Finite Group Databasebeta

← C137×#18568 #46867954

Subgroups — C137×#18568 #46867954

Order 35072.

1066 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508×C137C211yesmaximal, characteristic, complemented
2#3254×C137C211yesmaximal, characteristic, complemented
3#3151×C137C211yesmaximal, characteristic, complemented
4#3106×C137C211yesmaximal, characteristic, complemented
5#1631×C137C212yesmaximal, complemented
6#1630×C137C212yesmaximal, complemented
7#1621×C137C212yesmaximal, complemented
8#1620×C137C212yesmaximal, complemented
9#1545×C137C211yesmaximal, characteristic, complemented
10#1489×C137C211yesmaximal, characteristic, complemented
11#1485×C137C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C137—22no—
13#559×C137—21no—
14#559×C137C2212yes—
15#550×C137—21no—
16#550×C137C2212yes—
17#546×C137—22no—
18#546×C137—22no—
19#540×C137—22no—
20#535×C137—21no—
21#533×C137C2211yescharacteristic
22#532×C137—21no—
23#532×C137—21no—
24#532×C137C2211yescharacteristic
25C2×#127×C137C2211yescharacteristic
26#518×C137—21no—
27C4×#41×C137C2211yescharacteristic
28#496×C137—22no—
29#493×C137C2212yes—
30#492×C137C2212yes—
31#482×C137—21no—
32#477×C137C2211yescharacteristic, complemented
33#477×C137C2212yescomplemented
34#476×C137C2211yescharacteristic, complemented
35#474×C137—21no—
36#468×C137—22no—
37#461×C137—21no—
38#460×C137C2211yescharacteristic, complemented
39#460×C137C2212yescomplemented
40#459×C137C2211yescharacteristic, complemented
41#449×C137—21no—
42C4×SD16×C137—21no—
43C4×D8×C137—21no—
44#431×C137C2211yescharacteristic
45#423×C137C2211yescharacteristic
46C4×M16×C137C2211yescharacteristic
47#331×C137C2212yes—
48#331×C137C2212yes—
49Q8⋊C8×C137C2212yescomplemented
50Q8⋊C8×C137C2212yescomplemented
51Q8⋊C8×C137C2212yescomplemented
52D4⋊C8×C137C2212yescomplemented
53C2×#41×C137—22no—
54C2×#41×C137—22no—
55C2×#41×C137—21no—
56C2×#41×C137D411yescharacteristic
57C22×D4×C137—21no—
58C22×D4×C137—21no—
59C22×D4×C137D412yes—
60AGL(1,ℤ/8)×C137—48no—
61C2×SD16×C137—21no—
62C2×SD16×C137—24no—
63C2×SD16×C137—22no—
64C2×D8×C137—21no—
65C2×D8×C137—24no—
66C2×M16×C137—21no—
67C2×M16×C137—21no—
68C4⋊Q8×C137D411yescharacteristic
69C4⋊Q8×C137D411yescharacteristic
70C4⋊Q8×C137C2311yescharacteristic
71#127×C137—22no—
72#127×C137—21no—
73#127×C137—21no—
74#127×C137D412yes—
75#127×C137C2311yescharacteristic
76#127×C137C2312yes—
77#125×C137—21no—
78#124×C137—22no—
79#124×C137—22no—
80#124×C137—22no—
81#124×C137—21no—
82#123×C137—24no—
83#123×C137—24no—
84C22⋊Q8×C137—21no—
85C22⋊Q8×C137D412yes—
86#121×C137—22no—
87#121×C137—22no—
88#121×C137—22no—
89#121×C137—24no—
90#121×C137—24no—
91#121×C137—22no—
92#121×C137—21no—
93#121×C137—21no—
94#120×C137—24no—
95C4×Q8×C137—21no—
96C4×Q8×C137D411yescharacteristic
97C4×Q8×C137C2311yescharacteristic
98C4×Q8×C137C2312yes—
99C4×D4×C137—22no—
