Finite Group Databasebeta

← C151×#24189 #50329114

Subgroups — C151×#24189 #50329114

Order 38656.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C151C211yesmaximal, characteristic
2C2×#462×C151C211yesmaximal, characteristic, complemented
3C2×C4×SD16×C151C211yesmaximal, characteristic
4#2582×C151C211yesmaximal, characteristic, complemented
5#2042×C151C212yesmaximal
6#2015×C151C212yesmaximal
7#1984×C151C211yesmaximal, characteristic, complemented
8#1956×C151C211yesmaximal, characteristic, complemented
9#1878×C151C212yesmaximal, complemented
10#1865×C151C212yesmaximal, complemented
11#1802×C151C211yesmaximal, characteristic
12C22×Q16×C151C2212yes—
13C22×SD16×C151C2211yescharacteristic
14#556×C151—24no—
15#556×C151—24no—
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×#126×C151—22no—
18C2×#124×C151C2212yes—
19C2×C22⋊Q8×C151—21no—
20C2×C4×Q8×C151C2212yes—
21C2×C4×Q8×C151C2211yescharacteristic
22C2×C4×Q8×C151C2211yescharacteristic
23C2×C4×D4×C151C2211yescharacteristic
24C2×#117×C151—21no—
25#462×C151—24no—
26C4×SD16×C151—24no—
27C2×#106×C151C2211yescharacteristic
28C2×C4⋊C8×C151C2211yescharacteristic
29C2×#103×C151C2212yes—
30C2×#103×C151C2212yes—
31C2×#103×C151C2211yescharacteristic
32C2×#103×C151C2212yes—
33C2×#102×C151C2211yescharacteristic
34C2×#102×C151C2212yes—
35C2×C4×C8×C151C2211yescharacteristic
36#398×C151—22no—
37#391×C151—21no—
38#391×C151C2212yes—
39#389×C151—21no—
40#389×C151C2212yes—
41#384×C151C2211yescharacteristic
42#340×C151C2212yes—
43#336×C151C2212yes—
44#336×C151C2212yes—
45C22×Q8×C151D411yescharacteristic
46C22×Q8×C151C2312yes—
47C22×Q8×C151C2311yescharacteristic
48C22×D4×C151C2311yescharacteristic
49C23×C4×C151—21no—
50C2×Q16×C151—24no—
51C2×Q16×C151—24no—
52C2×Q16×C151—42no—
53C2×SD16×C151—22no—
54C2×SD16×C151—22no—
55C2×SD16×C151—22no—
56C22×C8×C151C2312yes—
57C22×C8×C151C2311yescharacteristic
58C22×C8×C151C2311yescharacteristic
59C4⋊Q8×C151—22no—
60C4⋊Q8×C151D414yes—
61#126×C151—216no—
62#124×C151—24no—
63#124×C151—24no—
64C22⋊Q8×C151—28no—
65C4×Q8×C151—24no—
66C4×Q8×C151—22no—
67C4×Q8×C151—22no—
68C4×Q8×C151—22no—
69C4×Q8×C151—22no—
70C4×Q8×C151—24no—
71C4×D4×C151—22no—
72C4×D4×C151D414yes—
73#117×C151—28no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—22no—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151—22no—
81C2×C4⋊C4×C151D411yescharacteristic
82C2×C4⋊C4×C151D412yes—
83C2×C4⋊C4×C151C2312yes—
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2311yescharacteristic
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—21no—
88C2×C22⋊C4×C151—22no—
89C2×C42×C151—22no—
90C2×C42×C151—21no—
91C2×C42×C151C2312yes—
92C2×C42×C151C2311yescharacteristic
93#106×C151—22no—
94C4⋊C8×C151—22no—
95#103×C151—24no—
96#103×C151—22no—
97#103×C151—24no—
98#103×C151—42no—
99#102×C151—22no—
100#102×C151—42no—
101C4×C8×C151—22no—
102#95×C151—22no—
103#95×C151—22no—
104#95×C151—22no—
105C24×C151—21no—
106C2×Q8×C151—22no—
107C2×Q8×C151—22no—
108C2×Q8×C151—22no—
109C2×Q8×C151—22no—
