Finite Group Databasebeta

← C151×#24752 #50329677

Subgroups — C151×#24752 #50329677

Order 38656.

992 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467×C151C212yesmaximal
2C2×#462×C151C211yesmaximal, characteristic, complemented
3C22×C4⋊C8×C151C211yesmaximal, characteristic
4#2857×C151C211yesmaximal, characteristic
5C4×C22⋊Q8×C151C211yesmaximal, characteristic
6#2068×C151C212yesmaximal, complemented
7#2056×C151C212yesmaximal, complemented
8#1981×C151C211yesmaximal, characteristic, complemented
9#1947×C151C211yesmaximal, characteristic, complemented
10#1892×C151C212yesmaximal
11#1803×C151C211yesmaximal, characteristic, complemented
12C22×Q16×C151C2212yes—
13C23×C8×C151C2212yes—
14C2×C4⋊Q8×C151C2211yescharacteristic
15C2×C22⋊Q8×C151C2212yes—
16C2×C22⋊Q8×C151C2211yescharacteristic
17C2×C4×Q8×C151C2211yescharacteristic
18C22×C42×C151C2211yescharacteristic
19#467×C151—28no—
20#462×C151—24no—
21C2×#107×C151C2212yes—
22C2×C4⋊C8×C151—24no—
23C2×C4⋊C8×C151C2211yescharacteristic
24C2×C4⋊C8×C151C2212yes—
25C2×C4⋊C8×C151C2211yescharacteristic
26C2×#103×C151C2212yes—
27C2×#103×C151C2212yes—
28C2×#103×C151C2212yes—
29C2×#103×C151C2212yes—
30#391×C151C2212yes—
31#391×C151C2211yescharacteristic
32#389×C151—21no—
33#389×C151—21no—
34#389×C151C2211yescharacteristic
35#387×C151—21no—
36#385×C151—21no—
37#384×C151C2211yescharacteristic
38#380×C151—21no—
39#380×C151—21no—
40#380×C151C2211yescharacteristic
41C4×C4⋊C4×C151—21no—
42C4×C4⋊C4×C151C2211yescharacteristic
43C4×C22⋊C4×C151—21no—
44#340×C151C2212yes—
45#340×C151C2212yes—
46C22×Q8×C151C2312yes—
47C22×Q8×C151C2311yescharacteristic
48C23×C4×C151D411yescharacteristic
49C23×C4×C151D411yescharacteristic
50C23×C4×C151C2311yescharacteristic
51C2×Q16×C151—24no—
52C2×Q16×C151—24no—
53C2×Q16×C151—42no—
54C22×C8×C151—24no—
55C22×C8×C151—24no—
56C22×C8×C151—42no—
57C22×C8×C151C2312yes—
58C22×C8×C151C2312yes—
59C4⋊Q8×C151—22no—
60C4⋊Q8×C151—22no—
61C22⋊Q8×C151—22no—
62C22⋊Q8×C151—24no—
63C22⋊Q8×C151—24no—
64C22⋊Q8×C151D414yes—
65C4×Q8×C151—22no—
66C4×Q8×C151—22no—
67C2×C4⋊C4×C151—21no—
68C2×C4⋊C4×C151—21no—
69C2×C4⋊C4×C151—21no—
70C2×C4⋊C4×C151—21no—
71C2×C4⋊C4×C151—21no—
72C2×C4⋊C4×C151—21no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—22no—
75C2×C4⋊C4×C151C2311yescharacteristic
76C2×C4⋊C4×C151C2311yescharacteristic
77C2×C4⋊C4×C151C2312yes—
78C2×C4⋊C4×C151C2311yescharacteristic
79C2×C22⋊C4×C151—21no—
80C2×C22⋊C4×C151—21no—
81C2×C22⋊C4×C151—22no—
82C2×C42×C151—22no—
83C2×C42×C151—22no—
84C2×C42×C151—21no—
85C2×C42×C151—21no—
86C2×C42×C151D411yescharacteristic
87C2×C42×C151D411yescharacteristic
88C2×C42×C151C2311yescharacteristic
89C2×C42×C151C2311yescharacteristic
90#107×C151—24no—
91C4⋊C8×C151—24no—
92C4⋊C8×C151—22no—
93C4⋊C8×C151D414yes—
94#103×C151—24no—
95#103×C151—24no—
96#103×C151—24no—
97#103×C151—42no—
98#95×C151—21no—
99#95×C151—21no—
100#95×C151—21no—
101#95×C151—21no—
102#95×C151—21no—
103#95×C151—22no—
104#95×C151—22no—
105C24×C151C2×D411yescharacteristic
106C2×Q8×C151—21no—
107C2×Q8×C151—21no—
108C2×Q8×C151—22no—
109C2×Q8×C151—22no—
110C2×Q8×C151—21no—
111C2×Q8×C151—22no—
112C2×Q8×C151—22no—
