Finite Group Databasebeta

← C137×#18853 #46868239

Subgroups — C137×#18853 #46868239

Order 35072.

874 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3518×C137C211yesmaximal, characteristic
2#3238×C137C211yesmaximal, characteristic, complemented
3#3209×C137C211yesmaximal, characteristic, complemented
4#3079×C137C211yesmaximal, characteristic, complemented
5#1638×C137C212yesmaximal, complemented
6#1637×C137C212yesmaximal, complemented
7#1634×C137C212yesmaximal, complemented
8#1633×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12Q82×C137C2211yescharacteristic
13#556×C137—21no—
14#556×C137—22no—
15#556×C137—21no—
16#556×C137C2211yescharacteristic
17#555×C137—21no—
18#555×C137C2211yescharacteristic
19D4×Q8×C137C2211yescharacteristic
20#548×C137—21no—
21#542×C137—22no—
22#542×C137—21no—
23#533×C137—21no—
24#533×C137C2211yescharacteristic
25C2×C4⋊Q8×C137C2211yescharacteristic
26#529×C137—21no—
27#529×C137C2211yescharacteristic
28#519×C137—21no—
29C4×#41×C137C2211yescharacteristic
30#501×C137—21no—
31#494×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#462×C137—22no—
43#451×C137—22no—
44#432×C137C2211yescharacteristic
45#431×C137C2211yescharacteristic
46#428×C137—21no—
47#423×C137C2211yescharacteristic
48#417×C137—21no—
49#333×C137C2212yes—
50#332×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52Q8⋊C8×C137C2212yes—
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137D411yescharacteristic
56C22×Q8×C137—21no—
57C22×Q8×C137—21no—
58C2×Q16×C137—24no—
59C2×M16×C137—21no—
60C4⋊Q8×C137—22no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137—21no—
67C4⋊Q8×C137C2312yes—
68C4⋊Q8×C137C2311yescharacteristic
69C4⋊Q8×C137C2311yescharacteristic
70#126×C137—22no—
71#126×C137—22no—
72#126×C137—22no—
73#126×C137—22no—
74#126×C137—21no—
75#125×C137—21no—
76#125×C137D411yescharacteristic
77#124×C137—22no—
78#124×C137—22no—
79#124×C137—22no—
80#124×C137—22no—
81#124×C137—21no—
82#124×C137D411yescharacteristic
83#123×C137—22no—
84#123×C137—21no—
85C22⋊Q8×C137—22no—
86C22⋊Q8×C137—22no—
87C22⋊Q8×C137—21no—
88C22⋊Q8×C137—21no—
89C22⋊Q8×C137—21no—
90C22⋊Q8×C137—22no—
91C22⋊Q8×C137D412yes—
92C22⋊Q8×C137D411yescharacteristic
93#121×C137—22no—
94#121×C137—21no—
95#121×C137—21no—
96#121×C137D411yescharacteristic
97C4×Q8×C137—22no—
98C4×Q8×C137—22no—
99C4×Q8×C137—22no—
100C4×Q8×C137—21no—
101C4×Q8×C137—21no—
102C4×Q8×C137—21no—
103C4×Q8×C137—21no—
104C4×Q8×C137D412yes—
105C4×Q8×C137D411yescharacteristic
106C4×Q8×C137C2311yescharacteristic
107C4×Q8×C137C2311yescharacteristic
108C4×D4×C137—22no—
109C4×D4×C137—21no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137D411yescharacteristic
114C4×D4×C137C2311yescharacteristic
115C4×D4×C137C2311yescharacteristic
116#117×C137—22no—
117#117×C137—22no—
118#117×C137—21no—
119#117×C137—21no—
120#117×C137—21no—
121#117×C137—21no—
122#117×C137—21no—
123#117×C137—21no—
124#117×C137D411yescharacteristic
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—
148C2×Q8×C137—22no—
149C2×Q8×C137—22no—
150C2×Q8×C137—21no—
151C2×Q8×C137—22no—
152C2×Q8×C137—42no—
153C2×Q8×C137—42no—
154C2×Q8×C137—41no—
