Finite Group Databasebeta

← C137×#18835 #46868221

Subgroups — C137×#18835 #46868221

Order 35072.

940 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514×C137C211yesmaximal, characteristic
2#3238×C137C211yesmaximal, characteristic, complemented
3#3206×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1638×C137C212yesmaximal, complemented
6#1637×C137C212yesmaximal, complemented
7#1630×C137C212yesmaximal, complemented
8#1629×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#501×C137—21no—
31#492×C137C2212yes—
32#489×C137—21no—
33#487×C137C2212yescomplemented
34#483×C137—21no—
35#480×C137—21no—
36#477×C137C2212yescomplemented
37#477×C137C2212yescomplemented
38#476×C137C2212yes—
39#476×C137C2212yes—
40#475×C137—21no—
41#474×C137—21no—
42#461×C137—22no—
43#448×C137—22no—
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×SD16×C137—24no—
62C2×M16×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137—21no—
67C4⋊Q8×C137C2312yes—
68C4⋊Q8×C137C2311yescharacteristic
69#127×C137C2311yescharacteristic
70#126×C137—22no—
71#126×C137—22no—
72#126×C137—21no—
73#125×C137D411yescharacteristic
74#124×C137—22no—
75#124×C137—22no—
76#124×C137—22no—
77#124×C137—21no—
78#124×C137—21no—
79#124×C137—21no—
80#124×C137—21no—
81#124×C137D411yescharacteristic
82#123×C137—22no—
83#123×C137—22no—
84#123×C137—22no—
85#123×C137—21no—
86C22⋊Q8×C137—22no—
87C22⋊Q8×C137—21no—
88C22⋊Q8×C137—21no—
89C22⋊Q8×C137—22no—
90C22⋊Q8×C137D411yescharacteristic
91#121×C137—22no—
92#121×C137—22no—
93#121×C137—21no—
94#121×C137—21no—
95#121×C137—21no—
96#121×C137D412yes—
97#121×C137D411yescharacteristic
98#120×C137—22no—
99#120×C137—22no—
100C4×Q8×C137—21no—
101C4×Q8×C137D411yescharacteristic
102C4×Q8×C137C2311yescharacteristic
103C4×Q8×C137C2311yescharacteristic
104C4×D4×C137—22no—
105C4×D4×C137—22no—
106C4×D4×C137—21no—
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×C137D412yes—
114C4×D4×C137D411yescharacteristic
115C4×D4×C137C2311yescharacteristic
116C4×D4×C137C2311yescharacteristic
117#117×C137—22no—
118#117×C137—21no—
119#117×C137—21no—
120#117×C137—21no—
121#117×C137D411yescharacteristic
122C2×C4⋊C4×C137—21no—
123C2×C4⋊C4×C137—21no—
124C2×C22⋊C4×C137—22no—
125C2×C22⋊C4×C137—22no—
126C2×C22⋊C4×C137—22no—
127C2×C42×C137—21no—
128C2×C42×C137C2311yescharacteristic
129#107×C137—22no—
130#107×C137—22no—
131#106×C137—22no—
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—24no—
140#102×C137—22no—
141#102×C137—22no—
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
182SD16×C137—84no—
183M16×C137—42no—
184C2×C8×C137—22no—
185C2×C8×C137—22no—
186C2×C8×C137—22no—
187C4⋊C4×C137—22no—
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—21no—
195C4⋊C4×C137—21no—
196C4⋊C4×C137—22no—
197C4⋊C4×C137—22no—
198C4⋊C4×C137—41no—
199C4⋊C4×C137—41no—
200C4⋊C4×C137C2×D412yes—
201C4⋊C4×C137C2×D411yescharacteristic
202C4⋊C4×C137C2×D411yescharacteristic
203C4⋊C4×C137C2×D411yescharacteristic
204C4⋊C4×C137C2×D411yescharacteristic
205C22⋊C4×C137—22no—
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—21no—
