Finite Group Databasebeta

← C137×#18857 #46868243

Subgroups — C137×#18857 #46868243

Order 35072.

956 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506×C137C211yesmaximal, characteristic, complemented
2#3253×C137C211yesmaximal, characteristic, complemented
3#3210×C137C211yesmaximal, characteristic, complemented
4#3109×C137C211yesmaximal, characteristic, complemented
5#1635×C137C212yesmaximal, complemented
6#1634×C137C212yesmaximal, complemented
7#1625×C137C212yesmaximal, complemented
8#1624×C137C212yesmaximal, complemented
9#1559×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic, complemented
11#1486×C137C211yesmaximal, characteristic, complemented
12#559×C137—22no—
13#556×C137—21no—
14#555×C137C2212yes—
15#553×C137—22no—
16#550×C137—21no—
17#547×C137C2212yes—
18#538×C137—22no—
19#535×C137—21no—
20#533×C137C2211yescharacteristic
21#532×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23#525×C137—22no—
24#520×C137—21no—
25C2×C4×D4×C137C2211yescharacteristic
26#494×C137C2211yescharacteristic, complemented
27#492×C137C2211yescharacteristic, complemented
28#487×C137C2211yescharacteristic, complemented
29#486×C137C2211yescharacteristic, complemented
30#481×C137—21no—
31#476×C137—21no—
32#476×C137C2212yes—
33#476×C137C2212yescomplemented
34#469×C137—22no—
35#464×C137—22no—
36#459×C137—21no—
37#459×C137C2212yes—
38#459×C137C2212yescomplemented
39#452×C137—22no—
40#447×C137—21no—
41#442×C137—22no—
42#432×C137C2211yescharacteristic
43#431×C137C2211yescharacteristic
44#423×C137C2211yescharacteristic
45#333×C137C2212yescomplemented
46#332×C137C2212yescomplemented
47D4⋊C8×C137C2212yes—
48D4⋊C8×C137C2212yes—
49D4⋊C8×C137C2212yescomplemented
50D4⋊C8×C137C2212yescomplemented
51C2×#41×C137—22no—
52C2×#41×C137—21no—
53C2×#41×C137—21no—
54C22×D4×C137D411yescharacteristic
55C23×C4×C137—21no—
56C2×Q16×C137—22no—
57C2×SD16×C137—22no—
58C2×D8×C137—22no—
59C2×D8×C137—22no—
60C2×M16×C137—21no—
61C4⋊Q8×C137D411yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63C4⋊Q8×C137C2311yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65#127×C137—22no—
66#127×C137D411yescharacteristic
67#127×C137C2311yescharacteristic
68#126×C137—24no—
69#125×C137—21no—
70#124×C137—22no—
71#124×C137—24no—
72#124×C137—21no—
73#124×C137—21no—
74#123×C137—24no—
75#123×C137—24no—
76C22⋊Q8×C137—21no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137D412yes—
79#121×C137—24no—
80#121×C137—22no—
81#121×C137—22no—
82#121×C137—22no—
83#121×C137—21no—
84#121×C137—21no—
85#121×C137—21no—
86#121×C137D412yes—
87#120×C137—24no—
88C4×Q8×C137—22no—
89C4×Q8×C137—21no—
90C4×Q8×C137D412yes—
91C4×D4×C137—24no—
92C4×D4×C137—22no—
93C4×D4×C137—24no—
94C4×D4×C137—21no—
95C4×D4×C137—21no—
96C4×D4×C137—21no—
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137D412yes—
100C4×D4×C137C2312yes—
101C4×D4×C137C2312yes—
102#117×C137—21no—
103#117×C137—22no—
104#117×C137—22no—
105C2×C4⋊C4×C137—22no—
106C2×C4⋊C4×C137D411yescharacteristic
107C2×C22⋊C4×C137—21no—
108C2×C42×C137C2311yescharacteristic
109#106×C137—22no—
110#106×C137—22no—
111C4⋊C8×C137—21no—
112C4⋊C8×C137—21no—
113C4⋊C8×C137C2312yes—
114C4⋊C8×C137C2312yes—
115#103×C137—22no—
116#102×C137—22no—
117#102×C137—22no—
118#102×C137—22no—
119#102×C137—22no—
120#102×C137—22no—
