Finite Group Databasebeta

← C151×#24123 #50329048

Subgroups — C151×#24123 #50329048

Order 38656.

1090 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C151C211yesmaximal, characteristic
2C2×#475×C151C211yesmaximal, characteristic, complemented
3C2×C4×Q16×C151C211yesmaximal, characteristic, complemented
4#2538×C151C211yesmaximal, characteristic
5#2035×C151C212yesmaximal, complemented
6#2001×C151C212yesmaximal, complemented
7#1990×C151C211yesmaximal, characteristic, complemented
8#1954×C151C211yesmaximal, characteristic
9#1873×C151C212yesmaximal, complemented
10#1854×C151C212yesmaximal, complemented
11#1803×C151C211yesmaximal, characteristic, complemented
12C22×Q16×C151C2211yescharacteristic
13#556×C151—24no—
14#556×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#126×C151—22no—
17C2×#124×C151—21no—
18C2×C22⋊Q8×C151C2212yes—
19C2×C22⋊Q8×C151C2212yes—
20C2×C4×Q8×C151—21no—
21C2×C4×Q8×C151C2211yescharacteristic
22C2×C4×Q8×C151C2211yescharacteristic
23C2×C4×D4×C151C2211yescharacteristic
24C2×#117×C151C2212yes—
25#475×C151—24no—
26C4×Q16×C151—24no—
27C2×#107×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×C4×C8×C151C2211yescharacteristic
37#396×C151—22no—
38#390×C151—21no—
39#389×C151—21no—
40#384×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×C151C2312yes—
50C2×Q16×C151—22no—
51C2×Q16×C151—22no—
52C2×Q16×C151—22no—
53C22×C8×C151C2311yescharacteristic
54C22×C8×C151C2312yes—
55C22×C8×C151C2311yescharacteristic
56C4⋊Q8×C151—22no—
57C4⋊Q8×C151D414yes—
58#126×C151—216no—
59#124×C151—28no—
60C22⋊Q8×C151—24no—
61C22⋊Q8×C151—24no—
62C22⋊Q8×C151—24no—
63C22⋊Q8×C151—42no—
64C4×Q8×C151—28no—
65C4×Q8×C151—22no—
66C4×Q8×C151—22no—
67C4×Q8×C151—22no—
68C4×Q8×C151D414yes—
69C4×D4×C151—22no—
70C4×D4×C151—22no—
71#117×C151—24no—
72#117×C151—24no—
73C2×C4⋊C4×C151—22no—
74C2×C4⋊C4×C151—21no—
75C2×C4⋊C4×C151—21no—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—22no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151D411yescharacteristic
81C2×C4⋊C4×C151C2312yes—
82C2×C4⋊C4×C151C2311yescharacteristic
83C2×C4⋊C4×C151C2312yes—
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C22⋊C4×C151—22no—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151D412yes—
90C2×C42×C151—22no—
91C2×C42×C151—21no—
92C2×C42×C151—21no—
93C2×C42×C151C2311yescharacteristic
94#107×C151—22no—
95#106×C151—24no—
96C4⋊C8×C151—22no—
97#103×C151—22no—
98#103×C151—24no—
99#103×C151—22no—
100#103×C151—42no—
101C22⋊C8×C151—24no—
102C22⋊C8×C151—42no—
103C4×C8×C151—22no—
104#95×C151—22no—
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—42no—
118C2×Q8×C151—42no—
119C2×Q8×C151C2×D412yes—
120C2×Q8×C151#4112yes—
121C2×D4×C151—22no—
122C2×D4×C151—22no—
123C2×D4×C151—22no—
124C22×C4×C151—21no—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
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—22no—
136C22×C4×C151C2×D412yes—
137C22×C4×C151C2×D411yescharacteristic
138C22×C4×C151C2×D411yescharacteristic
139C22×C4×C151C2411yescharacteristic
140Q16×C151—28no—
141C2×C8×C151—21no—
142C2×C8×C151—21no—
143C2×C8×C151—21no—
144C2×C8×C151—22no—
145C2×C8×C151—22no—
146C2×C8×C151—21no—
147C2×C8×C151—22no—
148C2×C8×C151—21no—
149C2×C8×C151#4112yes—
150C4⋊C4×C151—28no—
151C4⋊C4×C151—24no—
152C4⋊C4×C151—24no—
153C4⋊C4×C151—22no—
154C4⋊C4×C151—21no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—21no—
157C4⋊C4×C151—21no—
158C4⋊C4×C151—22no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—44no—
161C4⋊C4×C151—42no—
162C4⋊C4×C151—42no—
163C4⋊C4×C151—81no—
164C4⋊C4×C151C2×D414yes—
165C4⋊C4×C151C2×D412yes—
166C22⋊C4×C151—28no—
167C22⋊C4×C151—28no—
168C22⋊C4×C151—24no—
169C22⋊C4×C151—44no—
170C22⋊C4×C151—82no—
171C42×C151—28no—
172C42×C151—24no—
173C42×C151—42no—
174C42×C151C2×D412yes—