213C22⋊C4×C137—21no—
214C22⋊C4×C137—21no—
215C22⋊C4×C137—21no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—21no—
218C22⋊C4×C137—22no—
219C22⋊C4×C137—22no—
220C22⋊C4×C137—21no—
221C22⋊C4×C137—21no—
222C42×C137—22no—
223C42×C137—21no—
224C42×C137—21no—
225C42×C137—21no—
226C42×C137—21no—
227C42×C137C2×D411yescharacteristic
228C42×C137C2×D411yescharacteristic
229C42×C137C2×D411yescharacteristic
230C42×C137C2411yescharacteristic
231C23×C137—21no—
232C23×C137—21no—
233C23×C137—22no—
234C23×C137—44no—
235C23×C137—82no—
236C23×C137—82no—
237C23×C137#12011yescharacteristic
238Q8×C137—21no—
239Q8×C137—21no—
240Q8×C137—22no—
241Q8×C137—44no—
242Q8×C137—42no—
243Q8×C137—82no—
244Q8×C137—82no—
245D4×C137—21no—
246D4×C137—22no—
247D4×C137—42no—
248D4×C137—42no—
249D4×C137—44no—
250D4×C137—41no—
251D4×C137—41no—
252D4×C137—42no—
253D4×C137—81no—
254D4×C137—82no—
255D4×C137—82no—
256D4×C137#13712yes—
257C2×C4×C137—21no—
258C2×C4×C137—21no—
259C2×C4×C137—22no—
260C2×C4×C137—21no—
261C2×C4×C137—21no—
262C2×C4×C137—21no—
263C2×C4×C137—21no—
264C2×C4×C137—21no—
265C2×C4×C137—21no—
266C2×C4×C137—22no—
267C2×C4×C137—41no—
268C2×C4×C137—42no—
269C2×C4×C137—41no—
270C2×C4×C137—41no—
271C2×C4×C137—42no—
272C2×C4×C137—42no—
273C2×C4×C137—81no—
274C2×C4×C137—82no—
275C2×C4×C137—82no—
276C2×C4×C137—82no—
277C2×C4×C137—82no—
278C2×C4×C137#12011yescharacteristic
279C2×C4×C137#12011yescharacteristic
280C2×C4×C137#12011yescharacteristic
281C2×C4×C137C22×D411yescharacteristic
282C2×C4×C137C22×D411yescharacteristic
283C2×C4×C137C22×D411yescharacteristic
284C8×C137—42no—
285C8×C137—42no—
286C8×C137—42no—
287C22×C137—21no—
288C22×C137—22no—
289C22×C137—21no—
290C22×C137—21no—
291C22×C137—41no—
292C22×C137—41no—
293C22×C137—41no—
294C22×C137—42no—
295C22×C137—42no—
296C22×C137—41no—
297C22×C137—42no—
298C22×C137—41no—
299C22×C137—164no—
300C22×C137C2×#12011yescharacteristic
301C4×C137—21no—
302C4×C137—21no—
303C4×C137—21no—
304C4×C137—21no—
305C4×C137—41no—
306C4×C137—41no—
307C4×C137—41no—
308C4×C137—42no—
309C4×C137—41no—
310C4×C137—41no—
311C4×C137—41no—
312C4×C137—82no—
313C4×C137C2×#13711yescharacteristic
314C4×C137#57511yescharacteristic
315C4×C137#57712yes—
316C2×C137—41no—
317C2×C137—42no—
318C2×C137—82no—
319C2×C137—81no—
320C2×C137#300711yescharacteristic
321C2×C137#300911yescharacteristic
322C2×C137#302111yescharacteristic
323#18835C13711yesmaximal, characteristic, direct product factor, Sylow
324C137#1883511yescharacteristic, direct product factor, Sylow
325#3514C2×C13711yescharacteristic
326#3238C2×C13711yescharacteristic, complemented
327#3206C2×C13711yescharacteristic, complemented
328#3079C2×C13711yescharacteristic, complemented
329#1638C2×C13712yescomplemented
330#1637C2×C13712yescomplemented
331#1630C2×C13712yescomplemented
332#1629C2×C13712yescomplemented
333#1557C2×C13711yescharacteristic, complemented
334#1495C2×C13711yescharacteristic, complemented
335#1488C2×C13711yescharacteristic, complemented
336#559C22×C13711yescharacteristic
