Finite Group Databasebeta

← C151×#18873 #50323798

Subgroups — C151×#18873 #50323798

Order 38656.

900 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503×C151C211yesmaximal, characteristic
2#3249×C151C211yesmaximal, characteristic, complemented
3#3209×C151C211yesmaximal, characteristic, complemented
4#3076×C151C211yesmaximal, characteristic
5#1645×C151C212yesmaximal, complemented
6#1644×C151C212yesmaximal, complemented
7#1633×C151C212yesmaximal, complemented
8#1632×C151C212yesmaximal, complemented
9C2×#333×C151C211yesmaximal, characteristic, complemented
10#1481×C151C211yesmaximal, characteristic, complemented
11#1477×C151C211yesmaximal, characteristic
12Q82×C151C2212yes—
13#557×C151—21no—
14#554×C151—22no—
15#554×C151C2212yes—
16D4×Q8×C151—22no—
17D4×Q8×C151—21no—
18#544×C151—22no—
19#544×C151—21no—
20#533×C151—21no—
21#533×C151C2211yescharacteristic
22#533×C151C2211yescharacteristic
23C2×C4⋊Q8×C151—21no—
24C2×C4⋊Q8×C151C2211yescharacteristic
25#520×C151—21no—
26C4×#41×C151C2211yescharacteristic
27#494×C151C2212yes—
28#487×C151C2212yes—
29#479×C151—21no—
30#478×C151—21no—
31#477×C151—21no—
32#477×C151C2212yes—
33#477×C151C2211yescharacteristic
34#476×C151C2211yescharacteristic
35#475×C151C2211yescharacteristic
36#474×C151C2211yescharacteristic
37#467×C151—22no—
38#462×C151—22no—
39#462×C151—21no—
40#462×C151C2212yes—
41#451×C151—22no—
42#431×C151C2211yescharacteristic
43#427×C151—21no—
44C2×C4⋊C8×C151C2211yescharacteristic
45C2×C4⋊C8×C151C2211yescharacteristic
46#416×C151—21no—
47#333×C151C2212yes—
48#333×C151C2212yes—
49Q8⋊C8×C151C2212yes—
50Q8⋊C8×C151C2212yes—
51Q8⋊C8×C151C2212yes—
52D4⋊C8×C151C2212yes—
53C2×#41×C151D411yescharacteristic
54C22×Q8×C151—22no—
55C22×Q8×C151D412yes—
56C2×Q16×C151—24no—
57C2×Q16×C151—22no—
58C22×C8×C151—21no—
59C22×C8×C151—21no—
60C4⋊Q8×C151—22no—
61C4⋊Q8×C151—22no—
62C4⋊Q8×C151—22no—
63C4⋊Q8×C151—22no—
64C4⋊Q8×C151—21no—
65C4⋊Q8×C151—21no—
66C4⋊Q8×C151—24no—
67C4⋊Q8×C151D411yescharacteristic
68C4⋊Q8×C151D412yes—
69C4⋊Q8×C151D411yescharacteristic
70C4⋊Q8×C151C2311yescharacteristic
71C4⋊Q8×C151C2311yescharacteristic
72C4⋊Q8×C151C2312yes—
73#125×C151—22no—
74#125×C151—24no—
75#125×C151—22no—
76#125×C151—21no—
77#125×C151—21no—
78C22⋊Q8×C151—22no—
79C22⋊Q8×C151—24no—
80C22⋊Q8×C151—24no—
81C22⋊Q8×C151—21no—
82C22⋊Q8×C151—21no—
83C22⋊Q8×C151—21no—
84C22⋊Q8×C151—22no—
85C22⋊Q8×C151—24no—
86C22⋊Q8×C151D412yes—
87C4×Q8×C151—22no—
88C4×Q8×C151—21no—
89C4×Q8×C151—21no—
90C4×Q8×C151—21no—
91C4×Q8×C151—22no—
92C4×Q8×C151—22no—
93C4×Q8×C151D411yescharacteristic
94C4×Q8×C151C2312yes—
95C4×Q8×C151C2311yescharacteristic
96C4×D4×C151—24no—
97C4×D4×C151—22no—
98C4×D4×C151—21no—
99C4×D4×C151—21no—
100C4×D4×C151D411yescharacteristic
101C4×D4×C151C2311yescharacteristic
102#117×C151—22no—
103#117×C151—21no—
104#117×C151—21no—
105#117×C151—21no—
106#117×C151D411yescharacteristic
107C2×C4⋊C4×C151—22no—
108C2×C4⋊C4×C151—21no—
109C2×C4⋊C4×C151—21no—
110C2×C4⋊C4×C151—22no—
111C2×C42×C151—21no—
112C2×C42×C151C2311yescharacteristic
113#107×C151—21no—
114#107×C151—21no—
115#107×C151—22no—
116#106×C151—21no—
117#106×C151—21no—
118C4⋊C8×C151—21no—
119C4⋊C8×C151—21no—
120C4⋊C8×C151—21no—
121C4⋊C8×C151C2312yes—
