Finite Group Databasebeta

← C137×#18861 #46868247

Subgroups — C137×#18861 #46868247

Order 35072.

874 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3518×C137C211yesmaximal, characteristic
2#3229×C137C211yesmaximal, characteristic, complemented
3#3218×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1652×C137C212yesmaximal, complemented
6#1651×C137C212yesmaximal, complemented
7#1646×C137C212yesmaximal, complemented
8#1645×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12Q82×C137C2211yescharacteristic
13#556×C137—21no—
14#556×C137—22no—
15#556×C137—21no—
16#556×C137C2211yescharacteristic
17#555×C137—21no—
18#555×C137C2211yescharacteristic
19D4×Q8×C137C2211yescharacteristic
20#548×C137—21no—
21#542×C137—22no—
22#542×C137—21no—
23#533×C137—21no—
24#533×C137C2211yescharacteristic
25C2×C4⋊Q8×C137C2211yescharacteristic
26#529×C137—21no—
27#529×C137C2211yescharacteristic
28#519×C137—21no—
29C4×#41×C137C2211yescharacteristic
30#498×C137C2212yes—
31#497×C137—21no—
32#491×C137—21no—
33#487×C137C2212yes—
34#483×C137—22no—
35#477×C137—21no—
36#476×C137—21no—
37#475×C137—22no—
38#475×C137C2212yes—
39#474×C137C2212yes—
40#462×C137C2212yes—
41#461×C137C2212yes—
42#451×C137—21no—
43#448×C137—21no—
44#432×C137C2211yescharacteristic
45#431×C137C2211yescharacteristic
46#428×C137—21no—
47#423×C137C2211yescharacteristic
48#417×C137—21no—
49#333×C137C2212yes—
50#332×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52Q8⋊C8×C137C2212yes—
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137D411yescharacteristic
56C22×Q8×C137—21no—
57C22×Q8×C137—21no—
58C2×Q16×C137—22no—
59C2×SD16×C137—22no—
60C2×M16×C137—21no—
61C4⋊Q8×C137—22no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137—21no—
67C4⋊Q8×C137—21no—
68C4⋊Q8×C137C2312yes—
69C4⋊Q8×C137C2311yescharacteristic
70C4⋊Q8×C137C2311yescharacteristic
71#126×C137—22no—
72#126×C137—22no—
73#126×C137—22no—
74#126×C137—22no—
75#126×C137—21no—
76#125×C137—21no—
77#125×C137D411yescharacteristic
78#124×C137—22no—
79#124×C137—22no—
80#124×C137—22no—
81#124×C137—22no—
82#124×C137—21no—
83#124×C137D411yescharacteristic
84#123×C137—22no—
85#123×C137—21no—
86C22⋊Q8×C137—22no—
87C22⋊Q8×C137—22no—
88C22⋊Q8×C137—21no—
89C22⋊Q8×C137—21no—
90C22⋊Q8×C137—21no—
91C22⋊Q8×C137—22no—
92C22⋊Q8×C137D411yescharacteristic
93C22⋊Q8×C137D412yes—
94#121×C137—22no—
95#121×C137—21no—
96#121×C137—21no—
97#121×C137D411yescharacteristic
98C4×Q8×C137—22no—
99C4×Q8×C137—22no—
100C4×Q8×C137—22no—
101C4×Q8×C137—21no—
102C4×Q8×C137—21no—
103C4×Q8×C137—21no—
104C4×Q8×C137—21no—
105C4×Q8×C137D412yes—
106C4×Q8×C137D411yescharacteristic
107C4×Q8×C137C2311yescharacteristic
108C4×Q8×C137C2311yescharacteristic
109C4×D4×C137—22no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137D411yescharacteristic
115C4×D4×C137C2311yescharacteristic
116C4×D4×C137C2311yescharacteristic
117#117×C137—22no—
118#117×C137—22no—
119#117×C137—21no—
120#117×C137—21no—
121#117×C137—21no—
122#117×C137—21no—
123#117×C137—21no—
124#117×C137—21no—
125#117×C137D411yescharacteristic
126C2×C4⋊C4×C137—21no—
127C2×C4⋊C4×C137—21no—
128C2×C42×C137—21no—
129C2×C42×C137C2311yescharacteristic
130#107×C137—22no—
131#106×C137—22no—
132#106×C137—24no—
133C4⋊C8×C137—21no—
134C4⋊C8×C137—21no—
135C4⋊C8×C137C2312yes—
