Finite Group Databasebeta

← C151×#18351 #50323276

Subgroups — C151×#18351 #50323276

Order 38656.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3496×C151C211yesmaximal, characteristic
2#3193×C151C212yesmaximal, complemented
3#2638×C151C211yesmaximal, characteristic, complemented
4#1610×C151C212yesmaximal, complemented
5#1607×C151C212yesmaximal, complemented
6#1604×C151C212yesmaximal, complemented
7#1601×C151C212yesmaximal, complemented
8#1510×C151C212yesmaximal, complemented
9#1465×C151C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C151—21no—
11#547×C151—22no—
12#546×C151—24no—
13D42×C151—22no—
14#537×C151C2212yes—
15#534×C151—21no—
16#534×C151C2212yes—
17#532×C151—21no—
18C2×C22⋊Q8×C151C2211yescharacteristic, complemented
19C2×#121×C151—21no—
20C2×#121×C151C2212yes—
21C2×C4×D4×C151C2211yescharacteristic
22#484×C151—22no—
23#480×C151C2214yescomplemented
24#468×C151—24no—
25#450×C151—22no—
26#448×C151C2214yescomplemented
27#407×C151C2212yescomplemented
28#398×C151C2211yescharacteristic, complemented
29#397×C151—21no—
30#397×C151C2211yescharacteristic, complemented
31#395×C151C2211yescharacteristic, complemented
32#394×C151—21no—
33#393×C151C2211yescharacteristic, complemented
34#387×C151—21no—
35#385×C151—21no—
36#381×C151C2211yescharacteristic, complemented
37#327×C151C2214yescomplemented
38#327×C151C2214yescomplemented
39#324×C151C2212yescomplemented
40#323×C151C2212yescomplemented
41H(5,𝔽2)×C151—44no—
42H(5,𝔽2)×C151—44no—
43C2×#41×C151—24no—
44C2×#41×C151—22no—
45C22×Q8×C151C2311yescharacteristic
46C22×D4×C151—24no—
47C22×D4×C151—21no—
48C22×D4×C151—22no—
49C22×D4×C151—22no—
50C22×D4×C151D412yes—
51C22×D4×C151D411yescharacteristic
52C23×C4×C151D411yescharacteristic
53C23×C4×C151C2311yescharacteristic
54C2×SD16×C151—24no—
55C2×M16×C151—22no—
56C4⋊Q8×C151—21no—
57#127×C151—21no—
58#124×C151—22no—
59#123×C151—24no—
60#123×C151—24no—
61C22⋊Q8×C151—24no—
62C22⋊Q8×C151—22no—
63C22⋊Q8×C151—22no—
64#121×C151—22no—
65#121×C151—22no—
66#121×C151—24no—
67#121×C151—22no—
68#121×C151—22no—
69#121×C151—22no—
70#121×C151—22no—
71#121×C151D414yes—
72#121×C151C2314yes—
73#120×C151—24no—
74#120×C151—24no—
75C4×D4×C151—22no—
76C4×D4×C151—22no—
77C4×D4×C151—22no—
78C4×D4×C151—22no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151—21no—
82C2×C4⋊C4×C151—21no—
83C2×C4⋊C4×C151D412yes—
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2311yescharacteristic
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—21no—
89C2×C22⋊C4×C151—21no—
90C2×C22⋊C4×C151—21no—
91C2×C22⋊C4×C151—21no—
92C2×C22⋊C4×C151D411yescharacteristic
93C2×C22⋊C4×C151C2311yescharacteristic
94C2×C42×C151D411yescharacteristic
95#107×C151—24no—
96#103×C151—24no—
97#102×C151—24no—
98C22⋊C8×C151—22no—
99C22⋊C8×C151C2314yes—
100#95×C151—21no—
101#95×C151—21no—
102#95×C151—21no—
103#95×C151—21no—
104#95×C151—21no—
105#95×C151C2311yescharacteristic
106#95×C151C2311yescharacteristic
107C24×C151—22no—
108C24×C151—22no—
109C24×C151C2×D411yescharacteristic
110C24×C151C2×D411yescharacteristic
111#41×C151—44no—
