Finite Group Databasebeta

← C137×#18852 #46868238

Subgroups — C137×#18852 #46868238

Order 35072.

970 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3518×C137C211yesmaximal, characteristic, complemented
2#3217×C137C211yesmaximal, characteristic, complemented
3#3208×C137C211yesmaximal, characteristic, complemented
4#3038×C137C211yesmaximal, characteristic, complemented
5#1628×C137C212yesmaximal, complemented
6#1627×C137C212yesmaximal, complemented
7#1624×C137C212yesmaximal, complemented
8#1623×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C137—21no—
13Q82×C137C2211yescharacteristic, complemented
14#556×C137—21no—
15#556×C137—22no—
16#556×C137—21no—
17#556×C137C2211yescharacteristic, complemented
18#555×C137—21no—
19#555×C137C2211yescharacteristic, complemented
20D4×Q8×C137C2211yescharacteristic, complemented
21#548×C137—21no—
22#542×C137—22no—
23#542×C137—21no—
24#533×C137—21no—
25#533×C137C2211yescharacteristic
26C2×#127×C137C2211yescharacteristic
27#529×C137—21no—
28#529×C137C2211yescharacteristic
29#519×C137—21no—
30C4×#41×C137C2211yescharacteristic
31#496×C137—21no—
32#492×C137C2212yes—
33#490×C137—21no—
34#486×C137C2212yes—
35#460×C137—21no—
36#460×C137—22no—
37#460×C137C2212yes—
38#460×C137C2212yes—
39#459×C137—21no—
40#459×C137C2212yes—
41#459×C137C2212yes—
42#450×C137—22no—
43#450×C137—21no—
44#447×C137—21no—
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#423×C137C2211yescharacteristic
48#418×C137—21no—
49#333×C137C2212yescomplemented
50#332×C137C2212yescomplemented
51Q8⋊C8×C137C2212yescomplemented
52Q8⋊C8×C137C2212yescomplemented
53D4⋊C8×C137C2212yescomplemented
54D4⋊C8×C137C2212yescomplemented
55C2×#41×C137D411yescharacteristic
56C22×Q8×C137—21no—
57C22×D4×C137—21no—
58C22×D4×C137—21no—
59C22×D4×C137—21no—
60AGL(1,ℤ/8)×C137—44no—
61C2×SD16×C137—22no—
62C2×SD16×C137—24no—
63C2×SD16×C137—22no—
64C2×D8×C137—22no—
65C2×D8×C137—22no—
66C2×M16×C137—21no—
67C4⋊Q8×C137—22no—
68C4⋊Q8×C137—21no—
69C4⋊Q8×C137—21no—
70C4⋊Q8×C137—21no—
71C4⋊Q8×C137C2311yescharacteristic
72C4⋊Q8×C137C2311yescharacteristic
73#127×C137—21no—
74#127×C137—21no—
75#127×C137—21no—
76#127×C137C2312yes—
77#126×C137—22no—
78#126×C137—22no—
79#126×C137—22no—
80#126×C137—22no—
81#126×C137—21no—
82#125×C137—21no—
83#125×C137D411yescharacteristic
84#124×C137—22no—
85#124×C137—22no—
86#124×C137—22no—
87#124×C137—22no—
88#124×C137—21no—
89#124×C137D411yescharacteristic
90#123×C137—22no—
91#123×C137—21no—
92C22⋊Q8×C137—22no—
93C22⋊Q8×C137—22no—
94C22⋊Q8×C137—21no—
95C22⋊Q8×C137—21no—
96C22⋊Q8×C137—21no—
97C22⋊Q8×C137—22no—
98C22⋊Q8×C137D411yescharacteristic
99C22⋊Q8×C137D412yes—
100#121×C137—22no—
101#121×C137—21no—
102#121×C137—21no—
103#121×C137D411yescharacteristic
104C4×Q8×C137—22no—
105C4×Q8×C137—22no—
106C4×Q8×C137—22no—
107C4×Q8×C137—21no—
108C4×Q8×C137—21no—
109C4×Q8×C137—21no—
110C4×Q8×C137—21no—