110C2×Q8×C151—21no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—22no—
114C2×Q8×C151—21no—
115C2×Q8×C151—22no—
116C2×Q8×C151—22no—
117C2×Q8×C151—21no—
118C2×Q8×C151—22no—
119C2×Q8×C151C2×D414yes—
120C2×Q8×C151#4112yes—
121C2×D4×C151—21no—
122C2×D4×C151—21no—
123C2×D4×C151—21no—
124C2×D4×C151—21no—
125C2×D4×C151—21no—
126C2×D4×C151C2×D412yes—
127C22×C4×C151—21no—
128C22×C4×C151—21no—
129C22×C4×C151—22no—
130C22×C4×C151—24no—
131C22×C4×C151—22no—
132C22×C4×C151—21no—
133C22×C4×C151—42no—
134C22×C4×C151—42no—
135C22×C4×C151C2×D412yes—
136C22×C4×C151C2×D412yes—
137C22×C4×C151C2×D411yescharacteristic
138C22×C4×C151C2×D411yescharacteristic
139C22×C4×C151C2411yescharacteristic
140Q16×C151—48no—
141SD16×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—22no—
148C2×C8×C151—21no—
149C2×C8×C151—21no—
150C2×C8×C151#4112yes—
151C4⋊C4×C151—28no—
152C4⋊C4×C151—24no—
153C4⋊C4×C151—24no—
154C4⋊C4×C151—22no—
155C4⋊C4×C151—24no—
156C4⋊C4×C151—21no—
157C4⋊C4×C151—21no—
158C4⋊C4×C151—21no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—42no—
162C4⋊C4×C151—42no—
163C4⋊C4×C151—81no—
164C4⋊C4×C151—82no—
165C4⋊C4×C151C2×D412yes—
166C22⋊C4×C151—28no—
167C22⋊C4×C151—28no—
168C22⋊C4×C151—24no—
169C42×C151—28no—
170C42×C151—22no—
171C42×C151—22no—
172C42×C151—42no—
173C42×C151C2×D412yes—
174C42×C151C2×D412yes—
175C23×C151—22no—
176C23×C151—41no—
177C23×C151—41no—
178C23×C151—41no—
179C23×C151—41no—
180C23×C151—41no—
181C23×C151—41no—
182C23×C151C22×D411yescharacteristic
183Q8×C151—22no—
184Q8×C151—24no—
185Q8×C151—22no—
186Q8×C151—24no—
187Q8×C151—42no—
188Q8×C151—42no—
189Q8×C151—41no—
190Q8×C151—41no—
191Q8×C151—44no—
192D4×C151—22no—
193D4×C151—22no—
194D4×C151—41no—
195D4×C151—41no—
196C2×C4×C151—21no—
197C2×C4×C151—21no—
198C2×C4×C151—22no—
199C2×C4×C151—24no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—22no—
203C2×C4×C151—22no—
204C2×C4×C151—22no—
205C2×C4×C151—22no—
206C2×C4×C151—22no—
207C2×C4×C151—41no—
208C2×C4×C151—42no—
209C2×C4×C151—42no—
210C2×C4×C151—41no—
211C2×C4×C151—41no—
212C2×C4×C151—41no—
213C2×C4×C151—48no—
214C2×C4×C151—41no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—42no—
218C2×C4×C151—41no—
219C2×C4×C151—41no—
220C2×C4×C151#12012yes—
221C2×C4×C151#12012yes—
222C2×C4×C151#13512yes—
223C2×C4×C151#13712yes—
224C2×C4×C151C22×D411yescharacteristic
225C2×C4×C151C22×D411yescharacteristic
226C2×C4×C151C2×#4111yescharacteristic
227C2×C4×C151C2×#4111yescharacteristic
228C2×C4×C151H(5,𝔽2)11yescharacteristic
229C2×C4×C1512−1+411yescharacteristic
230C8×C151—22no—
231C8×C151—41no—
232C8×C151—42no—
233C22×C151—42no—
234C22×C151—42no—
235C22×C151—42no—
236C22×C151—81no—
237C22×C151—81no—
238C22×C151—81no—
239C22×C151—81no—
240C22×C151C2×#12011yescharacteristic
241C22×C151#54811yescharacteristic
242C22×C151#55011yescharacteristic
243C22×C151C2×#13511yescharacteristic
244C22×C151C2×#13711yescharacteristic
245C22×C151#57711yescharacteristic
246C22×C151#57811yescharacteristic
