Finite Group Databasebeta

← C137×#19024 #46868410

Subgroups — C137×#19024 #46868410

Order 35072.

948 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3558×C137C211yesmaximal, characteristic
2#3216×C137C211yesmaximal, characteristic, complemented
3#3172×C137C211yesmaximal, characteristic, complemented
4#3140×C137C211yesmaximal, characteristic, complemented
5#1662×C137C212yesmaximal, complemented
6#1661×C137C212yesmaximal, complemented
7#1659×C137C212yesmaximal, complemented
8#1659×C137C212yesmaximal, complemented
9#1529×C137C211yesmaximal, characteristic, complemented
10#1527×C137C211yesmaximal, characteristic, complemented
11#1525×C137C211yesmaximal, characteristic, complemented
12#562×C137C2212yes—
13H(3,𝔽4)×C137C2212yes—
14#559×C137—21no—
15#548×C137—21no—
16#546×C137—22no—
17#546×C137—21no—
18D42×C137—21no—
19#542×C137—21no—
20#538×C137—22no—
21#538×C137—21no—
22#535×C137—21no—
23#534×C137—21no—
24#533×C137C2211yescharacteristic, complemented
25#532×C137C2211yescharacteristic
26#529×C137C2211yescharacteristic
27C2×#124×C137C2211yescharacteristic
28#501×C137C2211yescharacteristic, complemented
29#497×C137C2211yescharacteristic, complemented
30#497×C137C2212yescomplemented
31#491×C137C2211yescharacteristic, complemented
32#489×C137C2212yescomplemented
33#489×C137C2212yescomplemented
34#489×C137C2211yescharacteristic, complemented
35#489×C137C2212yescomplemented
36#488×C137—22no—
37#470×C137—21no—
38#468×C137—21no—
39#462×C137—21no—
40#460×C137—21no—
41#452×C137—22no—
42C4×Q16×C137—21no—
43C4×SD16×C137—21no—
44#432×C137C2211yescharacteristic, complemented
45#419×C137—21no—
46#418×C137—21no—
47C4×M16×C137C2211yescharacteristic, complemented
48C4×M16×C137C2211yescharacteristic, complemented
49#330×C137C2212yescomplemented
50#329×C137C2212yescomplemented
51#329×C137C2212yescomplemented
52#329×C137C2212yescomplemented
53#329×C137C2212yescomplemented
54#329×C137C2212yescomplemented
55C2×#41×C137—21no—
56C2×#41×C137D412yes—
57C22×Q8×C137D411yescharacteristic
58C22×D4×C137—22no—
59C22×D4×C137—21no—
60C22×D4×C137D411yescharacteristic
61C2×Q16×C137—21no—
62C2×Q16×C137—22no—
63C2×SD16×C137—21no—
64C2×SD16×C137—22no—
65C2×M16×C137—21no—
66C2×M16×C137—21no—
67C4⋊Q8×C137D411yescharacteristic
68C4⋊Q8×C137C2311yescharacteristic
69C4⋊Q8×C137C2311yescharacteristic
70#127×C137—21no—
71#127×C137D411yescharacteristic
72#126×C137—22no—
73#126×C137—21no—
74#125×C137—21no—
75#125×C137C2311yescharacteristic
76#124×C137—21no—
77#124×C137—22no—
78#124×C137—22no—
79#124×C137—22no—
80#124×C137—21no—
81#124×C137—21no—
82#124×C137D412yes—
83#124×C137C2312yes—
84#124×C137C2311yescharacteristic
85#124×C137C2312yes—
86#123×C137—22no—
87#123×C137—22no—
88#123×C137—21no—
89#123×C137—22no—
90#123×C137—22no—
91C22⋊Q8×C137—21no—
92C22⋊Q8×C137—22no—
93C22⋊Q8×C137—22no—
94C22⋊Q8×C137—21no—
95C22⋊Q8×C137—21no—
96C22⋊Q8×C137D411yescharacteristic
97#121×C137—22no—
98#121×C137—22no—
99#121×C137—22no—
100#121×C137—21no—
101#121×C137—22no—
102#121×C137—22no—
103#121×C137—21no—
