Finite Group Databasebeta

← C151×#24134 #50329059

Subgroups — C151×#24134 #50329059

Order 38656.

1120 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
1C2×#554×C151C211yesmaximal, characteristic
2C2×#475×C151C211yesmaximal, characteristic, complemented
3C2×#441×C151C211yesmaximal, characteristic, complemented
4#2536×C151C211yesmaximal, characteristic
5#2035×C151C212yesmaximal, complemented
6#2001×C151C212yesmaximal, complemented
7#1993×C151C211yesmaximal, characteristic, complemented
8#1955×C151C211yesmaximal, characteristic
9#1872×C151C212yesmaximal, complemented
10#1853×C151C212yesmaximal, complemented
11#1805×C151C211yesmaximal, characteristic, complemented
12C22×Q16×C151C2211yescharacteristic
13#554×C151—24no—
14#554×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#125×C151—21no—
17C2×C22⋊Q8×C151—22no—
18C2×C22⋊Q8×C151C2212yes—
19C2×C22⋊Q8×C151C2212yes—
20C2×C4×Q8×C151C2211yescharacteristic
21C2×C4×Q8×C151C2211yescharacteristic
22C2×C4×D4×C151—21no—
23C2×C4×D4×C151C2211yescharacteristic
24C22×C4⋊C4×C151C2212yes—
25#475×C151—24no—
26#441×C151—24no—
27C2×#106×C151C2211yescharacteristic
28C2×#106×C151C2212yes—
29C2×C4⋊C8×C151C2211yescharacteristic
30C2×#103×C151C2211yescharacteristic
31C2×#103×C151C2212yes—
32C2×#103×C151C2211yescharacteristic
33C2×#103×C151C2212yes—
34C2×C22⋊C8×C151C2212yes—
35C2×C22⋊C8×C151C2212yes—
36C2×#97×C151C2211yescharacteristic
37#398×C151—21no—
38#394×C151—21no—
39#388×C151—22no—
40#382×C151C2211yescharacteristic
41#380×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45C22×Q8×C151—21no—
46C22×Q8×C151C2311yescharacteristic
47C22×Q8×C151C2311yescharacteristic
48C22×D4×C151D411yescharacteristic
49C23×C4×C151—22no—
50C23×C4×C151C2312yes—
51C2×Q16×C151—22no—
52C2×Q16×C151—22no—
53C2×Q16×C151—22no—
54C22×C8×C151C2311yescharacteristic
55C22×C8×C151C2312yes—
56C22×C8×C151C2311yescharacteristic
57C4⋊Q8×C151—22no—
58C4⋊Q8×C151D414yes—
59#125×C151—28no—
60C22⋊Q8×C151—216no—
61C22⋊Q8×C151—24no—
62C22⋊Q8×C151—24no—
63C22⋊Q8×C151—24no—
64C22⋊Q8×C151—42no—
65C4×Q8×C151—22no—
66C4×Q8×C151—22no—
67C4×Q8×C151—22no—
68C4×Q8×C151D414yes—
69C4×D4×C151—28no—
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C2×C4⋊C4×C151—22no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—22no—
76C2×C4⋊C4×C151—24no—
77C2×C4⋊C4×C151—24no—
78C2×C4⋊C4×C151—22no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151D411yescharacteristic
81C2×C4⋊C4×C151C2312yes—
82C2×C4⋊C4×C151C2311yescharacteristic
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2312yes—
85C2×C22⋊C4×C151—22no—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151D412yes—
89C2×C42×C151—21no—
90C2×C42×C151—21no—
91C2×C42×C151C2311yescharacteristic
92#106×C151—22no—
93#106×C151—24no—
94C4⋊C8×C151—22no—
95#103×C151—22no—
96#103×C151—24no—
97#103×C151—22no—
98#103×C151—42no—
99C22⋊C8×C151—24no—
100C22⋊C8×C151—42no—
101#97×C151—22no—
102#95×C151—22no—
103#95×C151—21no—
104#95×C151—21no—
105C24×C151C2×D412yes—
106C2×Q8×C151—24no—
107C2×Q8×C151—21no—
108C2×Q8×C151—21no—
109C2×Q8×C151—21no—
110C2×Q8×C151—21no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—21no—
114C2×Q8×C151—21no—
115C2×Q8×C151—21no—
116C2×Q8×C151—21no—
117C2×Q8×C151—82no—
118C2×Q8×C151C2×D412yes—
119C2×Q8×C151#4112yes—
120C2×D4×C151—24no—
121C2×D4×C151—22no—
122C22×C4×C151—21no—
123C22×C4×C151—21no—
124C22×C4×C151—21no—
125C22×C4×C151—22no—
126C22×C4×C151—28no—
127C22×C4×C151—22no—
128C22×C4×C151—22no—
129C22×C4×C151—22no—
130C22×C4×C151—22no—
131C22×C4×C151—22no—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—21no—
135C22×C4×C151—82no—
136C22×C4×C151—82no—
137C22×C4×C151C2×D412yes—
138C22×C4×C151C2×D411yescharacteristic
139C22×C4×C151C2×D411yescharacteristic
140C22×C4×C151C2411yescharacteristic
141Q16×C151—28no—
142C2×C8×C151—21no—
143C2×C8×C151—21no—
144C2×C8×C151—21no—
