Finite Group Databasebeta

← C137×#18854 #46868240

Subgroups — C137×#18854 #46868240

Order 35072.

1132 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3497×C137C211yesmaximal, characteristic, complemented
2#3253×C137C211yesmaximal, characteristic, complemented
3#3207×C137C211yesmaximal, characteristic, complemented
4#3112×C137C211yesmaximal, characteristic, complemented
5#1637×C137C212yesmaximal, complemented
6#1631×C137C212yesmaximal, complemented
7#1627×C137C212yesmaximal, complemented
8#1621×C137C212yesmaximal, complemented
9#1557×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic, complemented
11#1487×C137C211yesmaximal, characteristic, complemented
12C2×2−1+4×C137—21no—
13#550×C137—21no—
14#550×C137C2212yes—
15D4×Q8×C137—21no—
16D4×Q8×C137C2212yes—
17#548×C137—22no—
18#548×C137—22no—
19#537×C137—22no—
20#537×C137—21no—
21#532×C137—22no—
22#532×C137C2211yescharacteristic
23#532×C137C2211yescharacteristic
24#532×C137C2211yescharacteristic
25C2×C4×Q8×C137C2211yescharacteristic
26#493×C137C2211yescharacteristic
27#492×C137C2211yescharacteristic
28#487×C137C2211yescharacteristic
29#486×C137C2211yescharacteristic
30#480×C137—21no—
31#477×C137C2212yes—
32#476×C137—21no—
33#476×C137C2212yes—
34#469×C137—22no—
35#463×C137—22no—
36#460×C137C2212yes—
37#459×C137—21no—
38#459×C137C2212yes—
39#449×C137—22no—
40#448×C137—21no—
41#440×C137—22no—
42#432×C137C2211yescharacteristic
43#431×C137C2211yescharacteristic
44#423×C137C2211yescharacteristic
45#332×C137C2212yes—
46#331×C137C2212yes—
47Q8⋊C8×C137C2212yes—
48Q8⋊C8×C137C2212yes—
49D4⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
512−1+4×C137—48no—
52C2×#41×C137—24no—
53C2×#41×C137—22no—
54C2×#41×C137—22no—
55C2×#41×C137—21no—
56C2×#41×C137—21no—
57C2×#41×C137—21no—
58C22×Q8×C137—22no—
59C22×Q8×C137—22no—
60C22×Q8×C137D411yescharacteristic
61C2×SD16×C137—22no—
62C2×SD16×C137—22no—
63C2×D8×C137—22no—
64C2×D8×C137—22no—
65C2×M16×C137—21no—
66C4⋊Q8×C137—22no—
67C4⋊Q8×C137D411yescharacteristic
68C4⋊Q8×C137C2311yescharacteristic
69C4⋊Q8×C137C2311yescharacteristic
70#127×C137—22no—
71#127×C137D411yescharacteristic
72#127×C137C2311yescharacteristic
73#127×C137C2311yescharacteristic
74#124×C137—24no—
75#124×C137—21no—
76#124×C137—21no—
77#124×C137—21no—
78C22⋊Q8×C137—22no—
79C22⋊Q8×C137—24no—
80C22⋊Q8×C137—22no—
81C22⋊Q8×C137—24no—
82#121×C137—24no—
83#121×C137—22no—
84#121×C137—24no—
85#121×C137—24no—
86#121×C137—21no—
87#121×C137—21no—
88#121×C137—22no—
89#121×C137—21no—
90#121×C137—21no—
91#121×C137D412yes—
92#121×C137D412yes—
93C4×Q8×C137—21no—
94C4×Q8×C137—21no—
95C4×Q8×C137D412yes—
96C4×Q8×C137C2312yes—
97C4×D4×C137—24no—
98C4×D4×C137—22no—
99C4×D4×C137—22no—
100C4×D4×C137—21no—
101C4×D4×C137—21no—
102C4×D4×C137—21no—
103C4×D4×C137—21no—
104C4×D4×C137D412yes—
105C4×D4×C137C2312yes—
106C2×C4⋊C4×C137—21no—
107C2×C4⋊C4×C137D411yescharacteristic
108C2×C42×C137—21no—
109C2×C42×C137C2311yescharacteristic
110#107×C137—22no—
111#106×C137—22no—
112C4⋊C8×C137—21no—
113C4⋊C8×C137—21no—
114C4⋊C8×C137C2312yes—
115C4⋊C8×C137C2312yes—
116#103×C137—22no—