104#121×C137—21no—
105#121×C137D411yescharacteristic
106#120×C137—22no—
107#120×C137—22no—
108#120×C137—22no—
109#120×C137—22no—
110#120×C137—22no—
111C4×Q8×C137—21no—
112C4×Q8×C137—21no—
113C4×Q8×C137—21no—
114C4×Q8×C137D411yescharacteristic
115C4×D4×C137—22no—
116C4×D4×C137—22no—
117C4×D4×C137—21no—
118C4×D4×C137—21no—
119C4×D4×C137—21no—
120C4×D4×C137—21no—
121C4×D4×C137D411yescharacteristic
122#117×C137—21no—
123#117×C137—21no—
124C2×C22⋊C4×C137—22no—
125C2×C22⋊C4×C137—21no—
126C2×C22⋊C4×C137—21no—
127C2×C42×C137C2311yescharacteristic
128#107×C137—21no—
129#106×C137—21no—
130C4⋊C8×C137—21no—
131#103×C137—22no—
132#103×C137—22no—
133#103×C137—21no—
134#103×C137—22no—
135#103×C137—21no—
136#103×C137—21no—
137#102×C137—22no—
138#102×C137—22no—
139#102×C137—21no—
140#102×C137—22no—
141C22⋊C8×C137—21no—
142#97×C137C2312yes—
143#97×C137C2311yescharacteristic
144#97×C137C2311yescharacteristic
145#97×C137C2312yes—
146C4×C8×C137—21no—
147C4×C8×C137—21no—
148C24×C137—22no—
149C24×C137—21no—
150#41×C137—82no—
151#41×C137—82no—
152C2×Q8×C137—21no—
153C2×Q8×C137—21no—
154C2×Q8×C137—21no—
155C2×Q8×C137—42no—
156C2×Q8×C137C2×D412yes—
157C2×Q8×C137C2×D411yescharacteristic
158C2×Q8×C137C2×D412yes—
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—22no—
166C2×D4×C137—21no—
167C2×D4×C137—21no—
168C2×D4×C137—21no—
169C2×D4×C137—42no—
170C2×D4×C137—82no—
171C2×D4×C137—84no—
172C2×D4×C137C2×D412yes—
173C2×D4×C137C2×D411yescharacteristic
174C2×D4×C137C2×D412yes—
175C22×C4×C137—22no—
176C22×C4×C137—22no—
177C22×C4×C137—21no—
178C22×C4×C137—21no—
179C22×C4×C137—22no—
180C22×C4×C137C2×D411yescharacteristic
181C22×C4×C137C2×D411yescharacteristic
182C22×C4×C137C2×D411yescharacteristic
183Q16×C137—42no—
184Q16×C137—82no—
185SD16×C137—42no—
186SD16×C137—82no—
187M16×C137—42no—
188M16×C137—42no—
189C2×C8×C137—21no—
190C2×C8×C137—22no—
191C2×C8×C137—21no—
192C2×C8×C137—22no—
193C4⋊C4×C137—22no—
194C4⋊C4×C137—22no—
195C4⋊C4×C137—21no—
196C4⋊C4×C137—21no—
197C4⋊C4×C137—21no—
198C4⋊C4×C137—21no—
199C4⋊C4×C137—21no—
200C4⋊C4×C137—21no—
201C4⋊C4×C137—21no—
202C4⋊C4×C137—21no—
203C4⋊C4×C137—21no—
204C4⋊C4×C137—21no—
205C4⋊C4×C137—41no—
206C4⋊C4×C137—41no—
207C4⋊C4×C137C2×D411yescharacteristic
208C4⋊C4×C137C2×D411yescharacteristic
209C22⋊C4×C137—22no—
210C22⋊C4×C137—22no—
211C22⋊C4×C137—22no—
212C22⋊C4×C137—22no—
213C22⋊C4×C137—22no—
214C22⋊C4×C137—21no—
215C22⋊C4×C137—22no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—22no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—22no—
220C22⋊C4×C137—22no—
221C22⋊C4×C137—21no—
222C22⋊C4×C137—21no—
223C22⋊C4×C137—22no—
224C22⋊C4×C137—22no—
225C22⋊C4×C137—41no—
226C42×C137—22no—
227C42×C137—21no—
228C42×C137—41no—
229C42×C137C2×D411yescharacteristic
230C42×C137C2×D411yescharacteristic
231C42×C137C2×D411yescharacteristic
232C42×C137C2411yescharacteristic
233C23×C137—21no—
234C23×C137—22no—
235C23×C137—22no—
236C23×C137—42no—
237C23×C137—81no—
238C23×C137—81no—
239C23×C137—82no—
240C23×C137—82no—
241C23×C137—82no—
242C23×C137#12011yescharacteristic