100C4×D4×C137—22no—
101C4×D4×C137—22no—
102C4×D4×C137—21no—
103C4×D4×C137—21no—
104C4×D4×C137—22no—
105C4×D4×C137—22no—
106C4×D4×C137—22no—
107C4×D4×C137—21no—
108C4×D4×C137—21no—
109C4×D4×C137D411yescharacteristic
110C4×D4×C137C2311yescharacteristic
111#117×C137—21no—
112#117×C137—21no—
113#117×C137D411yescharacteristic
114C2×C22⋊C4×C137—24no—
115C2×C42×C137—21no—
116C2×C42×C137C2311yescharacteristic
117#107×C137—21no—
118#107×C137—22no—
119#106×C137—21no—
120C4⋊C8×C137—21no—
121C4⋊C8×C137C2312yes—
122C4⋊C8×C137C2311yescharacteristic
123C4⋊C8×C137C2311yescharacteristic
124#103×C137—21no—
125#103×C137—22no—
126#102×C137—21no—
127#102×C137—21no—
128#102×C137—21no—
129#102×C137—22no—
130C22⋊C8×C137—21no—
131#97×C137—21no—
132C4×C8×C137—21no—
133C4×C8×C137C2312yes—
134C24×C137—22no—
135#41×C137—41no—
136#41×C137—41no—
137#41×C137—44no—
138#41×C137—82no—
139#41×C137—82no—
140#41×C137—84no—
141C2×Q8×C137—21no—
142C2×Q8×C137—21no—
143C2×Q8×C137C2×D411yescharacteristic
144C2×Q8×C137C2×D412yes—
145C2×D4×C137—21no—
146C2×D4×C137—24no—
147C2×D4×C137—24no—
148C2×D4×C137—22no—
149C2×D4×C137—22no—
150C2×D4×C137—22no—
151C2×D4×C137—22no—
152C2×D4×C137—22no—
153C2×D4×C137—21no—
154C2×D4×C137—21no—
155C2×D4×C137—21no—
156C2×D4×C137—42no—
157C2×D4×C137—42no—
158C2×D4×C137—44no—
159C2×D4×C137—41no—
160C2×D4×C137—81no—
161C2×D4×C137—81no—
162C2×D4×C137C2×D411yescharacteristic
163C2×D4×C137C2×D414yes—
164C22×C4×C137—24no—
165C22×C4×C137—24no—
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
173SD16×C137—48no—
174SD16×C137—42no—
175SD16×C137—82no—
176D8×C137—48no—
177D8×C137—42no—
178M16×C137—24no—
179M16×C137—81no—
180C2×C8×C137—22no—
181C2×C8×C137—21no—
182C2×C8×C137—21no—
183C2×C8×C137—22no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—21no—
187C4⋊C4×C137—21no—
188C4⋊C4×C137—22no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—22no—
191C4⋊C4×C137—21no—
192C4⋊C4×C137—21no—
193C4⋊C4×C137C2×D411yescharacteristic
194C4⋊C4×C137C2×D411yescharacteristic
195C4⋊C4×C137C2×D412yes—
196C22⋊C4×C137—24no—
197C22⋊C4×C137—24no—
198C22⋊C4×C137—24no—
199C22⋊C4×C137—24no—
200C22⋊C4×C137—21no—
201C22⋊C4×C137—22no—
202C22⋊C4×C137—21no—
203C22⋊C4×C137—21no—
204C22⋊C4×C137—41no—
205C22⋊C4×C137—41no—
206C42×C137—21no—
207C42×C137—21no—
208C42×C137—21no—
209C42×C137—22no—
210C42×C137C2×D411yescharacteristic
211C42×C137C2×D411yescharacteristic
212C42×C137C2×D411yescharacteristic
213C42×C137C2411yescharacteristic
214C23×C137—24no—
215C23×C137—21no—
216C23×C137—44no—
217C23×C137—41no—
218C23×C137—82no—
219C23×C137—82no—
220C23×C137#12011yescharacteristic
221Q8×C137—21no—
222Q8×C137—21no—
223Q8×C137—22no—
224Q8×C137—22no—
225Q8×C137—42no—
226Q8×C137—42no—
227D4×C137—21no—
228D4×C137—21no—
229D4×C137—44no—
230D4×C137—44no—
231D4×C137—42no—
232D4×C137—42no—
233D4×C137—42no—
234D4×C137—42no—