112#41×C151—416no—
113#41×C151—44no—
114C2×Q8×C151—22no—
115C2×Q8×C151—22no—
116C2×Q8×C151—81no—
117C2×D4×C151—22no—
118C2×D4×C151—22no—
119C2×D4×C151—24no—
120C2×D4×C151—22no—
121C2×D4×C151—24no—
122C2×D4×C151—24no—
123C2×D4×C151—22no—
124C2×D4×C151—22no—
125C2×D4×C151—22no—
126C2×D4×C151—22no—
127C2×D4×C151—416no—
128C2×D4×C151—44no—
129C2×D4×C151—44no—
130C2×D4×C151—44no—
131C2×D4×C151—44no—
132C2×D4×C151—82no—
133C2×D4×C151—82no—
134C2×D4×C151C2×D414yes—
135C22×C4×C151—22no—
136C22×C4×C151—24no—
137C22×C4×C151—22no—
138C22×C4×C151—21no—
139C22×C4×C151—21no—
140C22×C4×C151—21no—
141C22×C4×C151—21no—
142C22×C4×C151—21no—
143C22×C4×C151—21no—
144C22×C4×C151—81no—
145C22×C4×C151—81no—
146C22×C4×C151C2×D411yescharacteristic
147C22×C4×C151C2×D411yescharacteristic
148C22×C4×C151C2×D411yescharacteristic
149C22×C4×C151C2×D411yescharacteristic
150C22×C4×C151C2×D412yes—
151C22×C4×C151C2×D412yes—
152C22×C4×C151C2411yescharacteristic
153SD16×C151—84no—
154M16×C151—82no—
155C2×C8×C151—24no—
156C4⋊C4×C151—22no—
157C4⋊C4×C151—22no—
158C4⋊C4×C151—22no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—41no—
162C4⋊C4×C151—41no—
163C4⋊C4×C151—42no—
164C4⋊C4×C151—81no—
165C4⋊C4×C151C2×D414yes—
166C22⋊C4×C151—24no—
167C22⋊C4×C151—24no—
168C22⋊C4×C151—22no—
169C22⋊C4×C151—24no—
170C22⋊C4×C151—24no—
171C22⋊C4×C151—22no—
172C22⋊C4×C151—42no—
173C22⋊C4×C151—42no—
174C22⋊C4×C151—81no—
175C22⋊C4×C151—81no—
176C42×C151—21no—
177C42×C151—21no—
178C23×C151—21no—
179C23×C151—21no—
180C23×C151—21no—
181C23×C151—21no—
182C23×C151—24no—
183C23×C151—41no—
184C23×C151—41no—
185C23×C151—41no—
186C23×C151—41no—
187C23×C151—44no—
188C23×C151—44no—
189C23×C151—82no—
190C23×C151—82no—
191C23×C151—82no—
192C23×C151—82no—
193C23×C151#12012yes—
194C23×C151AGL(1,ℤ/8)12yes—
195C23×C151C22×D411yescharacteristic
196Q8×C151—44no—
197Q8×C151—44no—
198D4×C151—28no—
199D4×C151—44no—
200D4×C151—44no—
201D4×C151—44no—
202D4×C151—48no—
203D4×C151—48no—
204D4×C151—44no—
205D4×C151—44no—
206D4×C151—82no—
207D4×C151—82no—
208C2×C4×C151—21no—
209C2×C4×C151—21no—
210C2×C4×C151—24no—
211C2×C4×C151—21no—
212C2×C4×C151—21no—
213C2×C4×C151—41no—
214C2×C4×C151—42no—
215C2×C4×C151—41no—
216C2×C4×C151—42no—
217C2×C4×C151—48no—
218C2×C4×C151—42no—
219C2×C4×C151—44no—
220C2×C4×C151—44no—
221C2×C4×C151—41no—
222C2×C4×C151—41no—
223C2×C4×C151—41no—
224C2×C4×C151—41no—
225C2×C4×C151—41no—
226C2×C4×C151—41no—
227C2×C4×C151—41no—
228C2×C4×C151—41no—
229C2×C4×C151—41no—
230C2×C4×C151—41no—
231C2×C4×C151—41no—
232C2×C4×C151—41no—
233C2×C4×C151—41no—
234C2×C4×C151—41no—
235C2×C4×C151—81no—
236C2×C4×C151#12011yescharacteristic
237C2×C4×C151#12011yescharacteristic
238C2×C4×C151AGL(1,ℤ/8)12yes—
239C2×C4×C151C22×D412yes—
240C8×C151—44no—
241C22×C151—21no—
242C22×C151—21no—
243C22×C151—22no—
244C22×C151—22no—
245C22×C151—22no—
246C22×C151—22no—
247C22×C151—42no—
248C22×C151—41no—
249C22×C151—41no—
250C22×C151—44no—
251C22×C151—44no—
252C22×C151—44no—
253C22×C151—81no—
254C22×C151—81no—
255C22×C151—81no—
256C22×C151—164no—
257C22×C151—164no—
