Finite Group Databasebeta

← C137×#18819 #46868205

Subgroups — C137×#18819 #46868205

Order 35072.

970 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508×C137C211yesmaximal, characteristic
2#3237×C137C211yesmaximal, characteristic, complemented
3#3208×C137C211yesmaximal, characteristic, complemented
4#3116×C137C211yesmaximal, characteristic, complemented
5#1642×C137C212yesmaximal, complemented
6#1640×C137C212yesmaximal, complemented
7#1631×C137C212yesmaximal, complemented
8#1630×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#533×C137C2211yescharacteristic
22#532×C137—21no—
23#532×C137—21no—
24C2×#127×C137C2211yescharacteristic
25#518×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#501×C137—21no—
28#492×C137C2212yes—
29#491×C137—21no—
30#486×C137C2212yes—
31#485×C137—22no—
32#485×C137—21no—
33#477×C137C2211yescharacteristic, complemented
34#477×C137C2212yescomplemented
35#476×C137C2211yescharacteristic, complemented
36#475×C137C2212yescomplemented
37#462×C137C2211yescharacteristic, complemented
38#461×C137C2211yescharacteristic, complemented
39#460×C137—22no—
40#452×C137—21no—
41#450×C137—22no—
42#441×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×C137D411yescharacteristic
56C22×D4×C137—21no—
57C22×D4×C137—21no—
58C22×D4×C137D412yes—
59C2×Q16×C137—21no—
60C2×SD16×C137—21no—
61C2×SD16×C137—24no—
62C4⋊Q8×C137D411yescharacteristic
63C4⋊Q8×C137D411yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65C4⋊Q8×C137C2311yescharacteristic
66#127×C137—22no—
67#127×C137—21no—
68#127×C137—21no—
69#127×C137D412yes—
70#127×C137C2312yes—
71#125×C137—21no—
72#125×C137—21no—
73#124×C137—22no—
74#124×C137—22no—
75#124×C137—22no—
76#123×C137—24no—
77#123×C137—24no—
78C22⋊Q8×C137—21no—
79C22⋊Q8×C137—21no—
80C22⋊Q8×C137—21no—
81C22⋊Q8×C137D412yes—
82#121×C137—22no—
83#121×C137—22no—
84#121×C137—22no—
85#121×C137—24no—
86#121×C137—24no—
87#121×C137—22no—
88#120×C137—24no—
89C4×Q8×C137—21no—
90C4×Q8×C137—21no—
91C4×Q8×C137—21no—
92C4×Q8×C137D411yescharacteristic
93C4×Q8×C137C2311yescharacteristic
94C4×Q8×C137C2312yes—
95C4×D4×C137—22no—
96C4×D4×C137—22no—
97C4×D4×C137—22no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137—22no—
101C4×D4×C137—22no—
102C4×D4×C137—22no—
103C4×D4×C137D411yescharacteristic
104C4×D4×C137C2311yescharacteristic
105#117×C137—21no—
106#117×C137—21no—
107#117×C137—21no—
108#117×C137D411yescharacteristic
109C2×C22⋊C4×C137—24no—
110C2×C42×C137—21no—
111C2×C42×C137C2311yescharacteristic
112#107×C137—21no—
113#107×C137—22no—
114#106×C137—21no—
115#106×C137—22no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2311yescharacteristic
119C4⋊C8×C137C2312yes—
120#103×C137—21no—
121#103×C137—22no—
122#103×C137—22no—
123#103×C137—21no—
124#103×C137—21no—
125#102×C137—21no—
126#102×C137—24no—
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—84no—
137#41×C137—84no—
138C2×Q8×C137—21no—
139C2×Q8×C137—21no—
140C2×Q8×C137—21no—
141C2×Q8×C137C2×D412yes—
142C2×Q8×C137C2×D411yescharacteristic
143C2×D4×C137—21no—
144C2×D4×C137—24no—
145C2×D4×C137—24no—
146C2×D4×C137—22no—
147C2×D4×C137—22no—
148C2×D4×C137—22no—
149C2×D4×C137—22no—
150C2×D4×C137—22no—
151C2×D4×C137—42no—