121C22⋊C8×C137—21no—
122C22⋊C8×C137—21no—
123#97×C137—21no—
124C4×C8×C137C2311yescharacteristic
125C4×C8×C137C2311yescharacteristic
126C24×C137—21no—
127#41×C137—44no—
128#41×C137—82no—
129#41×C137—82no—
130#41×C137—82no—
131C2×Q8×C137—21no—
132C2×Q8×C137—21no—
133C2×Q8×C137—21no—
134C2×Q8×C137C2×D412yes—
135C2×D4×C137—22no—
136C2×D4×C137—24no—
137C2×D4×C137—24no—
138C2×D4×C137—21no—
139C2×D4×C137—21no—
140C2×D4×C137—21no—
141C2×D4×C137—21no—
142C2×D4×C137—21no—
143C2×D4×C137—21no—
144C2×D4×C137—42no—
145C2×D4×C137—41no—
146C2×D4×C137C2×D412yes—
147C2×D4×C137C2×D412yes—
148C22×C4×C137—24no—
149C22×C4×C137—21no—
150C22×C4×C137—22no—
151C22×C4×C137—22no—
152C22×C4×C137—22no—
153C22×C4×C137—41no—
154C22×C4×C137—81no—
155C22×C4×C137—81no—
156C22×C4×C137C2×D411yescharacteristic
157C22×C4×C137C2×D411yescharacteristic
158C22×C4×C137C2×D411yescharacteristic
159Q16×C137—82no—
160SD16×C137—82no—
161D8×C137—44no—
162D8×C137—82no—
163M16×C137—81no—
164C2×C8×C137—21no—
165C2×C8×C137—21no—
166C2×C8×C137—22no—
167C2×C8×C137—22no—
168C4⋊C4×C137—24no—
169C4⋊C4×C137—21no—
170C4⋊C4×C137—22no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—22no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137C2×D412yes—
179C4⋊C4×C137C2×D412yes—
180C4⋊C4×C137C2×D412yes—
181C22⋊C4×C137—24no—
182C22⋊C4×C137—24no—
183C22⋊C4×C137—24no—
184C22⋊C4×C137—22no—
185C22⋊C4×C137—22no—
186C22⋊C4×C137—22no—
187C22⋊C4×C137—22no—
188C22⋊C4×C137—22no—
189C22⋊C4×C137—41no—
190C22⋊C4×C137—41no—
191C42×C137—24no—
192C42×C137—22no—
193C42×C137C2×D411yescharacteristic
194C42×C137C2×D411yescharacteristic
195C42×C137C2×D411yescharacteristic
196C42×C137C2411yescharacteristic
197C23×C137—22no—
198C23×C137—22no—
199C23×C137—42no—
200C23×C137—41no—
201C23×C137—81no—
202C23×C137—81no—
203C23×C137#12011yescharacteristic
204Q8×C137—42no—
205Q8×C137—42no—
206Q8×C137—42no—
207Q8×C137—41no—
208Q8×C137—41no—
209D4×C137—22no—
210D4×C137—22no—
211D4×C137—22no—
212D4×C137—22no—
213D4×C137—42no—
214D4×C137—42no—
215D4×C137—42no—
216D4×C137—44no—
217D4×C137—42no—
218D4×C137—41no—
219D4×C137—41no—
220D4×C137—81no—
221D4×C137—84no—
222D4×C137—84no—
223D4×C137—82no—
224C2×C4×C137—22no—
225C2×C4×C137—21no—
226C2×C4×C137—21no—
227C2×C4×C137—22no—
228C2×C4×C137—22no—
229C2×C4×C137—22no—
230C2×C4×C137—22no—
231C2×C4×C137—41no—
232C2×C4×C137—44no—
233C2×C4×C137—42no—
234C2×C4×C137—42no—
235C2×C4×C137—41no—
236C2×C4×C137—82no—
237C2×C4×C137—82no—
238C2×C4×C137—81no—
239C2×C4×C137—84no—
240C2×C4×C137#12011yescharacteristic
241C2×C4×C137#12011yescharacteristic
242C2×C4×C137#12011yescharacteristic
243C2×C4×C137C22×D411yescharacteristic
244C2×C4×C137C22×D411yescharacteristic
245C2×C4×C137C22×D411yescharacteristic
246C8×C137—41no—
247C8×C137—41no—
248C8×C137—42no—
249C8×C137—42no—
250C22×C137—21no—
251C22×C137—21no—
252C22×C137—21no—
253C22×C137—42no—
254C22×C137—42no—
255C22×C137—42no—
256C22×C137—42no—
257C22×C137—42no—
258C22×C137—42no—
259C22×C137—84no—
260C22×C137—81no—
261C22×C137—81no—
262C22×C137—81no—
263C22×C137C2×#12011yescharacteristic
264C4×C137—21no—
265C4×C137—21no—
266C4×C137—21no—
267C4×C137—21no—
268C4×C137—42no—
269C4×C137—42no—
270C4×C137—42no—
271C4×C137—42no—
272C4×C137—81no—
273C4×C137#57512yes—
274C4×C137#57611yescharacteristic