136C4⋊C8×C137C2312yes—
137#103×C137—22no—
138#103×C137—24no—
139#103×C137—22no—
140#103×C137—22no—
141#102×C137—22no—
142C22⋊C8×C137—21no—
143C22⋊C8×C137—21no—
144#97×C137—21no—
145C4×C8×C137C2312yes—
146#41×C137—22no—
147#41×C137—41no—
148C2×Q8×C137—22no—
149C2×Q8×C137—22no—
150C2×Q8×C137—21no—
151C2×Q8×C137—22no—
152C2×Q8×C137—42no—
153C2×Q8×C137—42no—
154C2×Q8×C137—41no—
155C2×Q8×C137—82no—
156C2×Q8×C137C2×D411yescharacteristic
157C2×Q8×C137C2×D411yescharacteristic
158C2×Q8×C137C2×D412yes—
159C2×D4×C137—21no—
160C2×D4×C137—22no—
161C2×D4×C137—21no—
162C2×D4×C137C2×D411yescharacteristic
163C2×D4×C137C2×D411yescharacteristic
164C22×C4×C137—22no—
165C22×C4×C137—21no—
166C22×C4×C137—21no—
167C22×C4×C137—21no—
168C22×C4×C137—21no—
169C22×C4×C137—41no—
170C22×C4×C137C2×D411yescharacteristic
171C22×C4×C137C2×D411yescharacteristic
172C22×C4×C137C2×D411yescharacteristic
173Q16×C137—82no—
174SD16×C137—82no—
175M16×C137—42no—
176C2×C8×C137—22no—
177C2×C8×C137—22no—
178C2×C8×C137—22no—
179C4⋊C4×C137—22no—
180C4⋊C4×C137—22no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—22no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137—22no—
186C4⋊C4×C137—21no—
187C4⋊C4×C137—21no—
188C4⋊C4×C137—22no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—21no—
191C4⋊C4×C137—21no—
192C4⋊C4×C137—21no—
193C4⋊C4×C137—21no—
194C4⋊C4×C137—22no—
195C4⋊C4×C137—22no—
196C4⋊C4×C137—41no—
197C4⋊C4×C137—41no—
198C4⋊C4×C137C2×D411yescharacteristic
199C4⋊C4×C137C2×D411yescharacteristic
200C4⋊C4×C137C2×D411yescharacteristic
201C4⋊C4×C137C2×D412yes—
202C4⋊C4×C137C2×D411yescharacteristic
203C22⋊C4×C137—22no—
204C22⋊C4×C137—22no—
205C22⋊C4×C137—22no—
206C22⋊C4×C137—22no—
207C22⋊C4×C137—22no—
208C22⋊C4×C137—22no—
209C22⋊C4×C137—21no—
210C22⋊C4×C137—21no—
211C22⋊C4×C137—21no—
212C22⋊C4×C137—21no—
213C22⋊C4×C137—21no—
214C22⋊C4×C137—21no—
215C22⋊C4×C137—21no—
216C22⋊C4×C137—21no—
217C42×C137—22no—
218C42×C137—22no—
219C42×C137—22no—
220C42×C137—21no—
221C42×C137—21no—
222C42×C137—22no—
223C42×C137—21no—
224C42×C137—21no—
225C42×C137C2×D411yescharacteristic
226C42×C137C2×D411yescharacteristic
227C42×C137C2×D411yescharacteristic
228C42×C137C2411yescharacteristic
229C23×C137—21no—
230C23×C137—21no—
231C23×C137#12011yescharacteristic
232Q8×C137—21no—
233Q8×C137—22no—
234Q8×C137—42no—
235Q8×C137—44no—
236Q8×C137—41no—
237Q8×C137—41no—
238Q8×C137—42no—
239Q8×C137—42no—
240Q8×C137—82no—
241Q8×C137—82no—
242Q8×C137#13712yes—
243D4×C137—21no—
244D4×C137—21no—
245D4×C137—22no—
246D4×C137—42no—
247D4×C137—81no—
248D4×C137—82no—
249C2×C4×C137—21no—
250C2×C4×C137—21no—
251C2×C4×C137—21no—
252C2×C4×C137—22no—
253C2×C4×C137—21no—
254C2×C4×C137—21no—
255C2×C4×C137—22no—
256C2×C4×C137—21no—
257C2×C4×C137—21no—
258C2×C4×C137—22no—
259C2×C4×C137—21no—
260C2×C4×C137—41no—
261C2×C4×C137—42no—
262C2×C4×C137—44no—
263C2×C4×C137—41no—
264C2×C4×C137—41no—
265C2×C4×C137—42no—
266C2×C4×C137—42no—
267C2×C4×C137—81no—
268C2×C4×C137—82no—
269C2×C4×C137#12011yescharacteristic
270C2×C4×C137#12011yescharacteristic
271C2×C4×C137#12011yescharacteristic
272C2×C4×C137C22×D411yescharacteristic
273C2×C4×C137C22×D411yescharacteristic
274C2×C4×C137C22×D411yescharacteristic
275C8×C137—42no—
276C8×C137—42no—
277C8×C137—42no—
278C22×C137—21no—