152C2×D4×C137—42no—
153C2×D4×C137—82no—
154C2×D4×C137—82no—
155C2×D4×C137C2×D411yescharacteristic
156C2×D4×C137C2×D414yes—
157C22×C4×C137—24no—
158C22×C4×C137—24no—
159C22×C4×C137—21no—
160C22×C4×C137—21no—
161C22×C4×C137—21no—
162C22×C4×C137C2×D411yescharacteristic
163C22×C4×C137C2×D411yescharacteristic
164C22×C4×C137C2×D411yescharacteristic
165Q16×C137—42no—
166SD16×C137—42no—
167SD16×C137—84no—
168C2×C8×C137—22no—
169C2×C8×C137—21no—
170C2×C8×C137—22no—
171C2×C8×C137—21no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—22no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—22no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—21no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—41no—
184C4⋊C4×C137—41no—
185C4⋊C4×C137C2×D411yescharacteristic
186C4⋊C4×C137C2×D411yescharacteristic
187C4⋊C4×C137C2×D412yes—
188C22⋊C4×C137—24no—
189C22⋊C4×C137—24no—
190C22⋊C4×C137—24no—
191C22⋊C4×C137—24no—
192C22⋊C4×C137—21no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—21no—
195C22⋊C4×C137—21no—
196C22⋊C4×C137—41no—
197C42×C137—21no—
198C42×C137—21no—
199C42×C137—21no—
200C42×C137—22no—
201C42×C137—41no—
202C42×C137C2×D411yescharacteristic
203C42×C137C2×D411yescharacteristic
204C42×C137C2×D411yescharacteristic
205C42×C137C2411yescharacteristic
206C23×C137—24no—
207C23×C137—21no—
208C23×C137—44no—
209C23×C137—82no—
210C23×C137—82no—
211C23×C137#12011yescharacteristic
212Q8×C137—21no—
213Q8×C137—21no—
214Q8×C137—24no—
215Q8×C137—42no—
216Q8×C137—42no—
217Q8×C137—41no—
218Q8×C137—41no—
219D4×C137—21no—
220D4×C137—21no—
221D4×C137—44no—
222D4×C137—44no—
223D4×C137—44no—
224D4×C137—42no—
225D4×C137—42no—
226D4×C137—42no—
227D4×C137—44no—
228D4×C137—82no—
229D4×C137—84no—
230D4×C137—84no—
231C2×C4×C137—21no—
232C2×C4×C137—21no—
233C2×C4×C137—22no—
234C2×C4×C137—21no—
235C2×C4×C137—21no—
236C2×C4×C137—21no—
237C2×C4×C137—22no—
238C2×C4×C137—22no—
239C2×C4×C137—41no—
240C2×C4×C137—42no—
241C2×C4×C137—42no—
242C2×C4×C137—42no—
243C2×C4×C137—41no—
244C2×C4×C137—41no—
245C2×C4×C137—84no—
246C2×C4×C137—84no—
247C2×C4×C137#12011yescharacteristic
248C2×C4×C137#12011yescharacteristic
249C2×C4×C137#12011yescharacteristic
250C2×C4×C137C22×D411yescharacteristic
251C2×C4×C137C22×D411yescharacteristic
252C2×C4×C137C22×D411yescharacteristic
253C8×C137—42no—
254C8×C137—41no—
255C8×C137—41no—
256C8×C137—42no—
257C22×C137—21no—
258C22×C137—21no—
259C22×C137—21no—
260C22×C137—41no—
261C22×C137—44no—
262C22×C137—41no—
263C22×C137—44no—
264C22×C137—41no—
265C22×C137—44no—
266C22×C137—164no—
267C22×C137C2×#12011yescharacteristic
268C4×C137—21no—
269C4×C137—21no—
270C4×C137—21no—
271C4×C137—21no—
272C4×C137—41no—
273C4×C137—41no—
274C4×C137—42no—
275C4×C137—41no—
276C4×C137—42no—
277C4×C137—81no—
278C4×C137—81no—
279C4×C137#57511yescharacteristic
280C4×C137#57511yescharacteristic
281C4×C137#57712yes—
282C2×C137—41no—
283C2×C137—81no—
284C2×C137—84no—
285C2×C137#300511yescharacteristic
286C2×C137#300911yescharacteristic
287C2×C137#301911yescharacteristic
288#18819C13711yesmaximal, characteristic, direct product factor, Sylow
289C137#1881911yescharacteristic, direct product factor, Sylow
