Finite Group Databasebeta

← C137×#18851 #46868237

Subgroups — C137×#18851 #46868237

Order 35072.

930 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3517×C137C211yesmaximal, characteristic
2#3238×C137C211yesmaximal, characteristic, complemented
3#3206×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1636×C137C212yesmaximal, complemented
6#1635×C137C212yesmaximal, complemented
7#1631×C137C212yesmaximal, complemented
8#1631×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#501×C137—21no—
31#492×C137C2212yes—
32#489×C137—21no—
33#487×C137C2212yes—
34#483×C137—21no—
35#480×C137—21no—
36#477×C137C2212yes—
37#477×C137C2212yes—
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
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×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×C137—22no—
70#127×C137—21no—
71#127×C137—21no—
72#127×C137C2311yescharacteristic
73#126×C137—22no—
74#126×C137—21no—
75#125×C137—22no—
76#125×C137—21no—
77#125×C137D411yescharacteristic
78#124×C137—21no—
79#124×C137D411yescharacteristic
80#123×C137—22no—
81#123×C137—22no—
82#123×C137—22no—
83#123×C137—21no—
84#123×C137—22no—
85C22⋊Q8×C137—21no—
86C22⋊Q8×C137D411yescharacteristic
87#121×C137—22no—
88#121×C137—22no—
89#121×C137—22no—
90#121×C137—22no—
91#121×C137—22no—
92#121×C137—21no—
93#121×C137—22no—
94#121×C137—22no—
95#121×C137—21no—
96#121×C137—21no—
97#121×C137—21no—
98#121×C137D412yes—
99#121×C137D411yescharacteristic
100C4×Q8×C137—21no—
101C4×Q8×C137—21no—
102C4×Q8×C137D411yescharacteristic
103C4×Q8×C137C2311yescharacteristic
104C4×Q8×C137C2311yescharacteristic
105C4×D4×C137—22no—
106C4×D4×C137—22no—
107C4×D4×C137—22no—
108C4×D4×C137—22no—
109C4×D4×C137—21no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137—21no—
115C4×D4×C137D412yes—
116C4×D4×C137D411yescharacteristic
117C4×D4×C137C2311yescharacteristic
118C4×D4×C137C2311yescharacteristic
119#117×C137—22no—
120#117×C137—21no—
121#117×C137—21no—
122#117×C137D411yescharacteristic
123C2×C4⋊C4×C137—22no—
124C2×C4⋊C4×C137—21no—
125C2×C4⋊C4×C137—21no—
126C2×C4⋊C4×C137—21no—
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—
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
184SD16×C137—84no—
185M16×C137—42no—
186C2×C8×C137—22no—
187C2×C8×C137—22no—
188C2×C8×C137—22no—
189C4⋊C4×C137—22no—
190C4⋊C4×C137—22no—
191C4⋊C4×C137—22no—
192C4⋊C4×C137—22no—
193C4⋊C4×C137—21no—
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—22no—
201C4⋊C4×C137—22no—
202C4⋊C4×C137—41no—
203C4⋊C4×C137—41no—
204C4⋊C4×C137C2×D412yes—
205C4⋊C4×C137C2×D411yescharacteristic
206C4⋊C4×C137C2×D411yescharacteristic
207C4⋊C4×C137C2×D411yescharacteristic
208C4⋊C4×C137C2×D411yescharacteristic
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—22no—
217C22⋊C4×C137—21no—
218C22⋊C4×C137—21no—
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#12011yescharacteristic
235Q8×C137—21no—
236Q8×C137—22no—
237Q8×C137—44no—
238Q8×C137—42no—
239Q8×C137—82no—
240Q8×C137#13712yes—
241D4×C137—21no—
242D4×C137—21no—
243D4×C137—22no—
244D4×C137—42no—
245D4×C137—42no—
246D4×C137—44no—
247D4×C137—41no—
248D4×C137—41no—
249D4×C137—42no—
250D4×C137—81no—
251D4×C137—82no—
252D4×C137—82no—
253D4×C137—82no—
254D4×C137—82no—
255D4×C137—82no—
256C2×C4×C137—21no—
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—22no—
265C2×C4×C137—21no—
266C2×C4×C137—41no—
267C2×C4×C137—42no—