155C2×Q8×C137—82no—
156C2×Q8×C137C2×D411yescharacteristic
157C2×Q8×C137C2×D411yescharacteristic
158C2×Q8×C137C2×D412yes—
159C2×D4×C137—21no—
160C2×D4×C137—22no—
161C2×D4×C137—21no—
162C2×D4×C137C2×D411yescharacteristic
163C2×D4×C137C2×D411yescharacteristic
164C22×C4×C137—22no—
165C22×C4×C137—21no—
166C22×C4×C137—21no—
167C22×C4×C137—21no—
168C22×C4×C137—21no—
169C22×C4×C137—41no—
170C22×C4×C137C2×D411yescharacteristic
171C22×C4×C137C2×D411yescharacteristic
172C22×C4×C137C2×D411yescharacteristic
173Q16×C137—84no—
174M16×C137—42no—
175C2×C8×C137—22no—
176C2×C8×C137—22no—
177C2×C8×C137—22no—
178C4⋊C4×C137—22no—
179C4⋊C4×C137—22no—
180C4⋊C4×C137—22no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—22no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—21no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—21no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—21no—
191C4⋊C4×C137—21no—
192C4⋊C4×C137—21no—
193C4⋊C4×C137—22no—
194C4⋊C4×C137—22no—
195C4⋊C4×C137—41no—
196C4⋊C4×C137—41no—
197C4⋊C4×C137C2×D412yes—
198C4⋊C4×C137C2×D411yescharacteristic
199C4⋊C4×C137C2×D411yescharacteristic
200C4⋊C4×C137C2×D411yescharacteristic
201C4⋊C4×C137C2×D411yescharacteristic
202C22⋊C4×C137—22no—
203C22⋊C4×C137—22no—
204C22⋊C4×C137—22no—
205C22⋊C4×C137—22no—
206C22⋊C4×C137—22no—
207C22⋊C4×C137—22no—
208C22⋊C4×C137—21no—
209C22⋊C4×C137—21no—
210C22⋊C4×C137—21no—
211C22⋊C4×C137—21no—
212C22⋊C4×C137—21no—
213C22⋊C4×C137—21no—
214C22⋊C4×C137—21no—
215C22⋊C4×C137—21no—
216C42×C137—22no—
217C42×C137—22no—
218C42×C137—22no—
219C42×C137—21no—
220C42×C137—21no—
221C42×C137—22no—
222C42×C137—21no—
223C42×C137—21no—
224C42×C137C2×D411yescharacteristic
225C42×C137C2×D411yescharacteristic
226C42×C137C2×D411yescharacteristic
227C42×C137C2411yescharacteristic
228C23×C137—21no—
229C23×C137—21no—
230C23×C137#12011yescharacteristic
231Q8×C137—21no—
232Q8×C137—22no—
233Q8×C137—42no—
234Q8×C137—44no—
235Q8×C137—44no—
236Q8×C137—41no—
237Q8×C137—41no—
238Q8×C137—82no—
239Q8×C137—82no—
240Q8×C137#13712yes—
241D4×C137—21no—
242D4×C137—21no—
243D4×C137—22no—
244D4×C137—42no—
245D4×C137—81no—
246D4×C137—82no—
247C2×C4×C137—21no—
248C2×C4×C137—21no—
249C2×C4×C137—21no—
250C2×C4×C137—22no—
251C2×C4×C137—21no—
252C2×C4×C137—21no—
253C2×C4×C137—22no—
254C2×C4×C137—21no—
255C2×C4×C137—21no—
256C2×C4×C137—22no—
257C2×C4×C137—21no—
258C2×C4×C137—41no—
259C2×C4×C137—42no—
260C2×C4×C137—44no—
261C2×C4×C137—41no—
262C2×C4×C137—41no—
263C2×C4×C137—42no—
264C2×C4×C137—42no—
265C2×C4×C137—81no—
266C2×C4×C137—82no—
267C2×C4×C137#12011yescharacteristic
268C2×C4×C137#12011yescharacteristic
269C2×C4×C137#12011yescharacteristic
270C2×C4×C137C22×D411yescharacteristic
271C2×C4×C137C22×D411yescharacteristic
272C2×C4×C137C22×D411yescharacteristic
273C8×C137—42no—
274C8×C137—42no—
275C8×C137—42no—
276C22×C137—21no—
277C22×C137—21no—
278C22×C137—21no—
279C22×C137—41no—
280C22×C137—41no—
281C22×C137—41no—
282C22×C137—41no—
283C22×C137—41no—
284C22×C137—41no—
285C22×C137C2×#12011yescharacteristic
286C4×C137—21no—
287C4×C137—22no—
288C4×C137—21no—
289C4×C137—21no—
290C4×C137—21no—
291C4×C137—42no—
292C4×C137—41no—