243Q8×C137—41no—
244Q8×C137—42no—
245Q8×C137—42no—
246Q8×C137—41no—
247Q8×C137—41no—
248Q8×C137—41no—
249Q8×C137—41no—
250Q8×C137—42no—
251D4×C137—41no—
252D4×C137—42no—
253D4×C137—42no—
254D4×C137—42no—
255D4×C137—44no—
256D4×C137—42no—
257D4×C137—42no—
258D4×C137—81no—
259D4×C137—82no—
260D4×C137—82no—
261D4×C137—82no—
262D4×C137—82no—
263D4×C137—82no—
264C2×C4×C137—22no—
265C2×C4×C137—21no—
266C2×C4×C137—22no—
267C2×C4×C137—22no—
268C2×C4×C137—21no—
269C2×C4×C137—21no—
270C2×C4×C137—41no—
271C2×C4×C137—41no—
272C2×C4×C137—42no—
273C2×C4×C137—41no—
274C2×C4×C137—41no—
275C2×C4×C137—82no—
276C2×C4×C137—81no—
277C2×C4×C137—82no—
278C2×C4×C137—82no—
279C2×C4×C137#12011yescharacteristic
280C2×C4×C137#12011yescharacteristic
281C2×C4×C137#12011yescharacteristic
282C2×C4×C137C22×D411yescharacteristic
283C2×C4×C137C22×D411yescharacteristic
284C2×C4×C137C22×D411yescharacteristic
285C8×C137—22no—
286C8×C137—42no—
287C8×C137—41no—
288C8×C137—42no—
289C22×C137—21no—
290C22×C137—21no—
291C22×C137—21no—
292C22×C137—41no—
293C22×C137—42no—
294C22×C137—42no—
295C22×C137—41no—
296C22×C137—42no—
297C22×C137—42no—
298C22×C137—42no—
299C22×C137—42no—
300C22×C137—41no—
301C22×C137—162no—
302C22×C137—164no—
303C22×C137C2×#12011yescharacteristic
304C4×C137—21no—
305C4×C137—21no—
306C4×C137—21no—
307C4×C137—21no—
308C4×C137—21no—
309C4×C137—41no—
310C4×C137—42no—
311C4×C137—41no—
312C4×C137—41no—
313C4×C137—42no—
314C4×C137—81no—
315C4×C137—81no—
316C4×C137#45512yes—
317C2×C137—41no—
318C2×C137—82no—
319C2×C137—81no—
320C2×C137—82no—
321C2×C137C2×#45511yescharacteristic
322C2×C137#301811yescharacteristic
323C2×C137#301911yescharacteristic
324#19024C13711yesmaximal, characteristic, direct product factor, Sylow
325C137#1902411yescharacteristic, direct product factor, Sylow
326#3558C2×C13711yescharacteristic
327#3216C2×C13711yescharacteristic, complemented
328#3172C2×C13711yescharacteristic, complemented
329#3140C2×C13711yescharacteristic, complemented
330#1662C2×C13712yescomplemented
331#1661C2×C13712yescomplemented
332#1659C2×C13712yescomplemented
333#1659C2×C13712yescomplemented
334#1529C2×C13711yescharacteristic, complemented
335#1527C2×C13711yescharacteristic, complemented
336#1525C2×C13711yescharacteristic, complemented
337#562C22×C13712yes—
338H(3,𝔽4)C22×C13712yes—
339#559—21no—
340#548—21no—
341#546—22no—
342#546—21no—
343D42—21no—
344#542—21no—
345#538—22no—
346#538—21no—
347#535—21no—
348#534—21no—
349#533C22×C13711yescharacteristic, complemented
350#532C22×C13711yescharacteristic
351#529C22×C13711yescharacteristic
352C2×#124C22×C13711yescharacteristic
353#501C22×C13711yescharacteristic, complemented
354#497C22×C13711yescharacteristic, complemented
355#497C22×C13712yescomplemented
356#491C22×C13711yescharacteristic, complemented
357#489C22×C13712yescomplemented
358#489C22×C13712yescomplemented
359#489C22×C13711yescharacteristic, complemented
360#489C22×C13712yescomplemented
361#488—22no—
362#470—21no—
363#468—21no—
364#462—21no—
365#460—21no—
