Finite Group Databasebeta

← C137×#18859 #46868245

Subgroups — C137×#18859 #46868245

Order 35072.

930 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3517×C137C211yesmaximal, characteristic
2#3226×C137C211yesmaximal, characteristic, complemented
3#3218×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1648×C137C212yesmaximal, complemented
6#1647×C137C212yesmaximal, complemented
7#1642×C137C212yesmaximal, complemented
8#1641×C137C212yesmaximal, complemented
9#1557×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12#557×C137C2211yescharacteristic
13#553×C137—22no—
14#553×C137—21no—
15#550×C137—21no—
16#550×C137—21no—
17#550×C137C2211yescharacteristic
18#550×C137C2211yescharacteristic
19#547×C137—21no—
20#547×C137C2211yescharacteristic
21#540×C137—22no—
22#540×C137—21no—
23#532×C137—21no—
24#532×C137C2211yescharacteristic
25C2×C4⋊Q8×C137C2211yescharacteristic
26#529×C137—21no—
27#529×C137C2211yescharacteristic
28#520×C137—21no—
29C4×#41×C137C2211yescharacteristic
30#500×C137C2212yes—
31#497×C137—21no—
32#491×C137—21no—
33#486×C137C2212yes—
34#480×C137—22no—
35#477×C137—21no—
36#476×C137—21no—
37#475×C137C2212yes—
38#474×C137—22no—
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#332×C137C2212yes—
50#331×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52Q8⋊C8×C137C2212yes—
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137—21no—
56C2×#41×C137—21no—
57C2×#41×C137—21no—
58C2×#41×C137—21no—
59C2×#41×C137D411yescharacteristic
60C22×Q8×C137—21no—
61C2×Q16×C137—22no—
62C2×SD16×C137—22no—
63C2×M16×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137—21no—
67C4⋊Q8×C137—21no—
68C4⋊Q8×C137C2312yes—
69C4⋊Q8×C137C2311yescharacteristic
70#127×C137—22no—
71#127×C137—21no—
72#127×C137—21no—
73#127×C137C2311yescharacteristic
74#126×C137—22no—
75#126×C137—21no—
76#125×C137—22no—
77#125×C137—21no—
78#125×C137D411yescharacteristic
79#124×C137—21no—
80#124×C137D411yescharacteristic
81#123×C137—22no—
82#123×C137—22no—
83#123×C137—22no—
84#123×C137—21no—
85#123×C137—22no—
86C22⋊Q8×C137—21no—
87C22⋊Q8×C137D411yescharacteristic
88#121×C137—22no—
89#121×C137—22no—
90#121×C137—22no—
91#121×C137—22no—
92#121×C137—22no—
93#121×C137—21no—
94#121×C137—22no—
95#121×C137—22no—
96#121×C137—21no—
97#121×C137—21no—
98#121×C137—21no—
99#121×C137D411yescharacteristic
100#121×C137D412yes—
101C4×Q8×C137—21no—
102C4×Q8×C137—21no—
103C4×Q8×C137D411yescharacteristic
104C4×Q8×C137C2311yescharacteristic
105C4×Q8×C137C2311yescharacteristic
106C4×D4×C137—22no—
107C4×D4×C137—22no—
108C4×D4×C137—22no—
109C4×D4×C137—22no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137—21no—
115C4×D4×C137—21no—
116C4×D4×C137D412yes—
117C4×D4×C137D411yescharacteristic
118C4×D4×C137C2311yescharacteristic
119C4×D4×C137C2311yescharacteristic
120#117×C137—22no—
121#117×C137—21no—
122#117×C137—21no—
123#117×C137D411yescharacteristic
124C2×C4⋊C4×C137—22no—
125C2×C4⋊C4×C137—21no—
126C2×C4⋊C4×C137—21no—
127C2×C4⋊C4×C137—21no—
128C2×C42×C137—21no—
129C2×C42×C137C2311yescharacteristic
130#107×C137—22no—
131#107×C137—24no—
132#106×C137—22no—
133C4⋊C8×C137—21no—
134C4⋊C8×C137—21no—
135C4⋊C8×C137C2312yes—
136C4⋊C8×C137C2312yes—
137#103×C137—22no—
138#103×C137—22no—
139#103×C137—22no—
140#102×C137—22no—
141#102×C137—24no—
142C22⋊C8×C137—21no—
143C22⋊C8×C137—21no—
144#97×C137—21no—
145C4×C8×C137C2312yes—
146#41×C137—22no—