293C4×C137—41no—
294C4×C137—41no—
295C4×C137—41no—
296C4×C137—42no—
297C4×C137—41no—
298C4×C137—82no—
299C4×C137C2×#13711yescharacteristic
300C4×C137#57511yescharacteristic
301C4×C137#57812yes—
302C2×C137—41no—
303C2×C137—81no—
304C2×C137—81no—
305C2×C137#300711yescharacteristic
306C2×C137#301111yescharacteristic
307C2×C137#302111yescharacteristic
308#18853C13711yesmaximal, characteristic, direct product factor, Sylow
309C137#1885311yescharacteristic, direct product factor, Sylow
310#3518C2×C13711yescharacteristic
311#3238C2×C13711yescharacteristic, complemented
312#3209C2×C13711yescharacteristic, complemented
313#3079C2×C13711yescharacteristic, complemented
314#1638C2×C13712yescomplemented
315#1637C2×C13712yescomplemented
316#1634C2×C13712yescomplemented
317#1633C2×C13712yescomplemented
318#1559C2×C13711yescharacteristic, complemented
319#1495C2×C13711yescharacteristic, complemented
320#1488C2×C13711yescharacteristic, complemented
321Q82C22×C13711yescharacteristic
322#556—21no—
323#556—22no—
324#556—21no—
325#556C22×C13711yescharacteristic
326#555—21no—
327#555C22×C13711yescharacteristic
328D4×Q8C22×C13711yescharacteristic
329#548—21no—
330#542—22no—
331#542—21no—
332#533—21no—
333#533C22×C13711yescharacteristic
334C2×C4⋊Q8C22×C13711yescharacteristic
335#529—21no—
336#529C22×C13711yescharacteristic
337#519—21no—
338C4×#41C22×C13711yescharacteristic
339#501—21no—
340#494C22×C13712yes—
341#489—21no—
342#487C22×C13712yes—
343#483—21no—
344#480—21no—
345#477C22×C13712yes—
346#477C22×C13712yes—
347#476C22×C13712yes—
348#476C22×C13712yes—
349#475—21no—
350#474—21no—
351#462—22no—
352#451—22no—
353#432C22×C13711yescharacteristic
354#431C22×C13711yescharacteristic
355#428—21no—
356#423C22×C13711yescharacteristic
357#417—21no—
358#333C22×C13712yes—
359#332C22×C13712yes—
360Q8⋊C8C22×C13712yes—
361Q8⋊C8C22×C13712yes—
362D4⋊C8C22×C13712yes—
363D4⋊C8C22×C13712yes—
364C2×#41D4×C13711yescharacteristic
365C22×Q8—21no—
366C22×Q8—21no—
367C2×Q16—24no—
368C2×M16—21no—
369C4⋊Q8—22no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8C23×C13712yes—
377C4⋊Q8C23×C13711yescharacteristic
378C4⋊Q8C23×C13711yescharacteristic
379#126—22no—
380#126—22no—
381#126—22no—
382#126—22no—
383#126—21no—
384#125—21no—
385#125D4×C13711yescharacteristic
386#124—22no—
387#124—22no—
388#124—22no—
389#124—22no—
390#124—21no—
391#124D4×C13711yescharacteristic
392#123—22no—
393#123—21no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8D4×C13711yescharacteristic
401C22⋊Q8D4×C13712yes—
402#121—22no—
403#121—21no—
404#121—21no—
405#121D4×C13711yescharacteristic
406C4×Q8—22no—
407C4×Q8—22no—
408C4×Q8—22no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8D4×C13711yescharacteristic
414C4×Q8D4×C13712yes—
415C4×Q8C23×C13711yescharacteristic
416C4×Q8C23×C13711yescharacteristic
417C4×D4—22no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4D4×C13711yescharacteristic
423C4×D4C23×C13711yescharacteristic
424C4×D4C23×C13711yescharacteristic
425#117—22no—
426#117—22no—
427#117—21no—
428#117—21no—
429#117—21no—
430#117—21no—
431#117—21no—
432#117—21no—
433#117D4×C13711yescharacteristic