275C4×C137#57711yescharacteristic
276C2×C137—41no—
277C2×C137—82no—
278C2×C137—82no—
279C2×C137—161no—
280C2×C137#300511yescharacteristic
281C2×C137#301211yescharacteristic
282C2×C137#302111yescharacteristic
283#18857C13711yesmaximal, characteristic, direct product factor, Sylow
284C137#1885711yescharacteristic, direct product factor, Sylow
285#3506C2×C13711yescharacteristic, complemented
286#3253C2×C13711yescharacteristic, complemented
287#3210C2×C13711yescharacteristic, complemented
288#3109C2×C13711yescharacteristic, complemented
289#1635C2×C13712yescomplemented
290#1634C2×C13712yescomplemented
291#1625C2×C13712yescomplemented
292#1624C2×C13712yescomplemented
293#1559C2×C13711yescharacteristic, complemented
294#1496C2×C13711yescharacteristic, complemented
295#1486C2×C13711yescharacteristic, complemented
296#559—22no—
297#556—21no—
298#555C22×C13712yes—
299#553—22no—
300#550—21no—
301#547C22×C13712yes—
302#538—22no—
303#535—21no—
304#533C22×C13711yescharacteristic
305#532C22×C13711yescharacteristic
306#532C22×C13711yescharacteristic
307#525—22no—
308#520—21no—
309C2×C4×D4C22×C13711yescharacteristic
310#494C22×C13711yescharacteristic, complemented
311#492C22×C13711yescharacteristic, complemented
312#487C22×C13711yescharacteristic, complemented
313#486C22×C13711yescharacteristic, complemented
314#481—21no—
315#476—21no—
316#476C22×C13712yes—
317#476C22×C13712yescomplemented
318#469—22no—
319#464—22no—
320#459—21no—
321#459C22×C13712yes—
322#459C22×C13712yescomplemented
323#452—22no—
324#447—21no—
325#442—22no—
326#432C22×C13711yescharacteristic
327#431C22×C13711yescharacteristic
328#423C22×C13711yescharacteristic
329#333C22×C13712yescomplemented
330#332C22×C13712yescomplemented
331D4⋊C8C22×C13712yes—
332D4⋊C8C22×C13712yes—
333D4⋊C8C22×C13712yescomplemented
334D4⋊C8C22×C13712yescomplemented
335C2×#41—22no—
336C2×#41—21no—
337C2×#41—21no—
338C22×D4D4×C13711yescharacteristic
339C23×C4—21no—
340C2×Q16—22no—
341C2×SD16—22no—
342C2×D8—22no—
343C2×D8—22no—
344C2×M16—21no—
345C4⋊Q8D4×C13711yescharacteristic
346C4⋊Q8C23×C13711yescharacteristic
347C4⋊Q8C23×C13711yescharacteristic
348C4⋊Q8C23×C13711yescharacteristic
349#127—22no—
350#127D4×C13711yescharacteristic
351#127C23×C13711yescharacteristic
352#126—24no—
353#125—21no—
354#124—22no—
355#124—24no—
356#124—21no—
357#124—21no—
358#123—24no—
359#123—24no—
360C22⋊Q8—21no—
361C22⋊Q8—22no—
362C22⋊Q8D4×C13712yes—
363#121—24no—
364#121—22no—
365#121—22no—
366#121—22no—
367#121—21no—
368#121—21no—
369#121—21no—
370#121D4×C13712yes—
371#120—24no—
372C4×Q8—22no—
373C4×Q8—21no—
374C4×Q8D4×C13712yes—
375C4×D4—24no—
376C4×D4—22no—
377C4×D4—24no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4D4×C13712yes—
384C4×D4C23×C13712yes—
385C4×D4C23×C13712yes—
386#117—21no—
387#117—22no—
388#117—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4D4×C13711yescharacteristic
391C2×C22⋊C4—21no—
392C2×C42C23×C13711yescharacteristic
393#106—22no—
394#106—22no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8C23×C13712yes—
398C4⋊C8C23×C13712yes—
399#103—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403#102—22no—
404#102—22no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407#97—21no—
408C4×C8C23×C13711yescharacteristic
409C4×C8C23×C13711yescharacteristic