111C4×Q8×C137D411yescharacteristic
112C4×Q8×C137D412yes—
113C4×Q8×C137C2311yescharacteristic
114C4×Q8×C137C2311yescharacteristic
115C4×D4×C137—22no—
116C4×D4×C137—21no—
117C4×D4×C137—21no—
118C4×D4×C137—21no—
119C4×D4×C137—21no—
120C4×D4×C137D411yescharacteristic
121C4×D4×C137C2311yescharacteristic
122C4×D4×C137C2311yescharacteristic
123#117×C137—22no—
124#117×C137—22no—
125#117×C137—21no—
126#117×C137—21no—
127#117×C137—21no—
128#117×C137—21no—
129#117×C137—21no—
130#117×C137—21no—
131#117×C137D411yescharacteristic
132C2×C42×C137—21no—
133C2×C42×C137C2311yescharacteristic
134C4⋊C8×C137—21no—
135C4⋊C8×C137—21no—
136C4⋊C8×C137C2312yes—
137C4⋊C8×C137C2312yes—
138#102×C137—22no—
139#102×C137—22no—
140#102×C137—22no—
141#102×C137—22no—
142#102×C137—24no—
143C22⋊C8×C137—21no—
144C22⋊C8×C137—21no—
145#97×C137—21no—
146C4×C8×C137C2312yes—
147C24×C137—41no—
148#41×C137—22no—
149#41×C137—41no—
150C2×Q8×C137—22no—
151C2×Q8×C137—22no—
152C2×Q8×C137—21no—
153C2×Q8×C137—42no—
154C2×Q8×C137—42no—
155C2×Q8×C137C2×D411yescharacteristic
156C2×Q8×C137C2×D411yescharacteristic
157C2×Q8×C137C2×D412yes—
158C2×D4×C137—21no—
159C2×D4×C137—22no—
160C2×D4×C137—21no—
161C2×D4×C137—22no—
162C2×D4×C137—22no—
163C2×D4×C137—22no—
164C2×D4×C137—42no—
165C2×D4×C137—42no—
166C2×D4×C137—41no—
167C2×D4×C137—41no—
168C2×D4×C137—41no—
169C2×D4×C137—82no—
170C2×D4×C137—82no—
171C2×D4×C137C2×D411yescharacteristic
172C2×D4×C137C2×D411yescharacteristic
173C22×C4×C137—22no—
174C22×C4×C137—21no—
175C22×C4×C137—21no—
176C22×C4×C137—21no—
177C22×C4×C137—21no—
178C22×C4×C137C2×D411yescharacteristic
179C22×C4×C137C2×D411yescharacteristic
180C22×C4×C137C2×D411yescharacteristic
181SD16×C137—44no—
182SD16×C137—82no—
183SD16×C137—84no—
184D8×C137—44no—
185D8×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—22no—
197C4⋊C4×C137—21no—
198C4⋊C4×C137—21no—
199C4⋊C4×C137—22no—
200C4⋊C4×C137—21no—
201C4⋊C4×C137—21no—
202C4⋊C4×C137—21no—
203C4⋊C4×C137—21no—
204C4⋊C4×C137—21no—
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—22no—
214C22⋊C4×C137—22no—
215C22⋊C4×C137—22no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—21no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—21no—
220C22⋊C4×C137—21no—
221C22⋊C4×C137—21no—
222C22⋊C4×C137—21no—
223C22⋊C4×C137—21no—
224C42×C137—22no—
225C42×C137—22no—
226C42×C137—22no—
227C42×C137—21no—
228C42×C137—21no—
229C42×C137—22no—
230C42×C137—21no—
231C42×C137—21no—
232C42×C137C2×D411yescharacteristic
233C42×C137C2×D411yescharacteristic
234C42×C137C2×D411yescharacteristic
235C42×C137C2411yescharacteristic
236C23×C137—21no—
237C23×C137—21no—
238C23×C137—42no—
239C23×C137—82no—
240C23×C137—82no—
241C23×C137—82no—
242C23×C137#12011yescharacteristic
243Q8×C137—21no—
244Q8×C137—22no—
245Q8×C137—42no—
246Q8×C137—44no—
247Q8×C137—41no—
248Q8×C137—41no—
249Q8×C137—82no—
250Q8×C137AGL(1,ℤ/8)12yes—
251D4×C137—21no—
252D4×C137—21no—