366#452—22no—
367C4×Q16—21no—
368C4×SD16—21no—
369#432C22×C13711yescharacteristic, complemented
370#419—21no—
371#418—21no—
372C4×M16C22×C13711yescharacteristic, complemented
373C4×M16C22×C13711yescharacteristic, complemented
374#330C22×C13712yescomplemented
375#329C22×C13712yescomplemented
376#329C22×C13712yescomplemented
377#329C22×C13712yescomplemented
378#329C22×C13712yescomplemented
379#329C22×C13712yescomplemented
380C2×#41—21no—
381C2×#41D4×C13712yes—
382C22×Q8D4×C13711yescharacteristic
383C22×D4—22no—
384C22×D4—21no—
385C22×D4D4×C13711yescharacteristic
386C2×Q16—21no—
387C2×Q16—22no—
388C2×SD16—21no—
389C2×SD16—22no—
390C2×M16—21no—
391C2×M16—21no—
392C4⋊Q8D4×C13711yescharacteristic
393C4⋊Q8C23×C13711yescharacteristic
394C4⋊Q8C23×C13711yescharacteristic
395#127—21no—
396#127D4×C13711yescharacteristic
397#126—22no—
398#126—21no—
399#125—21no—
400#125C23×C13711yescharacteristic
401#124—21no—
402#124—22no—
403#124—22no—
404#124—22no—
405#124—21no—
406#124—21no—
407#124D4×C13712yes—
408#124C23×C13712yes—
409#124C23×C13711yescharacteristic
410#124C23×C13712yes—
411#123—22no—
412#123—22no—
413#123—21no—
414#123—22no—
415#123—22no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8D4×C13711yescharacteristic
422#121—22no—
423#121—22no—
424#121—22no—
425#121—21no—
426#121—22no—
427#121—22no—
428#121—21no—
429#121—21no—
430#121D4×C13711yescharacteristic
431#120—22no—
432#120—22no—
433#120—22no—
434#120—22no—
435#120—22no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8D4×C13711yescharacteristic
440C4×D4—22no—
441C4×D4—22no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4D4×C13711yescharacteristic
447#117—21no—
448#117—21no—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C42C23×C13711yescharacteristic
453#107—21no—
454#106—21no—
455C4⋊C8—21no—
456#103—22no—
457#103—22no—
458#103—21no—
459#103—22no—
460#103—21no—
461#103—21no—
462#102—22no—
463#102—22no—
464#102—21no—
465#102—22no—
466C22⋊C8—21no—
467#97C23×C13711yescharacteristic
468#97C23×C13711yescharacteristic
469#97C23×C13712yes—
470#97C23×C13712yes—
471C4×C8—21no—
472C4×C8—21no—
473C24—22no—
474C24—21no—
475#41—82no—
476#41—82no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—42no—
481C2×Q8C2×D4×C13712yes—
482C2×Q8C2×D4×C13711yescharacteristic
483C2×Q8C2×D4×C13712yes—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—82no—
496C2×D4—84no—
497C2×D4C2×D4×C13712yes—
498C2×D4C2×D4×C13711yescharacteristic
499C2×D4C2×D4×C13712yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4C2×D4×C13711yescharacteristic
506C22×C4C2×D4×C13711yescharacteristic
507C22×C4C2×D4×C13711yescharacteristic
508Q16—42no—
509Q16—82no—
510SD16—42no—
511SD16—82no—
512M16—42no—
513M16—42no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—21no—
517C2×C8—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4C2×D4×C13711yescharacteristic
533C4⋊C4C2×D4×C13711yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—41no—
551C42—22no—
552C42—21no—
553C42—41no—
554C42C2×D4×C13711yescharacteristic