175C42×C151C2×D412yes—
176C23×C151—22no—
177C23×C151—22no—
178C23×C151—22no—
179C23×C151—22no—
180C23×C151—22no—
181C23×C151—22no—
182C23×C151—22no—
183C23×C151C22×D411yescharacteristic
184Q8×C151—22no—
185Q8×C151—22no—
186Q8×C151—22no—
187Q8×C151—22no—
188Q8×C151—41no—
189Q8×C151—41no—
190Q8×C151—41no—
191Q8×C151—41no—
192Q8×C151—48no—
193D4×C151—44no—
194C2×C4×C151—21no—
195C2×C4×C151—21no—
196C2×C4×C151—24no—
197C2×C4×C151—22no—
198C2×C4×C151—22no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—22no—
203C2×C4×C151—41no—
204C2×C4×C151—41no—
205C2×C4×C151—41no—
206C2×C4×C151—41no—
207C2×C4×C151—42no—
208C2×C4×C151—42no—
209C2×C4×C151—41no—
210C2×C4×C151—41no—
211C2×C4×C151—42no—
212C2×C4×C151—42no—
213C2×C4×C151—42no—
214C2×C4×C151—42no—
215C2×C4×C151—41no—
216C2×C4×C151—42no—
217C2×C4×C151—42no—
218C2×C4×C151—41no—
219C2×C4×C151—42no—
220C2×C4×C151—41no—
221C2×C4×C151#12012yes—
222C2×C4×C151#12012yes—
223C2×C4×C151#13512yes—
224C2×C4×C151#13712yes—
225C2×C4×C151C22×D411yescharacteristic
226C2×C4×C151C22×D411yescharacteristic
227C2×C4×C151C2×#4111yescharacteristic
228C2×C4×C151C2×#4111yescharacteristic
229C2×C4×C151H(5,𝔽2)11yescharacteristic
230C2×C4×C1512−1+411yescharacteristic
231C8×C151—22no—
232C8×C151—41no—
233C8×C151—42no—
234C22×C151—42no—
235C22×C151—42no—
236C22×C151—42no—
237C22×C151—42no—
238C22×C151—42no—
239C22×C151—42no—
240C22×C151—42no—
241C22×C151C2×#12011yescharacteristic
242C22×C151#54611yescharacteristic
243C22×C151#54711yescharacteristic
244C22×C151C2×#13511yescharacteristic
245C22×C151C2×#13711yescharacteristic
246C22×C151#57711yescharacteristic
247C22×C151#57811yescharacteristic
248C4×C151—22no—
249C4×C151—42no—
250C4×C151—42no—
251C4×C151—42no—
252C4×C151—44no—
253C4×C151—42no—
254C4×C151—81no—
255C4×C151—82no—
256C4×C151#45214yes—
257C4×C151#52511yescharacteristic
258C4×C151#53711yescharacteristic
259C4×C151#54811yescharacteristic
260C4×C151#55011yescharacteristic
261C2×C151—82no—
262C2×C151#240711yescharacteristic
263C2×C151C2×#45211yescharacteristic
264C2×C151#301311yescharacteristic
265C2×C151#330411yescharacteristic
266C2×C151#332811yescharacteristic
267C2×C151#334211yescharacteristic
268C2×C151#335611yescharacteristic
269#24123C15111yesmaximal, characteristic, direct product factor, Sylow
270C151#2412311yescharacteristic, direct product factor, Sylow
271C2×#556C2×C15111yescharacteristic
272C2×#475C2×C15111yescharacteristic, complemented
273C2×C4×Q16C2×C15111yescharacteristic, complemented
274#2538C2×C15111yescharacteristic
275#2035C2×C15112yescomplemented
276#2001C2×C15112yescomplemented
277#1990C2×C15111yescharacteristic, complemented
278#1954C2×C15111yescharacteristic
279#1873C2×C15112yescomplemented
280#1854C2×C15112yescomplemented
281#1803C2×C15111yescharacteristic, complemented
282C22×Q16C22×C15111yescharacteristic
283#556—24no—
284#556—24no—
285C2×C4⋊Q8C22×C15111yescharacteristic
286C2×#126—22no—
287C2×#124—21no—
288C2×C22⋊Q8C22×C15112yes—
289C2×C22⋊Q8C22×C15112yes—
290C2×C4×Q8—21no—
291C2×C4×Q8C22×C15111yescharacteristic
292C2×C4×Q8C22×C15111yescharacteristic
293C2×C4×D4C22×C15111yescharacteristic
294C2×#117C22×C15112yes—
295#475—24no—
296C4×Q16—24no—
297C2×#107C22×C15111yescharacteristic
298C2×#106C22×C15112yes—
299C2×C4⋊C8C22×C15111yescharacteristic
300C2×#103C22×C15112yes—
301C2×#103C22×C15111yescharacteristic
302C2×#103C22×C15111yescharacteristic
303C2×#103C22×C15112yes—
304C2×C22⋊C8C22×C15112yes—
305C2×C22⋊C8C22×C15112yes—
306C2×C4×C8C22×C15111yescharacteristic
307#396—22no—
308#390—21no—
309#389—21no—
310#384C22×C15111yescharacteristic
311#380C22×C15112yes—
312#340C22×C15112yes—
313#340C22×C15112yes—
314#340C22×C15112yes—
315C22×Q8—21no—
316C22×Q8C23×C15111yescharacteristic
317C22×Q8C23×C15111yescharacteristic
