Finite Group Databasebeta

← C151×#24226 #50329151

Subgroups — C151×#24226 #50329151

Order 38656.

1494 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3496×C151C211yesmaximal, characteristic, complemented
2#3036×C151C212yesmaximal, complemented
3C2×#453×C151C212yesmaximal, complemented
4#2328×C151C211yesmaximal, characteristic, complemented
5#2040×C151C212yesmaximal, complemented
6#2006×C151C212yesmaximal, complemented
7#1963×C151C212yesmaximal, complemented
8#1800×C151C211yesmaximal, characteristic, complemented
9#1792×C151C212yesmaximal, complemented
10C2×H(5,𝔽2)×C151C2212yes—
11C2×AGL(1,ℤ/8)×C151C2214yescomplemented
12#547×C151—22no—
13#546×C151—24no—
14D42×C151—22no—
15#537×C151C2212yes—
16#534×C151—22no—
17#532×C151C2212yes—
18C2×#127×C151C2211yescharacteristic
19C2×#121×C151—22no—
20C2×C4×D4×C151C2211yescharacteristic
21C2×C4×D4×C151C2211yescharacteristic, complemented
22#463×C151—24no—
23#461×C151—22no—
24#459×C151—22no—
25#453×C151—28no—
26#423×C151C2212yescomplemented
27C2×#104×C151C2214yescomplemented
28C2×#104×C151C2214yescomplemented
29#417×C151C2214yescomplemented
30C2×#100×C151C2212yes—
31#390×C151C2211yescharacteristic
32#390×C151C2211yescharacteristic, complemented
33#386×C151—22no—
34#386×C151—22no—
35#382×C151C2211yescharacteristic, complemented
36#376×C151C2211yescharacteristic, complemented
37#343×C151C2212yescomplemented
38H(5,𝔽2)×C151—24no—
39H(5,𝔽2)×C151—24no—
40H(5,𝔽2)×C151—44no—
41C2×#41×C151—24no—
42C2×#41×C151C2314yes—
43C22×D4×C151—22no—
44C22×D4×C151—24no—
45C22×D4×C151—24no—
46C22×D4×C151—22no—
47C22×D4×C151C2311yescharacteristic
48C22×D4×C151C2311yescharacteristic
49C23×C4×C151—21no—
50C23×C4×C151D412yes—
51AGL(1,ℤ/8)×C151—28no—
52AGL(1,ℤ/8)×C151—44no—
53C2×SD16×C151—24no—
54C2×D8×C151—24no—
55C2×M16×C151C2314yes—
56C4⋊Q8×C151D412yescomplemented
57#127×C151—22no—
58#127×C151D412yescomplemented
59#127×C151D414yescomplemented
60#124×C151D414yescomplemented
61#123×C151—28no—
62C22⋊Q8×C151—24no—
63#121×C151—24no—
64#121×C151—24no—
65#121×C151—24no—
66#121×C151—22no—
67#121×C151—24no—
68#121×C151—22no—
69#120×C151—28no—
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C4×D4×C151—22no—
73C4×D4×C151—22no—
74C4×D4×C151—22no—
75C4×D4×C151—22no—
76C2×C4⋊C4×C151—22no—
77C2×C4⋊C4×C151C2311yescharacteristic
78C2×C4⋊C4×C151C2311yescharacteristic
79C2×C22⋊C4×C151—24no—
80C2×C22⋊C4×C151—21no—
81C2×C22⋊C4×C151—22no—
82C2×C22⋊C4×C151—22no—
83C2×C22⋊C4×C151D412yes—
84C2×C42×C151C2311yescharacteristic
85C2×C42×C151C2311yescharacteristic
86C2×C42×C151C2311yescharacteristic
87C4⋊C8×C151—22no—
88C4⋊C8×C151—22no—
89#104×C151—28no—
90#104×C151—44no—
91#103×C151—24no—
92#102×C151—24no—
93#100×C151—24no—
94#95×C151—21no—
95#95×C151—21no—
96#95×C151—22no—
97C24×C151—24no—
98C24×C151—21no—
99C24×C151C2×D412yes—
100#41×C151—24no—
101#41×C151—24no—
102#41×C151—24no—
103#41×C151—24no—
104#41×C151—48no—
105#41×C151—48no—
106C2×Q8×C151C2×D414yes—
107C2×D4×C151—28no—
108C2×D4×C151—24no—
109C2×D4×C151—24no—
110C2×D4×C151—28no—
111C2×D4×C151—22no—
112C2×D4×C151—22no—
113C2×D4×C151—22no—
114C2×D4×C151—22no—
115C2×D4×C151—21no—