117#103×C137—22no—
118#102×C137—22no—
119#102×C137—22no—
120#102×C137—22no—
121#102×C137—22no—
122C22⋊C8×C137—21no—
123C22⋊C8×C137—21no—
124#97×C137—21no—
125C4×C8×C137C2311yescharacteristic
126C4×C8×C137C2311yescharacteristic
127#41×C137—416no—
128#41×C137—44no—
129#41×C137—44no—
130#41×C137—48no—
131#41×C137—44no—
132#41×C137—44no—
133#41×C137—82no—
134C2×Q8×C137—24no—
135C2×Q8×C137—24no—
136C2×Q8×C137—21no—
137C2×Q8×C137—21no—
138C2×Q8×C137—21no—
139C2×Q8×C137—21no—
140C2×Q8×C137—48no—
141C2×Q8×C137—44no—
142C2×Q8×C137—44no—
143C2×Q8×C137C2×D412yes—
144C2×D4×C137—24no—
145C2×D4×C137—22no—
146C2×D4×C137—22no—
147C2×D4×C137—24no—
148C2×D4×C137—21no—
149C2×D4×C137—21no—
150C2×D4×C137—21no—
151C2×D4×C137—21no—
152C2×D4×C137—21no—
153C2×D4×C137—21no—
154C2×D4×C137—21no—
155C2×D4×C137—41no—
156C2×D4×C137C2×D412yes—
157C2×D4×C137C2×D412yes—
158C22×C4×C137—21no—
159C22×C4×C137—24no—
160C22×C4×C137—24no—
161C22×C4×C137—21no—
162C22×C4×C137—22no—
163C22×C4×C137—22no—
164C22×C4×C137—41no—
165C22×C4×C137C2×D411yescharacteristic
166C22×C4×C137C2×D411yescharacteristic
167C22×C4×C137C2×D411yescharacteristic
168SD16×C137—44no—
169SD16×C137—82no—
170D8×C137—82no—
171D8×C137—82no—
172M16×C137—81no—
173C2×C8×C137—21no—
174C2×C8×C137—21no—
175C2×C8×C137—22no—
176C2×C8×C137—22no—
177C4⋊C4×C137—24no—
178C4⋊C4×C137—22no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137C2×D412yes—
186C4⋊C4×C137C2×D412yes—
187C4⋊C4×C137C2×D412yes—
188C22⋊C4×C137—24no—
189C22⋊C4×C137—24no—
190C22⋊C4×C137—22no—
191C22⋊C4×C137—22no—
192C22⋊C4×C137—22no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—41no—
195C22⋊C4×C137—41no—
196C42×C137—22no—
197C42×C137—22no—
198C42×C137C2×D411yescharacteristic
199C42×C137C2×D411yescharacteristic
200C42×C137C2×D411yescharacteristic
201C42×C137C2411yescharacteristic
202C23×C137—22no—
203C23×C137—22no—
204C23×C137—41no—
205C23×C137#12011yescharacteristic
206Q8×C137—24no—
207Q8×C137—22no—
208Q8×C137—48no—
209Q8×C137—42no—
210Q8×C137—42no—
211Q8×C137—42no—
212Q8×C137AGL(1,ℤ/8)14yes—
213D4×C137—22no—
214D4×C137—22no—
215D4×C137—42no—
216D4×C137—48no—
217D4×C137—42no—
218D4×C137—42no—
219D4×C137—44no—
220D4×C137—44no—
221D4×C137—42no—
222D4×C137—42no—
223D4×C137—41no—
224D4×C137—41no—
225D4×C137—41no—
226D4×C137—41no—
227D4×C137—81no—
228D4×C137—84no—
229D4×C137—82no—
230C2×C4×C137—24no—
231C2×C4×C137—22no—
232C2×C4×C137—22no—
233C2×C4×C137—21no—
234C2×C4×C137—21no—
235C2×C4×C137—22no—
236C2×C4×C137—22no—
237C2×C4×C137—22no—
238C2×C4×C137—41no—
239C2×C4×C137—48no—
240C2×C4×C137—48no—
241C2×C4×C137—44no—
242C2×C4×C137—42no—
243C2×C4×C137—42no—
244C2×C4×C137—41no—
245C2×C4×C137—82no—
246C2×C4×C137—81no—
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—41no—
254C8×C137—41no—
255C8×C137—42no—
256C8×C137—42no—
257C22×C137—21no—
258C22×C137—21no—
259C22×C137—21no—
260C22×C137—42no—
261C22×C137—42no—
262C22×C137—42no—
263C22×C137—42no—
264C22×C137—42no—
265C22×C137—42no—
266C22×C137—81no—
267C22×C137—81no—
268C22×C137—81no—
269C22×C137C2×#12011yescharacteristic
270C4×C137—24no—
271C4×C137—21no—
272C4×C137—21no—