147#41×C137—41no—
148#41×C137—42no—
149#41×C137—42no—
150#41×C137—82no—
151#41×C137—82no—
152#41×C137—82no—
153C2×Q8×C137—21no—
154C2×Q8×C137—22no—
155C2×Q8×C137—41no—
156C2×Q8×C137—82no—
157C2×Q8×C137C2×D411yescharacteristic
158C2×Q8×C137C2×D411yescharacteristic
159C2×D4×C137—22no—
160C2×D4×C137—21no—
161C2×D4×C137—22no—
162C2×D4×C137—22no—
163C2×D4×C137—22no—
164C2×D4×C137—22no—
165C2×D4×C137—21no—
166C2×D4×C137—22no—
167C2×D4×C137—21no—
168C2×D4×C137—21no—
169C2×D4×C137C2×D411yescharacteristic
170C2×D4×C137C2×D411yescharacteristic
171C2×D4×C137C2×D412yes—
172C22×C4×C137—22no—
173C22×C4×C137—22no—
174C22×C4×C137—22no—
175C22×C4×C137—21no—
176C22×C4×C137—21no—
177C22×C4×C137—21no—
178C22×C4×C137—21no—
179C22×C4×C137—22no—
180C22×C4×C137—41no—
181C22×C4×C137C2×D411yescharacteristic
182C22×C4×C137C2×D411yescharacteristic
183C22×C4×C137C2×D411yescharacteristic
184Q16×C137—82no—
185SD16×C137—82no—
186M16×C137—42no—
187C2×C8×C137—22no—
188C2×C8×C137—22no—
189C2×C8×C137—22no—
190C4⋊C4×C137—22no—
191C4⋊C4×C137—22no—
192C4⋊C4×C137—22no—
193C4⋊C4×C137—22no—
194C4⋊C4×C137—21no—
195C4⋊C4×C137—21no—
196C4⋊C4×C137—21no—
197C4⋊C4×C137—21no—
198C4⋊C4×C137—21no—
199C4⋊C4×C137—21no—
200C4⋊C4×C137—21no—
201C4⋊C4×C137—22no—
202C4⋊C4×C137—22no—
203C4⋊C4×C137—41no—
204C4⋊C4×C137—41no—
205C4⋊C4×C137C2×D411yescharacteristic
206C4⋊C4×C137C2×D411yescharacteristic
207C4⋊C4×C137C2×D411yescharacteristic
208C4⋊C4×C137C2×D411yescharacteristic
209C4⋊C4×C137C2×D412yes—
210C22⋊C4×C137—22no—
211C22⋊C4×C137—22no—
212C22⋊C4×C137—22no—
213C22⋊C4×C137—21no—
214C22⋊C4×C137—21no—
215C22⋊C4×C137—21no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—22no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—21no—
220C22⋊C4×C137—22no—
221C22⋊C4×C137—21no—
222C22⋊C4×C137—21no—
223C42×C137—22no—
224C42×C137—21no—
225C42×C137—21no—
226C42×C137—21no—
227C42×C137—21no—
228C42×C137C2×D411yescharacteristic
229C42×C137C2×D411yescharacteristic
230C42×C137C2×D411yescharacteristic
231C42×C137C2411yescharacteristic
232C23×C137—21no—
233C23×C137—21no—
234C23×C137—22no—
235C23×C137#12011yescharacteristic
236Q8×C137—21no—
237Q8×C137—22no—
238Q8×C137—42no—
239Q8×C137—42no—
240Q8×C137—42no—
241Q8×C137—82no—
242Q8×C137#13712yes—
243D4×C137—21no—
244D4×C137—21no—
245D4×C137—22no—
246D4×C137—42no—
247D4×C137—42no—
248D4×C137—44no—
249D4×C137—41no—
250D4×C137—41no—
251D4×C137—42no—
252D4×C137—81no—
253D4×C137—82no—
254D4×C137—82no—
255D4×C137—82no—
256D4×C137—82no—
257D4×C137—82no—
258C2×C4×C137—21no—
259C2×C4×C137—21no—
260C2×C4×C137—21no—
261C2×C4×C137—22no—
262C2×C4×C137—21no—
263C2×C4×C137—21no—
264C2×C4×C137—21no—
265C2×C4×C137—21no—
266C2×C4×C137—22no—
267C2×C4×C137—21no—
268C2×C4×C137—41no—
269C2×C4×C137—42no—
270C2×C4×C137—44no—
271C2×C4×C137—41no—
272C2×C4×C137—41no—
273C2×C4×C137—42no—
274C2×C4×C137—42no—
275C2×C4×C137—81no—
276C2×C4×C137—82no—
277C2×C4×C137—82no—
278C2×C4×C137—82no—
279C2×C4×C137—82no—
280C2×C4×C137#12011yescharacteristic
281C2×C4×C137#12011yescharacteristic
282C2×C4×C137#12011yescharacteristic
283C2×C4×C137C22×D411yescharacteristic
284C2×C4×C137C22×D411yescharacteristic
285C2×C4×C137C22×D411yescharacteristic
286C8×C137—42no—
287C8×C137—42no—
288C8×C137—42no—
289C22×C137—21no—
290C22×C137—21no—
291C22×C137—21no—
292C22×C137—41no—
293C22×C137—42no—
294C22×C137—41no—
295C22×C137—41no—
296C22×C137—41no—