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C42—21no—
437C2×C42C23×C13711yescharacteristic
438#107—22no—
439#107—22no—
440#106—22no—
441#106—22no—
442C4⋊C8—21no—
443C4⋊C8—21no—
444C4⋊C8C23×C13712yes—
445C4⋊C8C23×C13712yes—
446#103—22no—
447#103—22no—
448#103—24no—
449#102—22no—
450#102—22no—
451C22⋊C8—21no—
452C22⋊C8—21no—
453#97—21no—
454C4×C8C23×C13712yes—
455#41—22no—
456#41—41no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—41no—
464C2×Q8—82no—
465C2×Q8C2×D4×C13711yescharacteristic
466C2×Q8C2×D4×C13711yescharacteristic
467C2×Q8C2×D4×C13712yes—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4C2×D4×C13711yescharacteristic
472C2×D4C2×D4×C13711yescharacteristic
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4C2×D4×C13711yescharacteristic
480C22×C4C2×D4×C13711yescharacteristic
481C22×C4C2×D4×C13711yescharacteristic
482Q16—84no—
483M16—42no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4C2×D4×C13712yes—
507C4⋊C4C2×D4×C13711yescharacteristic
508C4⋊C4C2×D4×C13711yescharacteristic
509C4⋊C4C2×D4×C13711yescharacteristic
510C4⋊C4C2×D4×C13711yescharacteristic
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42C2×D4×C13711yescharacteristic
534C42C2×D4×C13711yescharacteristic
535C42C2×D4×C13711yescharacteristic
536C42C24×C13711yescharacteristic
537C23—21no—
538C23—21no—
539C23#120×C13711yescharacteristic
540Q8—21no—
541Q8—22no—
542Q8—42no—
543Q8—44no—
544Q8—44no—
545Q8—41no—
546Q8—41no—
547Q8—82no—
548Q8—82no—
549Q8#137×C13712yes—
550D4—21no—
551D4—21no—
552D4—22no—
553D4—42no—
554D4—81no—
555D4—82no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4#120×C13711yescharacteristic
577C2×C4#120×C13711yescharacteristic
578C2×C4#120×C13711yescharacteristic
579C2×C4C22×D4×C13711yescharacteristic
580C2×C4C22×D4×C13711yescharacteristic
581C2×C4C22×D4×C13711yescharacteristic
582C8—42no—
583C8—42no—
584C8—42no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22C2×#120×C13711yescharacteristic
595C4—21no—
596C4—22no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—41no—
607C4—82no—
608C4C2×#137×C13711yescharacteristic
609C4#575×C13711yescharacteristic
610C4#578×C13712yes—
611C2—41no—
612C2—81no—
613C2—81no—
614C2#3007×C13711yescharacteristic
615C2#3011×C13711yescharacteristic
616C2#3021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3518×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. #3209×C137C2semidirect
4. #3079×C137C2semidirect
5. #1638×C137C2semidirect
6. #1637×C137C2semidirect
7. #1634×C137C2semidirect
8. #1633×C137C2semidirect
9. #1559×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
308. #18853C137direct (N × H)
309. C137#18853direct (N × H)
311. #3238C2×C137semidirect
312. #3209C2×C137semidirect
313. #3079C2×C137semidirect
314. #1638C2×C137semidirect
315. #1637C2×C137semidirect
316. #1634C2×C137semidirect
317. #1633C2×C137semidirect
318. #1559C2×C137semidirect
319. #1495C2×C137semidirect
320. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3518×C1372
#3238×C1372
#3209×C1372
#3079×C1372
#1638×C1372
#1637×C1372
#1634×C1372
#1633×C1372
#1559×C1372
#1495×C1372
#1488×C1372
#18853137