290#3508C2×C13711yescharacteristic
291#3237C2×C13711yescharacteristic, complemented
292#3208C2×C13711yescharacteristic, complemented
293#3116C2×C13711yescharacteristic, complemented
294#1642C2×C13712yescomplemented
295#1640C2×C13712yescomplemented
296#1631C2×C13712yescomplemented
297#1630C2×C13712yescomplemented
298#1555C2×C13711yescharacteristic, complemented
299#1497C2×C13711yescharacteristic, complemented
300#1488C2×C13711yescharacteristic, complemented
301#559—21no—
302#559C22×C13712yes—
303#550—21no—
304#550C22×C13712yes—
305#546—22no—
306#546—22no—
307#540—22no—
308#535—21no—
309#533C22×C13711yescharacteristic
310#533C22×C13711yescharacteristic
311#532—21no—
312#532—21no—
313C2×#127C22×C13711yescharacteristic
314#518—21no—
315C4×#41C22×C13711yescharacteristic
316#501—21no—
317#492C22×C13712yes—
318#491—21no—
319#486C22×C13712yes—
320#485—22no—
321#485—21no—
322#477C22×C13711yescharacteristic, complemented
323#477C22×C13712yescomplemented
324#476C22×C13711yescharacteristic, complemented
325#475C22×C13712yescomplemented
326#462C22×C13711yescharacteristic, complemented
327#461C22×C13711yescharacteristic, complemented
328#460—22no—
329#452—21no—
330#450—22no—
331#441—21no—
332#440—21no—
333#432C22×C13711yescharacteristic
334#432C22×C13711yescharacteristic
335#431C22×C13711yescharacteristic
336#331C22×C13712yes—
337#331C22×C13712yes—
338Q8⋊C8C22×C13712yescomplemented
339Q8⋊C8C22×C13712yescomplemented
340Q8⋊C8C22×C13712yescomplemented
341D4⋊C8C22×C13712yescomplemented
342C2×#41—22no—
343C2×#41—22no—
344C2×#41D4×C13711yescharacteristic
345C22×D4—21no—
346C22×D4—21no—
347C22×D4D4×C13712yes—
348C2×Q16—21no—
349C2×SD16—21no—
350C2×SD16—24no—
351C4⋊Q8D4×C13711yescharacteristic
352C4⋊Q8D4×C13711yescharacteristic
353C4⋊Q8C23×C13711yescharacteristic
354C4⋊Q8C23×C13711yescharacteristic
355#127—22no—
356#127—21no—
357#127—21no—
358#127D4×C13712yes—
359#127C23×C13712yes—
360#125—21no—
361#125—21no—
362#124—22no—
363#124—22no—
364#124—22no—
365#123—24no—
366#123—24no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8D4×C13712yes—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—24no—
375#121—24no—
376#121—22no—
377#120—24no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8D4×C13711yescharacteristic
382C4×Q8C23×C13711yescharacteristic
383C4×Q8C23×C13712yes—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4D4×C13711yescharacteristic
393C4×D4C23×C13711yescharacteristic
394#117—21no—
395#117—21no—
396#117—21no—
397#117D4×C13711yescharacteristic
398C2×C22⋊C4—24no—
399C2×C42—21no—
400C2×C42C23×C13711yescharacteristic
401#107—21no—
402#107—22no—
403#106—21no—
404#106—22no—
405C4⋊C8—21no—
406C4⋊C8C23×C13712yes—
407C4⋊C8C23×C13711yescharacteristic
408C4⋊C8C23×C13711yescharacteristic
409#103—21no—
410#103—22no—
411#103—22no—
412#103—21no—
413#103—21no—
414#102—21no—
415#102—24no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419#97—21no—
420#97—21no—
421C4×C8C23×C13712yes—
422C24—22no—
423#41—41no—
424#41—41no—
425#41—84no—
426#41—84no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8C2×D4×C13712yes—
431C2×Q8C2×D4×C13711yescharacteristic
432C2×D4—21no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4C2×D4×C13711yescharacteristic