297C22×C137—42no—
298C22×C137—41no—
299C22×C137—42no—
300C22×C137—41no—
301C22×C137C2×#12011yescharacteristic
302C4×C137—21no—
303C4×C137—22no—
304C4×C137—21no—
305C4×C137—21no—
306C4×C137—21no—
307C4×C137—41no—
308C4×C137—41no—
309C4×C137—41no—
310C4×C137—42no—
311C4×C137—41no—
312C4×C137—41no—
313C4×C137—82no—
314C4×C137C2×#13711yescharacteristic
315C4×C137#57511yescharacteristic
316C4×C137#57612yes—
317C2×C137—41no—
318C2×C137—81no—
319C2×C137—81no—
320C2×C137—82no—
321C2×C137#300711yescharacteristic
322C2×C137#300811yescharacteristic
323C2×C137#302211yescharacteristic
324#18859C13711yesmaximal, characteristic, direct product factor, Sylow
325C137#1885911yescharacteristic, direct product factor, Sylow
326#3517C2×C13711yescharacteristic
327#3226C2×C13711yescharacteristic, complemented
328#3218C2×C13711yescharacteristic, complemented
329#3079C2×C13711yescharacteristic, complemented
330#1648C2×C13712yescomplemented
331#1647C2×C13712yescomplemented
332#1642C2×C13712yescomplemented
333#1641C2×C13712yescomplemented
334#1557C2×C13711yescharacteristic, complemented
335#1495C2×C13711yescharacteristic, complemented
336#1488C2×C13711yescharacteristic, complemented
337#557C22×C13711yescharacteristic
338#553—22no—
339#553—21no—
340#550—21no—
341#550—21no—
342#550C22×C13711yescharacteristic
343#550C22×C13711yescharacteristic
344#547—21no—
345#547C22×C13711yescharacteristic
346#540—22no—
347#540—21no—
348#532—21no—
349#532C22×C13711yescharacteristic
350C2×C4⋊Q8C22×C13711yescharacteristic
351#529—21no—
352#529C22×C13711yescharacteristic
353#520—21no—
354C4×#41C22×C13711yescharacteristic
355#500C22×C13712yes—
356#497—21no—
357#491—21no—
358#486C22×C13712yes—
359#480—22no—
360#477—21no—
361#476—21no—
362#475C22×C13712yes—
363#474—22no—
364#474C22×C13712yes—
365#462C22×C13712yes—
366#461C22×C13712yes—
367#451—21no—
368#448—21no—
369#432C22×C13711yescharacteristic
370#431C22×C13711yescharacteristic
371#428—21no—
372#423C22×C13711yescharacteristic
373#417—21no—
374#332C22×C13712yes—
375#331C22×C13712yes—
376Q8⋊C8C22×C13712yes—
377Q8⋊C8C22×C13712yes—
378D4⋊C8C22×C13712yes—
379D4⋊C8C22×C13712yes—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41D4×C13711yescharacteristic
385C22×Q8—21no—
386C2×Q16—22no—
387C2×SD16—22no—
388C2×M16—21no—
389C4⋊Q8—21no—
390C4⋊Q8—21no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393C4⋊Q8C23×C13712yes—
394C4⋊Q8C23×C13711yescharacteristic
395#127—22no—
396#127—21no—
397#127—21no—
398#127C23×C13711yescharacteristic
399#126—22no—
400#126—21no—
401#125—22no—
402#125—21no—
403#125D4×C13711yescharacteristic
404#124—21no—
405#124D4×C13711yescharacteristic
406#123—22no—
407#123—22no—
408#123—22no—
409#123—21no—
410#123—22no—
411C22⋊Q8—21no—
412C22⋊Q8D4×C13711yescharacteristic
413#121—22no—
414#121—22no—
415#121—22no—
416#121—22no—
417#121—22no—
418#121—21no—
419#121—22no—
420#121—22no—
421#121—21no—
422#121—21no—
423#121—21no—
424#121D4×C13711yescharacteristic
425#121D4×C13712yes—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8D4×C13711yescharacteristic
429C4×Q8C23×C13711yescharacteristic
430C4×Q8C23×C13711yescharacteristic
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4D4×C13711yescharacteristic
442C4×D4D4×C13712yes—
443C4×D4C23×C13711yescharacteristic
444C4×D4C23×C13711yescharacteristic
445#117—22no—
446#117—21no—
447#117—21no—
448#117D4×C13711yescharacteristic