116C2×D4×C151—22no—
117C2×D4×C151—22no—
118C2×D4×C151—48no—
119C2×D4×C151—48no—
120C2×D4×C151—48no—
121C2×D4×C151—48no—
122C2×D4×C151—84no—
123C2×D4×C151—82no—
124C2×D4×C151—82no—
125C2×D4×C151C2×D414yes—
126C2×D4×C151C2×D412yes—
127C2×D4×C151C2×D412yescomplemented
128C2×D4×C151#4112yes—
129C22×C4×C151—24no—
130C22×C4×C151—21no—
131C22×C4×C151—22no—
132C22×C4×C151—24no—
133C22×C4×C151—22no—
134C22×C4×C151—21no—
135C22×C4×C151—21no—
136C22×C4×C151—42no—
137C22×C4×C151—82no—
138C22×C4×C151—82no—
139C22×C4×C151C2×D412yes—
140C22×C4×C151C2×D411yescharacteristic
141C22×C4×C151C2×D411yescharacteristic
142C22×C4×C151C2411yescharacteristic
143SD16×C151—48no—
144D8×C151—48no—
145M16×C151—24no—
146M16×C151—24no—
147C2×C8×C151—24no—
148C4⋊C4×C151—24no—
149C4⋊C4×C151—22no—
150C4⋊C4×C151—22no—
151C4⋊C4×C151—22no—
152C4⋊C4×C151—22no—
153C22⋊C4×C151—28no—
154C22⋊C4×C151—24no—
155C22⋊C4×C151—24no—
156C22⋊C4×C151—24no—
157C22⋊C4×C151—41no—
158C22⋊C4×C151—41no—
159C22⋊C4×C151—82no—
160C22⋊C4×C151—82no—
161C42×C151—21no—
162C42×C151—22no—
163C42×C151C2×D412yes—
164C42×C151C2×D412yes—
165C42×C151C2×D412yescomplemented
166C23×C151—22no—
167C23×C151—22no—
168C23×C151—24no—
169C23×C151—22no—
170C23×C151—42no—
171C23×C151—42no—
172C23×C151—42no—
173C23×C151—42no—
174C23×C151—48no—
175C23×C151—41no—
176C23×C151—41no—
177C23×C151—84no—
178C23×C151—84no—
179C23×C151#12114yes—
180C23×C151C22×D411yescharacteristic
181Q8×C151—24no—
182Q8×C151—24no—
183D4×C151—24no—
184D4×C151—24no—
185D4×C151—24no—
186D4×C151—24no—
187D4×C151—24no—
188D4×C151—44no—
189D4×C151—416no—
190D4×C151—44no—
191D4×C151—44no—
192D4×C151—42no—
193D4×C151—82no—
194D4×C151—88no—
195D4×C151—84no—
196C2×C4×C151—24no—
197C2×C4×C151—22no—
198C2×C4×C151—22no—
199C2×C4×C151—41no—
200C2×C4×C151—41no—
201C2×C4×C151—44no—
202C2×C4×C151—48no—
203C2×C4×C151—41no—
204C2×C4×C151—41no—
205C2×C4×C151—44no—
206C2×C4×C151—44no—
207C2×C4×C151—42no—
208C2×C4×C151—41no—
209C2×C4×C151—41no—
210C2×C4×C151—42no—
211C2×C4×C151—41no—
212C2×C4×C151—41no—
213C2×C4×C151—82no—
214C2×C4×C151—84no—
215C2×C4×C151#12011yescharacteristic
216C2×C4×C151#12012yes—
217C2×C4×C151#12011yescharacteristic
218C2×C4×C151C22×D411yescharacteristic
219C2×C4×C151C22×D412yes—
220C2×C4×C151C2×#4111yescharacteristic
221C2×C4×C151H(5,𝔽2)12yes—
222C8×C151—44no—
223C22×C151—21no—
224C22×C151—21no—
225C22×C151—24no—
226C22×C151—44no—
227C22×C151—44no—
228C22×C151—42no—
229C22×C151—42no—
230C22×C151—44no—
231C22×C151—42no—
232C22×C151—44no—
233C22×C151—44no—
234C22×C151—82no—
235C22×C151—82no—
236C22×C151—81no—
237C22×C151—81no—
238C22×C151—168no—
239C22×C151C2×#12011yescharacteristic
240C22×C151C2×#12111yescharacteristic
241C22×C151#53511yescharacteristic
242C4×C151—21no—
243C4×C151—21no—
244C4×C151—22no—
245C4×C151—42no—
246C4×C151—44no—
247C4×C151—42no—
248C4×C151—81no—
249C4×C151—81no—
250C4×C151D4212yes—
251C4×C151#54612yes—
252C2×C151—21no—
253C2×C151—21no—
254C2×C151—44no—
255C2×C151—82no—
256C2×C151—84no—
257C2×C151#240811yescharacteristic
258C2×C151#301911yescharacteristic
259C2×C151#302111yescharacteristic
260#24226C15111yesmaximal, characteristic, direct product factor, Sylow