253D4×C137—22no—
254D4×C137—42no—
255D4×C137—44no—
256D4×C137—42no—
257D4×C137—42no—
258D4×C137—42no—
259D4×C137—42no—
260D4×C137—82no—
261D4×C137—82no—
262D4×C137—82no—
263D4×C137—81no—
264D4×C137—82no—
265C2×C4×C137—21no—
266C2×C4×C137—21no—
267C2×C4×C137—21no—
268C2×C4×C137—22no—
269C2×C4×C137—21no—
270C2×C4×C137—21no—
271C2×C4×C137—22no—
272C2×C4×C137—21no—
273C2×C4×C137—21no—
274C2×C4×C137—22no—
275C2×C4×C137—21no—
276C2×C4×C137—41no—
277C2×C4×C137—42no—
278C2×C4×C137—44no—
279C2×C4×C137—41no—
280C2×C4×C137—41no—
281C2×C4×C137—42no—
282C2×C4×C137—81no—
283C2×C4×C137#12011yescharacteristic
284C2×C4×C137#12011yescharacteristic
285C2×C4×C137#12011yescharacteristic
286C2×C4×C137C22×D411yescharacteristic
287C2×C4×C137C22×D411yescharacteristic
288C2×C4×C137C22×D411yescharacteristic
289C8×C137—42no—
290C8×C137—42no—
291C8×C137—42no—
292C22×C137—21no—
293C22×C137—21no—
294C22×C137—21no—
295C22×C137—41no—
296C22×C137—41no—
297C22×C137—41no—
298C22×C137—41no—
299C22×C137—41no—
300C22×C137—41no—
301C22×C137—82no—
302C22×C137—82no—
303C22×C137—82no—
304C22×C137—164no—
305C22×C137C2×#12011yescharacteristic
306C4×C137—21no—
307C4×C137—22no—
308C4×C137—21no—
309C4×C137—21no—
310C4×C137—21no—
311C4×C137—42no—
312C4×C137—41no—
313C4×C137—41no—
314C4×C137—41no—
315C4×C137—41no—
316C4×C137—42no—
317C4×C137—41no—
318C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
319C4×C137#57511yescharacteristic
320C4×C137#57712yes—
321C2×C137—41no—
322C2×C137—81no—
323C2×C137—81no—
324C2×C137—162no—
325C2×C137#300611yescharacteristic
326C2×C137#301011yescharacteristic
327C2×C137#302111yescharacteristic
328#18852C13711yesmaximal, characteristic, direct product factor, Sylow
329C137#1885211yescharacteristic, direct product factor, Sylow
330#3518C2×C13711yescharacteristic, complemented
331#3217C2×C13711yescharacteristic, complemented
332#3208C2×C13711yescharacteristic, complemented
333#3038C2×C13711yescharacteristic, complemented
334#1628C2×C13712yescomplemented
335#1627C2×C13712yescomplemented
336#1624C2×C13712yescomplemented
337#1623C2×C13712yescomplemented
338#1559C2×C13711yescharacteristic, complemented
339#1495C2×C13711yescharacteristic, complemented
340#1488C2×C13711yescharacteristic, complemented
341C2×AGL(1,ℤ/8)—21no—
342Q82C22×C13711yescharacteristic, complemented
343#556—21no—
344#556—22no—
345#556—21no—
346#556C22×C13711yescharacteristic, complemented
347#555—21no—
348#555C22×C13711yescharacteristic, complemented
349D4×Q8C22×C13711yescharacteristic, complemented
350#548—21no—
351#542—22no—
352#542—21no—
353#533—21no—
354#533C22×C13711yescharacteristic
355C2×#127C22×C13711yescharacteristic
356#529—21no—
357#529C22×C13711yescharacteristic
358#519—21no—
359C4×#41C22×C13711yescharacteristic
360#496—21no—
361#492C22×C13712yes—
362#490—21no—
363#486C22×C13712yes—
364#460—22no—
365#460—21no—
366#460C22×C13712yes—
367#460C22×C13712yes—