122C4⋊C8×C151C2311yescharacteristic
123C4⋊C8×C151C2311yescharacteristic
124#103×C151—22no—
125#103×C151—21no—
126#103×C151—21no—
127#103×C151—24no—
128#103×C151—22no—
129#102×C151—21no—
130#102×C151—21no—
131C22⋊C8×C151—21no—
132C4×C8×C151C2312yes—
133#41×C151—22no—
134#41×C151Q1614yescomplemented
135C2×Q8×C151—24no—
136C2×Q8×C151—22no—
137C2×Q8×C151—21no—
138C2×Q8×C151—21no—
139C2×Q8×C151—21no—
140C2×Q8×C151—84no—
141C2×Q8×C151—82no—
142C2×Q8×C151C2×D412yes—
143C2×Q8×C151C2×D411yescharacteristic
144C2×Q8×C151C2×D414yes—
145C2×D4×C151—21no—
146C2×D4×C151C2×D411yescharacteristic
147C22×C4×C151—21no—
148C22×C4×C151—24no—
149C22×C4×C151—22no—
150C22×C4×C151—21no—
151C22×C4×C151—21no—
152C22×C4×C151C2×D411yescharacteristic
153C22×C4×C151C2×D411yescharacteristic
154C22×C4×C151C2×D411yescharacteristic
155Q16×C151—82no—
156Q16×C151—84no—
157C2×C8×C151—22no—
158C2×C8×C151—21no—
159C2×C8×C151—21no—
160C2×C8×C151—22no—
161C2×C8×C151—42no—
162C2×C8×C151—81no—
163C4⋊C4×C151—24no—
164C4⋊C4×C151—24no—
165C4⋊C4×C151—24no—
166C4⋊C4×C151—24no—
167C4⋊C4×C151—21no—
168C4⋊C4×C151—22no—
169C4⋊C4×C151—22no—
170C4⋊C4×C151—21no—
171C4⋊C4×C151—21no—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—22no—
174C4⋊C4×C151—22no—
175C4⋊C4×C151—22no—
176C4⋊C4×C151—21no—
177C4⋊C4×C151—22no—
178C4⋊C4×C151—21no—
179C4⋊C4×C151—21no—
180C4⋊C4×C151—21no—
181C4⋊C4×C151—21no—
182C4⋊C4×C151—41no—
183C4⋊C4×C151—41no—
184C4⋊C4×C151C2×D412yes—
185C4⋊C4×C151C2×D411yescharacteristic
186C4⋊C4×C151C2×D411yescharacteristic
187C22⋊C4×C151—24no—
188C22⋊C4×C151—22no—
189C22⋊C4×C151—21no—
190C22⋊C4×C151—21no—
191C22⋊C4×C151—21no—
192C22⋊C4×C151—41no—
193C42×C151—24no—
194C42×C151—21no—
195C42×C151—21no—
196C42×C151—21no—
197C42×C151—22no—
198C42×C151—41no—
199C42×C151C2×D411yescharacteristic
200C42×C151C2×D411yescharacteristic
201C42×C151C2×D411yescharacteristic
202C42×C151C2411yescharacteristic
203C23×C151—21no—
204C23×C151#12011yescharacteristic
205Q8×C151—21no—
206Q8×C151—22no—
207Q8×C151—22no—
208Q8×C151—44no—
209Q8×C151—44no—
210Q8×C151—42no—
211Q8×C151—42no—
212Q8×C151—41no—
213Q8×C151—41no—
214Q8×C151—84no—
215Q8×C151C2×Q1612yes—
216D4×C151—24no—
217D4×C151—21no—
218D4×C151C2×Q1612yes—
219C2×C4×C151—21no—
220C2×C4×C151—21no—
221C2×C4×C151—21no—
222C2×C4×C151—24no—
223C2×C4×C151—21no—
224C2×C4×C151—22no—
225C2×C4×C151—24no—
226C2×C4×C151—22no—
227C2×C4×C151—21no—
228C2×C4×C151—41no—
229C2×C4×C151—42no—
230C2×C4×C151—41no—
231C2×C4×C151—41no—
232C2×C4×C151—41no—
233C2×C4×C151—41no—
234C2×C4×C151—82no—
235C2×C4×C151—84no—
236C2×C4×C151#12011yescharacteristic
237C2×C4×C151#12011yescharacteristic
238C2×C4×C151#12011yescharacteristic
239C2×C4×C151C2×Q1612yes—
240C2×C4×C151#13712yes—
241C2×C4×C151C22×D411yescharacteristic
242C2×C4×C151C22×D411yescharacteristic
243C2×C4×C151C22×D411yescharacteristic
244C8×C151—42no—
245C8×C151—41no—
246C8×C151—41no—
247C8×C151—42no—
248C22×C151—21no—
249C22×C151—21no—
250C22×C151—22no—
251C22×C151—41no—
252C22×C151—41no—
253C22×C151#45112yes—
254C22×C151C2×#12011yescharacteristic
255C4×C151—22no—
256C4×C151—21no—
257C4×C151—21no—
258C4×C151—21no—
259C4×C151—41no—
260C4×C151—42no—
261C4×C151—41no—
262C4×C151—44no—