337#556C22×C13711yescharacteristic
338#555—21no—
339#551—22no—
340D4×Q8C22×C13711yescharacteristic
341#548—21no—
342#546—21no—
343#546—21no—
344D42C22×C13711yescharacteristic
345#542—21no—
346#538—22no—
347#532C22×C13711yescharacteristic
348C2×C4⋊Q8C22×C13711yescharacteristic, complemented
349#529—21no—
350#529C22×C13711yescharacteristic
351C2×#124—21no—
352#518—21no—
353C4×#41C22×C13711yescharacteristic
354#501—21no—
355#492C22×C13712yes—
356#489—21no—
357#487C22×C13712yescomplemented
358#483—21no—
359#480—21no—
360#477C22×C13712yescomplemented
361#477C22×C13712yescomplemented
362#476C22×C13712yes—
363#476C22×C13712yes—
364#475—21no—
365#474—21no—
366#461—22no—
367#448—22no—
368#432C22×C13711yescharacteristic, complemented
369#431C22×C13711yescharacteristic, complemented
370#428—21no—
371#423C22×C13711yescharacteristic, complemented
372#417—21no—
373#332C22×C13712yescomplemented
374#331C22×C13712yes—
375Q8⋊C8C22×C13712yescomplemented
376Q8⋊C8C22×C13712yescomplemented
377D4⋊C8C22×C13712yes—
378D4⋊C8C22×C13712yes—
379C2×#41—21no—
380C2×#41D4×C13711yescharacteristic
381C22×Q8—21no—
382C22×Q8—21no—
383C22×D4—21no—
384C22×D4—21no—
385C2×SD16—24no—
386C2×M16—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8—21no—
391C4⋊Q8C23×C13712yes—
392C4⋊Q8C23×C13711yescharacteristic
393#127C23×C13711yescharacteristic
394#126—22no—
395#126—22no—
396#126—21no—
397#125D4×C13711yescharacteristic
398#124—22no—
399#124—22no—
400#124—22no—
401#124—21no—
402#124—21no—
403#124—21no—
404#124—21no—
405#124D4×C13711yescharacteristic
406#123—22no—
407#123—22no—
408#123—22no—
409#123—21no—
410C22⋊Q8—22no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—22no—
414C22⋊Q8D4×C13711yescharacteristic
415#121—22no—
416#121—22no—
417#121—21no—
418#121—21no—
419#121—21no—
420#121D4×C13711yescharacteristic
421#121D4×C13712yes—
422#120—22no—
423#120—22no—
424C4×Q8—21no—
425C4×Q8D4×C13711yescharacteristic
426C4×Q8C23×C13711yescharacteristic
427C4×Q8C23×C13711yescharacteristic
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4D4×C13711yescharacteristic
438C4×D4D4×C13712yes—
439C4×D4C23×C13711yescharacteristic
440C4×D4C23×C13711yescharacteristic
441#117—22no—
442#117—21no—
443#117—21no—
444#117—21no—
445#117D4×C13711yescharacteristic
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—22no—
451C2×C42—21no—
452C2×C42C23×C13711yescharacteristic
453#107—22no—
454#107—22no—
455#106—22no—
456#106—22no—
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8C23×C13712yes—
460C4⋊C8C23×C13712yes—
461#103—22no—
462#103—22no—
463#103—24no—
464#102—22no—
465#102—22no—
466C22⋊C8—21no—
467C22⋊C8—21no—
468#97—21no—
469C4×C8C23×C13712yes—
470C24—22no—
471#41—22no—
472#41—41no—
473#41—82no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—41no—
478C2×Q8—82no—
479C2×Q8—82no—
480C2×Q8C2×D4×C13711yescharacteristic
481C2×Q8C2×D4×C13711yescharacteristic
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—82no—
492C2×D4C2×D4×C13711yescharacteristic
493C2×D4C2×D4×C13711yescharacteristic