247C4×C151—22no—
248C4×C151—44no—
249C4×C151—42no—
250C4×C151—42no—
251C4×C151—42no—
252C4×C151—42no—
253C4×C151—82no—
254C4×C151—81no—
255C4×C151#45214yes—
256C4×C151#52511yescharacteristic
257C4×C151#53711yescharacteristic
258C4×C151#54611yescharacteristic
259C4×C151#54711yescharacteristic
260C2×C151—82no—
261C2×C151#240711yescharacteristic
262C2×C151C2×#45211yescharacteristic
263C2×C151#301311yescharacteristic
264C2×C151#336411yescharacteristic
265C2×C151#336911yescharacteristic
266C2×C151#339411yescharacteristic
267C2×C151#339811yescharacteristic
268#24189C15111yesmaximal, characteristic, direct product factor, Sylow
269C151#2418911yescharacteristic, direct product factor, Sylow
270C2×#556C2×C15111yescharacteristic
271C2×#462C2×C15111yescharacteristic, complemented
272C2×C4×SD16C2×C15111yescharacteristic
273#2582C2×C15111yescharacteristic, complemented
274#2042C2×C15112yes—
275#2015C2×C15112yes—
276#1984C2×C15111yescharacteristic, complemented
277#1956C2×C15111yescharacteristic, complemented
278#1878C2×C15112yescomplemented
279#1865C2×C15112yescomplemented
280#1802C2×C15111yescharacteristic
281C22×Q16C22×C15112yes—
282C22×SD16C22×C15111yescharacteristic
283#556—24no—
284#556—24no—
285C2×C4⋊Q8C22×C15111yescharacteristic
286C2×#126—22no—
287C2×#124C22×C15112yes—
288C2×C22⋊Q8—21no—
289C2×C4×Q8C22×C15111yescharacteristic
290C2×C4×Q8C22×C15112yes—
291C2×C4×Q8C22×C15111yescharacteristic
292C2×C4×D4C22×C15111yescharacteristic
293C2×#117—21no—
294#462—24no—
295C4×SD16—24no—
296C2×#106C22×C15111yescharacteristic
297C2×C4⋊C8C22×C15111yescharacteristic
298C2×#103C22×C15112yes—
299C2×#103C22×C15112yes—
300C2×#103C22×C15111yescharacteristic
301C2×#103C22×C15112yes—
302C2×#102C22×C15111yescharacteristic
303C2×#102C22×C15112yes—
304C2×C4×C8C22×C15111yescharacteristic
305#398—22no—
306#391—21no—
307#391C22×C15112yes—
308#389—21no—
309#389C22×C15112yes—
310#384C22×C15111yescharacteristic
311#340C22×C15112yes—
312#336C22×C15112yes—
313#336C22×C15112yes—
314C22×Q8D4×C15111yescharacteristic
315C22×Q8C23×C15111yescharacteristic
316C22×Q8C23×C15112yes—
317C22×D4C23×C15111yescharacteristic
318C23×C4—21no—
319C2×Q16—24no—
320C2×Q16—24no—
321C2×Q16—42no—
322C2×SD16—22no—
323C2×SD16—22no—
324C2×SD16—22no—
325C22×C8C23×C15112yes—
326C22×C8C23×C15111yescharacteristic
327C22×C8C23×C15111yescharacteristic
328C4⋊Q8—22no—
329C4⋊Q8D4×C15114yes—
330#126—216no—
331#124—24no—
332#124—24no—
333C22⋊Q8—28no—
334C4×Q8—24no—
335C4×Q8—24no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×D4—22no—
341C4×D4D4×C15114yes—
342#117—28no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4D4×C15111yescharacteristic
351C2×C4⋊C4D4×C15112yes—
352C2×C4⋊C4C23×C15111yescharacteristic
353C2×C4⋊C4C23×C15112yes—
354C2×C4⋊C4C23×C15111yescharacteristic
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42C23×C15112yes—
361C2×C42C23×C15111yescharacteristic
362#106—22no—
363C4⋊C8—22no—
364#103—24no—
365#103—24no—
366#103—22no—
367#103—42no—
368#102—22no—