268C2×C4×C137—44no—
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—21no—
289C22×C137—21no—
290C22×C137—41no—
291C22×C137—42no—
292C22×C137—41no—
293C22×C137—41no—
294C22×C137—41no—
295C22×C137—42no—
296C22×C137—41no—
297C22×C137—42no—
298C22×C137—41no—
299C22×C137C2×#12011yescharacteristic
300C4×C137—21no—
301C4×C137—22no—
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—82no—
312C4×C137C2×#13711yescharacteristic
313C4×C137#57511yescharacteristic
314C4×C137#57712yes—
315C2×C137—41no—
316C2×C137—81no—
317C2×C137—81no—
318C2×C137—82no—
319C2×C137#300711yescharacteristic
320C2×C137#300911yescharacteristic
321C2×C137#302111yescharacteristic
322#18851C13711yesmaximal, characteristic, direct product factor, Sylow
323C137#1885111yescharacteristic, direct product factor, Sylow
324#3517C2×C13711yescharacteristic
325#3238C2×C13711yescharacteristic, complemented
326#3206C2×C13711yescharacteristic, complemented
327#3079C2×C13711yescharacteristic, complemented
328#1636C2×C13712yescomplemented
329#1635C2×C13712yescomplemented
330#1631C2×C13712yescomplemented
331#1631C2×C13712yescomplemented
332#1557C2×C13711yescharacteristic, complemented
333#1495C2×C13711yescharacteristic, complemented
334#1488C2×C13711yescharacteristic, complemented
335#557C22×C13711yescharacteristic
336#553—22no—
337#553—21no—
338#550—21no—
339#550—21no—
340#550C22×C13711yescharacteristic
341#550C22×C13711yescharacteristic
342#547—21no—
343#547C22×C13711yescharacteristic
344#540—22no—
345#540—21no—
346#532—21no—
347#532C22×C13711yescharacteristic
348C2×C4⋊Q8C22×C13711yescharacteristic
349#529—21no—
350#529C22×C13711yescharacteristic
351#520—21no—
352C4×#41C22×C13711yescharacteristic
353#501—21no—
354#492C22×C13712yes—
355#489—21no—
356#487C22×C13712yes—
357#483—21no—
358#480—21no—
359#477C22×C13712yes—
360#477C22×C13712yes—
361#476C22×C13712yes—
362#476C22×C13712yes—
363#475—21no—
364#474—21no—
365#461—22no—
366#448—22no—
367#432C22×C13711yescharacteristic
368#431C22×C13711yescharacteristic
369#428—21no—
370#423C22×C13711yescharacteristic
371#417—21no—
372#332C22×C13712yes—
373#331C22×C13712yes—
374Q8⋊C8C22×C13712yes—
375Q8⋊C8C22×C13712yes—
376D4⋊C8C22×C13712yes—
377D4⋊C8C22×C13712yes—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41D4×C13711yescharacteristic
383C22×Q8—21no—
384C2×SD16—24no—
385C2×M16—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C23×C13712yes—
391C4⋊Q8C23×C13711yescharacteristic
392#127—22no—
393#127—21no—
394#127—21no—
395#127C23×C13711yescharacteristic
396#126—22no—
397#126—21no—
398#125—22no—
399#125—21no—
400#125D4×C13711yescharacteristic
401#124—21no—
402#124D4×C13711yescharacteristic
403#123—22no—
404#123—22no—
405#123—22no—
406#123—21no—
407#123—22no—
408C22⋊Q8—21no—
409C22⋊Q8D4×C13711yescharacteristic
410#121—22no—
411#121—22no—
412#121—22no—
413#121—22no—
414#121—22no—
415#121—21no—
416#121—22no—
417#121—22no—
418#121—21no—
419#121—21no—
420#121—21no—
421#121D4×C13711yescharacteristic
422#121D4×C13712yes—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C13711yescharacteristic
426C4×Q8C23×C13711yescharacteristic
427C4×Q8C23×C13711yescharacteristic
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4D4×C13711yescharacteristic
439C4×D4D4×C13712yes—
440C4×D4C23×C13711yescharacteristic
441C4×D4C23×C13711yescharacteristic
442#117—22no—
443#117—21no—
444#117—21no—
445#117D4×C13711yescharacteristic
