Finite Group Databasebeta

← C137×#18843 #46868229

Subgroups — C137×#18843 #46868229

Order 35072.

940 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514×C137C211yesmaximal, characteristic
2#3226×C137C211yesmaximal, characteristic, complemented
3#3218×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1650×C137C212yesmaximal, complemented
6#1649×C137C212yesmaximal, complemented
7#1640×C137C212yesmaximal, complemented
8#1639×C137C212yesmaximal, complemented
9#1557×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12#559×C137C2211yescharacteristic
13#556×C137C2211yescharacteristic
14#555×C137—21no—
15#551×C137—22no—
16D4×Q8×C137C2211yescharacteristic
17#548×C137—21no—
18#546×C137—21no—
19#546×C137—21no—
20D42×C137C2211yescharacteristic
21#542×C137—21no—
22#538×C137—22no—
23#532×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic, complemented
25#529×C137—21no—
26#529×C137C2211yescharacteristic
27C2×#124×C137—21no—
28#518×C137—21no—
29C4×#41×C137C2211yescharacteristic
30#500×C137C2212yescomplemented
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×C137C2212yescomplemented
38#474×C137—22no—
39#474×C137C2212yes—
40#462×C137C2212yescomplemented
41#461×C137C2212yes—
42#451×C137—21no—
43#448×C137—21no—
44#432×C137C2211yescharacteristic, complemented
45#431×C137C2211yescharacteristic, complemented
46#428×C137—21no—
47#423×C137C2211yescharacteristic, complemented
48#417×C137—21no—
49#332×C137C2212yescomplemented
50#331×C137C2212yes—
51Q8⋊C8×C137C2212yescomplemented
52Q8⋊C8×C137C2212yescomplemented
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137—21no—
56C2×#41×C137D411yescharacteristic
57C22×Q8×C137—21no—
58C22×Q8×C137—21no—
59C22×D4×C137—21no—
60C22×D4×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×C137C2311yescharacteristic
71#126×C137—22no—
72#126×C137—22no—
73#126×C137—21no—
74#125×C137D411yescharacteristic
75#124×C137—22no—
76#124×C137—22no—
77#124×C137—22no—
78#124×C137—21no—
79#124×C137—21no—
80#124×C137—21no—
81#124×C137—21no—
82#124×C137D411yescharacteristic
83#123×C137—22no—
84#123×C137—22no—
85#123×C137—22no—
86#123×C137—21no—
87C22⋊Q8×C137—22no—
88C22⋊Q8×C137—21no—
89C22⋊Q8×C137—21no—
90C22⋊Q8×C137—22no—
91C22⋊Q8×C137D411yescharacteristic
92#121×C137—22no—
93#121×C137—22no—
94#121×C137—21no—
95#121×C137—21no—
96#121×C137—21no—
97#121×C137D411yescharacteristic
98#121×C137D412yes—
99#120×C137—22no—
100#120×C137—22no—
101C4×Q8×C137—21no—
102C4×Q8×C137D411yescharacteristic
103C4×Q8×C137C2311yescharacteristic
104C4×Q8×C137C2311yescharacteristic
105C4×D4×C137—22no—
106C4×D4×C137—22no—
107C4×D4×C137—21no—
108C4×D4×C137—21no—
109C4×D4×C137—21no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137D412yes—
115C4×D4×C137D411yescharacteristic
116C4×D4×C137C2311yescharacteristic
117C4×D4×C137C2311yescharacteristic
118#117×C137—22no—
119#117×C137—21no—
120#117×C137—21no—
121#117×C137—21no—
122#117×C137D411yescharacteristic
123C2×C4⋊C4×C137—21no—
124C2×C4⋊C4×C137—21no—
125C2×C22⋊C4×C137—22no—
126C2×C22⋊C4×C137—22no—
127C2×C22⋊C4×C137—22no—
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—
146C24×C137—22no—
147#41×C137—22no—
148#41×C137—41no—
149#41×C137—82no—
150C2×Q8×C137—22no—