113C2×Q8×C151—21no—
114C2×Q8×C151—22no—
115C2×Q8×C151—21no—
116C2×Q8×C151—22no—
117C2×Q8×C151C2×D412yes—
118C22×C4×C151—22no—
119C22×C4×C151—22no—
120C22×C4×C151—21no—
121C22×C4×C151—21no—
122C22×C4×C151—21no—
123C22×C4×C151—21no—
124C22×C4×C151—22no—
125C22×C4×C151—21no—
126C22×C4×C151—21no—
127C22×C4×C151—22no—
128C22×C4×C151—21no—
129C22×C4×C151—21no—
130C22×C4×C151—22no—
131C22×C4×C151—22no—
132C22×C4×C151—41no—
133C22×C4×C151Q1614yescomplemented
134C22×C4×C151C2×D411yescharacteristic
135C22×C4×C151C2×D411yescharacteristic
136C22×C4×C151C2×D411yescharacteristic
137C22×C4×C151C2×D412yes—
138C22×C4×C151C2×D411yescharacteristic
139C22×C4×C151C2×D411yescharacteristic
140C22×C4×C151#4112yes—
141C22×C4×C151C2411yescharacteristic
142Q16×C151—48no—
143C2×C8×C151—22no—
144C2×C8×C151—22no—
145C2×C8×C151—22no—
146C2×C8×C151—22no—
147C2×C8×C151—22no—
148C2×C8×C151—48no—
149C2×C8×C151C2×D414yes—
150C4⋊C4×C151—24no—
151C4⋊C4×C151—21no—
152C4⋊C4×C151—21no—
153C4⋊C4×C151—22no—
154C4⋊C4×C151—21no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—21no—
157C4⋊C4×C151—21no—
158C4⋊C4×C151—42no—
159C4⋊C4×C151—42no—
160C4⋊C4×C151—42no—
161C4⋊C4×C151—44no—
162C4⋊C4×C151—42no—
163C4⋊C4×C151—42no—
164C4⋊C4×C151—42no—
165C4⋊C4×C151C2×D412yes—
166C22⋊C4×C151—24no—
167C22⋊C4×C151—44no—
168C22⋊C4×C151—42no—
169C42×C151—22no—
170C42×C151—22no—
171C42×C151—21no—
172C42×C151—21no—
173C42×C151—42no—
174C42×C151—42no—
175C42×C151C2×D412yes—
176C42×C151#4112yes—
177C23×C151—21no—
178C23×C151—21no—
179C23×C151—21no—
180C23×C151—21no—
181C23×C151#12112yes—
182C23×C151C2×Q1612yes—
183C23×C151#13712yes—
184C23×C151C22×D411yescharacteristic
185Q8×C151—22no—
186Q8×C151—22no—
187Q8×C151—41no—
188Q8×C151—41no—
189Q8×C151—42no—
190Q8×C151—42no—
191Q8×C151—42no—
192Q8×C151—42no—
193C2×C4×C151—21no—
194C2×C4×C151—21no—
195C2×C4×C151—21no—
196C2×C4×C151—21no—
197C2×C4×C151—21no—
198C2×C4×C151—21no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—22no—
203C2×C4×C151—28no—
204C2×C4×C151—22no—
205C2×C4×C151—21no—
206C2×C4×C151—22no—
207C2×C4×C151—21no—
208C2×C4×C151—21no—
209C2×C4×C151—44no—
210C2×C4×C151—41no—
211C2×C4×C151—41no—
212C2×C4×C151—41no—
213C2×C4×C151—41no—
214C2×C4×C151—42no—
215C2×C4×C151—42no—
216C2×C4×C151—41no—
217C2×C4×C151—42no—
218C2×C4×C151—42no—
219C2×C4×C151—41no—
220C2×C4×C151—41no—
221C2×C4×C151—42no—
222C2×C4×C151—41no—
223C2×C4×C151—41no—
224C2×C4×C151—42no—
225C2×C4×C151—41no—
226C2×C4×C151#12112yes—
227C2×C4×C151#12112yes—
228C2×C4×C151#12112yes—
229C2×C4×C151C2×Q1612yes—
230C2×C4×C151C2×Q1612yes—
231C2×C4×C151#13512yes—
232C2×C4×C151C22×D411yescharacteristic
233C2×C4×C151C22×D411yescharacteristic
234C2×C4×C151C2×#4111yescharacteristic
235C2×C4×C151C2×#4111yescharacteristic
236C2×C4×C151H(5,𝔽2)11yescharacteristic
237C2×C4×C1512−1+411yescharacteristic
238C8×C151—24no—
239C8×C151—42no—
240C22×C151—22no—
241C22×C151—22no—
242C22×C151—22no—
243C22×C151—22no—
244C22×C151—22no—
245C22×C151—22no—
246C22×C151#46214yescomplemented
247C22×C151C2×#12111yescharacteristic
248C22×C151C2×#12111yescharacteristic
249C22×C151#53711yescharacteristic
250C22×C151C22×Q1611yescharacteristic