273C4×C137—21no—
274C4×C137—21no—
275C4×C137—42no—
276C4×C137—42no—
277C4×C137—42no—
278C4×C137—81no—
279C4×C137C2×AGL(1,ℤ/8)12yes—
280C4×C137#57711yescharacteristic
281C4×C137#57811yescharacteristic
282C2×C137—41no—
283C2×C137—82no—
284C2×C137—82no—
285C2×C137—161no—
286C2×C137#300311yescharacteristic
287C2×C137#301311yescharacteristic
288C2×C137#302111yescharacteristic
289#18854C13711yesmaximal, characteristic, direct product factor, Sylow
290C137#1885411yescharacteristic, direct product factor, Sylow
291#3497C2×C13711yescharacteristic, complemented
292#3253C2×C13711yescharacteristic, complemented
293#3207C2×C13711yescharacteristic, complemented
294#3112C2×C13711yescharacteristic, complemented
295#1637C2×C13712yescomplemented
296#1631C2×C13712yescomplemented
297#1627C2×C13712yescomplemented
298#1621C2×C13712yescomplemented
299#1557C2×C13711yescharacteristic, complemented
300#1496C2×C13711yescharacteristic, complemented
301#1487C2×C13711yescharacteristic, complemented
302C2×2−1+4—21no—
303#550—21no—
304#550C22×C13712yes—
305D4×Q8—21no—
306D4×Q8C22×C13712yes—
307#548—22no—
308#548—22no—
309#537—22no—
310#537—21no—
311#532—22no—
312#532C22×C13711yescharacteristic
313#532C22×C13711yescharacteristic
314#532C22×C13711yescharacteristic
315C2×C4×Q8C22×C13711yescharacteristic
316#493C22×C13711yescharacteristic
317#492C22×C13711yescharacteristic
318#487C22×C13711yescharacteristic
319#486C22×C13711yescharacteristic
320#480—21no—
321#477C22×C13712yes—
322#476—21no—
323#476C22×C13712yes—
324#469—22no—
325#463—22no—
326#460C22×C13712yes—
327#459—21no—
328#459C22×C13712yes—
329#449—22no—
330#448—21no—
331#440—22no—
332#432C22×C13711yescharacteristic
333#431C22×C13711yescharacteristic
334#423C22×C13711yescharacteristic
335#332C22×C13712yes—
336#331C22×C13712yes—
337Q8⋊C8C22×C13712yes—
338Q8⋊C8C22×C13712yes—
339D4⋊C8C22×C13712yes—
340D4⋊C8C22×C13712yes—
3412−1+4—48no—
342C2×#41—24no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—21no—
346C2×#41—21no—
347C2×#41—21no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8D4×C13711yescharacteristic
351C2×SD16—22no—
352C2×SD16—22no—
353C2×D8—22no—
354C2×D8—22no—
355C2×M16—21no—
356C4⋊Q8—22no—
357C4⋊Q8D4×C13711yescharacteristic
358C4⋊Q8C23×C13711yescharacteristic
359C4⋊Q8C23×C13711yescharacteristic
360#127—22no—
361#127D4×C13711yescharacteristic
362#127C23×C13711yescharacteristic
363#127C23×C13711yescharacteristic
364#124—24no—
365#124—21no—
366#124—21no—
367#124—21no—
368C22⋊Q8—22no—
369C22⋊Q8—24no—
370C22⋊Q8—22no—
371C22⋊Q8—24no—
372#121—24no—
373#121—22no—
374#121—24no—
375#121—24no—
376#121—21no—
377#121—21no—
378#121—22no—
379#121—21no—
380#121—21no—
381#121D4×C13712yes—
382#121D4×C13712yes—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8D4×C13712yes—
386C4×Q8C23×C13712yes—
387C4×D4—24no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4D4×C13712yes—
395C4×D4C23×C13712yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4D4×C13711yescharacteristic
398C2×C42—21no—
399C2×C42C23×C13711yescharacteristic
400#107—22no—
401#106—22no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8C23×C13712yes—
