Finite Group Databasebeta

← C151×#22996 #50327921

Subgroups — C151×#22996 #50327921

Order 38656.

1030 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#461×C151C211yesmaximal, characteristic
2C2×C4×SD16×C151C212yesmaximal
3#2365×C151C211yesmaximal, characteristic, complemented
4D4×C4⋊C4×C151C211yesmaximal, characteristic
5#1956×C151C211yesmaximal, characteristic, complemented
6#1948×C151C211yesmaximal, characteristic, complemented
7#1933×C151C212yesmaximal, complemented
8#1877×C151C212yesmaximal
9#1865×C151C212yesmaximal, complemented
10C8×C4⋊C4×C151C211yesmaximal, characteristic, complemented
11C4×#102×C151C211yesmaximal, characteristic
12C22×SD16×C151C2212yes—
13C2×C4⋊Q8×C151C2211yescharacteristic
14C2×C4×Q8×C151C2212yes—
15C2×C4×D4×C151C2212yes—
16C2×C4×D4×C151C2211yescharacteristic
17C22×C4⋊C4×C151—21no—
18#461×C151C418yescomplemented
19C4×SD16×C151—28no—
20C2×#106×C151C2212yes—
21C2×C4⋊C8×C151C2211yescharacteristic
22C2×#103×C151C2212yes—
23C2×#103×C151C2212yes—
24C2×#103×C151C2212yes—
25C2×#102×C151C2211yescharacteristic
26C2×#102×C151C2211yescharacteristic
27C2×C4×C8×C151C2212yes—
28C2×C4×C8×C151C2211yescharacteristic
29#391×C151C2212yes—
30#389×C151—21no—
31#389×C151C2212yes—
32#387×C151—22no—
33#385×C151—22no—
34#384×C151C2211yescharacteristic
35#380×C151—21no—
36C4×C4⋊C4×C151—21no—
37C4×C4⋊C4×C151C2211yescharacteristic
38C4×C4⋊C4×C151C2211yescharacteristic
39#340×C151C2211yescharacteristic
40#340×C151C2212yes—
41#340×C151C2211yescharacteristic
42#336×C151C2212yes—
43C22×Q8×C151C2312yes—
44C22×D4×C151C2311yescharacteristic
45C23×C4×C151—21no—
46C23×C4×C151—22no—
47C2×SD16×C151—24no—
48C2×SD16×C151—24no—
49C2×SD16×C151C2×C418yes—
50C22×C8×C151C2311yescharacteristic
51C22×C8×C151C2312yes—
52C22×C8×C151C2311yescharacteristic
53C4⋊Q8×C151—22no—
54C4⋊Q8×C151C2×C414yescomplemented
55C4×Q8×C151—24no—
56C4×Q8×C151—42no—
57C4×D4×C151—24no—
58C4×D4×C151—22no—
59C4×D4×C151—42no—
60C4×D4×C151C2×C414yes—
61C2×C4⋊C4×C151—21no—
62C2×C4⋊C4×C151—21no—
63C2×C4⋊C4×C151—21no—
64C2×C4⋊C4×C151—22no—
65C2×C4⋊C4×C151—22no—
66C2×C4⋊C4×C151—21no—
67C2×C4⋊C4×C151—44no—
68C2×C4⋊C4×C151D411yescharacteristic
69C2×C4⋊C4×C151D411yescharacteristic
70C2×C4⋊C4×C151C2311yescharacteristic
71C2×C4⋊C4×C151C2312yes—
72C2×C4⋊C4×C151C2311yescharacteristic
73C2×C4⋊C4×C151C2311yescharacteristic
74C2×C22⋊C4×C151—21no—
75C2×C22⋊C4×C151—22no—
76C2×C42×C151—21no—
77C2×C42×C151—22no—
78C2×C42×C151C2312yes—
79C2×C42×C151C2311yescharacteristic
80#106×C151—24no—
81C4⋊C8×C151C2×C414yescomplemented
82#103×C151—24no—
83#103×C151—42no—
84#103×C151C2×C418yescomplemented
85#102×C151—22no—
86#102×C151D414yes—
87C4×C8×C151—24no—
88C4×C8×C151—22no—
89#95×C151—22no—
90#95×C151—21no—
91#95×C151—22no—
92#95×C151—22no—
93#95×C151D412yes—
94C24×C151—21no—
95C2×Q8×C151—22no—
96C2×Q8×C151—22no—
97C2×Q8×C151—22no—
98C2×Q8×C151—22no—
99C2×Q8×C151—22no—
100C2×Q8×C151C22×C414yes—
101C2×D4×C151—21no—
102C2×D4×C151—21no—
103C2×D4×C151—21no—
104C2×D4×C151C22×C412yes—
105C2×D4×C151C2×D412yes—
106C2×D4×C151#4112yes—