251C22×C151C2×#13511yescharacteristic
252C22×C151C2×#13711yescharacteristic
253C22×C151#57511yescharacteristic
254C4×C151—22no—
255C4×C151—22no—
256C4×C151—22no—
257C4×C151—42no—
258C4×C151—42no—
259C4×C151—41no—
260C4×C151—82no—
261C4×C151—81no—
262C4×C151—82no—
263C4×C151—81no—
264C4×C151#46714yescomplemented
265C4×C151D4211yescharacteristic
266C4×C151#54711yescharacteristic
267C4×C151#54811yescharacteristic
268C4×C151#55311yescharacteristic
269C2×C151—24no—
270C2×C151#241611yescharacteristic
271C2×C151C2×#46211yescharacteristic
272C2×C151#303511yescharacteristic
273C2×C151C2×#46711yescharacteristic
274C2×C151#305611yescharacteristic
275C2×C151#324811yescharacteristic
276C2×C151#324911yescharacteristic
277#24752C15111yesmaximal, characteristic, direct product factor, Sylow
278C151#2475211yescharacteristic, direct product factor, Sylow
279C2×#467C2×C15112yes—
280C2×#462C2×C15111yescharacteristic, complemented
281C22×C4⋊C8C2×C15111yescharacteristic
282#2857C2×C15111yescharacteristic
283C4×C22⋊Q8C2×C15111yescharacteristic
284#2068C2×C15112yescomplemented
285#2056C2×C15112yescomplemented
286#1981C2×C15111yescharacteristic, complemented
287#1947C2×C15111yescharacteristic, complemented
288#1892C2×C15112yes—
289#1803C2×C15111yescharacteristic, complemented
290C22×Q16C22×C15112yes—
291C23×C8C22×C15112yes—
292C2×C4⋊Q8C22×C15111yescharacteristic
293C2×C22⋊Q8C22×C15112yes—
294C2×C22⋊Q8C22×C15111yescharacteristic
295C2×C4×Q8C22×C15111yescharacteristic
296C22×C42C22×C15111yescharacteristic
297#467—28no—
298#462—24no—
299C2×#107C22×C15112yes—
300C2×C4⋊C8—24no—
301C2×C4⋊C8C22×C15112yes—
302C2×C4⋊C8C22×C15111yescharacteristic
303C2×C4⋊C8C22×C15111yescharacteristic
304C2×#103C22×C15112yes—
305C2×#103C22×C15112yes—
306C2×#103C22×C15112yes—
307C2×#103C22×C15112yes—
308#391C22×C15111yescharacteristic
309#391C22×C15112yes—
310#389—21no—
311#389—21no—
312#389C22×C15111yescharacteristic
313#387—21no—
314#385—21no—
315#384C22×C15111yescharacteristic
316#380—21no—
317#380—21no—
318#380C22×C15111yescharacteristic
319C4×C4⋊C4—21no—
320C4×C4⋊C4C22×C15111yescharacteristic
321C4×C22⋊C4—21no—
322#340C22×C15112yes—
323#340C22×C15112yes—
324C22×Q8C23×C15112yes—
325C22×Q8C23×C15111yescharacteristic
326C23×C4D4×C15111yescharacteristic
327C23×C4D4×C15111yescharacteristic
328C23×C4C23×C15111yescharacteristic
329C2×Q16—24no—
330C2×Q16—24no—
331C2×Q16—42no—
332C22×C8—24no—
333C22×C8—24no—
334C22×C8—42no—
335C22×C8C23×C15112yes—
336C22×C8C23×C15112yes—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—24no—
341C22⋊Q8—24no—
342C22⋊Q8D4×C15114yes—
343C4×Q8—22no—
344C4×Q8—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C23×C15111yescharacteristic
354C2×C4⋊C4C23×C15112yes—
355C2×C4⋊C4C23×C15111yescharacteristic
356C2×C4⋊C4C23×C15111yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42D4×C15111yescharacteristic
365C2×C42D4×C15111yescharacteristic
366C2×C42C23×C15111yescharacteristic
367C2×C42C23×C15111yescharacteristic
368#107—24no—
369C4⋊C8—24no—
370C4⋊C8—22no—
371C4⋊C8D4×C15114yes—
372#103—24no—
373#103—24no—
374#103—24no—
375#103—42no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383C24C2×D4×C15111yescharacteristic