235D4×C137—41no—
236D4×C137—41no—
237D4×C137—42no—
238D4×C137—41no—
239D4×C137—42no—
240D4×C137—41no—
241D4×C137—44no—
242D4×C137—82no—
243D4×C137—84no—
244D4×C137—84no—
245D4×C137—82no—
246C2×C4×C137—21no—
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—21no—
254C2×C4×C137—22no—
255C2×C4×C137—22no—
256C2×C4×C137—42no—
257C2×C4×C137—41no—
258C2×C4×C137—41no—
259C2×C4×C137—42no—
260C2×C4×C137—41no—
261C2×C4×C137—84no—
262C2×C4×C137—84no—
263C2×C4×C137—82no—
264C2×C4×C137#12011yescharacteristic
265C2×C4×C137#12011yescharacteristic
266C2×C4×C137#12011yescharacteristic
267C2×C4×C137C22×D411yescharacteristic
268C2×C4×C137C22×D411yescharacteristic
269C2×C4×C137C22×D411yescharacteristic
270C8×C137—24no—
271C8×C137—41no—
272C8×C137—41no—
273C8×C137—42no—
274C22×C137—21no—
275C22×C137—21no—
276C22×C137—21no—
277C22×C137—41no—
278C22×C137—44no—
279C22×C137—41no—
280C22×C137—44no—
281C22×C137—41no—
282C22×C137—44no—
283C22×C137—81no—
284C22×C137—81no—
285C22×C137—81no—
286C22×C137—164no—
287C22×C137C2×#12011yescharacteristic
288C4×C137—21no—
289C4×C137—21no—
290C4×C137—21no—
291C4×C137—41no—
292C4×C137—41no—
293C4×C137—42no—
294C4×C137—41no—
295C4×C137—42no—
296C4×C137—81no—
297C4×C137#45312yes—
298C4×C137#57511yescharacteristic
299C4×C137#57511yescharacteristic
300C4×C137#57611yescharacteristic
301C4×C137#57711yescharacteristic
302C2×C137—41no—
303C2×C137—81no—
304C2×C137—84no—
305C2×C137—161no—
306C2×C137#300511yescharacteristic
307C2×C137#301211yescharacteristic
308C2×C137C2×#45311yescharacteristic
309#18568C13711yesmaximal, characteristic, direct product factor, Sylow
310C137#1856811yescharacteristic, direct product factor, Sylow
311#3508C2×C13711yescharacteristic, complemented
312#3254C2×C13711yescharacteristic, complemented
313#3151C2×C13711yescharacteristic, complemented
314#3106C2×C13711yescharacteristic, complemented
315#1631C2×C13712yescomplemented
316#1630C2×C13712yescomplemented
317#1621C2×C13712yescomplemented
318#1620C2×C13712yescomplemented
319#1545C2×C13711yescharacteristic, complemented
320#1489C2×C13711yescharacteristic, complemented
321#1485C2×C13711yescharacteristic, complemented
322C2×AGL(1,ℤ/8)—22no—
323#559—21no—
324#559C22×C13712yes—
325#550—21no—
326#550C22×C13712yes—
327#546—22no—
328#546—22no—
329#540—22no—
330#535—21no—
331#533C22×C13711yescharacteristic
332#532—21no—
333#532—21no—
334#532C22×C13711yescharacteristic
335C2×#127C22×C13711yescharacteristic
336#518—21no—
337C4×#41C22×C13711yescharacteristic
338#496—22no—
339#493C22×C13712yes—
340#492C22×C13712yes—
341#482—21no—
342#477C22×C13711yescharacteristic, complemented
343#477C22×C13712yescomplemented
344#476C22×C13711yescharacteristic, complemented
345#474—21no—
346#468—22no—
347#461—21no—
348#460C22×C13711yescharacteristic, complemented
349#460C22×C13712yescomplemented