369#102—42no—
370C4×C8—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374C24—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8C2×D4×C15114yes—
389C2×Q8#41×C15112yes—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C2×D4×C15112yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C2×D4×C15112yes—
405C22×C4C2×D4×C15112yes—
406C22×C4C2×D4×C15111yescharacteristic
407C22×C4C2×D4×C15111yescharacteristic
408C22×C4C24×C15111yescharacteristic
409Q16—48no—
410SD16—28no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8#41×C15112yes—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—81no—
433C4⋊C4—82no—
434C4⋊C4C2×D4×C15112yes—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C42—28no—
439C42—22no—
440C42—22no—
441C42—42no—
442C42C2×D4×C15112yes—
443C42C2×D4×C15112yes—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C22×D4×C15111yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—24no—
455Q8—24no—
456Q8—44no—
457Q8—42no—
458Q8—42no—
459Q8—41no—
460Q8—41no—
461D4—22no—
462D4—22no—
463D4—41no—
464D4—41no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—48no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4#120×C15112yes—
490C2×C4#120×C15112yes—
491C2×C4#135×C15112yes—
492C2×C4#137×C15112yes—
493C2×C4C22×D4×C15111yescharacteristic
494C2×C4C22×D4×C15111yescharacteristic
495C2×C4C2×#41×C15111yescharacteristic
496C2×C4C2×#41×C15111yescharacteristic
497C2×C4H(5,𝔽2)×C15111yescharacteristic
498C2×C42−1+4×C15111yescharacteristic
499C8—22no—
500C8—41no—
501C8—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22C2×#120×C15111yescharacteristic
510C22#548×C15111yescharacteristic
511C22#550×C15111yescharacteristic
512C22C2×#135×C15111yescharacteristic
513C22C2×#137×C15111yescharacteristic
514C22#577×C15111yescharacteristic
515C22#578×C15111yescharacteristic
516C4—22no—
517C4—44no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—81no—
524C4#452×C15114yes—
525C4#525×C15111yescharacteristic
526C4#537×C15111yescharacteristic
527C4#546×C15111yescharacteristic
528C4#547×C15111yescharacteristic
529C2—82no—
530C2#2407×C15111yescharacteristic
531C2C2×#452×C15111yescharacteristic
532C2#3013×C15111yescharacteristic
533C2#3364×C15111yescharacteristic
534C2#3369×C15111yescharacteristic
535C2#3394×C15111yescharacteristic
536C2#3398×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×Q8×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#462×C151C2semidirect
4. #2582×C151C2semidirect
7. #1984×C151C2semidirect
8. #1956×C151C2semidirect
9. #1878×C151C2semidirect
10. #1865×C151C2semidirect
268. #24189C151direct (N × H)
269. C151#24189direct (N × H)
271. C2×#462C2×C151semidirect
273. #2582C2×C151semidirect
276. #1984C2×C151semidirect
277. #1956C2×C151semidirect
278. #1878C2×C151semidirect
279. #1865C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#556×C1512
C2×#462×C1512
C2×C4×SD16×C1512
#2582×C1512
#2042×C1512
#2015×C1512
#1984×C1512
#1956×C1512
#1878×C1512
#1865×C1512
#1802×C1512
#24189151