Finite Group Databasebeta

← C137×#18844 #46868230

Subgroups — C137×#18844 #46868230

Order 35072.

896 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3515×C137C211yesmaximal, characteristic
2#3239×C137C211yesmaximal, characteristic, complemented
3#3230×C137C211yesmaximal, characteristic, complemented
4#3039×C137C211yesmaximal, characteristic, complemented
5#1648×C137C212yesmaximal, complemented
6#1647×C137C212yesmaximal, complemented
7#1644×C137C212yesmaximal, complemented
8#1643×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12C2×#137×C137—21no—
13#557×C137C2211yescharacteristic
14#554×C137C2211yescharacteristic
15#554×C137C2211yescharacteristic
16#553×C137—21no—
17#552×C137—21no—
18#552×C137—22no—
19#552×C137—21no—
20#547×C137—21no—
21#547×C137C2211yescharacteristic
22#541×C137—21no—
23#539×C137—22no—
24#533×C137C2211yescharacteristic
25C2×C4⋊Q8×C137C2211yescharacteristic
26#529×C137—21no—
27#529×C137C2211yescharacteristic
28C2×#125×C137—21no—
29#520×C137—21no—
30C4×#41×C137C2211yescharacteristic
31#501×C137—21no—
32#500×C137C2212yes—
33#489×C137—21no—
34#487×C137C2212yes—
35#484×C137—22no—
36#484×C137—21no—
37#481×C137—21no—
38#477×C137—22no—
39#475×C137C2212yes—
40#474×C137C2212yes—
41#462×C137—21no—
42#462×C137C2212yes—
43#461×C137—21no—
44#461×C137C2212yes—
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#423×C137C2211yescharacteristic
48#419×C137—21no—
49#333×C137C2212yes—
50#332×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52Q8⋊C8×C137C2212yes—
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137—21no—
56C2×#41×C137D411yescharacteristic
57C22×Q8×C137—21no—
58#137×C137—44no—
59C2×Q16×C137—22no—
60C2×SD16×C137—22no—
61C2×M16×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137C2312yes—
67C4⋊Q8×C137C2311yescharacteristic
68C4⋊Q8×C137C2311yescharacteristic
69#126×C137—22no—
70#126×C137—22no—
71#126×C137—22no—
72#126×C137—21no—
73#125×C137—22no—
74#125×C137—22no—
75#125×C137—22no—
76#125×C137—21no—
77#125×C137—21no—
78#125×C137—22no—
79#125×C137—21no—
80#125×C137—21no—
81#125×C137D411yescharacteristic
82#124×C137D411yescharacteristic
83#123×C137—22no—
84#123×C137—22no—
85#123×C137—22no—
86#123×C137—21no—
87#123×C137—22no—
88C22⋊Q8×C137—21no—
89C22⋊Q8×C137—22no—
90C22⋊Q8×C137—22no—
91C22⋊Q8×C137—21no—
92C22⋊Q8×C137—21no—
93C22⋊Q8×C137—21no—
94C22⋊Q8×C137D411yescharacteristic
95C22⋊Q8×C137D412yes—
96#121×C137—22no—
97#121×C137—21no—
98#121×C137D411yescharacteristic
99C4×Q8×C137—21no—
100C4×Q8×C137—21no—
101C4×Q8×C137—21no—
102C4×Q8×C137D411yescharacteristic
103C4×Q8×C137D412yes—
104C4×Q8×C137C2311yescharacteristic
105C4×Q8×C137C2311yescharacteristic
106C4×D4×C137—22no—
107C4×D4×C137—22no—
108C4×D4×C137—21no—
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×C137D411yescharacteristic
122C2×C4⋊C4×C137—22no—
123C2×C4⋊C4×C137—22no—
124C2×C4⋊C4×C137—21no—
125C2×C4⋊C4×C137—21no—
126C2×C4⋊C4×C137—21no—
127C2×C4⋊C4×C137—21no—
128C2×C4⋊C4×C137—21no—
129C2×C42×C137—21no—
130C2×C42×C137C2311yescharacteristic
131#107×C137—22no—
132#107×C137—24no—
133#106×C137—22no—
134C4⋊C8×C137—21no—
135C4⋊C8×C137—21no—