410C24—21no—
411#41—44no—
412#41—82no—
413#41—82no—
414#41—82no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×D4×C13712yes—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4C2×D4×C13712yes—
431C2×D4C2×D4×C13712yes—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442C22×C4C2×D4×C13711yescharacteristic
443Q16—82no—
444SD16—82no—
445D8—44no—
446D8—82no—
447M16—81no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4C2×D4×C13712yes—
463C4⋊C4C2×D4×C13712yes—
464C4⋊C4C2×D4×C13712yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C42—24no—
476C42—22no—
477C42C2×D4×C13711yescharacteristic
478C42C2×D4×C13711yescharacteristic
479C42C2×D4×C13711yescharacteristic
480C42C24×C13711yescharacteristic
481C23—22no—
482C23—22no—
483C23—42no—
484C23—41no—
485C23—81no—
486C23—81no—
487C23#120×C13711yescharacteristic
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—41no—
492Q8—41no—
493D4—22no—
494D4—22no—
495D4—22no—
496D4—22no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—44no—
501D4—42no—
502D4—41no—
503D4—41no—
504D4—81no—
505D4—84no—
506D4—84no—
507D4—82no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—84no—
524C2×C4#120×C13711yescharacteristic
525C2×C4#120×C13711yescharacteristic
526C2×C4#120×C13711yescharacteristic
527C2×C4C22×D4×C13711yescharacteristic
528C2×C4C22×D4×C13711yescharacteristic
529C2×C4C22×D4×C13711yescharacteristic
530C8—41no—
531C8—41no—
532C8—42no—
533C8—42no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—84no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#120×C13711yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—81no—
557C4#575×C13712yes—
558C4#576×C13711yescharacteristic
559C4#577×C13711yescharacteristic
560C2—41no—
561C2—82no—
562C2—82no—
563C2—161no—
564C2#3005×C13711yescharacteristic
565C2#3012×C13711yescharacteristic
566C2#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. #3506×C137C2semidirect
2. #3253×C137C2semidirect
3. #3210×C137C2semidirect
4. #3109×C137C2semidirect
5. #1635×C137C2semidirect
6. #1634×C137C2semidirect
7. #1625×C137C2semidirect
8. #1624×C137C2semidirect
9. #1559×C137C2semidirect
10. #1496×C137C2semidirect
11. #1486×C137C2semidirect
26. #494×C137C22semidirect
27. #492×C137C22semidirect
28. #487×C137C22semidirect
29. #486×C137C22semidirect
33. #476×C137C22semidirect
38. #459×C137C22semidirect
45. #333×C137C22semidirect
46. #332×C137C22semidirect
49. D4⋊C8×C137C22semidirect
50. D4⋊C8×C137C22semidirect
283. #18857C137direct (N × H)
284. C137#18857direct (N × H)
285. #3506C2×C137semidirect
286. #3253C2×C137semidirect
287. #3210C2×C137semidirect
288. #3109C2×C137semidirect
289. #1635C2×C137semidirect
290. #1634C2×C137semidirect
291. #1625C2×C137semidirect
292. #1624C2×C137semidirect
293. #1559C2×C137semidirect
294. #1496C2×C137semidirect
295. #1486C2×C137semidirect
310. #494C22×C137semidirect
311. #492C22×C137semidirect
312. #487C22×C137semidirect
313. #486C22×C137semidirect
317. #476C22×C137semidirect
322. #459C22×C137semidirect
329. #333C22×C137semidirect
330. #332C22×C137semidirect
333. D4⋊C8C22×C137semidirect
334. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3506×C1372
#3253×C1372
#3210×C1372
#3109×C1372
#1635×C1372
#1634×C1372
#1625×C1372
#1624×C1372
#1559×C1372
#1496×C1372
#1486×C1372
#18857137