107C22×C4×C151—22no—
108C22×C4×C151—21no—
109C22×C4×C151—21no—
110C22×C4×C151—24no—
111C22×C4×C151—21no—
112C22×C4×C151—22no—
113C22×C4×C151—44no—
114C22×C4×C151—42no—
115C22×C4×C151—42no—
116C22×C4×C151—44no—
117C22×C4×C151—82no—
118C22×C4×C151C2×D412yes—
119C22×C4×C151C2×D411yescharacteristic
120C22×C4×C151C2×D412yes—
121C22×C4×C151C2×D411yescharacteristic
122C22×C4×C151C2411yescharacteristic
123SD16×C151—216no—
124C2×C8×C151—22no—
125C2×C8×C151—21no—
126C2×C8×C151—22no—
127C2×C8×C151—21no—
128C2×C8×C151—21no—
129C2×C8×C151—21no—
130C2×C8×C151—21no—
131C2×C8×C151C22×C414yes—
132C2×C8×C151C2×D412yes—
133C4⋊C4×C151—24no—
134C4⋊C4×C151—24no—
135C4⋊C4×C151—22no—
136C4⋊C4×C151—21no—
137C4⋊C4×C151—21no—
138C4⋊C4×C151—22no—
139C4⋊C4×C151—21no—
140C4⋊C4×C151—22no—
141C4⋊C4×C151—21no—
142C4⋊C4×C151—44no—
143C4⋊C4×C151—44no—
144C4⋊C4×C151—42no—
145C4⋊C4×C151—81no—
146C4⋊C4×C151SD1614yescomplemented
147C4⋊C4×C151C22×C412yes—
148C4⋊C4×C151C2×D412yes—
149C22⋊C4×C151—24no—
150C22⋊C4×C151—44no—
151C42×C151—22no—
152C42×C151—22no—
153C42×C151—44no—
154C42×C151—42no—
155C42×C151C22×C412yes—
156C42×C151#4112yes—
157C23×C151—22no—
158C23×C151—22no—
159C23×C151—22no—
160C23×C151—41no—
161C23×C151—41no—
162C23×C151—41no—
163C23×C151—41no—
164C23×C151C22×D411yescharacteristic
165Q8×C151—24no—
166Q8×C151—24no—
167Q8×C151—42no—
168Q8×C151—42no—
169D4×C151—22no—
170D4×C151—22no—
171D4×C151—22no—
172D4×C151C4×D414yes—
173C2×C4×C151—22no—
174C2×C4×C151—21no—
175C2×C4×C151—22no—
176C2×C4×C151—21no—
177C2×C4×C151—24no—
178C2×C4×C151—22no—
179C2×C4×C151—22no—
180C2×C4×C151—22no—
181C2×C4×C151—22no—
182C2×C4×C151—22no—
183C2×C4×C151—42no—
184C2×C4×C151—42no—
185C2×C4×C151—41no—
186C2×C4×C151—41no—
187C2×C4×C151—41no—
188C2×C4×C151—41no—
189C2×C4×C151—48no—
190C2×C4×C151—42no—
191C2×C4×C151—41no—
192C2×C4×C151—42no—
193C2×C4×C151—41no—
194C2×C4×C151—42no—
195C2×C4×C151—41no—
196C2×C4×C151—88no—
197C2×C4×C151C4×D412yes—
198C2×C4×C151C4×D412yes—
199C2×C4×C151C2×SD1614yes—
200C2×C4×C151C2×SD1612yes—
201C2×C4×C151#13512yes—
202C2×C4×C151C23×C411yescharacteristic
203C2×C4×C151C22×D411yescharacteristic
204C2×C4×C151C2×#4111yescharacteristic
205C2×C4×C151C2×#4111yescharacteristic
206C2×C4×C151H(5,𝔽2)11yescharacteristic
207C2×C4×C1512−1+411yescharacteristic
208C8×C151—24no—
209C8×C151—22no—
210C8×C151—41no—
211C22×C151—24no—
212C22×C151—42no—
213C22×C151—42no—
214C22×C151—42no—
215C22×C151—42no—
216C22×C151—42no—
217C22×C151—42no—
218C22×C151C2×C4×D411yescharacteristic
219C22×C151D4211yescharacteristic
220C22×C151#54811yescharacteristic
221C22×C151C22×SD1611yescharacteristic
222C22×C151C2×#13511yescharacteristic
223C22×C151#57711yescharacteristic
224C22×C151#57811yescharacteristic
225C4×C151—22no—
226C4×C151—24no—
227C4×C151—44no—
228C4×C151—42no—
229C4×C151—42no—
230C4×C151—42no—
231C4×C151—81no—
232C4×C151—82no—
233C4×C151C4×SD1614yescomplemented
234C4×C151C2×C4×D411yescharacteristic
235C4×C151#52011yescharacteristic
236C4×C151#54711yescharacteristic
237C4×C151#55311yescharacteristic
238C2×C151—44no—