368#459—21no—
369#459C22×C13712yes—
370#459C22×C13712yes—
371#450—22no—
372#450—21no—
373#447—21no—
374#432C22×C13711yescharacteristic
375#431C22×C13711yescharacteristic
376#423C22×C13711yescharacteristic
377#418—21no—
378#333C22×C13712yescomplemented
379#332C22×C13712yescomplemented
380Q8⋊C8C22×C13712yescomplemented
381Q8⋊C8C22×C13712yescomplemented
382D4⋊C8C22×C13712yescomplemented
383D4⋊C8C22×C13712yescomplemented
384C2×#41D4×C13711yescharacteristic
385C22×Q8—21no—
386C22×D4—21no—
387C22×D4—21no—
388C22×D4—21no—
389AGL(1,ℤ/8)—44no—
390C2×SD16—22no—
391C2×SD16—24no—
392C2×SD16—22no—
393C2×D8—22no—
394C2×D8—22no—
395C2×M16—21no—
396C4⋊Q8—22no—
397C4⋊Q8—21no—
398C4⋊Q8—21no—
399C4⋊Q8—21no—
400C4⋊Q8C23×C13711yescharacteristic
401C4⋊Q8C23×C13711yescharacteristic
402#127—21no—
403#127—21no—
404#127—21no—
405#127C23×C13712yes—
406#126—22no—
407#126—22no—
408#126—22no—
409#126—22no—
410#126—21no—
411#125—21no—
412#125D4×C13711yescharacteristic
413#124—22no—
414#124—22no—
415#124—22no—
416#124—22no—
417#124—21no—
418#124D4×C13711yescharacteristic
419#123—22no—
420#123—21no—
421C22⋊Q8—22no—
422C22⋊Q8—22no—
423C22⋊Q8—21no—
424C22⋊Q8—21no—
425C22⋊Q8—21no—
426C22⋊Q8—22no—
427C22⋊Q8D4×C13711yescharacteristic
428C22⋊Q8D4×C13712yes—
429#121—22no—
430#121—21no—
431#121—21no—
432#121D4×C13711yescharacteristic
433C4×Q8—22no—
434C4×Q8—22no—
435C4×Q8—22no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8—21no—
440C4×Q8D4×C13711yescharacteristic
441C4×Q8D4×C13712yes—
442C4×Q8C23×C13711yescharacteristic
443C4×Q8C23×C13711yescharacteristic
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4D4×C13711yescharacteristic
450C4×D4C23×C13711yescharacteristic
451C4×D4C23×C13711yescharacteristic
452#117—22no—
453#117—22no—
454#117—21no—
455#117—21no—
456#117—21no—
457#117—21no—
458#117—21no—
459#117—21no—
460#117D4×C13711yescharacteristic
461C2×C42—21no—
462C2×C42C23×C13711yescharacteristic
463C4⋊C8—21no—
464C4⋊C8—21no—
465C4⋊C8C23×C13712yes—
466C4⋊C8C23×C13712yes—
467#102—22no—
468#102—22no—
469#102—22no—
470#102—22no—
471#102—24no—
472C22⋊C8—21no—
473C22⋊C8—21no—
474#97—21no—
475C4×C8C23×C13712yes—
476C24—41no—
477#41—22no—
478#41—41no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8C2×D4×C13711yescharacteristic
485C2×Q8C2×D4×C13711yescharacteristic
486C2×Q8C2×D4×C13712yes—
487C2×D4—21no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—41no—
496C2×D4—41no—
497C2×D4—41no—
498C2×D4—82no—
499C2×D4—82no—
500C2×D4C2×D4×C13711yescharacteristic
501C2×D4C2×D4×C13711yescharacteristic
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4C2×D4×C13711yescharacteristic
508C22×C4C2×D4×C13711yescharacteristic
509C22×C4C2×D4×C13711yescharacteristic
510SD16—44no—
511SD16—82no—
512SD16—84no—
513D8—44no—
514D8—82no—
515M16—42no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4C2×D4×C13711yescharacteristic
535C4⋊C4C2×D4×C13711yescharacteristic