151C2×Q8×C137—21no—
152C2×Q8×C137—22no—
153C2×Q8×C137—41no—
154C2×Q8×C137—82no—
155C2×Q8×C137—82no—
156C2×Q8×C137C2×D411yescharacteristic
157C2×Q8×C137C2×D411yescharacteristic
158C2×D4×C137—22no—
159C2×D4×C137—21no—
160C2×D4×C137—22no—
161C2×D4×C137—21no—
162C2×D4×C137—22no—
163C2×D4×C137—21no—
164C2×D4×C137—21no—
165C2×D4×C137—42no—
166C2×D4×C137—42no—
167C2×D4×C137—82no—
168C2×D4×C137C2×D411yescharacteristic
169C2×D4×C137C2×D411yescharacteristic
170C2×D4×C137C2×D412yes—
171C22×C4×C137—22no—
172C22×C4×C137—22no—
173C22×C4×C137—21no—
174C22×C4×C137—21no—
175C22×C4×C137—21no—
176C22×C4×C137—21no—
177C22×C4×C137—22no—
178C22×C4×C137—41no—
179C22×C4×C137C2×D411yescharacteristic
180C22×C4×C137C2×D411yescharacteristic
181C22×C4×C137C2×D411yescharacteristic
182Q16×C137—82no—
183SD16×C137—82no—
184M16×C137—42no—
185C2×C8×C137—22no—
186C2×C8×C137—22no—
187C2×C8×C137—22no—
188C4⋊C4×C137—22no—
189C4⋊C4×C137—22no—
190C4⋊C4×C137—21no—
191C4⋊C4×C137—21no—
192C4⋊C4×C137—21no—
193C4⋊C4×C137—21no—
194C4⋊C4×C137—21no—
195C4⋊C4×C137—21no—
196C4⋊C4×C137—21no—
197C4⋊C4×C137—22no—
198C4⋊C4×C137—22no—
199C4⋊C4×C137—41no—
200C4⋊C4×C137—41no—
201C4⋊C4×C137C2×D411yescharacteristic
202C4⋊C4×C137C2×D411yescharacteristic
203C4⋊C4×C137C2×D411yescharacteristic
204C4⋊C4×C137C2×D411yescharacteristic
205C4⋊C4×C137C2×D412yes—
206C22⋊C4×C137—22no—
207C22⋊C4×C137—22no—
208C22⋊C4×C137—22no—
209C22⋊C4×C137—22no—
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—21no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—22no—
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—44no—
236C23×C137—82no—
237C23×C137—82no—
238C23×C137#12011yescharacteristic
239Q8×C137—21no—
240Q8×C137—21no—
241Q8×C137—22no—
242Q8×C137—42no—
243Q8×C137—42no—
244Q8×C137—42no—
245Q8×C137—82no—
246Q8×C137—82no—
247D4×C137—21no—
248D4×C137—22no—
249D4×C137—42no—
250D4×C137—42no—
251D4×C137—44no—
252D4×C137—41no—
253D4×C137—41no—
254D4×C137—42no—
255D4×C137—81no—
256D4×C137—82no—
257D4×C137—82no—
258D4×C137#13712yes—
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—21no—
267C2×C4×C137—21no—
268C2×C4×C137—22no—
269C2×C4×C137—41no—
270C2×C4×C137—42no—
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—22no—
291C22×C137—21no—
292C22×C137—21no—
293C22×C137—41no—
294C22×C137—41no—
295C22×C137—41no—
296C22×C137—42no—
297C22×C137—42no—
298C22×C137—41no—
299C22×C137—42no—
300C22×C137—41no—
301C22×C137—164no—
302C22×C137C2×#12011yescharacteristic
303C4×C137—21no—
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—41no—
314C4×C137—82no—
315C4×C137C2×#13711yescharacteristic
316C4×C137#57511yescharacteristic
317C4×C137#57612yes—
318C2×C137—41no—
319C2×C137—42no—
320C2×C137—82no—
321C2×C137—81no—
322C2×C137#300711yescharacteristic
323C2×C137#300811yescharacteristic
324C2×C137#302211yescharacteristic
325#18843C13711yesmaximal, characteristic, direct product factor, Sylow
326C137#1884311yescharacteristic, direct product factor, Sylow
327#3514C2×C13711yescharacteristic
328#3226C2×C13711yescharacteristic, complemented
329#3218C2×C13711yescharacteristic, complemented
330#3079C2×C13711yescharacteristic, complemented
331#1650C2×C13712yescomplemented
332#1649C2×C13712yescomplemented
