Finite Group Databasebeta

← C137×#18818 #46868204

Subgroups — C137×#18818 #46868204

Order 35072.

1004 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508×C137C211yesmaximal, characteristic, complemented
2#3216×C137C211yesmaximal, characteristic, complemented
3#3205×C137C211yesmaximal, characteristic, complemented
4#3115×C137C211yesmaximal, characteristic, complemented
5#1641×C137C212yesmaximal, complemented
6#1640×C137C212yesmaximal, complemented
7#1621×C137C212yesmaximal, complemented
8#1620×C137C212yesmaximal, complemented
9#1555×C137C211yesmaximal, characteristic, complemented
10#1497×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12#559×C137—21no—
13#559×C137C2212yes—
14#550×C137—21no—
15#550×C137C2212yes—
16#546×C137—22no—
17#546×C137—22no—
18#540×C137—22no—
19#535×C137—21no—
20#533×C137C2211yescharacteristic
21#532×C137—21no—
22#532×C137—21no—
23#532×C137C2211yescharacteristic
24C2×#127×C137C2211yescharacteristic
25#518×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#497×C137—21no—
28#493×C137C2212yes—
29#489×C137—21no—
30#486×C137C2212yes—
31#482×C137—21no—
32#475×C137C2211yescharacteristic, complemented
33#474×C137C2211yescharacteristic, complemented
34#462×C137C2212yescomplemented
35#460×C137C2211yescharacteristic, complemented
36#460×C137C2212yescomplemented
37#459×C137—22no—
38#459×C137C2211yescharacteristic, complemented
39#452×C137—22no—
40#449×C137—21no—
41#447×C137—22no—
42#442×C137—21no—
43#440×C137—21no—
44#432×C137C2211yescharacteristic
45#432×C137C2211yescharacteristic
46#431×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—
60C2×Q16×C137—22no—
61C2×SD16×C137—21no—
62C2×SD16×C137—22no—
63C2×D8×C137—21no—
64C2×D8×C137—24no—
65C4⋊Q8×C137D411yescharacteristic
66C4⋊Q8×C137D411yescharacteristic
67C4⋊Q8×C137C2311yescharacteristic
68#127×C137—22no—
69#127×C137—21no—
70#127×C137—21no—
71#127×C137D412yes—
72#127×C137C2311yescharacteristic
73#127×C137C2312yes—
74#125×C137—21no—
75#124×C137—22no—
76#124×C137—22no—
77#124×C137—22no—
78#124×C137—21no—
79#123×C137—24no—
80#123×C137—24no—
81C22⋊Q8×C137—21no—
82C22⋊Q8×C137D412yes—
83#121×C137—22no—
84#121×C137—22no—
85#121×C137—22no—
86#121×C137—24no—
87#121×C137—24no—
88#121×C137—22no—
89#121×C137—21no—
90#121×C137—21no—
91#120×C137—24no—
92C4×Q8×C137—21no—
93C4×Q8×C137D411yescharacteristic
94C4×Q8×C137C2312yes—
95C4×Q8×C137C2311yescharacteristic
96C4×D4×C137—22no—
97C4×D4×C137—22no—
98C4×D4×C137—22no—
99C4×D4×C137—21no—
100C4×D4×C137—21no—
101C4×D4×C137—22no—
102C4×D4×C137—22no—
103C4×D4×C137—22no—
104C4×D4×C137—21no—
105C4×D4×C137—21no—
106C4×D4×C137D411yescharacteristic
107C4×D4×C137C2311yescharacteristic
108#117×C137—21no—
109#117×C137—21no—
110#117×C137D411yescharacteristic
111C2×C22⋊C4×C137—24no—
112C2×C42×C137—21no—
113C2×C42×C137C2311yescharacteristic
114#107×C137—21no—
115#106×C137—21no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2312yes—
119C4⋊C8×C137C2311yescharacteristic
120#103×C137—21no—
121#103×C137—22no—
122#102×C137—21no—
123#102×C137—24no—
124#102×C137—21no—
125#102×C137—21no—
126#102×C137—22no—
127C22⋊C8×C137—21no—
128C22⋊C8×C137—21no—
129C22⋊C8×C137—21no—
130#97×C137—21no—
131#97×C137—21no—
132C4×C8×C137C2312yes—
133C24×C137—22no—
134#41×C137—41no—
135#41×C137—41no—
136#41×C137—82no—
137#41×C137—82no—
138#41×C137—82no—
139#41×C137—84no—
140C2×Q8×C137—21no—