261C151#2422611yescharacteristic, direct product factor, Sylow
262#3496C2×C15111yescharacteristic, complemented
263#3036C2×C15112yescomplemented
264C2×#453C2×C15112yescomplemented
265#2328C2×C15111yescharacteristic, complemented
266#2040C2×C15112yescomplemented
267#2006C2×C15112yescomplemented
268#1963C2×C15112yescomplemented
269#1800C2×C15111yescharacteristic, complemented
270#1792C2×C15112yescomplemented
271C2×H(5,𝔽2)C22×C15112yes—
272C2×AGL(1,ℤ/8)C22×C15114yescomplemented
273#547—22no—
274#546—24no—
275D42—22no—
276#537C22×C15112yes—
277#534—22no—
278#532C22×C15112yes—
279C2×#127C22×C15111yescharacteristic
280C2×#121—22no—
281C2×C4×D4C22×C15111yescharacteristic
282C2×C4×D4C22×C15111yescharacteristic, complemented
283#463—24no—
284#461—22no—
285#459—22no—
286#453—28no—
287#423C22×C15112yescomplemented
288C2×#104C22×C15114yescomplemented
289C2×#104C22×C15114yescomplemented
290#417C22×C15114yescomplemented
291C2×#100C22×C15112yes—
292#390C22×C15111yescharacteristic
293#390C22×C15111yescharacteristic, complemented
294#386—22no—
295#386—22no—
296#382C22×C15111yescharacteristic, complemented
297#376C22×C15111yescharacteristic, complemented
298#343C22×C15112yescomplemented
299H(5,𝔽2)—24no—
300H(5,𝔽2)—24no—
301H(5,𝔽2)—44no—
302C2×#41—24no—
303C2×#41C23×C15114yes—
304C22×D4—24no—
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—24no—
308C22×D4C23×C15111yescharacteristic
309C22×D4C23×C15111yescharacteristic
310C23×C4—21no—
311C23×C4D4×C15112yes—
312AGL(1,ℤ/8)—28no—
313AGL(1,ℤ/8)—44no—
314C2×SD16—24no—
315C2×D8—24no—
316C2×M16C23×C15114yes—
317C4⋊Q8D4×C15112yescomplemented
318#127—22no—
319#127D4×C15112yescomplemented
320#127D4×C15114yescomplemented
321#124D4×C15114yescomplemented
322#123—28no—
323C22⋊Q8—24no—
324#121—24no—
325#121—24no—
326#121—24no—
327#121—22no—
328#121—24no—
329#121—22no—
330#120—28no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4C23×C15111yescharacteristic
339C2×C4⋊C4C23×C15111yescharacteristic
340C2×C22⋊C4—24no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4D4×C15112yes—
345C2×C42C23×C15111yescharacteristic
346C2×C42C23×C15111yescharacteristic
347C2×C42C23×C15111yescharacteristic
348C4⋊C8—22no—
349C4⋊C8—22no—
350#104—28no—
351#104—44no—
352#103—24no—
353#102—24no—
354#100—24no—
355#95—21no—
356#95—21no—
357#95—22no—
358C24—24no—
359C24—21no—
360C24C2×D4×C15112yes—
361#41—24no—
362#41—24no—
363#41—24no—
364#41—24no—
365#41—48no—
366#41—48no—
367C2×Q8C2×D4×C15114yes—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—28no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—84no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×D4×C15114yes—
387C2×D4C2×D4×C15112yes—
388C2×D4C2×D4×C15112yescomplemented
389C2×D4#41×C15112yes—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C2×D4×C15112yes—
401C22×C4C2×D4×C15111yescharacteristic
402C22×C4C2×D4×C15111yescharacteristic
403C22×C4C24×C15111yescharacteristic
404SD16—48no—
405D8—48no—
406M16—24no—
407M16—24no—
408C2×C8—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C22⋊C4—28no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C42—21no—