258C22×C151C2×#12011yescharacteristic
259C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
261C4×C151—22no—
262C4×C151—22no—
263C4×C151—21no—
264C4×C151—21no—
265C4×C151—44no—
266C4×C151—41no—
267C4×C151—41no—
268C4×C151—81no—
269C4×C151—81no—
270C4×C151—81no—
271C4×C151—81no—
272C2×C151—21no—
273C2×C151—21no—
274C2×C151—42no—
275C2×C151—42no—
276C2×C151—84no—
277C2×C151#300311yescharacteristic
278C2×C151#302211yescharacteristic
279C2×C151#302611yescharacteristic
280#18351C15111yesmaximal, characteristic, direct product factor, Sylow
281C151#1835111yescharacteristic, direct product factor, Sylow
282#3496C2×C15111yescharacteristic
283#3193C2×C15112yescomplemented
284#2638C2×C15111yescharacteristic, complemented
285#1610C2×C15112yescomplemented
286#1607C2×C15112yescomplemented
287#1604C2×C15112yescomplemented
288#1601C2×C15112yescomplemented
289#1510C2×C15112yescomplemented
290#1465C2×C15111yescharacteristic, complemented
291C2×H(5,𝔽2)—21no—
292#547—22no—
293#546—24no—
294D42—22no—
295#537C22×C15112yes—
296#534—21no—
297#534C22×C15112yes—
298#532—21no—
299C2×C22⋊Q8C22×C15111yescharacteristic, complemented
300C2×#121—21no—
301C2×#121C22×C15112yes—
302C2×C4×D4C22×C15111yescharacteristic
303#484—22no—
304#480C22×C15114yescomplemented
305#468—24no—
306#450—22no—
307#448C22×C15114yescomplemented
308#407C22×C15112yescomplemented
309#398C22×C15111yescharacteristic, complemented
310#397—21no—
311#397C22×C15111yescharacteristic, complemented
312#395C22×C15111yescharacteristic, complemented
313#394—21no—
314#393C22×C15111yescharacteristic, complemented
315#387—21no—
316#385—21no—
317#381C22×C15111yescharacteristic, complemented
318#327C22×C15114yescomplemented
319#327C22×C15114yescomplemented
320#324C22×C15112yescomplemented
321#323C22×C15112yescomplemented
322H(5,𝔽2)—44no—
323H(5,𝔽2)—44no—
324C2×#41—24no—
325C2×#41—22no—
326C22×Q8C23×C15111yescharacteristic
327C22×D4—24no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—21no—
331C22×D4D4×C15111yescharacteristic
332C22×D4D4×C15112yes—
333C23×C4D4×C15111yescharacteristic
334C23×C4C23×C15111yescharacteristic
335C2×SD16—24no—
336C2×M16—22no—
337C4⋊Q8—21no—
338#127—21no—
339#124—22no—
340#123—24no—
341#123—24no—
342C22⋊Q8—24no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345#121—22no—
346#121—22no—
347#121—24no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121—22no—
352#121D4×C15114yes—
353#121C23×C15114yes—
354#120—24no—
355#120—24no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4D4×C15112yes—
365C2×C4⋊C4C23×C15111yescharacteristic
366C2×C4⋊C4C23×C15111yescharacteristic
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4D4×C15111yescharacteristic
374C2×C22⋊C4C23×C15111yescharacteristic
375C2×C42D4×C15111yescharacteristic
376#107—24no—
377#103—24no—
378#102—24no—
379C22⋊C8—22no—
380C22⋊C8C23×C15114yes—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95C23×C15111yescharacteristic
387#95C23×C15111yescharacteristic
388C24—22no—
389C24—22no—
390C24C2×D4×C15111yescharacteristic
391C24C2×D4×C15111yescharacteristic