333#1640C2×C13712yescomplemented
334#1639C2×C13712yescomplemented
335#1557C2×C13711yescharacteristic, complemented
336#1495C2×C13711yescharacteristic, complemented
337#1488C2×C13711yescharacteristic, complemented
338#559C22×C13711yescharacteristic
339#556C22×C13711yescharacteristic
340#555—21no—
341#551—22no—
342D4×Q8C22×C13711yescharacteristic
343#548—21no—
344#546—21no—
345#546—21no—
346D42C22×C13711yescharacteristic
347#542—21no—
348#538—22no—
349#532C22×C13711yescharacteristic
350C2×C4⋊Q8C22×C13711yescharacteristic, complemented
351#529—21no—
352#529C22×C13711yescharacteristic
353C2×#124—21no—
354#518—21no—
355C4×#41C22×C13711yescharacteristic
356#500C22×C13712yescomplemented
357#497—21no—
358#491—21no—
359#486C22×C13712yes—
360#480—22no—
361#477—21no—
362#476—21no—
363#475C22×C13712yescomplemented
364#474—22no—
365#474C22×C13712yes—
366#462C22×C13712yescomplemented
367#461C22×C13712yes—
368#451—21no—
369#448—21no—
370#432C22×C13711yescharacteristic, complemented
371#431C22×C13711yescharacteristic, complemented
372#428—21no—
373#423C22×C13711yescharacteristic, complemented
374#417—21no—
375#332C22×C13712yescomplemented
376#331C22×C13712yes—
377Q8⋊C8C22×C13712yescomplemented
378Q8⋊C8C22×C13712yescomplemented
379D4⋊C8C22×C13712yes—
380D4⋊C8C22×C13712yes—
381C2×#41—21no—
382C2×#41D4×C13711yescharacteristic
383C22×Q8—21no—
384C22×Q8—21no—
385C22×D4—21no—
386C22×D4—21no—
387C2×Q16—22no—
388C2×SD16—22no—
389C2×M16—21no—
390C4⋊Q8—21no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393C4⋊Q8—21no—
394C4⋊Q8C23×C13712yes—
395C4⋊Q8C23×C13711yescharacteristic
396#127C23×C13711yescharacteristic
397#126—22no—
398#126—22no—
399#126—21no—
400#125D4×C13711yescharacteristic
401#124—22no—
402#124—22no—
403#124—22no—
404#124—21no—
405#124—21no—
406#124—21no—
407#124—21no—
408#124D4×C13711yescharacteristic
409#123—22no—
410#123—22no—
411#123—22no—
412#123—21no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8D4×C13711yescharacteristic
418#121—22no—
419#121—22no—
420#121—21no—
421#121—21no—
422#121—21no—
423#121D4×C13711yescharacteristic
424#121D4×C13712yes—
425#120—22no—
426#120—22no—
427C4×Q8—21no—
428C4×Q8D4×C13711yescharacteristic
429C4×Q8C23×C13711yescharacteristic
430C4×Q8C23×C13711yescharacteristic
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4D4×C13711yescharacteristic
441C4×D4D4×C13712yes—
442C4×D4C23×C13711yescharacteristic
443C4×D4C23×C13711yescharacteristic
444#117—22no—
445#117—21no—
446#117—21no—
447#117—21no—
448#117D4×C13711yescharacteristic
449C2×C4⋊C4—21no—
450C2×C4⋊C4—21no—
451C2×C22⋊C4—22no—
452C2×C22⋊C4—22no—
453C2×C22⋊C4—22no—
454C2×C42—21no—
455C2×C42C23×C13711yescharacteristic
456#107—22no—
457#107—24no—
458#106—22no—
459C4⋊C8—21no—
460C4⋊C8—21no—
461C4⋊C8C23×C13712yes—
462C4⋊C8C23×C13712yes—
463#103—22no—
464#103—22no—
465#103—22no—
466#102—22no—
467#102—24no—
468C22⋊C8—21no—
469C22⋊C8—21no—
470#97—21no—
471C4×C8C23×C13712yes—
472C24—22no—
473#41—22no—
474#41—41no—
475#41—82no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—22no—
479C2×Q8—41no—
480C2×Q8—82no—
481C2×Q8—82no—
482C2×Q8C2×D4×C13711yescharacteristic
483C2×Q8C2×D4×C13711yescharacteristic
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—82no—
