Finite Group Databasebeta

← C137×#18840 #46868226

Subgroups — C137×#18840 #46868226

Order 35072.

892 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507×C137C211yesmaximal, characteristic, complemented
2#3254×C137C211yesmaximal, characteristic, complemented
3#3207×C137C211yesmaximal, characteristic
4#3112×C137C211yesmaximal, characteristic, complemented
5#1636×C137C212yesmaximal, complemented
6#1630×C137C212yesmaximal, complemented
7#1626×C137C212yesmaximal, complemented
8#1620×C137C212yesmaximal, complemented
9#1557×C137C211yesmaximal, characteristic, complemented
10#1497×C137C211yesmaximal, characteristic, complemented
11#1487×C137C211yesmaximal, characteristic, complemented
12#559×C137C2212yes—
13#557×C137C2212yes—
14#556×C137—21no—
15#556×C137—22no—
16#554×C137—21no—
17#552×C137—22no—
18#539×C137—22no—
19#536×C137—21no—
20#533×C137C2211yescharacteristic
21#532×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23#529×C137—22no—
24#520×C137—21no—
25C2×C4×Q8×C137C2211yescharacteristic
26#493×C137C2211yescharacteristic
27#492×C137C2211yescharacteristic
28#487×C137C2211yescharacteristic
29#486×C137C2211yescharacteristic
30#480×C137—21no—
31#477×C137C2212yes—
32#477×C137C2212yes—
33#474×C137—21no—
34#468×C137—22no—
35#463×C137—22no—
36#461×C137—21no—
37#460×C137C2212yes—
38#460×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—
49Q8⋊C8×C137C2212yes—
50Q8⋊C8×C137C2212yes—
51C2×#41×C137—21no—
52C2×#41×C137—21no—
53C22×Q8×C137D411yescharacteristic
54C2×SD16×C137—22no—
55C2×SD16×C137—22no—
56C2×SD16×C137—22no—
57C2×D8×C137—22no—
58C2×M16×C137—21no—
59C4⋊Q8×C137—22no—
60C4⋊Q8×C137D411yescharacteristic
61C4⋊Q8×C137D411yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63C4⋊Q8×C137C2311yescharacteristic
64#127×C137C2311yescharacteristic
65#127×C137C2311yescharacteristic
66#126×C137—24no—
67#126×C137—24no—
68#125×C137—22no—
69#125×C137—24no—
70#125×C137—21no—
71#124×C137—22no—
72#124×C137—22no—
73#124×C137—21no—
74#124×C137—21no—
75#123×C137—24no—
76#123×C137—24no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137—22no—
79C22⋊Q8×C137—24no—
80C22⋊Q8×C137—21no—
81C22⋊Q8×C137D412yes—
82#121×C137—21no—
83#121×C137—21no—
84#121×C137—21no—
85#121×C137D412yes—
86C4×Q8×C137—22no—
87C4×Q8×C137—22no—
88C4×Q8×C137—21no—
89C4×Q8×C137—21no—
90C4×Q8×C137—21no—
91C4×Q8×C137D412yes—
92C4×Q8×C137C2312yes—
93C4×Q8×C137C2312yes—
94C4×D4×C137—24no—
95C4×D4×C137—22no—
96C4×D4×C137—22no—
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137D412yes—
101#117×C137—22no—
102#117×C137—24no—
103#117×C137—21no—
104C2×C4⋊C4×C137—22no—
105C2×C4⋊C4×C137—21no—
106C2×C4⋊C4×C137D411yescharacteristic
107C2×C42×C137—21no—
108C2×C42×C137C2311yescharacteristic
109#107×C137—22no—
110#107×C137—22no—
111C4⋊C8×C137—21no—
112C4⋊C8×C137—21no—
113C4⋊C8×C137C2312yes—
114C4⋊C8×C137C2312yes—
115#103×C137—22no—
116#103×C137—22no—
117#103×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
126#41×C137—44no—
127#41×C137—82no—
128C2×Q8×C137—21no—
129C2×Q8×C137—21no—
130C2×Q8×C137—21no—
131C2×Q8×C137—21no—
132C2×Q8×C137—42no—
133C2×Q8×C137C2×D412yes—
134C2×Q8×C137C2×D412yes—
135C2×D4×C137—22no—
136C2×D4×C137—21no—
137C2×D4×C137—21no—
138C2×D4×C137—21no—
139C2×D4×C137—21no—
140C2×D4×C137—21no—
141C2×D4×C137—41no—
142C2×D4×C137C2×D412yes—
143C22×C4×C137—21no—
144C22×C4×C137—24no—
145C22×C4×C137—21no—
146C22×C4×C137—22no—
147C22×C4×C137—41no—
148C22×C4×C137C2×D411yescharacteristic
149C22×C4×C137C2×D411yescharacteristic
150C22×C4×C137C2×D411yescharacteristic
151SD16×C137—44no—
152SD16×C137—82no—
153SD16×C137—82no—
154D8×C137—82no—
155M16×C137—81no—
156C2×C8×C137—21no—
157C2×C8×C137—21no—
158C2×C8×C137—22no—
159C2×C8×C137—22no—
160C4⋊C4×C137—24no—
161C4⋊C4×C137—24no—
162C4⋊C4×C137—24no—
163C4⋊C4×C137—21no—
164C4⋊C4×C137—21no—
165C4⋊C4×C137—22no—
166C4⋊C4×C137—21no—