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8C2×D4×C15112yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—41no—
411C22×C4Q16×C15114yescomplemented
412C22×C4C2×D4×C15111yescharacteristic
413C22×C4C2×D4×C15111yescharacteristic
414C22×C4C2×D4×C15111yescharacteristic
415C22×C4C2×D4×C15112yes—
416C22×C4C2×D4×C15111yescharacteristic
417C22×C4C2×D4×C15111yescharacteristic
418C22×C4#41×C15112yes—
419C22×C4C24×C15111yescharacteristic
420Q16—48no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—48no—
427C2×C8C2×D4×C15114yes—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4C2×D4×C15112yes—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—21no—
451C42—42no—
452C42—42no—
453C42C2×D4×C15112yes—
454C42#41×C15112yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23#121×C15112yes—
460C23C2×Q16×C15112yes—
461C23#137×C15112yes—
462C23C22×D4×C15111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4#121×C15112yes—
505C2×C4#121×C15112yes—
506C2×C4#121×C15112yes—
507C2×C4C2×Q16×C15112yes—
508C2×C4C2×Q16×C15112yes—
509C2×C4#135×C15112yes—
510C2×C4C22×D4×C15111yescharacteristic
511C2×C4C22×D4×C15111yescharacteristic
512C2×C4C2×#41×C15111yescharacteristic
513C2×C4C2×#41×C15111yescharacteristic
514C2×C4H(5,𝔽2)×C15111yescharacteristic
515C2×C42−1+4×C15111yescharacteristic
516C8—24no—
517C8—42no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22#462×C15114yescomplemented
525C22C2×#121×C15111yescharacteristic
526C22C2×#121×C15111yescharacteristic
527C22#537×C15111yescharacteristic
528C22C22×Q16×C15111yescharacteristic
529C22C2×#135×C15111yescharacteristic
530C22C2×#137×C15111yescharacteristic
531C22#575×C15111yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—41no—
538C4—82no—
539C4—81no—
540C4—82no—
541C4—81no—
542C4#467×C15114yescomplemented
543C4D42×C15111yescharacteristic
544C4#547×C15111yescharacteristic
545C4#548×C15111yescharacteristic
546C4#553×C15111yescharacteristic
547C2—24no—
548C2#2416×C15111yescharacteristic
549C2C2×#462×C15111yescharacteristic
550C2#3035×C15111yescharacteristic
551C2C2×#467×C15111yescharacteristic
552C2#3056×C15111yescharacteristic
553C2#3248×C15111yescharacteristic
554C2#3249×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC24×C151C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C22×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#462×C151C2semidirect
6. #2068×C151C2semidirect
7. #2056×C151C2semidirect
8. #1981×C151C2semidirect
9. #1947×C151C2semidirect
11. #1803×C151C2semidirect
133. C22×C4×C151Q16semidirect
246. C22×C151#462semidirect
264. C4×C151#467semidirect
277. #24752C151direct (N × H)
278. C151#24752direct (N × H)
280. C2×#462C2×C151semidirect
284. #2068C2×C151semidirect
285. #2056C2×C151semidirect
286. #1981C2×C151semidirect
287. #1947C2×C151semidirect
289. #1803C2×C151semidirect
411. C22×C4Q16×C151semidirect
524. C22#462×C151semidirect
542. C4#467×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#467×C1512
C2×#462×C1512
C22×C4⋊C8×C1512
#2857×C1512
C4×C22⋊Q8×C1512
#2068×C1512
#2056×C1512
#1981×C1512
#1947×C1512
#1892×C1512
#1803×C1512
#24752151