141C2×Q8×C137—21no—
142C2×Q8×C137C2×D411yescharacteristic
143C2×Q8×C137C2×D412yes—
144C2×D4×C137—21no—
145C2×D4×C137—24no—
146C2×D4×C137—24no—
147C2×D4×C137—22no—
148C2×D4×C137—22no—
149C2×D4×C137—22no—
150C2×D4×C137—22no—
151C2×D4×C137—22no—
152C2×D4×C137—21no—
153C2×D4×C137—21no—
154C2×D4×C137—21no—
155C2×D4×C137—42no—
156C2×D4×C137—42no—
157C2×D4×C137—41no—
158C2×D4×C137—82no—
159C2×D4×C137—82no—
160C2×D4×C137C2×D411yescharacteristic
161C2×D4×C137C2×D414yes—
162C22×C4×C137—24no—
163C22×C4×C137—24no—
164C22×C4×C137—21no—
165C22×C4×C137—21no—
166C22×C4×C137—21no—
167C22×C4×C137—41no—
168C22×C4×C137C2×D411yescharacteristic
169C22×C4×C137C2×D411yescharacteristic
170C22×C4×C137C2×D411yescharacteristic
171Q16×C137—82no—
172SD16×C137—42no—
173SD16×C137—82no—
174D8×C137—42no—
175D8×C137—84no—
176C2×C8×C137—22no—
177C2×C8×C137—21no—
178C2×C8×C137—21no—
179C2×C8×C137—22no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—21no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—22no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—21no—
188C4⋊C4×C137—21no—
189C4⋊C4×C137C2×D412yes—
190C4⋊C4×C137C2×D411yescharacteristic
191C4⋊C4×C137C2×D411yescharacteristic
192C22⋊C4×C137—24no—
193C22⋊C4×C137—24no—
194C22⋊C4×C137—24no—
195C22⋊C4×C137—24no—
196C22⋊C4×C137—21no—
197C22⋊C4×C137—22no—
198C22⋊C4×C137—21no—
199C22⋊C4×C137—21no—
200C22⋊C4×C137—41no—
201C22⋊C4×C137—41no—
202C42×C137—21no—
203C42×C137—21no—
204C42×C137—21no—
205C42×C137—22no—
206C42×C137C2×D411yescharacteristic
207C42×C137C2×D411yescharacteristic
208C42×C137C2×D411yescharacteristic
209C42×C137C2411yescharacteristic
210C23×C137—24no—
211C23×C137—21no—
212C23×C137—44no—
213C23×C137—41no—
214C23×C137—82no—
215C23×C137—82no—
216C23×C137#12011yescharacteristic
217Q8×C137—21no—
218Q8×C137—21no—
219Q8×C137—22no—
220Q8×C137—22no—
221Q8×C137—42no—
222Q8×C137—42no—
223D4×C137—21no—
224D4×C137—21no—
225D4×C137—44no—
226D4×C137—44no—
227D4×C137—44no—
228D4×C137—42no—
229D4×C137—42no—
230D4×C137—42no—
231D4×C137—42no—
232D4×C137—41no—
233D4×C137—41no—
234D4×C137—42no—
235D4×C137—42no—
236D4×C137—42no—
237D4×C137—82no—
238D4×C137—84no—
239D4×C137—84no—
240D4×C137—82no—
241C2×C4×C137—21no—
242C2×C4×C137—21no—
243C2×C4×C137—22no—
244C2×C4×C137—21no—
245C2×C4×C137—21no—
246C2×C4×C137—21no—
247C2×C4×C137—22no—
248C2×C4×C137—22no—
249C2×C4×C137—41no—
250C2×C4×C137—42no—
251C2×C4×C137—42no—
252C2×C4×C137—42no—
253C2×C4×C137—41no—
254C2×C4×C137—82no—
255C2×C4×C137—84no—
256C2×C4×C137—84no—
257C2×C4×C137#12011yescharacteristic
258C2×C4×C137#12011yescharacteristic
259C2×C4×C137#12011yescharacteristic
260C2×C4×C137C22×D411yescharacteristic
261C2×C4×C137C22×D411yescharacteristic
262C2×C4×C137C22×D411yescharacteristic
263C8×C137—42no—
264C8×C137—41no—
265C8×C137—41no—
266C8×C137—42no—
267C22×C137—21no—
268C22×C137—21no—
269C22×C137—21no—
270C22×C137—41no—
271C22×C137—44no—
272C22×C137—41no—
273C22×C137—44no—
274C22×C137—41no—
275C22×C137—44no—
276C22×C137—81no—
277C22×C137—81no—
278C22×C137—81no—
279C22×C137—164no—
280C22×C137C2×#12011yescharacteristic
281C4×C137—21no—
282C4×C137—21no—
283C4×C137—21no—
284C4×C137—21no—
285C4×C137—41no—
286C4×C137—41no—
287C4×C137—42no—
288C4×C137—41no—
289C4×C137—42no—
290C4×C137—81no—
291C4×C137#57511yescharacteristic
292C4×C137#57511yescharacteristic
293C4×C137#57612yes—
294C2×C137—41no—
295C2×C137—81no—