445C2×D4C2×D4×C13714yes—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4C2×D4×C13711yescharacteristic
452C22×C4C2×D4×C13711yescharacteristic
453C22×C4C2×D4×C13711yescharacteristic
454Q16—42no—
455SD16—42no—
456SD16—84no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4C2×D4×C13711yescharacteristic
475C4⋊C4C2×D4×C13711yescharacteristic
476C4⋊C4C2×D4×C13712yes—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—22no—
490C42—41no—
491C42C2×D4×C13711yescharacteristic
492C42C2×D4×C13711yescharacteristic
493C42C2×D4×C13711yescharacteristic
494C42C24×C13711yescharacteristic
495C23—24no—
496C23—21no—
497C23—44no—
498C23—82no—
499C23—82no—
500C23#120×C13711yescharacteristic
501Q8—21no—
502Q8—21no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506Q8—41no—
507Q8—41no—
508D4—21no—
509D4—21no—
510D4—44no—
511D4—44no—
512D4—44no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—44no—
517D4—82no—
518D4—84no—
519D4—84no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4#120×C13711yescharacteristic
537C2×C4#120×C13711yescharacteristic
538C2×C4#120×C13711yescharacteristic
539C2×C4C22×D4×C13711yescharacteristic
540C2×C4C22×D4×C13711yescharacteristic
541C2×C4C22×D4×C13711yescharacteristic
542C8—42no—
543C8—41no—
544C8—41no—
545C8—42no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—44no—
551C22—41no—
552C22—44no—
553C22—41no—
554C22—44no—
555C22—164no—
556C22C2×#120×C13711yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4#575×C13711yescharacteristic
569C4#575×C13711yescharacteristic
570C4#577×C13712yes—
571C2—41no—
572C2—84no—
573C2—81no—
574C2#3005×C13711yescharacteristic
575C2#3009×C13711yescharacteristic
576C2#3019×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3508×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. #3237×C137C2semidirect
3. #3208×C137C2semidirect
4. #3116×C137C2semidirect
5. #1642×C137C2semidirect
6. #1640×C137C2semidirect
7. #1631×C137C2semidirect
8. #1630×C137C2semidirect
9. #1555×C137C2semidirect
10. #1497×C137C2semidirect
11. #1488×C137C2semidirect
33. #477×C137C22semidirect
34. #477×C137C22semidirect
35. #476×C137C22semidirect
36. #475×C137C22semidirect
37. #462×C137C22semidirect
38. #461×C137C22semidirect
49. Q8⋊C8×C137C22semidirect
50. Q8⋊C8×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. D4⋊C8×C137C22semidirect
288. #18819C137direct (N × H)
289. C137#18819direct (N × H)
291. #3237C2×C137semidirect
292. #3208C2×C137semidirect
293. #3116C2×C137semidirect
294. #1642C2×C137semidirect
295. #1640C2×C137semidirect
296. #1631C2×C137semidirect
297. #1630C2×C137semidirect
298. #1555C2×C137semidirect
299. #1497C2×C137semidirect
300. #1488C2×C137semidirect
322. #477C22×C137semidirect
323. #477C22×C137semidirect
324. #476C22×C137semidirect
325. #475C22×C137semidirect
326. #462C22×C137semidirect
327. #461C22×C137semidirect
338. Q8⋊C8C22×C137semidirect
339. Q8⋊C8C22×C137semidirect
340. Q8⋊C8C22×C137semidirect
341. 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
#3237×C1372
#3208×C1372
#3116×C1372
#1642×C1372
#1640×C1372
#1631×C1372
#1630×C1372
#1555×C1372
#1497×C1372
#1488×C1372
#18819137