Finite Group Databasebeta

← C137×#18877 #46868263

Subgroups — C137×#18877 #46868263

Order 35072.

922 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511×C137C211yesmaximal, characteristic
2#3246×C137C211yesmaximal, characteristic, complemented
3#3209×C137C211yesmaximal, characteristic, complemented
4#3076×C137C211yesmaximal, characteristic, complemented
5#1646×C137C212yesmaximal, complemented
6#1643×C137C212yesmaximal, complemented
7#1634×C137C212yesmaximal, complemented
8#1632×C137C212yesmaximal, complemented
9C2×#333×C137C211yesmaximal, characteristic, complemented
10#1480×C137C211yesmaximal, characteristic, complemented
11#1477×C137C211yesmaximal, characteristic, complemented
12#556×C137—21no—
13#555×C137C2212yes—
14#554×C137C2212yes—
15#553×C137—22no—
16#548×C137—22no—
17#547×C137—21no—
18#542×C137—22no—
19#537×C137—21no—
20#533×C137—21no—
21#533×C137C2211yescharacteristic
22#532×C137—21no—
23#532×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic
25#520×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#494×C137C2212yes—
28#487×C137C2212yes—
29#479×C137—21no—
30#478×C137—21no—
31#477×C137C2211yescharacteristic
32#476×C137C2211yescharacteristic
33#476×C137C2212yes—
34#475×C137—21no—
35#475×C137C2211yescharacteristic
36#474×C137C2211yescharacteristic
37#465×C137—22no—
38#462×C137—22no—
39#461×C137C2212yes—
40#460×C137—21no—
41#451×C137—22no—
42#431×C137C2211yescharacteristic
43#427×C137—21no—
44C2×C4⋊C8×C137C2211yescharacteristic
45C2×C4⋊C8×C137C2211yescharacteristic
46#416×C137—21no—
47#333×C137C2212yes—
48#333×C137C2212yes—
49Q8⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
51D4⋊C8×C137C2212yes—
52D4⋊C8×C137C2212yes—
53C2×#41×C137—22no—
54C2×#41×C137—21no—
55C2×#41×C137D411yescharacteristic
56C22×Q8×C137D412yes—
57C2×Q16×C137—24no—
58C2×SD16×C137—22no—
59C22×C8×C137—21no—
60C22×C8×C137—21no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137D412yes—
64C4⋊Q8×C137D411yescharacteristic
65C4⋊Q8×C137C2311yescharacteristic
66C4⋊Q8×C137C2311yescharacteristic
67C4⋊Q8×C137C2312yes—
68#127×C137D411yescharacteristic
69#126×C137—24no—
70#125×C137—22no—
71#125×C137—21no—
72#124×C137—22no—
73#124×C137—24no—
74#124×C137—21no—
75#123×C137—24no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137—22no—
79C22⋊Q8×C137—24no—
80C22⋊Q8×C137—21no—
81C22⋊Q8×C137—21no—
82#121×C137—22no—
83#121×C137—24no—
84#121×C137—22no—
85#121×C137—22no—
86#121×C137—21no—
87#121×C137D412yes—
88C4×Q8×C137—22no—
89C4×Q8×C137—22no—
90C4×Q8×C137—21no—
91C4×Q8×C137—21no—
92C4×Q8×C137D411yescharacteristic
93C4×Q8×C137C2311yescharacteristic
94C4×D4×C137—22no—
95C4×D4×C137—24no—
96C4×D4×C137—22no—
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137D411yescharacteristic
101C4×D4×C137C2312yes—