296C2×C137—84no—
297C2×C137—161no—
298C2×C137#300511yescharacteristic
299C2×C137#300811yescharacteristic
300C2×C137#301911yescharacteristic
301#18818C13711yesmaximal, characteristic, direct product factor, Sylow
302C137#1881811yescharacteristic, direct product factor, Sylow
303#3508C2×C13711yescharacteristic, complemented
304#3216C2×C13711yescharacteristic, complemented
305#3205C2×C13711yescharacteristic, complemented
306#3115C2×C13711yescharacteristic, complemented
307#1641C2×C13712yescomplemented
308#1640C2×C13712yescomplemented
309#1621C2×C13712yescomplemented
310#1620C2×C13712yescomplemented
311#1555C2×C13711yescharacteristic, complemented
312#1497C2×C13711yescharacteristic, complemented
313#1488C2×C13711yescharacteristic, complemented
314#559—21no—
315#559C22×C13712yes—
316#550—21no—
317#550C22×C13712yes—
318#546—22no—
319#546—22no—
320#540—22no—
321#535—21no—
322#533C22×C13711yescharacteristic
323#532—21no—
324#532—21no—
325#532C22×C13711yescharacteristic
326C2×#127C22×C13711yescharacteristic
327#518—21no—
328C4×#41C22×C13711yescharacteristic
329#497—21no—
330#493C22×C13712yes—
331#489—21no—
332#486C22×C13712yes—
333#482—21no—
334#475C22×C13711yescharacteristic, complemented
335#474C22×C13711yescharacteristic, complemented
336#462C22×C13712yescomplemented
337#460C22×C13711yescharacteristic, complemented
338#460C22×C13712yescomplemented
339#459—22no—
340#459C22×C13711yescharacteristic, complemented
341#452—22no—
342#449—21no—
343#447—22no—
344#442—21no—
345#440—21no—
346#432C22×C13711yescharacteristic
347#432C22×C13711yescharacteristic
348#431C22×C13711yescharacteristic
349#331C22×C13712yes—
350#331C22×C13712yes—
351Q8⋊C8C22×C13712yescomplemented
352Q8⋊C8C22×C13712yescomplemented
353Q8⋊C8C22×C13712yescomplemented
354D4⋊C8C22×C13712yescomplemented
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—21no—
358C2×#41D4×C13711yescharacteristic
359C22×D4—21no—
360C22×D4—21no—
361C22×D4D4×C13712yes—
362C2×Q16—22no—
363C2×SD16—21no—
364C2×SD16—22no—
365C2×D8—21no—
366C2×D8—24no—
367C4⋊Q8D4×C13711yescharacteristic
368C4⋊Q8D4×C13711yescharacteristic
369C4⋊Q8C23×C13711yescharacteristic
370#127—22no—
371#127—21no—
372#127—21no—
373#127D4×C13712yes—
374#127C23×C13711yescharacteristic
375#127C23×C13712yes—
376#125—21no—
377#124—22no—
378#124—22no—
379#124—22no—
380#124—21no—
381#123—24no—
382#123—24no—
383C22⋊Q8—21no—
384C22⋊Q8D4×C13712yes—
385#121—22no—
386#121—22no—
387#121—22no—
388#121—24no—
389#121—24no—
390#121—22no—
391#121—21no—
392#121—21no—
393#120—24no—
394C4×Q8—21no—
395C4×Q8D4×C13711yescharacteristic
396C4×Q8C23×C13711yescharacteristic
397C4×Q8C23×C13712yes—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4D4×C13711yescharacteristic
409C4×D4C23×C13711yescharacteristic
410#117—21no—
411#117—21no—
412#117D4×C13711yescharacteristic
413C2×C22⋊C4—24no—
414C2×C42—21no—
415C2×C42C23×C13711yescharacteristic
416#107—21no—
417#106—21no—
418C4⋊C8—21no—
419C4⋊C8C23×C13712yes—
420C4⋊C8C23×C13711yescharacteristic
421C4⋊C8C23×C13711yescharacteristic
422#103—21no—
423#103—22no—
424#102—21no—
425#102—24no—
426#102—21no—
427#102—21no—
428#102—22no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432#97—21no—
433#97—21no—
434C4×C8C23×C13712yes—
435C24—22no—
436#41—41no—
437#41—41no—
438#41—82no—
439#41—82no—
440#41—82no—
441#41—84no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×D4×C13711yescharacteristic