392#41—44no—
393#41—416no—
394#41—44no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—81no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—416no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C2×D4×C15114yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4C2×D4×C15111yescharacteristic
428C22×C4C2×D4×C15111yescharacteristic
429C22×C4C2×D4×C15111yescharacteristic
430C22×C4C2×D4×C15111yescharacteristic
431C22×C4C2×D4×C15112yes—
432C22×C4C2×D4×C15112yes—
433C22×C4C24×C15111yescharacteristic
434SD16—84no—
435M16—82no—
436C2×C8—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4C2×D4×C15114yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—44no—
469C23—44no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23#120×C15112yes—
475C23AGL(1,ℤ/8)×C15112yes—
476C23C22×D4×C15111yescharacteristic
477Q8—44no—
478Q8—44no—
479D4—28no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—48no—
484D4—48no—
485D4—44no—
486D4—44no—
487D4—82no—
488D4—82no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4#120×C15111yescharacteristic
518C2×C4#120×C15111yescharacteristic
519C2×C4AGL(1,ℤ/8)×C15112yes—
520C2×C4C22×D4×C15112yes—
521C8—44no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—164no—
538C22—164no—
539C22C2×#120×C15111yescharacteristic
540C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
541C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C2—21no—
554C2—21no—
555C2—42no—
556C2—42no—
557C2—84no—
558C2#3003×C15111yescharacteristic
559C2#3022×C15111yescharacteristic
560C2#3026×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3496×C151C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3193×C151C2semidirect
3. #2638×C151C2semidirect
4. #1610×C151C2semidirect
5. #1607×C151C2semidirect
6. #1604×C151C2semidirect
7. #1601×C151C2semidirect
8. #1510×C151C2semidirect
9. #1465×C151C2semidirect
18. C2×C22⋊Q8×C151C22semidirect
23. #480×C151C22semidirect
26. #448×C151C22semidirect
27. #407×C151C22semidirect
28. #398×C151C22semidirect
30. #397×C151C22semidirect
31. #395×C151C22semidirect
33. #393×C151C22semidirect
36. #381×C151C22semidirect
37. #327×C151C22semidirect
38. #327×C151C22semidirect
39. #324×C151C22semidirect
40. #323×C151C22semidirect
280. #18351C151direct (N × H)
281. C151#18351direct (N × H)
283. #3193C2×C151semidirect
284. #2638C2×C151semidirect
285. #1610C2×C151semidirect
286. #1607C2×C151semidirect
287. #1604C2×C151semidirect
288. #1601C2×C151semidirect
289. #1510C2×C151semidirect
290. #1465C2×C151semidirect
299. C2×C22⋊Q8C22×C151semidirect
304. #480C22×C151semidirect
307. #448C22×C151semidirect
308. #407C22×C151semidirect
309. #398C22×C151semidirect
311. #397C22×C151semidirect
312. #395C22×C151semidirect
314. #393C22×C151semidirect
317. #381C22×C151semidirect
318. #327C22×C151semidirect
319. #327C22×C151semidirect
320. #324C22×C151semidirect
321. #323C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3496×C1512
#3193×C1512
#2638×C1512
#1610×C1512
#1607×C1512
#1604×C1512
#1601×C1512
#1510×C1512
#1465×C1512
#18351151