136C4⋊C8×C137C2312yes—
137C4⋊C8×C137C2312yes—
138#103×C137—22no—
139#103×C137—24no—
140#103×C137—22no—
141#103×C137—22no—
142#102×C137—22no—
143C22⋊C8×C137—21no—
144C22⋊C8×C137—21no—
145#97×C137—21no—
146C4×C8×C137C2312yes—
147#41×C137—22no—
148#41×C137—41no—
149#41×C137—42no—
150#41×C137—42no—
151C2×Q8×C137—21no—
152C2×Q8×C137—22no—
153C2×Q8×C137—42no—
154C2×Q8×C137—42no—
155C2×Q8×C137—41no—
156C2×Q8×C137C2×D411yescharacteristic
157C2×Q8×C137C2×D411yescharacteristic
158C2×Q8×C137C2×D412yes—
159C2×D4×C137—22no—
160C2×D4×C137—21no—
161C2×D4×C137—21no—
162C2×D4×C137C2×D411yescharacteristic
163C2×D4×C137C2×D411yescharacteristic
164C22×C4×C137—22no—
165C22×C4×C137—22no—
166C22×C4×C137—22no—
167C22×C4×C137—21no—
168C22×C4×C137—21no—
169C22×C4×C137—21no—
170C22×C4×C137—21no—
171C22×C4×C137—41no—
172C22×C4×C137C2×D411yescharacteristic
173C22×C4×C137C2×D411yescharacteristic
174C22×C4×C137C2×D411yescharacteristic
175Q16×C137—44no—
176SD16×C137—44no—
177M16×C137—42no—
178C2×C8×C137—22no—
179C2×C8×C137—22no—
180C2×C8×C137—22no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—22no—
183C4⋊C4×C137—22no—
184C4⋊C4×C137—22no—
185C4⋊C4×C137—22no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—22no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—21no—
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×D411yescharacteristic
205C4⋊C4×C137C2×D411yescharacteristic
206C4⋊C4×C137C2×D411yescharacteristic
207C4⋊C4×C137C2×D411yescharacteristic
208C4⋊C4×C137C2×D412yes—
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—21no—
220C22⋊C4×C137—21no—
221C42×C137—22no—
222C42×C137—21no—
223C42×C137—21no—
224C42×C137—22no—
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#12011yescharacteristic
234Q8×C137—21no—
235Q8×C137—21no—
236Q8×C137—22no—
237Q8×C137—42no—
238Q8×C137—41no—
239Q8×C137—41no—
240Q8×C137—42no—
241Q8×C137—42no—
242Q8×C137—82no—
243D4×C137—21no—
244D4×C137—22no—
245D4×C137—42no—
246D4×C137—44no—
247D4×C137—81no—
248D4×C137AGL(1,ℤ/8)12yes—
249C2×C4×C137—21no—
250C2×C4×C137—21no—
251C2×C4×C137—22no—
252C2×C4×C137—21no—
253C2×C4×C137—21no—
254C2×C4×C137—21no—
255C2×C4×C137—21no—
256C2×C4×C137—22no—
257C2×C4×C137—22no—
258C2×C4×C137—21no—
259C2×C4×C137—21no—
260C2×C4×C137—41no—
261C2×C4×C137—42no—
262C2×C4×C137—44no—
263C2×C4×C137—41no—
264C2×C4×C137—41no—
265C2×C4×C137—42no—
266C2×C4×C137—42no—
267C2×C4×C137—81no—
268C2×C4×C137—82no—
269C2×C4×C137—82no—
270C2×C4×C137—82no—
271C2×C4×C137#12011yescharacteristic
272C2×C4×C137#12011yescharacteristic
273C2×C4×C137#12011yescharacteristic
274C2×C4×C137C22×D411yescharacteristic
275C2×C4×C137C22×D411yescharacteristic
276C2×C4×C137C22×D411yescharacteristic
277C8×C137—42no—
278C8×C137—42no—
279C8×C137—42no—
280C22×C137—21no—
281C22×C137—22no—
282C22×C137—21no—
283C22×C137—21no—
284C22×C137—41no—
285C22×C137—41no—
286C22×C137—41no—
287C22×C137—41no—
288C22×C137—41no—
289C22×C137C2×#12011yescharacteristic
290C4×C137—21no—
291C4×C137—21no—
292C4×C137—21no—
293C4×C137—21no—
294C4×C137—41no—
295C4×C137—41no—
296C4×C137—42no—
297C4×C137—41no—
298C4×C137—41no—
299C4×C137—41no—
300C4×C137—41no—