494C2×D4C2×D4×C13712yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—41no—
503C22×C4C2×D4×C13711yescharacteristic
504C22×C4C2×D4×C13711yescharacteristic
505C22×C4C2×D4×C13711yescharacteristic
506SD16—84no—
507M16—42no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4C2×D4×C13712yes—
525C4⋊C4C2×D4×C13711yescharacteristic
526C4⋊C4C2×D4×C13711yescharacteristic
527C4⋊C4C2×D4×C13711yescharacteristic
528C4⋊C4C2×D4×C13711yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42C2×D4×C13711yescharacteristic
552C42C2×D4×C13711yescharacteristic
553C42C2×D4×C13711yescharacteristic
554C42C24×C13711yescharacteristic
555C23—21no—
556C23—21no—
557C23—22no—
558C23—44no—
559C23—82no—
560C23—82no—
561C23#120×C13711yescharacteristic
562Q8—21no—
563Q8—21no—
564Q8—22no—
565Q8—44no—
566Q8—42no—
567Q8—82no—
568Q8—82no—
569D4—21no—
570D4—22no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—41no—
575D4—41no—
576D4—42no—
577D4—82no—
578D4—81no—
579D4—82no—
580D4#137×C13712yes—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4#120×C13711yescharacteristic
603C2×C4#120×C13711yescharacteristic
604C2×C4#120×C13711yescharacteristic
605C2×C4C22×D4×C13711yescharacteristic
606C2×C4C22×D4×C13711yescharacteristic
607C2×C4C22×D4×C13711yescharacteristic
608C8—42no—
609C8—42no—
610C8—42no—
611C22—21no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—164no—
624C22C2×#120×C13711yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—82no—
637C4C2×#137×C13711yescharacteristic
638C4#575×C13711yescharacteristic
639C4#577×C13712yes—
640C2—41no—
641C2—42no—
642C2—81no—
643C2—82no—
644C2#3007×C13711yescharacteristic
645C2#3009×C13711yescharacteristic
646C2#3021×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. #3238×C137C2semidirect
3. #3206×C137C2semidirect
4. #3079×C137C2semidirect
5. #1638×C137C2semidirect
6. #1637×C137C2semidirect
7. #1630×C137C2semidirect
8. #1629×C137C2semidirect
9. #1557×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
24. C2×C4⋊Q8×C137C22semidirect
33. #487×C137C22semidirect
36. #477×C137C22semidirect
37. #477×C137C22semidirect
44. #432×C137C22semidirect
45. #431×C137C22semidirect
47. #423×C137C22semidirect
49. #332×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. Q8⋊C8×C137C22semidirect
323. #18835C137direct (N × H)
324. C137#18835direct (N × H)
326. #3238C2×C137semidirect
327. #3206C2×C137semidirect
328. #3079C2×C137semidirect
329. #1638C2×C137semidirect
330. #1637C2×C137semidirect
331. #1630C2×C137semidirect
332. #1629C2×C137semidirect
333. #1557C2×C137semidirect
334. #1495C2×C137semidirect
335. #1488C2×C137semidirect
348. C2×C4⋊Q8C22×C137semidirect
357. #487C22×C137semidirect
360. #477C22×C137semidirect
361. #477C22×C137semidirect
368. #432C22×C137semidirect
369. #431C22×C137semidirect
371. #423C22×C137semidirect
373. #332C22×C137semidirect
375. Q8⋊C8C22×C137semidirect
376. 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
#3238×C1372
#3206×C1372
#3079×C1372
#1638×C1372
#1637×C1372
#1630×C1372
#1629×C1372
#1557×C1372
#1495×C1372
#1488×C1372
#18835137