145C2×C8×C151—22no—
146C2×C8×C151—22no—
147C2×C8×C151—21no—
148C2×C8×C151—22no—
149C2×C8×C151—21no—
150C2×C8×C151#4112yes—
151C4⋊C4×C151—28no—
152C4⋊C4×C151—28no—
153C4⋊C4×C151—24no—
154C4⋊C4×C151—24no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—21no—
157C4⋊C4×C151—22no—
158C4⋊C4×C151—21no—
159C4⋊C4×C151—21no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—22no—
162C4⋊C4×C151—44no—
163C4⋊C4×C151—42no—
164C4⋊C4×C151C2×D412yes—
165C4⋊C4×C151C2×D414yes—
166C22⋊C4×C151—28no—
167C22⋊C4×C151—24no—
168C22⋊C4×C151—44no—
169C22⋊C4×C151—44no—
170C42×C151—24no—
171C42×C151—42no—
172C42×C151C2×D412yes—
173C42×C151C2×D412yes—
174C23×C151—22no—
175C23×C151—22no—
176C23×C151—22no—
177C23×C151—22no—
178C23×C151—22no—
179C23×C151—22no—
180C23×C151#13714yes—
181C23×C151C22×D411yescharacteristic
182Q8×C151—22no—
183Q8×C151—22no—
184Q8×C151—22no—
185Q8×C151—22no—
186Q8×C151—41no—
187Q8×C151—41no—
188Q8×C151—41no—
189Q8×C151—41no—
190Q8×C151—84no—
191D4×C151—28no—
192C2×C4×C151—22no—
193C2×C4×C151—22no—
194C2×C4×C151—24no—
195C2×C4×C151—22no—
196C2×C4×C151—22no—
197C2×C4×C151—22no—
198C2×C4×C151—22no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—41no—
202C2×C4×C151—41no—
203C2×C4×C151—41no—
204C2×C4×C151—41no—
205C2×C4×C151—42no—
206C2×C4×C151—42no—
207C2×C4×C151—42no—
208C2×C4×C151—42no—
209C2×C4×C151—42no—
210C2×C4×C151—42no—
211C2×C4×C151—42no—
212C2×C4×C151—41no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—88no—
218C2×C4×C151#12012yes—
219C2×C4×C151#12012yes—
220C2×C4×C151C22×D411yescharacteristic
221C2×C4×C151C22×D411yescharacteristic
222C2×C4×C151C2×#4111yescharacteristic
223C2×C4×C151C2×#4111yescharacteristic
224C2×C4×C151H(5,𝔽2)11yescharacteristic
225C2×C4×C1512−1+411yescharacteristic
226C8×C151—22no—
227C8×C151—41no—
228C8×C151—42no—
229C22×C151—24no—
230C22×C151—24no—
231C22×C151—24no—
232C22×C151—42no—
233C22×C151—42no—
234C22×C151—42no—
235C22×C151—42no—
236C22×C151C2×#12011yescharacteristic
237C22×C151#54611yescharacteristic
238C22×C151#54711yescharacteristic
239C22×C151C2×#13712yes—
240C22×C151#57712yes—
241C4×C151—22no—
242C4×C151—22no—
243C4×C151—42no—
244C4×C151—42no—
245C4×C151—42no—
246C4×C151—42no—
247C4×C151—44no—
248C4×C151—81no—
249C4×C151—82no—
250C4×C151#52511yescharacteristic
251C4×C151#53711yescharacteristic
252C4×C151#54811yescharacteristic
253C4×C151#55011yescharacteristic
254C2×C151—44no—
255C2×C151#240711yescharacteristic
256C2×C151#300411yescharacteristic
257C2×C151#300911yescharacteristic
258C2×C151#331612yes—
259C2×C151#334412yes—
260#24134C15111yesmaximal, characteristic, direct product factor, Sylow
261C151#2413411yescharacteristic, direct product factor, Sylow
262C2×#554C2×C15111yescharacteristic
263C2×#475C2×C15111yescharacteristic, complemented
264C2×#441C2×C15111yescharacteristic, complemented
265#2536C2×C15111yescharacteristic
266#2035C2×C15112yescomplemented
267#2001C2×C15112yescomplemented
268#1993C2×C15111yescharacteristic, complemented
269#1955C2×C15111yescharacteristic
270#1872C2×C15112yescomplemented
271#1853C2×C15112yescomplemented
272#1805C2×C15111yescharacteristic, complemented
273C22×Q16C22×C15111yescharacteristic
274#554—24no—
275#554—24no—
276C2×C4⋊Q8C22×C15111yescharacteristic
277C2×#125—21no—
278C2×C22⋊Q8—22no—
279C2×C22⋊Q8C22×C15112yes—
280C2×C22⋊Q8C22×C15112yes—
281C2×C4×Q8C22×C15111yescharacteristic
282C2×C4×Q8C22×C15111yescharacteristic
283C2×C4×D4—21no—
284C2×C4×D4C22×C15111yescharacteristic
285C22×C4⋊C4C22×C15112yes—
286#475—24no—
287#441—24no—
288C2×#106C22×C15112yes—
289C2×#106C22×C15111yescharacteristic
290C2×C4⋊C8C22×C15111yescharacteristic
291C2×#103C22×C15112yes—
292C2×#103C22×C15111yescharacteristic
293C2×#103C22×C15111yescharacteristic
294C2×#103C22×C15112yes—
295C2×C22⋊C8C22×C15112yes—
296C2×C22⋊C8C22×C15112yes—
297C2×#97C22×C15111yescharacteristic
298#398—21no—