301C4×C137—42no—
302C4×C137—82no—
303C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
304C4×C137#57511yescharacteristic
305C4×C137#57812yes—
306C2×C137—41no—
307C2×C137—42no—
308C2×C137—81no—
309C2×C137#300611yescharacteristic
310C2×C137#301111yescharacteristic
311C2×C137#302211yescharacteristic
312#18844C13711yesmaximal, characteristic, direct product factor, Sylow
313C137#1884411yescharacteristic, direct product factor, Sylow
314#3515C2×C13711yescharacteristic
315#3239C2×C13711yescharacteristic, complemented
316#3230C2×C13711yescharacteristic, complemented
317#3039C2×C13711yescharacteristic, complemented
318#1648C2×C13712yescomplemented
319#1647C2×C13712yescomplemented
320#1644C2×C13712yescomplemented
321#1643C2×C13712yescomplemented
322#1559C2×C13711yescharacteristic, complemented
323#1495C2×C13711yescharacteristic, complemented
324#1488C2×C13711yescharacteristic, complemented
325C2×#137—21no—
326#557C22×C13711yescharacteristic
327#554C22×C13711yescharacteristic
328#554C22×C13711yescharacteristic
329#553—21no—
330#552—21no—
331#552—22no—
332#552—21no—
333#547—21no—
334#547C22×C13711yescharacteristic
335#541—21no—
336#539—22no—
337#533C22×C13711yescharacteristic
338C2×C4⋊Q8C22×C13711yescharacteristic
339#529—21no—
340#529C22×C13711yescharacteristic
341C2×#125—21no—
342#520—21no—
343C4×#41C22×C13711yescharacteristic
344#501—21no—
345#500C22×C13712yes—
346#489—21no—
347#487C22×C13712yes—
348#484—22no—
349#484—21no—
350#481—21no—
351#477—22no—
352#475C22×C13712yes—
353#474C22×C13712yes—
354#462—21no—
355#462C22×C13712yes—
356#461—21no—
357#461C22×C13712yes—
358#432C22×C13711yescharacteristic
359#431C22×C13711yescharacteristic
360#423C22×C13711yescharacteristic
361#419—21no—
362#333C22×C13712yes—
363#332C22×C13712yes—
364Q8⋊C8C22×C13712yes—
365Q8⋊C8C22×C13712yes—
366D4⋊C8C22×C13712yes—
367D4⋊C8C22×C13712yes—
368C2×#41—21no—
369C2×#41D4×C13711yescharacteristic
370C22×Q8—21no—
371#137—44no—
372C2×Q16—22no—
373C2×SD16—22no—
374C2×M16—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8C23×C13712yes—
380C4⋊Q8C23×C13711yescharacteristic
381C4⋊Q8C23×C13711yescharacteristic
382#126—22no—
383#126—22no—
384#126—22no—
385#126—21no—
386#125—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125—21no—
391#125—22no—
392#125—21no—
393#125—21no—
394#125D4×C13711yescharacteristic
395#124D4×C13711yescharacteristic
396#123—22no—
397#123—22no—
398#123—22no—
399#123—21no—
400#123—22no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8D4×C13711yescharacteristic
408C22⋊Q8D4×C13712yes—
409#121—22no—
410#121—21no—
411#121D4×C13711yescharacteristic
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8D4×C13711yescharacteristic
416C4×Q8D4×C13712yes—
417C4×Q8C23×C13711yescharacteristic
418C4×Q8C23×C13711yescharacteristic
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4D4×C13711yescharacteristic
427C4×D4C23×C13711yescharacteristic
428C4×D4C23×C13711yescharacteristic
429#117—22no—
430#117—22no—
431#117—21no—
432#117—21no—
433#117—21no—
434#117D4×C13711yescharacteristic
435C2×C4⋊C4—22no—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C42—21no—
