, complemented
283#529C22×C15112yes—
284C2×#124C22×C15111yescharacteristic
285#501C22×C15114yescomplemented
286#497—21no—
287#497—21no—
288#497C22×C15112yes—
289#497C22×C15112yescomplemented
290#489C22×C15114yescomplemented
291#483—22no—
292#480—22no—
293#477—22no—
294#476—22no—
295#432C22×C15112yescomplemented
296C2×#97C22×C15111yescharacteristic, complemented
297#330C22×C15114yescomplemented
298#329C22×C15112yes—
299#329C22×C15114yescomplemented
300#329C22×C15112yescomplemented
301C2×#41—21no—
302C22×Q8—21no—
303C22×Q8D4×C15111yescharacteristic
304C22×D4—21no—
305C22×D4D4×C15111yescharacteristic
306C2×Q16—22no—
307C2×Q16—22no—
308C2×Q16—44no—
309C2×SD16—22no—
310C2×SD16—22no—
311C2×SD16—44no—
312C22×C8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—22no—
316C4⋊Q8—21no—
317C4⋊Q8C23×C15112yes—
318#126—22no—
319#126—22no—
320#126—22no—
321#125C23×C15112yes—
322#124—21no—
323#124—22no—
324#124—21no—
325#124—22no—
326#124—22no—
327#124D4×C15111yescharacteristic
328#124D4×C15111yescharacteristic
329#124C23×C15111yescharacteristic
330#124C23×C15111yescharacteristic
331#124C23×C15112yes—
332#123—21no—
333#123—22no—
334#123—22no—
335#123—21no—
336#123—22no—
337#123—22no—
338#123—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8D4×C15112yes—
344#121—22no—
345#121—22no—
346#121—22no—
347#121—22no—
348#121D4×C15112yes—
349#120—21no—
350#120—22no—
351#120—21no—
352#120—22no—
353C4×Q8D4×C15112yes—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4D4×C15112yes—
358#117—21no—
359#117—22no—
360#117—22no—
361C2×C4⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42C23×C15111yescharacteristic
366#107—24no—
367#106—24no—
368C4⋊C8—22no—
369#103—24no—
370#102—24no—
371C22⋊C8—22no—
372#97—21no—
373#97C23×C15114yes—
374#97C23×C15112yes—
375C24—21no—
376C24—21no—
377#41—82no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8C2×D4×C15112yes—
386C2×Q8C2×D4×C15111yescharacteristic
387C2×Q8C2×D4×C15111yescharacteristic
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4C2×D4×C15112yes—
397C2×D4C2×D4×C15111yescharacteristic
398C2×D4C2×D4×C15111yescharacteristic
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4C2×D4×C15111yescharacteristic
406C22×C4C2×D4×C15112yes—
407Q16—44no—
408Q16—44no—
409SD16—44no—
410SD16—44no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4C2×D4×C15112yes—
424C4⋊C4C2×D4×C15112yes—
425C22⋊C4—22no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C42—21no—
441C42—21no—
442C42—22no—
443C42C2×D4×C15111yescharacteristic
444C42C2×D4×C15112yes—
445C42C24×C15111yescharacteristic
446C23—21no—
447C23—22no—
448C23—21no—
449C23—42no—
450C23—82no—
451C23—81no—
452C23—82no—
453C23#120×C15111yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—81no—
460Q8—82no—
461Q8—81no—
462D4—41no—
463D4—41no—
464D4—42no—
465D4—44no—
466D4—81no—
467D4—82no—
468D4—81no—
469D4—82no—
470D4—82no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#120×C15112yes—
486C2×C4#120×C15111yescharacteristic
487C2×C4C22×D4×C15112yes—
488C2×C4C22×D4×C15111yescharacteristic
489C8—24no—
490C8—44no—
491C22—22no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—162no—
500C22—162no—
501C22#453×C15112yes—
502C22C2×#120×C15111yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—41no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C2—22no—
514C2—81no—
515C2—82no—
516C2—81no—
517C2C2×#453×C15111yescharacteristic
518C2#3022×C15112yes—
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radical#3545×C151C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3238×C151C2semidirect
3. C2×#497×C151C2semidirect
4. #1661×C151C2semidirect
5. #1660×C151C2semidirect
6. #1659×C151C2semidirect
7. #1533×C151C2semidirect
8. #1532×C151C2semidirect
22. C2×C4⋊Q8×C151C22semidirect
25. #501×C151C22semidirect
29. #497×C151C22semidirect
30. #489×C151C22semidirect
35. #432×C151C22semidirect
36. C2×#97×C151C22semidirect
37. #330×C151C22semidirect
39. #329×C151C22semidirect
40. #329×C151C22semidirect
259. #19051C151direct (N × H)
260. C151#19051direct (N × H)
262. #3238C2×C151semidirect
263. C2×#497C2×C151semidirect
264. #1661C2×C151semidirect
265. #1660C2×C151semidirect
266. #1659C2×C151semidirect
267. #1533C2×C151semidirect
268. #1532C2×C151semidirect
282. C2×C4⋊Q8C22×C151semidirect
285. #501C22×C151semidirect
289. #497C22×C151semidirect
290. #489C22×C151semidirect
295. #432C22×C151semidirect
296. C2×#97C22×C151semidirect
297. #330C22×C151semidirect
299. #329C22×C151semidirect
300. #329C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3545×C1512
#3238×C1512
C2×#497×C1512
#1661×C1512
#1660×C1512
#1659×C1512
#1533×C1512
#1532×C1512
#19051151