279C22×C137—21no—
280C22×C137—21no—
281C22×C137—41no—
282C22×C137—41no—
283C22×C137—41no—
284C22×C137—41no—
285C22×C137—41no—
286C22×C137—41no—
287C22×C137C2×#12011yescharacteristic
288C4×C137—21no—
289C4×C137—22no—
290C4×C137—21no—
291C4×C137—21no—
292C4×C137—21no—
293C4×C137—42no—
294C4×C137—41no—
295C4×C137—41no—
296C4×C137—41no—
297C4×C137—41no—
298C4×C137—42no—
299C4×C137—41no—
300C4×C137—82no—
301C4×C137C2×#13711yescharacteristic
302C4×C137#57511yescharacteristic
303C4×C137#57712yes—
304C2×C137—41no—
305C2×C137—81no—
306C2×C137—81no—
307C2×C137#300711yescharacteristic
308C2×C137#301011yescharacteristic
309C2×C137#302211yescharacteristic
310#18861C13711yesmaximal, characteristic, direct product factor, Sylow
311C137#1886111yescharacteristic, direct product factor, Sylow
312#3518C2×C13711yescharacteristic
313#3229C2×C13711yescharacteristic, complemented
314#3218C2×C13711yescharacteristic, complemented
315#3079C2×C13711yescharacteristic, complemented
316#1652C2×C13712yescomplemented
317#1651C2×C13712yescomplemented
318#1646C2×C13712yescomplemented
319#1645C2×C13712yescomplemented
320#1559C2×C13711yescharacteristic, complemented
321#1495C2×C13711yescharacteristic, complemented
322#1488C2×C13711yescharacteristic, complemented
323Q82C22×C13711yescharacteristic
324#556—21no—
325#556—22no—
326#556—21no—
327#556C22×C13711yescharacteristic
328#555—21no—
329#555C22×C13711yescharacteristic
330D4×Q8C22×C13711yescharacteristic
331#548—21no—
332#542—22no—
333#542—21no—
334#533—21no—
335#533C22×C13711yescharacteristic
336C2×C4⋊Q8C22×C13711yescharacteristic
337#529—21no—
338#529C22×C13711yescharacteristic
339#519—21no—
340C4×#41C22×C13711yescharacteristic
341#498C22×C13712yes—
342#497—21no—
343#491—21no—
344#487C22×C13712yes—
345#483—22no—
346#477—21no—
347#476—21no—
348#475—22no—
349#475C22×C13712yes—
350#474C22×C13712yes—
351#462C22×C13712yes—
352#461C22×C13712yes—
353#451—21no—
354#448—21no—
355#432C22×C13711yescharacteristic
356#431C22×C13711yescharacteristic
357#428—21no—
358#423C22×C13711yescharacteristic
359#417—21no—
360#333C22×C13712yes—
361#332C22×C13712yes—
362Q8⋊C8C22×C13712yes—
363Q8⋊C8C22×C13712yes—
364D4⋊C8C22×C13712yes—
365D4⋊C8C22×C13712yes—
366C2×#41D4×C13711yescharacteristic
367C22×Q8—21no—
368C22×Q8—21no—
369C2×Q16—22no—
370C2×SD16—22no—
371C2×M16—21no—
372C4⋊Q8—22no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8C23×C13712yes—
380C4⋊Q8C23×C13711yescharacteristic
381C4⋊Q8C23×C13711yescharacteristic
382#126—22no—
383#126—22no—
384#126—22no—
385#126—22no—
386#126—21no—
387#125—21no—
388#125D4×C13711yescharacteristic
389#124—22no—
390#124—22no—
391#124—22no—
392#124—22no—
393#124—21no—
394#124D4×C13711yescharacteristic
395#123—22no—
396#123—21no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8D4×C13711yescharacteristic
404C22⋊Q8D4×C13712yes—
405#121—22no—
406#121—21no—
407#121—21no—
408#121D4×C13711yescharacteristic
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8D4×C13711yescharacteristic
417C4×Q8D4×C13712yes—
418C4×Q8C23×C13711yescharacteristic
419C4×Q8C23×C13711yescharacteristic
420C4×D4—22no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4D4×C13711yescharacteristic
426C4×D4C23×C13711yescharacteristic
427C4×D4C23×C13711yescharacteristic