350#459C22×C13711yescharacteristic, complemented
351#449—21no—
352C4×SD16—21no—
353C4×D8—21no—
354#431C22×C13711yescharacteristic
355#423C22×C13711yescharacteristic
356C4×M16C22×C13711yescharacteristic
357#331C22×C13712yes—
358#331C22×C13712yes—
359Q8⋊C8C22×C13712yescomplemented
360Q8⋊C8C22×C13712yescomplemented
361Q8⋊C8C22×C13712yescomplemented
362D4⋊C8C22×C13712yescomplemented
363C2×#41—22no—
364C2×#41—22no—
365C2×#41—21no—
366C2×#41D4×C13711yescharacteristic
367C22×D4—21no—
368C22×D4—21no—
369C22×D4D4×C13712yes—
370AGL(1,ℤ/8)—48no—
371C2×SD16—21no—
372C2×SD16—24no—
373C2×SD16—22no—
374C2×D8—24no—
375C2×D8—21no—
376C2×M16—21no—
377C2×M16—21no—
378C4⋊Q8D4×C13711yescharacteristic
379C4⋊Q8D4×C13711yescharacteristic
380C4⋊Q8C23×C13711yescharacteristic
381#127—22no—
382#127—21no—
383#127—21no—
384#127D4×C13712yes—
385#127C23×C13711yescharacteristic
386#127C23×C13712yes—
387#125—21no—
388#124—22no—
389#124—22no—
390#124—22no—
391#124—21no—
392#123—24no—
393#123—24no—
394C22⋊Q8—21no—
395C22⋊Q8D4×C13712yes—
396#121—22no—
397#121—22no—
398#121—22no—
399#121—24no—
400#121—24no—
401#121—22no—
402#121—21no—
403#121—21no—
404#120—24no—
405C4×Q8—21no—
406C4×Q8D4×C13711yescharacteristic
407C4×Q8C23×C13711yescharacteristic
408C4×Q8C23×C13712yes—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4D4×C13711yescharacteristic
420C4×D4C23×C13711yescharacteristic
421#117—21no—
422#117—21no—
423#117D4×C13711yescharacteristic
424C2×C22⋊C4—24no—
425C2×C42—21no—
426C2×C42C23×C13711yescharacteristic
427#107—21no—
428#107—22no—
429#106—21no—
430C4⋊C8—21no—
431C4⋊C8C23×C13712yes—
432C4⋊C8C23×C13711yescharacteristic
433C4⋊C8C23×C13711yescharacteristic
434#103—21no—
435#103—22no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—22no—
440C22⋊C8—21no—
441#97—21no—
442C4×C8—21no—
443C4×C8C23×C13712yes—
444C24—22no—
445#41—41no—
446#41—41no—
447#41—44no—
448#41—82no—
449#41—82no—
450#41—84no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8C2×D4×C13711yescharacteristic
454C2×Q8C2×D4×C13712yes—
455C2×D4—21no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—44no—
469C2×D4—41no—
470C2×D4—81no—
471C2×D4—81no—
472C2×D4C2×D4×C13711yescharacteristic
473C2×D4C2×D4×C13714yes—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4C2×D4×C13711yescharacteristic
481C22×C4C2×D4×C13711yescharacteristic
482C22×C4C2×D4×C13711yescharacteristic
483SD16—48no—
484SD16—42no—
485SD16—82no—
486D8—48no—
487D8—42no—
488M16—24no—
489M16—81no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4C2×D4×C13711yescharacteristic
504C4⋊C4C2×D4×C13711yescharacteristic
505C4⋊C4C2×D4×C13712yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—21no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—22no—
520C42C2×D4×C13711yescharacteristic
521C42C2×D4×C13711yescharacteristic
522C42C2×D4×C13711yescharacteristic
523C42C24×C13711yescharacteristic