239C2×C151#236111yescharacteristic
240C2×C151C2×C4×SD1611yescharacteristic
241C2×C151#294711yescharacteristic
242C2×C151D4×SD1611yescharacteristic
243C2×C151#328311yescharacteristic
244C2×C151#335911yescharacteristic
245C2×C151#336011yescharacteristic
246#22996C15111yesmaximal, characteristic, direct product factor, Sylow
247C151#2299611yescharacteristic, direct product factor, Sylow
248C2×#461C2×C15111yescharacteristic
249C2×C4×SD16C2×C15112yes—
250#2365C2×C15111yescharacteristic, complemented
251D4×C4⋊C4C2×C15111yescharacteristic
252#1956C2×C15111yescharacteristic, complemented
253#1948C2×C15111yescharacteristic, complemented
254#1933C2×C15112yescomplemented
255#1877C2×C15112yes—
256#1865C2×C15112yescomplemented
257C8×C4⋊C4C2×C15111yescharacteristic, complemented
258C4×#102C2×C15111yescharacteristic
259C22×SD16C22×C15112yes—
260C2×C4⋊Q8C22×C15111yescharacteristic
261C2×C4×Q8C22×C15112yes—
262C2×C4×D4C22×C15112yes—
263C2×C4×D4C22×C15111yescharacteristic
264C22×C4⋊C4—21no—
265#461C4×C15118yescomplemented
266C4×SD16—28no—
267C2×#106C22×C15112yes—
268C2×C4⋊C8C22×C15111yescharacteristic
269C2×#103C22×C15112yes—
270C2×#103C22×C15112yes—
271C2×#103C22×C15112yes—
272C2×#102C22×C15111yescharacteristic
273C2×#102C22×C15111yescharacteristic
274C2×C4×C8C22×C15112yes—
275C2×C4×C8C22×C15111yescharacteristic
276#391C22×C15112yes—
277#389—21no—
278#389C22×C15112yes—
279#387—22no—
280#385—22no—
281#384C22×C15111yescharacteristic
282#380—21no—
283C4×C4⋊C4—21no—
284C4×C4⋊C4C22×C15111yescharacteristic
285C4×C4⋊C4C22×C15111yescharacteristic
286#340C22×C15111yescharacteristic
287#340C22×C15112yes—
288#340C22×C15111yescharacteristic
289#336C22×C15112yes—
290C22×Q8C23×C15112yes—
291C22×D4C23×C15111yescharacteristic
292C23×C4—21no—
293C23×C4—22no—
294C2×SD16—24no—
295C2×SD16—24no—
296C2×SD16C2×C4×C15118yes—
297C22×C8C23×C15111yescharacteristic
298C22×C8C23×C15112yes—
299C22×C8C23×C15111yescharacteristic
300C4⋊Q8—22no—
301C4⋊Q8C2×C4×C15114yescomplemented
302C4×Q8—24no—
303C4×Q8—42no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—42no—
307C4×D4C2×C4×C15114yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4D4×C15111yescharacteristic
316C2×C4⋊C4D4×C15111yescharacteristic
317C2×C4⋊C4C23×C15112yes—
318C2×C4⋊C4C23×C15111yescharacteristic
319C2×C4⋊C4C23×C15111yescharacteristic
320C2×C4⋊C4C23×C15111yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42C23×C15112yes—
326C2×C42C23×C15111yescharacteristic
327#106—24no—
328C4⋊C8C2×C4×C15114yescomplemented
329#103—24no—
330#103—42no—
331#103C2×C4×C15118yescomplemented
332#102—22no—
333#102D4×C15114yes—
334C4×C8—24no—
335C4×C8—22no—
336#95—22no—
337#95—21no—
338#95—22no—
339#95—22no—
340#95D4×C15112yes—
341C24—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8C22×C4×C15114yes—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4C22×C4×C15112yes—
352C2×D4C2×D4×C15112yes—
353C2×D4#41×C15112yes—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4C2×D4×C15112yes—
366C22×C4C2×D4×C15111yescharacteristic
367C22×C4C2×D4×C15112yes—
368C22×C4C2×D4×C15111yescharacteristic