428#117—22no—
429#117—22no—
430#117—21no—
431#117—21no—
432#117—21no—
433#117—21no—
434#117—21no—
435#117—21no—
436#117D4×C13711yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C42—21no—
440C2×C42C23×C13711yescharacteristic
441#107—22no—
442#106—22no—
443#106—24no—
444C4⋊C8—21no—
445C4⋊C8—21no—
446C4⋊C8C23×C13712yes—
447C4⋊C8C23×C13712yes—
448#103—22no—
449#103—24no—
450#103—22no—
451#103—22no—
452#102—22no—
453C22⋊C8—21no—
454C22⋊C8—21no—
455#97—21no—
456C4×C8C23×C13712yes—
457#41—22no—
458#41—41no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—41no—
466C2×Q8—82no—
467C2×Q8C2×D4×C13711yescharacteristic
468C2×Q8C2×D4×C13711yescharacteristic
469C2×Q8C2×D4×C13712yes—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4C2×D4×C13711yescharacteristic
474C2×D4C2×D4×C13711yescharacteristic
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4C2×D4×C13711yescharacteristic
482C22×C4C2×D4×C13711yescharacteristic
483C22×C4C2×D4×C13711yescharacteristic
484Q16—82no—
485SD16—82no—
486M16—42no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4C2×D4×C13711yescharacteristic
510C4⋊C4C2×D4×C13711yescharacteristic
511C4⋊C4C2×D4×C13711yescharacteristic
512C4⋊C4C2×D4×C13712yes—
513C4⋊C4C2×D4×C13711yescharacteristic
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42C2×D4×C13711yescharacteristic
537C42C2×D4×C13711yescharacteristic
538C42C2×D4×C13711yescharacteristic
539C42C24×C13711yescharacteristic
540C23—21no—
541C23—21no—
542C23#120×C13711yescharacteristic
543Q8—21no—
544Q8—22no—
545Q8—42no—
546Q8—44no—
547Q8—41no—
548Q8—41no—
549Q8—42no—
550Q8—42no—
551Q8—82no—
552Q8—82no—
553Q8#137×C13712yes—
554D4—21no—
555D4—21no—
556D4—22no—
557D4—42no—
558D4—81no—
559D4—82no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4#120×C13711yescharacteristic
581C2×C4#120×C13711yescharacteristic
582C2×C4#120×C13711yescharacteristic
583C2×C4C22×D4×C13711yescharacteristic
584C2×C4C22×D4×C13711yescharacteristic
585C2×C4C22×D4×C13711yescharacteristic
586C8—42no—
587C8—42no—
588C8—42no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C2×#120×C13711yescharacteristic
599C4—21no—
600C4—22no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—82no—
612C4C2×#137×C13711yescharacteristic
613C4#575×C13711yescharacteristic
614C4#577×C13712yes—
615C2—41no—
616C2—81no—
617C2—81no—
618C2#3007×C13711yescharacteristic
619C2#3010×C13711yescharacteristic
620C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3518×C137C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3229×C137C2semidirect
3. #3218×C137C2semidirect
4. #3079×C137C2semidirect
5. #1652×C137C2semidirect
6. #1651×C137C2semidirect
7. #1646×C137C2semidirect
8. #1645×C137C2semidirect
9. #1559×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
310. #18861C137direct (N × H)
311. C137#18861direct (N × H)
313. #3229C2×C137semidirect
314. #3218C2×C137semidirect
315. #3079C2×C137semidirect
316. #1652C2×C137semidirect
317. #1651C2×C137semidirect
318. #1646C2×C137semidirect
319. #1645C2×C137semidirect
320. #1559C2×C137semidirect
321. #1495C2×C137semidirect
322. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3518×C1372
#3229×C1372
#3218×C1372
#3079×C1372
#1652×C1372
#1651×C1372
#1646×C1372
#1645×C1372
#1559×C1372
#1495×C1372
#1488×C1372
#18861137