167C4⋊C4×C137—22no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—22no—
170C4⋊C4×C137—21no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137C2×D412yes—
175C4⋊C4×C137C2×D412yes—
176C4⋊C4×C137C2×D412yes—
177C22⋊C4×C137—24no—
178C22⋊C4×C137—24no—
179C22⋊C4×C137—24no—
180C22⋊C4×C137—22no—
181C22⋊C4×C137—22no—
182C22⋊C4×C137—22no—
183C22⋊C4×C137—41no—
184C22⋊C4×C137—41no—
185C42×C137—24no—
186C42×C137—22no—
187C42×C137—22no—
188C42×C137—22no—
189C42×C137C2×D411yescharacteristic
190C42×C137C2×D411yescharacteristic
191C42×C137C2×D411yescharacteristic
192C42×C137C2411yescharacteristic
193C23×C137—22no—
194C23×C137—41no—
195C23×C137#12011yescharacteristic
196Q8×C137—22no—
197Q8×C137—22no—
198Q8×C137—22no—
199Q8×C137—22no—
200Q8×C137—42no—
201Q8×C137—42no—
202Q8×C137—42no—
203Q8×C137—81no—
204D4×C137—42no—
205D4×C137—42no—
206D4×C137—44no—
207D4×C137—42no—
208D4×C137—42no—
209D4×C137—41no—
210D4×C137—41no—
211D4×C137—41no—
212D4×C137—41no—
213D4×C137—82no—
214C2×C4×C137—22no—
215C2×C4×C137—21no—
216C2×C4×C137—21no—
217C2×C4×C137—22no—
218C2×C4×C137—22no—
219C2×C4×C137—22no—
220C2×C4×C137—22no—
221C2×C4×C137—22no—
222C2×C4×C137—41no—
223C2×C4×C137—44no—
224C2×C4×C137—42no—
225C2×C4×C137—41no—
226C2×C4×C137—82no—
227C2×C4×C137—81no—
228C2×C4×C137—84no—
229C2×C4×C137#12011yescharacteristic
230C2×C4×C137#12011yescharacteristic
231C2×C4×C137#12011yescharacteristic
232C2×C4×C137C22×D411yescharacteristic
233C2×C4×C137C22×D411yescharacteristic
234C2×C4×C137C22×D411yescharacteristic
235C8×C137—41no—
236C8×C137—41no—
237C8×C137—42no—
238C8×C137—42no—
239C22×C137—21no—
240C22×C137—21no—
241C22×C137—21no—
242C22×C137—42no—
243C22×C137—42no—
244C22×C137—42no—
245C22×C137—81no—
246C22×C137—81no—
247C22×C137—81no—
248C22×C137C2×#12011yescharacteristic
249C4×C137—21no—
250C4×C137—21no—
251C4×C137—21no—
252C4×C137—21no—
253C4×C137—42no—
254C4×C137—42no—
255C4×C137—42no—
256C4×C137—42no—
257C4×C137—42no—
258C4×C137—81no—
259C4×C137#57512yes—
260C4×C137#57711yescharacteristic
261C4×C137#57811yescharacteristic
262C2×C137—41no—
263C2×C137—82no—
264C2×C137—161no—
265C2×C137#300511yescharacteristic
266C2×C137#301311yescharacteristic
267C2×C137#302111yescharacteristic
268#18840C13711yesmaximal, characteristic, direct product factor, Sylow
269C137#1884011yescharacteristic, direct product factor, Sylow
270#3507C2×C13711yescharacteristic, complemented
271#3254C2×C13711yescharacteristic, complemented
272#3207C2×C13711yescharacteristic
273#3112C2×C13711yescharacteristic, complemented
274#1636C2×C13712yescomplemented
275#1630C2×C13712yescomplemented
276#1626C2×C13712yescomplemented
277#1620C2×C13712yescomplemented
278#1557C2×C13711yescharacteristic, complemented
279#1497C2×C13711yescharacteristic, complemented
280#1487C2×C13711yescharacteristic, complemented
281#559C22×C13712yes—
282#557C22×C13712yes—
283#556—21no—
284#556—22no—
285#554—21no—
286#552—22no—
287#539—22no—
288#536—21no—
289#533C22×C13711yescharacteristic
290#532C22×C13711yescharacteristic
291#532C22×C13711yescharacteristic
292#529—22no—
293#520—21no—
294C2×C4×Q8C22×C13711yescharacteristic
295#493C22×C13711yescharacteristic
296#492C22×C13711yescharacteristic
297#487C22×C13711yescharacteristic
298#486C22×C13711yescharacteristic
299#480—21no—
300#477C22×C13712yes—
301#477C22×C13712yes—
302#474—21no—
303#468—22no—
304#463—22no—
305#461—21no—
306#460C22×C13712yes—
307#460C22×C13712yes—
308#449—22no—
309#448—21no—
310#440—22no—
311#432C22×C13711yescharacteristic
312#431C22×C13711yescharacteristic
313#423C22×C13711yescharacteristic
314#332C22×C13712yes—
315#331C22×C13712yes—
316Q8⋊C8C22×C13712yes—
317Q8⋊C8C22×C13712yes—
318Q8⋊C8C22×C13712yes—
319Q8⋊C8C22×C13712yes—
320C2×#41—21no—
321C2×#41—21no—
322C22×Q8D4×C13711yescharacteristic
323C2×SD16—22no—
324C2×SD16—22no—
325C2×SD16—22no—