449C2×C4⋊C4—22no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C42—21no—
454C2×C42C23×C13711yescharacteristic
455#107—22no—
456#107—24no—
457#106—22no—
458C4⋊C8—21no—
459C4⋊C8—21no—
460C4⋊C8C23×C13712yes—
461C4⋊C8C23×C13712yes—
462#103—22no—
463#103—22no—
464#103—22no—
465#102—22no—
466#102—24no—
467C22⋊C8—21no—
468C22⋊C8—21no—
469#97—21no—
470C4×C8C23×C13712yes—
471#41—22no—
472#41—41no—
473#41—42no—
474#41—42no—
475#41—82no—
476#41—82no—
477#41—82no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—41no—
481C2×Q8—82no—
482C2×Q8C2×D4×C13711yescharacteristic
483C2×Q8C2×D4×C13711yescharacteristic
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×D4×C13711yescharacteristic
495C2×D4C2×D4×C13711yescharacteristic
496C2×D4C2×D4×C13712yes—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—41no—
506C22×C4C2×D4×C13711yescharacteristic
507C22×C4C2×D4×C13711yescharacteristic
508C22×C4C2×D4×C13711yescharacteristic
509Q16—82no—
510SD16—82no—
511M16—42no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4C2×D4×C13711yescharacteristic
531C4⋊C4C2×D4×C13711yescharacteristic
532C4⋊C4C2×D4×C13711yescharacteristic
533C4⋊C4C2×D4×C13711yescharacteristic
534C4⋊C4C2×D4×C13712yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42C2×D4×C13711yescharacteristic
554C42C2×D4×C13711yescharacteristic
555C42C2×D4×C13711yescharacteristic
556C42C24×C13711yescharacteristic
557C23—21no—
558C23—21no—
559C23—22no—
560C23#120×C13711yescharacteristic
561Q8—21no—
562Q8—22no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—82no—
567Q8#137×C13712yes—
568D4—21no—
569D4—21no—
570D4—22no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—41no—
575D4—41no—
576D4—42no—
577D4—82no—
578D4—81no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—82no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4#120×C13711yescharacteristic
606C2×C4#120×C13711yescharacteristic
607C2×C4#120×C13711yescharacteristic
608C2×C4C22×D4×C13711yescharacteristic
609C2×C4C22×D4×C13711yescharacteristic
610C2×C4C22×D4×C13711yescharacteristic
611C8—42no—
612C8—42no—
613C8—42no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—41no—
624C22—42no—
625C22—41no—
626C22C2×#120×C13711yescharacteristic
627C4—21no—
628C4—22no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4C2×#137×C13711yescharacteristic
640C4#575×C13711yescharacteristic
641C4#576×C13712yes—
642C2—41no—
643C2—81no—
644C2—81no—
645C2—82no—
646C2#3007×C13711yescharacteristic
647C2#3008×C13711yescharacteristic
648C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3517×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. #3226×C137C2semidirect
3. #3218×C137C2semidirect
4. #3079×C137C2semidirect
5. #1648×C137C2semidirect
6. #1647×C137C2semidirect
7. #1642×C137C2semidirect
8. #1641×C137C2semidirect
9. #1557×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
324. #18859C137direct (N × H)
325. C137#18859direct (N × H)
327. #3226C2×C137semidirect
328. #3218C2×C137semidirect
329. #3079C2×C137semidirect
330. #1648C2×C137semidirect
331. #1647C2×C137semidirect
332. #1642C2×C137semidirect
333. #1641C2×C137semidirect
334. #1557C2×C137semidirect
335. #1495C2×C137semidirect
336. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3517×C1372
#3226×C1372
#3218×C1372
#3079×C1372
#1648×C1372
#1647×C1372
#1642×C1372
#1641×C1372
#1557×C1372
#1495×C1372
#1488×C1372
#18859137