443C2×C42C23×C13711yescharacteristic
444#107—22no—
445#107—24no—
446#106—22no—
447C4⋊C8—21no—
448C4⋊C8—21no—
449C4⋊C8C23×C13712yes—
450C4⋊C8C23×C13712yes—
451#103—22no—
452#103—24no—
453#103—22no—
454#103—22no—
455#102—22no—
456C22⋊C8—21no—
457C22⋊C8—21no—
458#97—21no—
459C4×C8C23×C13712yes—
460#41—22no—
461#41—41no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×Q8—22no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—41no—
469C2×Q8C2×D4×C13711yescharacteristic
470C2×Q8C2×D4×C13711yescharacteristic
471C2×Q8C2×D4×C13712yes—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4C2×D4×C13711yescharacteristic
476C2×D4C2×D4×C13711yescharacteristic
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—41no—
485C22×C4C2×D4×C13711yescharacteristic
486C22×C4C2×D4×C13711yescharacteristic
487C22×C4C2×D4×C13711yescharacteristic
488Q16—44no—
489SD16—44no—
490M16—42no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4C2×D4×C13711yescharacteristic
518C4⋊C4C2×D4×C13711yescharacteristic
519C4⋊C4C2×D4×C13711yescharacteristic
520C4⋊C4C2×D4×C13711yescharacteristic
521C4⋊C4C2×D4×C13712yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42C2×D4×C13711yescharacteristic
541C42C2×D4×C13711yescharacteristic
542C42C2×D4×C13711yescharacteristic
543C42C24×C13711yescharacteristic
544C23—21no—
545C23—21no—
546C23#120×C13711yescharacteristic
547Q8—21no—
548Q8—21no—
549Q8—22no—
550Q8—42no—
551Q8—41no—
552Q8—41no—
553Q8—42no—
554Q8—42no—
555Q8—82no—
556D4—21no—
557D4—22no—
558D4—42no—
559D4—44no—
560D4—81no—
561D4AGL(1,ℤ/8)×C13712yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4#120×C13711yescharacteristic
585C2×C4#120×C13711yescharacteristic
586C2×C4#120×C13711yescharacteristic
587C2×C4C22×D4×C13711yescharacteristic
588C2×C4C22×D4×C13711yescharacteristic
589C2×C4C22×D4×C13711yescharacteristic
590C8—42no—
591C8—42no—
592C8—42no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22C2×#120×C13711yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—82no—
616C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
617C4#575×C13711yescharacteristic
618C4#578×C13712yes—
619C2—41no—
620C2—42no—
621C2—81no—
622C2#3006×C13711yescharacteristic
623C2#3011×C13711yescharacteristic
624C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3515×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. #3239×C137C2semidirect
3. #3230×C137C2semidirect
4. #3039×C137C2semidirect
5. #1648×C137C2semidirect
6. #1647×C137C2semidirect
7. #1644×C137C2semidirect
8. #1643×C137C2semidirect
9. #1559×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
312. #18844C137direct (N × H)
313. C137#18844direct (N × H)
315. #3239C2×C137semidirect
316. #3230C2×C137semidirect
317. #3039C2×C137semidirect
318. #1648C2×C137semidirect
319. #1647C2×C137semidirect
320. #1644C2×C137semidirect
321. #1643C2×C137semidirect
322. #1559C2×C137semidirect
323. #1495C2×C137semidirect
324. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3515×C1372
#3239×C1372
#3230×C1372
#3039×C1372
#1648×C1372
#1647×C1372
#1644×C1372
#1643×C1372
#1559×C1372
#1495×C1372
#1488×C1372
#18844137