423C42—22no—
424C42C2×D4×C15112yes—
425C42C2×D4×C15112yes—
426C42C2×D4×C15112yescomplemented
427C23—22no—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—48no—
436C23—41no—
437C23—41no—
438C23—84no—
439C23—84no—
440C23#121×C15114yes—
441C23C22×D4×C15111yescharacteristic
442Q8—24no—
443Q8—24no—
444D4—24no—
445D4—24no—
446D4—24no—
447D4—24no—
448D4—24no—
449D4—44no—
450D4—416no—
451D4—44no—
452D4—44no—
453D4—42no—
454D4—82no—
455D4—88no—
456D4—84no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4#120×C15111yescharacteristic
477C2×C4#120×C15112yes—
478C2×C4#120×C15111yescharacteristic
479C2×C4C22×D4×C15111yescharacteristic
480C2×C4C22×D4×C15112yes—
481C2×C4C2×#41×C15111yescharacteristic
482C2×C4H(5,𝔽2)×C15112yes—
483C8—44no—
484C22—21no—
485C22—21no—
486C22—24no—
487C22—44no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—42no—
493C22—44no—
494C22—44no—
495C22—82no—
496C22—82no—
497C22—81no—
498C22—81no—
499C22—168no—
500C22C2×#120×C15111yescharacteristic
501C22C2×#121×C15111yescharacteristic
502C22#535×C15111yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—42no—
507C4—44no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4D42×C15112yes—
512C4#546×C15112yes—
513C2—21no—
514C2—21no—
515C2—44no—
516C2—82no—
517C2—84no—
518C2#2408×C15111yescharacteristic
519C2#3019×C15111yescharacteristic
520C2#3021×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3496×C151C2semidirect
2. #3036×C151C2semidirect
3. C2×#453×C151C2semidirect
4. #2328×C151C2semidirect
5. #2040×C151C2semidirect
6. #2006×C151C2semidirect
7. #1963×C151C2semidirect
8. #1800×C151C2semidirect
9. #1792×C151C2semidirect
11. C2×AGL(1,ℤ/8)×C151C22semidirect
21. C2×C4×D4×C151C22semidirect
26. #423×C151C22semidirect
27. C2×#104×C151C22semidirect
28. C2×#104×C151C22semidirect
29. #417×C151C22semidirect
32. #390×C151C22semidirect
35. #382×C151C22semidirect
36. #376×C151C22semidirect
37. #343×C151C22semidirect
56. C4⋊Q8×C151D4semidirect
58. #127×C151D4semidirect
59. #127×C151D4semidirect
60. #124×C151D4semidirect
127. C2×D4×C151C2×D4semidirect
165. C42×C151C2×D4semidirect
260. #24226C151direct (N × H)
261. C151#24226direct (N × H)
262. #3496C2×C151semidirect
263. #3036C2×C151semidirect
264. C2×#453C2×C151semidirect
265. #2328C2×C151semidirect
266. #2040C2×C151semidirect
267. #2006C2×C151semidirect
268. #1963C2×C151semidirect
269. #1800C2×C151semidirect
270. #1792C2×C151semidirect
272. C2×AGL(1,ℤ/8)C22×C151semidirect
282. C2×C4×D4C22×C151semidirect
287. #423C22×C151semidirect
288. C2×#104C22×C151semidirect
289. C2×#104C22×C151semidirect
290. #417C22×C151semidirect
293. #390C22×C151semidirect
296. #382C22×C151semidirect
297. #376C22×C151semidirect
298. #343C22×C151semidirect
317. C4⋊Q8D4×C151semidirect
319. #127D4×C151semidirect
320. #127D4×C151semidirect
321. #124D4×C151semidirect
388. C2×D4C2×D4×C151semidirect
426. C42C2×D4×C151semidirect

Maximal subgroups

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

subgroupindex
#3496×C1512
#3036×C1512
C2×#453×C1512
#2328×C1512
#2040×C1512
#2006×C1512
#1963×C1512
#1800×C1512
#1792×C1512
#24226151