524C23—24no—
525C23—21no—
526C23—44no—
527C23—41no—
528C23—82no—
529C23—82no—
530C23#120×C13711yescharacteristic
531Q8—21no—
532Q8—21no—
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—42no—
537D4—21no—
538D4—21no—
539D4—44no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—44no—
544D4—42no—
545D4—42no—
546D4—41no—
547D4—41no—
548D4—42no—
549D4—41no—
550D4—42no—
551D4—41no—
552D4—82no—
553D4—84no—
554D4—84no—
555D4—82no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—82no—
574C2×C4#120×C13711yescharacteristic
575C2×C4#120×C13711yescharacteristic
576C2×C4#120×C13711yescharacteristic
577C2×C4C22×D4×C13711yescharacteristic
578C2×C4C22×D4×C13711yescharacteristic
579C2×C4C22×D4×C13711yescharacteristic
580C8—24no—
581C8—41no—
582C8—41no—
583C8—42no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—41no—
588C22—44no—
589C22—41no—
590C22—44no—
591C22—41no—
592C22—44no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—164no—
597C22C2×#120×C13711yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4#453×C13712yes—
608C4#575×C13711yescharacteristic
609C4#575×C13711yescharacteristic
610C4#576×C13711yescharacteristic
611C4#577×C13711yescharacteristic
612C2—41no—
613C2—84no—
614C2—81no—
615C2—161no—
616C2#3005×C13711yescharacteristic
617C2#3012×C13711yescharacteristic
618C2C2×#453×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
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
1. #3508×C137C2semidirect
2. #3254×C137C2semidirect
3. #3151×C137C2semidirect
4. #3106×C137C2semidirect
5. #1631×C137C2semidirect
6. #1630×C137C2semidirect
7. #1621×C137C2semidirect
8. #1620×C137C2semidirect
9. #1545×C137C2semidirect
10. #1489×C137C2semidirect
11. #1485×C137C2semidirect
32. #477×C137C22semidirect
33. #477×C137C22semidirect
34. #476×C137C22semidirect
38. #460×C137C22semidirect
39. #460×C137C22semidirect
40. #459×C137C22semidirect
49. Q8⋊C8×C137C22semidirect
50. Q8⋊C8×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. D4⋊C8×C137C22semidirect
309. #18568C137direct (N × H)
310. C137#18568direct (N × H)
311. #3508C2×C137semidirect
312. #3254C2×C137semidirect
313. #3151C2×C137semidirect
314. #3106C2×C137semidirect
315. #1631C2×C137semidirect
316. #1630C2×C137semidirect
317. #1621C2×C137semidirect
318. #1620C2×C137semidirect
319. #1545C2×C137semidirect
320. #1489C2×C137semidirect
321. #1485C2×C137semidirect
342. #477C22×C137semidirect
343. #477C22×C137semidirect
344. #476C22×C137semidirect
348. #460C22×C137semidirect
349. #460C22×C137semidirect
350. #459C22×C137semidirect
359. Q8⋊C8C22×C137semidirect
360. Q8⋊C8C22×C137semidirect
361. Q8⋊C8C22×C137semidirect
362. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3508×C1372
#3254×C1372
#3151×C1372
#3106×C1372
#1631×C1372
#1630×C1372
#1621×C1372
#1620×C1372
#1545×C1372
#1489×C1372
#1485×C1372
#18568137