555C42C2×D4×C13711yescharacteristic
556C42C2×D4×C13711yescharacteristic
557C42C24×C13711yescharacteristic
558C23—21no—
559C23—22no—
560C23—22no—
561C23—42no—
562C23—81no—
563C23—81no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23#120×C13711yescharacteristic
568Q8—41no—
569Q8—42no—
570Q8—42no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—42no—
576D4—41no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—42no—
582D4—42no—
583D4—81no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—82no—
588D4—82no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4#120×C13711yescharacteristic
605C2×C4#120×C13711yescharacteristic
606C2×C4#120×C13711yescharacteristic
607C2×C4C22×D4×C13711yescharacteristic
608C2×C4C22×D4×C13711yescharacteristic
609C2×C4C22×D4×C13711yescharacteristic
610C8—22no—
611C8—42no—
612C8—41no—
613C8—42no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—162no—
627C22—164no—
628C22C2×#120×C13711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4#455×C13712yes—
642C2—41no—
643C2—81no—
644C2—82no—
645C2—82no—
646C2C2×#455×C13711yescharacteristic
647C2#3018×C13711yescharacteristic
648C2#3019×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3558×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. #3216×C137C2semidirect
3. #3172×C137C2semidirect
4. #3140×C137C2semidirect
5. #1662×C137C2semidirect
6. #1661×C137C2semidirect
7. #1659×C137C2semidirect
8. #1659×C137C2semidirect
9. #1529×C137C2semidirect
10. #1527×C137C2semidirect
11. #1525×C137C2semidirect
24. #533×C137C22semidirect
28. #501×C137C22semidirect
29. #497×C137C22semidirect
30. #497×C137C22semidirect
31. #491×C137C22semidirect
32. #489×C137C22semidirect
33. #489×C137C22semidirect
34. #489×C137C22semidirect
35. #489×C137C22semidirect
44. #432×C137C22semidirect
47. C4×M16×C137C22semidirect
48. C4×M16×C137C22semidirect
49. #330×C137C22semidirect
50. #329×C137C22semidirect
51. #329×C137C22semidirect
52. #329×C137C22semidirect
53. #329×C137C22semidirect
54. #329×C137C22semidirect
324. #19024C137direct (N × H)
325. C137#19024direct (N × H)
327. #3216C2×C137semidirect
328. #3172C2×C137semidirect
329. #3140C2×C137semidirect
330. #1662C2×C137semidirect
331. #1661C2×C137semidirect
332. #1659C2×C137semidirect
333. #1659C2×C137semidirect
334. #1529C2×C137semidirect
335. #1527C2×C137semidirect
336. #1525C2×C137semidirect
349. #533C22×C137semidirect
353. #501C22×C137semidirect
354. #497C22×C137semidirect
355. #497C22×C137semidirect
356. #491C22×C137semidirect
357. #489C22×C137semidirect
358. #489C22×C137semidirect
359. #489C22×C137semidirect
360. #489C22×C137semidirect
369. #432C22×C137semidirect
372. C4×M16C22×C137semidirect
373. C4×M16C22×C137semidirect
374. #330C22×C137semidirect
375. #329C22×C137semidirect
376. #329C22×C137semidirect
377. #329C22×C137semidirect
378. #329C22×C137semidirect
379. #329C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3558×C1372
#3216×C1372
#3172×C1372
#3140×C1372
#1662×C1372
#1661×C1372
#1659×C1372
#1529×C1372
#1527×C1372
#1525×C1372
#19024137