494C2×D4C2×D4×C13711yescharacteristic
495C2×D4C2×D4×C13711yescharacteristic
496C2×D4C2×D4×C13712yes—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—41no—
505C22×C4C2×D4×C13711yescharacteristic
506C22×C4C2×D4×C13711yescharacteristic
507C22×C4C2×D4×C13711yescharacteristic
508Q16—82no—
509SD16—82no—
510M16—42no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4C2×D4×C13711yescharacteristic
528C4⋊C4C2×D4×C13711yescharacteristic
529C4⋊C4C2×D4×C13711yescharacteristic
530C4⋊C4C2×D4×C13711yescharacteristic
531C4⋊C4C2×D4×C13712yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42C2×D4×C13711yescharacteristic
555C42C2×D4×C13711yescharacteristic
556C42C2×D4×C13711yescharacteristic
557C42C24×C13711yescharacteristic
558C23—21no—
559C23—21no—
560C23—22no—
561C23—44no—
562C23—82no—
563C23—82no—
564C23#120×C13711yescharacteristic
565Q8—21no—
566Q8—21no—
567Q8—22no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—82no—
572Q8—82no—
573D4—21no—
574D4—22no—
575D4—42no—
576D4—42no—
577D4—44no—
578D4—41no—
579D4—41no—
580D4—42no—
581D4—82no—
582D4—81no—
583D4—82no—
584D4#137×C13712yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4#120×C13711yescharacteristic
607C2×C4#120×C13711yescharacteristic
608C2×C4#120×C13711yescharacteristic
609C2×C4C22×D4×C13711yescharacteristic
610C2×C4C22×D4×C13711yescharacteristic
611C2×C4C22×D4×C13711yescharacteristic
612C8—42no—
613C8—42no—
614C8—42no—
615C22—21no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—41no—
627C22—164no—
628C22C2×#120×C13711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—82no—
641C4C2×#137×C13711yescharacteristic
642C4#575×C13711yescharacteristic
643C4#576×C13712yes—
644C2—41no—
645C2—42no—
646C2—81no—
647C2—82no—
648C2#3007×C13711yescharacteristic
649C2#3008×C13711yescharacteristic
650C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3514×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. #1650×C137C2semidirect
6. #1649×C137C2semidirect
7. #1640×C137C2semidirect
8. #1639×C137C2semidirect
9. #1557×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
24. C2×C4⋊Q8×C137C22semidirect
30. #500×C137C22semidirect
37. #475×C137C22semidirect
40. #462×C137C22semidirect
44. #432×C137C22semidirect
45. #431×C137C22semidirect
47. #423×C137C22semidirect
49. #332×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. Q8⋊C8×C137C22semidirect
325. #18843C137direct (N × H)
326. C137#18843direct (N × H)
328. #3226C2×C137semidirect
329. #3218C2×C137semidirect
330. #3079C2×C137semidirect
331. #1650C2×C137semidirect
332. #1649C2×C137semidirect
333. #1640C2×C137semidirect
334. #1639C2×C137semidirect
335. #1557C2×C137semidirect
336. #1495C2×C137semidirect
337. #1488C2×C137semidirect
350. C2×C4⋊Q8C22×C137semidirect
356. #500C22×C137semidirect
363. #475C22×C137semidirect
366. #462C22×C137semidirect
370. #432C22×C137semidirect
371. #431C22×C137semidirect
373. #423C22×C137semidirect
375. #332C22×C137semidirect
377. Q8⋊C8C22×C137semidirect
378. Q8⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3514×C1372
#3226×C1372
#3218×C1372
#3079×C1372
#1650×C1372
#1649×C1372
#1640×C1372
#1639×C1372
#1557×C1372
#1495×C1372
#1488×C1372
#18843137