263C4×C151—42no—
264C4×C151—81no—
265C4×C151—81no—
266C4×C151#45112yes—
267C4×C151C22×Q1611yescharacteristic
268C4×C151C2×#13711yescharacteristic
269C4×C151#57711yescharacteristic
270C4×C151#57811yescharacteristic
271C2×C151—22no—
272C2×C151—42no—
273C2×C151C2×#45111yescharacteristic
274C2×C151#301311yescharacteristic
275C2×C151#301911yescharacteristic
276#18873C15111yesmaximal, characteristic, direct product factor, Sylow
277C151#1887311yescharacteristic, direct product factor, Sylow
278#3503C2×C15111yescharacteristic
279#3249C2×C15111yescharacteristic, complemented
280#3209C2×C15111yescharacteristic, complemented
281#3076C2×C15111yescharacteristic
282#1645C2×C15112yescomplemented
283#1644C2×C15112yescomplemented
284#1633C2×C15112yescomplemented
285#1632C2×C15112yescomplemented
286C2×#333C2×C15111yescharacteristic, complemented
287#1481C2×C15111yescharacteristic, complemented
288#1477C2×C15111yescharacteristic
289Q82C22×C15112yes—
290#557—21no—
291#554—22no—
292#554C22×C15112yes—
293D4×Q8—22no—
294D4×Q8—21no—
295#544—22no—
296#544—21no—
297#533—21no—
298#533C22×C15111yescharacteristic
299#533C22×C15111yescharacteristic
300C2×C4⋊Q8—21no—
301C2×C4⋊Q8C22×C15111yescharacteristic
302#520—21no—
303C4×#41C22×C15111yescharacteristic
304#494C22×C15112yes—
305#487C22×C15112yes—
306#479—21no—
307#478—21no—
308#477—21no—
309#477C22×C15111yescharacteristic
310#477C22×C15112yes—
311#476C22×C15111yescharacteristic
312#475C22×C15111yescharacteristic
313#474C22×C15111yescharacteristic
314#467—22no—
315#462—22no—
316#462—21no—
317#462C22×C15112yes—
318#451—22no—
319#431C22×C15111yescharacteristic
320#427—21no—
321C2×C4⋊C8C22×C15111yescharacteristic
322C2×C4⋊C8C22×C15111yescharacteristic
323#416—21no—
324#333C22×C15112yes—
325#333C22×C15112yes—
326Q8⋊C8C22×C15112yes—
327Q8⋊C8C22×C15112yes—
328Q8⋊C8C22×C15112yes—
329D4⋊C8C22×C15112yes—
330C2×#41D4×C15111yescharacteristic
331C22×Q8—22no—
332C22×Q8D4×C15112yes—
333C2×Q16—24no—
334C2×Q16—22no—
335C22×C8—21no—
336C22×C8—21no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—24no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8D4×C15112yes—
345C4⋊Q8D4×C15111yescharacteristic
346C4⋊Q8D4×C15111yescharacteristic
347C4⋊Q8C23×C15111yescharacteristic
348C4⋊Q8C23×C15111yescharacteristic
349C4⋊Q8C23×C15112yes—
350#125—22no—
351#125—24no—
352#125—22no—
353#125—21no—
354#125—21no—
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—22no—
359C22⋊Q8—24no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8D4×C15112yes—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×Q8—22no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8D4×C15111yescharacteristic
371C4×Q8C23×C15111yescharacteristic
372C4×Q8C23×C15112yes—
373C4×D4—24no—
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4D4×C15111yescharacteristic
378C4×D4C23×C15111yescharacteristic
379#117—22no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117D4×C15111yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C42—21no—
389C2×C42C23×C15111yescharacteristic
390#107—21no—
391#107—21no—
392#107—22no—
393#106—21no—
394#106—21no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8C23×C15112yes—
399C4⋊C8C23×C15111yescharacteristic