318C22×D4D4×C15111yescharacteristic
319C23×C4C23×C15112yes—
320C2×Q16—22no—
321C2×Q16—22no—
322C2×Q16—22no—
323C22×C8C23×C15111yescharacteristic
324C22×C8C23×C15112yes—
325C22×C8C23×C15111yescharacteristic
326C4⋊Q8—22no—
327C4⋊Q8D4×C15114yes—
328#126—216no—
329#124—28no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333C22⋊Q8—42no—
334C4×Q8—28no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8D4×C15114yes—
339C4×D4—22no—
340C4×D4—22no—
341#117—24no—
342#117—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4D4×C15111yescharacteristic
351C2×C4⋊C4C23×C15111yescharacteristic
352C2×C4⋊C4C23×C15112yes—
353C2×C4⋊C4C23×C15111yescharacteristic
354C2×C4⋊C4C23×C15112yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4D4×C15112yes—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42C23×C15111yescharacteristic
364#107—22no—
365#106—24no—
366C4⋊C8—22no—
367#103—22no—
368#103—22no—
369#103—24no—
370#103—42no—
371C22⋊C8—24no—
372C22⋊C8—42no—
373C4×C8—22no—
374#95—22no—
375C24C2×D4×C15112yes—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8C2×D4×C15112yes—
390C2×Q8#41×C15112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4C2×D4×C15112yes—
407C22×C4C2×D4×C15111yescharacteristic
408C22×C4C2×D4×C15111yescharacteristic
409C22×C4C24×C15111yescharacteristic
410Q16—28no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8#41×C15112yes—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4C2×D4×C15114yes—
435C4⋊C4C2×D4×C15112yes—
436C22⋊C4—28no—
437C22⋊C4—28no—
438C22⋊C4—24no—
439C22⋊C4—44no—
440C22⋊C4—82no—
441C42—28no—
442C42—24no—
443C42—42no—
444C42C2×D4×C15112yes—
445C42C2×D4×C15112yes—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23C22×D4×C15111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—22no—
457Q8—22no—
458Q8—48no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463D4—44no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4#120×C15112yes—
492C2×C4#120×C15112yes—
493C2×C4#135×C15112yes—
494C2×C4#137×C15112yes—
495C2×C4C22×D4×C15111yescharacteristic
496C2×C4C22×D4×C15111yescharacteristic
497C2×C4C2×#41×C15111yescharacteristic
498C2×C4C2×#41×C15111yescharacteristic
499C2×C4H(5,𝔽2)×C15111yescharacteristic
500C2×C42−1+4×C15111yescharacteristic
501C8—22no—
502C8—41no—
503C8—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22C2×#120×C15111yescharacteristic
512C22#546×C15111yescharacteristic
513C22#547×C15111yescharacteristic
514C22C2×#135×C15111yescharacteristic
515C22C2×#137×C15111yescharacteristic
516C22#577×C15111yescharacteristic
517C22#578×C15111yescharacteristic
518C4—22no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—81no—
525C4—82no—
526C4#452×C15114yes—
527C4#525×C15111yescharacteristic
528C4#537×C15111yescharacteristic
529C4#548×C15111yescharacteristic
530C4#550×C15111yescharacteristic
531C2—82no—
532C2#2407×C15111yescharacteristic
533C2C2×#452×C15111yescharacteristic
534C2#3013×C15111yescharacteristic
535C2#3304×C15111yescharacteristic
536C2#3328×C15111yescharacteristic
537C2#3342×C15111yescharacteristic
538C2#3356×C15111yescharacteristic
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×C4×Q16×C151C2semidirect
5. #2035×C151C2semidirect
6. #2001×C151C2semidirect
7. #1990×C151C2semidirect
9. #1873×C151C2semidirect
10. #1854×C151C2semidirect
11. #1803×C151C2semidirect
269. #24123C151direct (N × H)
270. C151#24123direct (N × H)
272. C2×#475C2×C151semidirect
273. C2×C4×Q16C2×C151semidirect
275. #2035C2×C151semidirect
276. #2001C2×C151semidirect
277. #1990C2×C151semidirect
279. #1873C2×C151semidirect
280. #1854C2×C151semidirect
281. #1803C2×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×#475×C1512
C2×C4×Q16×C1512
#2538×C1512
#2035×C1512
#2001×C1512
#1990×C1512
#1954×C1512
#1873×C1512
#1854×C1512
#1803×C1512
#24123151