299#394—21no—
300#388—22no—
301#382C22×C15111yescharacteristic
302#380C22×C15112yes—
303#340C22×C15112yes—
304#340C22×C15112yes—
305#340C22×C15112yes—
306C22×Q8—21no—
307C22×Q8C23×C15111yescharacteristic
308C22×Q8C23×C15111yescharacteristic
309C22×D4D4×C15111yescharacteristic
310C23×C4—22no—
311C23×C4C23×C15112yes—
312C2×Q16—22no—
313C2×Q16—22no—
314C2×Q16—22no—
315C22×C8C23×C15111yescharacteristic
316C22×C8C23×C15112yes—
317C22×C8C23×C15111yescharacteristic
318C4⋊Q8—22no—
319C4⋊Q8D4×C15114yes—
320#125—28no—
321C22⋊Q8—216no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—42no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8D4×C15114yes—
330C4×D4—28no—
331C4×D4—22no—
332C4×D4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4D4×C15111yescharacteristic
342C2×C4⋊C4C23×C15111yescharacteristic
343C2×C4⋊C4C23×C15112yes—
344C2×C4⋊C4C23×C15111yescharacteristic
345C2×C4⋊C4C23×C15112yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4D4×C15112yes—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42C23×C15111yescharacteristic
353#106—22no—
354#106—24no—
355C4⋊C8—22no—
356#103—22no—
357#103—22no—
358#103—24no—
359#103—42no—
360C22⋊C8—24no—
361C22⋊C8—42no—
362#97—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366C24C2×D4×C15112yes—
367C2×Q8—24no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—82no—
379C2×Q8C2×D4×C15112yes—
380C2×Q8#41×C15112yes—
381C2×D4—24no—
382C2×D4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4C2×D4×C15112yes—
399C22×C4C2×D4×C15111yescharacteristic
400C22×C4C2×D4×C15111yescharacteristic
401C22×C4C24×C15111yescharacteristic
402Q16—28no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8#41×C15112yes—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4C2×D4×C15112yes—
426C4⋊C4C2×D4×C15114yes—
427C22⋊C4—28no—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C42—24no—
432C42—42no—
433C42C2×D4×C15112yes—
434C42C2×D4×C15112yes—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23#137×C15114yes—
442C23C22×D4×C15111yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451Q8—84no—
452D4—28no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—88no—
479C2×C4#120×C15112yes—
480C2×C4#120×C15112yes—
481C2×C4C22×D4×C15111yescharacteristic
482C2×C4C22×D4×C15111yescharacteristic
483C2×C4C2×#41×C15111yescharacteristic
484C2×C4C2×#41×C15111yescharacteristic
485C2×C4H(5,𝔽2)×C15111yescharacteristic
486C2×C42−1+4×C15111yescharacteristic
487C8—22no—
488C8—41no—
489C8—42no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22C2×#120×C15111yescharacteristic
498C22#546×C15111yescharacteristic
499C22#547×C15111yescharacteristic
500C22C2×#137×C15112yes—
501C22#577×C15112yes—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—44no—
509C4—81no—
510C4—82no—
511C4#525×C15111yescharacteristic
512C4#537×C15111yescharacteristic
513C4#548×C15111yescharacteristic
514C4#550×C15111yescharacteristic
515C2—44no—
516C2#2407×C15111yescharacteristic
517C2#3004×C15111yescharacteristic
518C2#3009×C15111yescharacteristic
519C2#3316×C15112yes—
520C2#3344×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×D4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#475×C151C2semidirect
3. C2×#441×C151C2semidirect
5. #2035×C151C2semidirect
6. #2001×C151C2semidirect
7. #1993×C151C2semidirect
9. #1872×C151C2semidirect
10. #1853×C151C2semidirect
11. #1805×C151C2semidirect
260. #24134C151direct (N × H)
261. C151#24134direct (N × H)
263. C2×#475C2×C151semidirect
264. C2×#441C2×C151semidirect
266. #2035C2×C151semidirect
267. #2001C2×C151semidirect
268. #1993C2×C151semidirect
270. #1872C2×C151semidirect
271. #1853C2×C151semidirect
272. #1805C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#554×C1512
C2×#475×C1512
C2×#441×C1512
#2536×C1512
#2035×C1512
#2001×C1512
#1993×C1512
#1955×C1512
#1872×C1512
#1853×C1512
#1805×C1512
#24134151