400C4⋊C8C23×C15111yescharacteristic
401#103—21no—
402#103—21no—
403#103—24no—
404#103—22no—
405#103—22no—
406#102—21no—
407#102—21no—
408C22⋊C8—21no—
409C4×C8C23×C15112yes—
410#41—22no—
411#41Q16×C15114yescomplemented
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—84no—
418C2×Q8—82no—
419C2×Q8C2×D4×C15112yes—
420C2×Q8C2×D4×C15111yescharacteristic
421C2×Q8C2×D4×C15114yes—
422C2×D4—21no—
423C2×D4C2×D4×C15111yescharacteristic
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4C2×D4×C15111yescharacteristic
430C22×C4C2×D4×C15111yescharacteristic
431C22×C4C2×D4×C15111yescharacteristic
432Q16—82no—
433Q16—84no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—42no—
439C2×C8—81no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4C2×D4×C15112yes—
462C4⋊C4C2×D4×C15111yescharacteristic
463C4⋊C4C2×D4×C15111yescharacteristic
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—41no—
470C42—24no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—41no—
476C42C2×D4×C15111yescharacteristic
477C42C2×D4×C15111yescharacteristic
478C42C2×D4×C15111yescharacteristic
479C42C24×C15111yescharacteristic
480C23—21no—
481C23#120×C15111yescharacteristic
482Q8—21no—
483Q8—22no—
484Q8—22no—
485Q8—44no—
486Q8—44no—
487Q8—42no—
488Q8—42no—
489Q8—41no—
490Q8—41no—
491Q8—84no—
492Q8C2×Q16×C15112yes—
493D4—24no—
494D4—21no—
495D4C2×Q16×C15112yes—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4#120×C15111yescharacteristic
514C2×C4#120×C15111yescharacteristic
515C2×C4#120×C15111yescharacteristic
516C2×C4C2×Q16×C15112yes—
517C2×C4#137×C15112yes—
518C2×C4C22×D4×C15111yescharacteristic
519C2×C4C22×D4×C15111yescharacteristic
520C2×C4C22×D4×C15111yescharacteristic
521C8—42no—
522C8—41no—
523C8—41no—
524C8—42no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—41no—
529C22—41no—
530C22#451×C15112yes—
531C22C2×#120×C15111yescharacteristic
532C4—22no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—41no—
537C4—42no—
538C4—41no—
539C4—44no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4#451×C15112yes—
544C4C22×Q16×C15111yescharacteristic
545C4C2×#137×C15111yescharacteristic
546C4#577×C15111yescharacteristic
547C4#578×C15111yescharacteristic
548C2—22no—
549C2—42no—
550C2C2×#451×C15111yescharacteristic
551C2#3013×C15111yescharacteristic
552C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radicalC2×#41×C151D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3249×C151C2semidirect
3. #3209×C151C2semidirect
5. #1645×C151C2semidirect
6. #1644×C151C2semidirect
7. #1633×C151C2semidirect
8. #1632×C151C2semidirect
9. C2×#333×C151C2semidirect
10. #1481×C151C2semidirect
134. #41×C151Q16semidirect
276. #18873C151direct (N × H)
277. C151#18873direct (N × H)
279. #3249C2×C151semidirect
280. #3209C2×C151semidirect
282. #1645C2×C151semidirect
283. #1644C2×C151semidirect
284. #1633C2×C151semidirect
285. #1632C2×C151semidirect
286. C2×#333C2×C151semidirect
287. #1481C2×C151semidirect
411. #41Q16×C151semidirect

Maximal subgroups

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

subgroupindex
#3503×C1512
#3249×C1512
#3209×C1512
#3076×C1512
#1645×C1512
#1644×C1512
#1633×C1512
#1632×C1512
C2×#333×C1512
#1481×C1512
#1477×C1512
#18873151