446C2×C4⋊C4—22no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—21no—
450C2×C42—21no—
451C2×C42C23×C13711yescharacteristic
452#107—22no—
453#107—22no—
454#106—22no—
455#106—22no—
456C4⋊C8—21no—
457C4⋊C8—21no—
458C4⋊C8C23×C13712yes—
459C4⋊C8C23×C13712yes—
460#103—22no—
461#103—22no—
462#103—24no—
463#102—22no—
464#102—22no—
465C22⋊C8—21no—
466C22⋊C8—21no—
467#97—21no—
468C4×C8C23×C13712yes—
469#41—22no—
470#41—41no—
471#41—42no—
472#41—42no—
473#41—82no—
474#41—82no—
475#41—82no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—41no—
479C2×Q8—82no—
480C2×Q8C2×D4×C13711yescharacteristic
481C2×Q8C2×D4×C13711yescharacteristic
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4C2×D4×C13711yescharacteristic
493C2×D4C2×D4×C13711yescharacteristic
494C2×D4C2×D4×C13712yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—41no—
504C22×C4C2×D4×C13711yescharacteristic
505C22×C4C2×D4×C13711yescharacteristic
506C22×C4C2×D4×C13711yescharacteristic
507SD16—84no—
508M16—42no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C4⋊C4—22no—
513C4⋊C4—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×C13712yes—
528C4⋊C4C2×D4×C13711yescharacteristic
529C4⋊C4C2×D4×C13711yescharacteristic
530C4⋊C4C2×D4×C13711yescharacteristic
531C4⋊C4C2×D4×C13711yescharacteristic
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C42—22no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42C2×D4×C13711yescharacteristic
551C42C2×D4×C13711yescharacteristic
552C42C2×D4×C13711yescharacteristic
553C42C24×C13711yescharacteristic
554C23—21no—
555C23—21no—
556C23—22no—
557C23#120×C13711yescharacteristic
558Q8—21no—
559Q8—22no—
560Q8—44no—
561Q8—42no—
562Q8—82no—
563Q8#137×C13712yes—
564D4—21no—
565D4—21no—
566D4—22no—
567D4—42no—
568D4—42no—
569D4—44no—
570D4—41no—
571D4—41no—
572D4—42no—
573D4—82no—
574D4—81no—
575D4—82no—
576D4—82no—
577D4—82no—
578D4—82no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4#120×C13711yescharacteristic
602C2×C4#120×C13711yescharacteristic
603C2×C4#120×C13711yescharacteristic
604C2×C4C22×D4×C13711yescharacteristic
605C2×C4C22×D4×C13711yescharacteristic
606C2×C4C22×D4×C13711yescharacteristic
607C8—42no—
608C8—42no—
609C8—42no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22C2×#120×C13711yescharacteristic
623C4—21no—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—82no—
635C4C2×#137×C13711yescharacteristic
636C4#575×C13711yescharacteristic
637C4#577×C13712yes—
638C2—41no—
639C2—81no—
640C2—81no—
641C2—82no—
642C2#3007×C13711yescharacteristic
643C2#3009×C13711yescharacteristic
644C2#3021×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. #3238×C137C2semidirect
3. #3206×C137C2semidirect
4. #3079×C137C2semidirect
5. #1636×C137C2semidirect
6. #1635×C137C2semidirect
7. #1631×C137C2semidirect
8. #1631×C137C2semidirect
9. #1557×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
322. #18851C137direct (N × H)
323. C137#18851direct (N × H)
325. #3238C2×C137semidirect
326. #3206C2×C137semidirect
327. #3079C2×C137semidirect
328. #1636C2×C137semidirect
329. #1635C2×C137semidirect
330. #1631C2×C137semidirect
331. #1631C2×C137semidirect
332. #1557C2×C137semidirect
333. #1495C2×C137semidirect
334. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3517×C1372
#3238×C1372
#3206×C1372
#3079×C1372
#1636×C1372
#1635×C1372
#1631×C1372
#1557×C1372
#1495×C1372
#1488×C1372
#18851137