536C4⋊C4C2×D4×C13711yescharacteristic
537C4⋊C4C2×D4×C13711yescharacteristic
538C4⋊C4C2×D4×C13712yes—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42C2×D4×C13711yescharacteristic
562C42C2×D4×C13711yescharacteristic
563C42C2×D4×C13711yescharacteristic
564C42C24×C13711yescharacteristic
565C23—21no—
566C23—21no—
567C23—42no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23#120×C13711yescharacteristic
572Q8—21no—
573Q8—22no—
574Q8—42no—
575Q8—44no—
576Q8—41no—
577Q8—41no—
578Q8—82no—
579Q8AGL(1,ℤ/8)×C13712yes—
580D4—21no—
581D4—21no—
582D4—22no—
583D4—42no—
584D4—44no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—82no—
590D4—82no—
591D4—82no—
592D4—81no—
593D4—82no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—81no—
612C2×C4#120×C13711yescharacteristic
613C2×C4#120×C13711yescharacteristic
614C2×C4#120×C13711yescharacteristic
615C2×C4C22×D4×C13711yescharacteristic
616C2×C4C22×D4×C13711yescharacteristic
617C2×C4C22×D4×C13711yescharacteristic
618C8—42no—
619C8—42no—
620C8—42no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—164no—
634C22C2×#120×C13711yescharacteristic
635C4—21no—
636C4—22no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
648C4#575×C13711yescharacteristic
649C4#577×C13712yes—
650C2—41no—
651C2—81no—
652C2—81no—
653C2—162no—
654C2#3006×C13711yescharacteristic
655C2#3010×C13711yescharacteristic
656C2#3021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
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
1. #3518×C137C2semidirect
2. #3217×C137C2semidirect
3. #3208×C137C2semidirect
4. #3038×C137C2semidirect
5. #1628×C137C2semidirect
6. #1627×C137C2semidirect
7. #1624×C137C2semidirect
8. #1623×C137C2semidirect
9. #1559×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
13. Q82×C137C22semidirect
17. #556×C137C22semidirect
19. #555×C137C22semidirect
20. D4×Q8×C137C22semidirect
49. #333×C137C22semidirect
50. #332×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. Q8⋊C8×C137C22semidirect
53. D4⋊C8×C137C22semidirect
54. D4⋊C8×C137C22semidirect
328. #18852C137direct (N × H)
329. C137#18852direct (N × H)
330. #3518C2×C137semidirect
331. #3217C2×C137semidirect
332. #3208C2×C137semidirect
333. #3038C2×C137semidirect
334. #1628C2×C137semidirect
335. #1627C2×C137semidirect
336. #1624C2×C137semidirect
337. #1623C2×C137semidirect
338. #1559C2×C137semidirect
339. #1495C2×C137semidirect
340. #1488C2×C137semidirect
342. Q82C22×C137semidirect
346. #556C22×C137semidirect
348. #555C22×C137semidirect
349. D4×Q8C22×C137semidirect
378. #333C22×C137semidirect
379. #332C22×C137semidirect
380. Q8⋊C8C22×C137semidirect
381. Q8⋊C8C22×C137semidirect
382. D4⋊C8C22×C137semidirect
383. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3518×C1372
#3217×C1372
#3208×C1372
#3038×C1372
#1628×C1372
#1627×C1372
#1624×C1372
#1623×C1372
#1559×C1372
#1495×C1372
#1488×C1372
#18852137