326C2×D8—22no—
327C2×M16—21no—
328C4⋊Q8—22no—
329C4⋊Q8D4×C13711yescharacteristic
330C4⋊Q8D4×C13711yescharacteristic
331C4⋊Q8C23×C13711yescharacteristic
332C4⋊Q8C23×C13711yescharacteristic
333#127C23×C13711yescharacteristic
334#127C23×C13711yescharacteristic
335#126—24no—
336#126—24no—
337#125—22no—
338#125—24no—
339#125—21no—
340#124—22no—
341#124—22no—
342#124—21no—
343#124—21no—
344#123—24no—
345#123—24no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—24no—
349C22⋊Q8—21no—
350C22⋊Q8D4×C13712yes—
351#121—21no—
352#121—21no—
353#121—21no—
354#121D4×C13712yes—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8D4×C13712yes—
361C4×Q8C23×C13712yes—
362C4×Q8C23×C13712yes—
363C4×D4—24no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4D4×C13712yes—
370#117—22no—
371#117—24no—
372#117—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4D4×C13711yescharacteristic
376C2×C42—21no—
377C2×C42C23×C13711yescharacteristic
378#107—22no—
379#107—22no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8C23×C13712yes—
383C4⋊C8C23×C13712yes—
384#103—22no—
385#103—22no—
386#103—22no—
387#102—22no—
388#102—22no—
389#102—22no—
390C22⋊C8—21no—
391C22⋊C8—21no—
392#97—21no—
393C4×C8C23×C13711yescharacteristic
394C4×C8C23×C13711yescharacteristic
395#41—44no—
396#41—82no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×Q8C2×D4×C13712yes—
403C2×Q8C2×D4×C13712yes—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4C2×D4×C13712yes—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4C2×D4×C13711yescharacteristic
418C22×C4C2×D4×C13711yescharacteristic
419C22×C4C2×D4×C13711yescharacteristic
420SD16—44no—
421SD16—82no—
422SD16—82no—
423D8—82no—
424M16—81no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4C2×D4×C13712yes—
444C4⋊C4C2×D4×C13712yes—
445C4⋊C4C2×D4×C13712yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42C2×D4×C13711yescharacteristic
459C42C2×D4×C13711yescharacteristic
460C42C2×D4×C13711yescharacteristic
461C42C24×C13711yescharacteristic
462C23—22no—
463C23—41no—
464C23#120×C13711yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—22no—
468Q8—22no—
469Q8—42no—
470Q8—42no—
471Q8—42no—
472Q8—81no—
473D4—42no—
474D4—42no—
475D4—44no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—82no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—84no—
498C2×C4#120×C13711yescharacteristic
499C2×C4#120×C13711yescharacteristic
500C2×C4#120×C13711yescharacteristic
501C2×C4C22×D4×C13711yescharacteristic
502C2×C4C22×D4×C13711yescharacteristic
503C2×C4C22×D4×C13711yescharacteristic
504C8—41no—
505C8—41no—
506C8—42no—
507C8—42no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C2×#120×C13711yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—81no—
528C4#575×C13712yes—
529C4#577×C13711yescharacteristic
530C4#578×C13711yescharacteristic
531C2—41no—
532C2—82no—
533C2—161no—
534C2#3005×C13711yescharacteristic
535C2#3013×C13711yescharacteristic
536C2#3021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3207×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
1. #3507×C137C2semidirect
2. #3254×C137C2semidirect
4. #3112×C137C2semidirect
5. #1636×C137C2semidirect
6. #1630×C137C2semidirect
7. #1626×C137C2semidirect
8. #1620×C137C2semidirect
9. #1557×C137C2semidirect
10. #1497×C137C2semidirect
11. #1487×C137C2semidirect
268. #18840C137direct (N × H)
269. C137#18840direct (N × H)
270. #3507C2×C137semidirect
271. #3254C2×C137semidirect
273. #3112C2×C137semidirect
274. #1636C2×C137semidirect
275. #1630C2×C137semidirect
276. #1626C2×C137semidirect
277. #1620C2×C137semidirect
278. #1557C2×C137semidirect
279. #1497C2×C137semidirect
280. #1487C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3507×C1372
#3254×C1372
#3207×C1372
#3112×C1372
#1636×C1372
#1630×C1372
#1626×C1372
#1620×C1372
#1557×C1372
#1497×C1372
#1487×C1372
#18840137