369C22×C4C24×C15111yescharacteristic
370SD16—216no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8C22×C4×C15114yes—
379C2×C8C2×D4×C15112yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—81no—
393C4⋊C4SD16×C15114yescomplemented
394C4⋊C4C22×C4×C15112yes—
395C4⋊C4C2×D4×C15112yes—
396C22⋊C4—24no—
397C22⋊C4—44no—
398C42—22no—
399C42—22no—
400C42—44no—
401C42—42no—
402C42C22×C4×C15112yes—
403C42#41×C15112yes—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23C22×D4×C15111yescharacteristic
412Q8—24no—
413Q8—24no—
414Q8—42no—
415Q8—42no—
416D4—22no—
417D4—22no—
418D4—22no—
419D4C4×D4×C15114yes—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—88no—
444C2×C4C4×D4×C15112yes—
445C2×C4C4×D4×C15112yes—
446C2×C4C2×SD16×C15114yes—
447C2×C4C2×SD16×C15112yes—
448C2×C4#135×C15112yes—
449C2×C4C23×C4×C15111yescharacteristic
450C2×C4C22×D4×C15111yescharacteristic
451C2×C4C2×#41×C15111yescharacteristic
452C2×C4C2×#41×C15111yescharacteristic
453C2×C4H(5,𝔽2)×C15111yescharacteristic
454C2×C42−1+4×C15111yescharacteristic
455C8—24no—
456C8—22no—
457C8—41no—
458C22—24no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22C2×C4×D4×C15111yescharacteristic
466C22D42×C15111yescharacteristic
467C22#548×C15111yescharacteristic
468C22C22×SD16×C15111yescharacteristic
469C22C2×#135×C15111yescharacteristic
470C22#577×C15111yescharacteristic
471C22#578×C15111yescharacteristic
472C4—22no—
473C4—24no—
474C4—44no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4C4×SD16×C15114yescomplemented
481C4C2×C4×D4×C15111yescharacteristic
482C4#520×C15111yescharacteristic
483C4#547×C15111yescharacteristic
484C4#553×C15111yescharacteristic
485C2—44no—
486C2#2361×C15111yescharacteristic
487C2C2×C4×SD16×C15111yescharacteristic
488C2#2947×C15111yescharacteristic
489C2D4×SD16×C15111yescharacteristic
490C2#3283×C15111yescharacteristic
491C2#3359×C15111yescharacteristic
492C2#3360×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutatorC2×C4C23×C4×C151
Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C4⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
3. #2365×C151C2semidirect
5. #1956×C151C2semidirect
6. #1948×C151C2semidirect
7. #1933×C151C2semidirect
9. #1865×C151C2semidirect
10. C8×C4⋊C4×C151C2semidirect
18. #461×C151C4semidirect
54. C4⋊Q8×C151C2×C4semidirect
81. C4⋊C8×C151C2×C4semidirect
84. #103×C151C2×C4semidirect
146. C4⋊C4×C151SD16semidirect
233. C4×C151C4×SD16semidirect
246. #22996C151direct (N × H)
247. C151#22996direct (N × H)
250. #2365C2×C151semidirect
252. #1956C2×C151semidirect
253. #1948C2×C151semidirect
254. #1933C2×C151semidirect
256. #1865C2×C151semidirect
257. C8×C4⋊C4C2×C151semidirect
265. #461C4×C151semidirect
301. C4⋊Q8C2×C4×C151semidirect
328. C4⋊C8C2×C4×C151semidirect
331. #103C2×C4×C151semidirect
393. C4⋊C4SD16×C151semidirect
480. C4C4×SD16×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#461×C1512
C2×C4×SD16×C1512
#2365×C1512
D4×C4⋊C4×C1512
#1956×C1512
#1948×C1512
#1933×C1512
#1877×C1512
#1865×C1512
C8×C4⋊C4×C1512
C4×#102×C1512
#22996151