445C2×Q8C2×D4×C13712yes—
446C2×D4—21no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4C2×D4×C13711yescharacteristic
463C2×D4C2×D4×C13714yes—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4C2×D4×C13711yescharacteristic
471C22×C4C2×D4×C13711yescharacteristic
472C22×C4C2×D4×C13711yescharacteristic
473Q16—82no—
474SD16—42no—
475SD16—82no—
476D8—42no—
477D8—84no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4C2×D4×C13712yes—
492C4⋊C4C2×D4×C13711yescharacteristic
493C4⋊C4C2×D4×C13711yescharacteristic
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—22no—
508C42C2×D4×C13711yescharacteristic
509C42C2×D4×C13711yescharacteristic
510C42C2×D4×C13711yescharacteristic
511C42C24×C13711yescharacteristic
512C23—24no—
513C23—21no—
514C23—44no—
515C23—41no—
516C23—82no—
517C23—82no—
518C23#120×C13711yescharacteristic
519Q8—21no—
520Q8—21no—
521Q8—22no—
522Q8—22no—
523Q8—42no—
524Q8—42no—
525D4—21no—
526D4—21no—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—41no—
535D4—41no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—82no—
540D4—84no—
541D4—84no—
542D4—82no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4#120×C13711yescharacteristic
560C2×C4#120×C13711yescharacteristic
561C2×C4#120×C13711yescharacteristic
562C2×C4C22×D4×C13711yescharacteristic
563C2×C4C22×D4×C13711yescharacteristic
564C2×C4C22×D4×C13711yescharacteristic
565C8—42no—
566C8—41no—
567C8—41no—
568C8—42no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—44no—
574C22—41no—
575C22—44no—
576C22—41no—
577C22—44no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—164no—
582C22C2×#120×C13711yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—81no—
593C4#575×C13711yescharacteristic
594C4#575×C13711yescharacteristic
595C4#576×C13712yes—
596C2—41no—
597C2—84no—
598C2—81no—
599C2—161no—
600C2#3005×C13711yescharacteristic
601C2#3008×C13711yescharacteristic
602C2#3019×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. #3216×C137C2semidirect
3. #3205×C137C2semidirect
4. #3115×C137C2semidirect
5. #1641×C137C2semidirect
6. #1640×C137C2semidirect
7. #1621×C137C2semidirect
8. #1620×C137C2semidirect
9. #1555×C137C2semidirect
10. #1497×C137C2semidirect
11. #1488×C137C2semidirect
32. #475×C137C22semidirect
33. #474×C137C22semidirect
34. #462×C137C22semidirect
35. #460×C137C22semidirect
36. #460×C137C22semidirect
38. #459×C137C22semidirect
49. Q8⋊C8×C137C22semidirect
50. Q8⋊C8×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. D4⋊C8×C137C22semidirect
301. #18818C137direct (N × H)
302. C137#18818direct (N × H)
303. #3508C2×C137semidirect
304. #3216C2×C137semidirect
305. #3205C2×C137semidirect
306. #3115C2×C137semidirect
307. #1641C2×C137semidirect
308. #1640C2×C137semidirect
309. #1621C2×C137semidirect
310. #1620C2×C137semidirect
311. #1555C2×C137semidirect
312. #1497C2×C137semidirect
313. #1488C2×C137semidirect
334. #475C22×C137semidirect
335. #474C22×C137semidirect
336. #462C22×C137semidirect
337. #460C22×C137semidirect
338. #460C22×C137semidirect
340. #459C22×C137semidirect
351. Q8⋊C8C22×C137semidirect
352. Q8⋊C8C22×C137semidirect
353. Q8⋊C8C22×C137semidirect
354. 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
#3216×C1372
#3205×C1372
#3115×C1372
#1641×C1372
#1640×C1372
#1621×C1372
#1620×C1372
#1555×C1372
#1497×C1372
#1488×C1372
#18818137