405C4⋊C8C23×C13712yes—
406#103—22no—
407#103—22no—
408#102—22no—
409#102—22no—
410#102—22no—
411#102—22no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414#97—21no—
415C4×C8C23×C13711yescharacteristic
416C4×C8C23×C13711yescharacteristic
417#41—416no—
418#41—44no—
419#41—44no—
420#41—48no—
421#41—44no—
422#41—44no—
423#41—82no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8C2×D4×C13712yes—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—41no—
446C2×D4C2×D4×C13712yes—
447C2×D4C2×D4×C13712yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4C2×D4×C13711yescharacteristic
456C22×C4C2×D4×C13711yescharacteristic
457C22×C4C2×D4×C13711yescharacteristic
458SD16—44no—
459SD16—82no—
460D8—82no—
461D8—82no—
462M16—81no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4C2×D4×C13712yes—
476C4⋊C4C2×D4×C13712yes—
477C4⋊C4C2×D4×C13712yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C42—22no—
487C42—22no—
488C42C2×D4×C13711yescharacteristic
489C42C2×D4×C13711yescharacteristic
490C42C2×D4×C13711yescharacteristic
491C42C24×C13711yescharacteristic
492C23—22no—
493C23—22no—
494C23—41no—
495C23#120×C13711yescharacteristic
496Q8—24no—
497Q8—22no—
498Q8—48no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8AGL(1,ℤ/8)×C13714yes—
503D4—22no—
504D4—22no—
505D4—42no—
506D4—48no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—44no—
511D4—42no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—81no—
518D4—84no—
519D4—82no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4#120×C13711yescharacteristic
538C2×C4#120×C13711yescharacteristic
539C2×C4#120×C13711yescharacteristic
540C2×C4C22×D4×C13711yescharacteristic
541C2×C4C22×D4×C13711yescharacteristic
542C2×C4C22×D4×C13711yescharacteristic
543C8—41no—
544C8—41no—
545C8—42no—
546C8—42no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#120×C13711yescharacteristic
560C4—24no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4C2×AGL(1,ℤ/8)×C13712yes—
570C4#577×C13711yescharacteristic
571C4#578×C13711yescharacteristic
572C2—41no—
573C2—82no—
574C2—82no—
575C2—161no—
576C2#3003×C13711yescharacteristic
577C2#3013×C13711yescharacteristic
578C2#3021×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. #3497×C137C2semidirect
2. #3253×C137C2semidirect
3. #3207×C137C2semidirect
4. #3112×C137C2semidirect
5. #1637×C137C2semidirect
6. #1631×C137C2semidirect
7. #1627×C137C2semidirect
8. #1621×C137C2semidirect
9. #1557×C137C2semidirect
10. #1496×C137C2semidirect
11. #1487×C137C2semidirect
289. #18854C137direct (N × H)
290. C137#18854direct (N × H)
291. #3497C2×C137semidirect
292. #3253C2×C137semidirect
293. #3207C2×C137semidirect
294. #3112C2×C137semidirect
295. #1637C2×C137semidirect
296. #1631C2×C137semidirect
297. #1627C2×C137semidirect
298. #1621C2×C137semidirect
299. #1557C2×C137semidirect
300. #1496C2×C137semidirect
301. #1487C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3497×C1372
#3253×C1372
#3207×C1372
#3112×C1372
#1637×C1372
#1631×C1372
#1627×C1372
#1621×C1372
#1557×C1372
#1496×C1372
#1487×C1372
#18854137