102C4×D4×C137C2311yescharacteristic
103#117×C137—22no—
104#117×C137—22no—
105#117×C137—21no—
106#117×C137—21no—
107#117×C137D411yescharacteristic
108C2×C4⋊C4×C137—21no—
109C2×C4⋊C4×C137—21no—
110C2×C4⋊C4×C137—22no—
111C2×C42×C137—21no—
112C2×C42×C137C2311yescharacteristic
113#107×C137—21no—
114#107×C137—21no—
115#106×C137—21no—
116#106×C137—21no—
117#106×C137—22no—
118C4⋊C8×C137—21no—
119C4⋊C8×C137—21no—
120C4⋊C8×C137—21no—
121C4⋊C8×C137C2312yes—
122C4⋊C8×C137C2311yescharacteristic
123C4⋊C8×C137C2311yescharacteristic
124#103×C137—21no—
125#103×C137—21no—
126#103×C137—24no—
127#103×C137—22no—
128#102×C137—21no—
129#102×C137—21no—
130#102×C137—22no—
131C22⋊C8×C137—21no—
132C4×C8×C137C2312yes—
133#41×C137—22no—
134#41×C137—22no—
135#41×C137—42no—
136#41×C137—42no—
137#41×C137—84no—
138C2×Q8×C137—21no—
139C2×Q8×C137—22no—
140C2×Q8×C137—21no—
141C2×Q8×C137—21no—
142C2×Q8×C137—82no—
143C2×Q8×C137C2×D411yescharacteristic
144C2×Q8×C137C2×D414yes—
145C2×D4×C137—21no—
146C2×D4×C137—22no—
147C2×D4×C137—24no—
148C2×D4×C137—21no—
149C2×D4×C137—21no—
150C2×D4×C137C2×D411yescharacteristic
151C2×D4×C137C2×D412yes—
152C22×C4×C137—24no—
153C22×C4×C137—22no—
154C22×C4×C137—21no—
155C22×C4×C137—21no—
156C22×C4×C137—21no—
157C22×C4×C137—22no—
158C22×C4×C137C2×D411yescharacteristic
159C22×C4×C137C2×D411yescharacteristic
160C22×C4×C137C2×D411yescharacteristic
161Q16×C137—84no—
162SD16×C137—82no—
163C2×C8×C137—22no—
164C2×C8×C137—21no—
165C2×C8×C137—22no—
166C2×C8×C137—21no—
167C2×C8×C137—42no—
168C2×C8×C137—81no—
169C4⋊C4×C137—24no—
170C4⋊C4×C137—24no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—22no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—22no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—21no—
182C4⋊C4×C137—41no—
183C4⋊C4×C137—41no—
184C4⋊C4×C137C2×D412yes—
185C4⋊C4×C137C2×D411yescharacteristic
186C4⋊C4×C137C2×D411yescharacteristic
187C22⋊C4×C137—24no—
188C22⋊C4×C137—24no—
189C22⋊C4×C137—24no—
190C22⋊C4×C137—21no—
191C22⋊C4×C137—21no—
192C22⋊C4×C137—22no—
193C22⋊C4×C137—21no—
194C22⋊C4×C137—22no—
195C22⋊C4×C137—41no—
196C42×C137—24no—
197C42×C137—21no—
198C42×C137—21no—
199C42×C137—21no—
200C42×C137—41no—
201C42×C137C2×D411yescharacteristic
202C42×C137C2×D411yescharacteristic
203C42×C137C2×D411yescharacteristic
204C42×C137C2411yescharacteristic
205C23×C137—21no—
206C23×C137—22no—
207C23×C137#12011yescharacteristic
208Q8×C137—21no—
209Q8×C137—21no—
210Q8×C137—44no—
211Q8×C137—44no—
212Q8×C137—42no—
213Q8×C137—42no—
214Q8×C137—41no—
215Q8×C137—41no—
216D4×C137—21no—
217D4×C137—21no—
218D4×C137—22no—
219D4×C137—22no—
220D4×C137—42no—
221D4×C137—42no—
222D4×C137—44no—
223D4×C137—42no—
224D4×C137—84no—
225C2×C4×C137—21no—
226C2×C4×C137—21no—
227C2×C4×C137—21no—
228C2×C4×C137—21no—
229C2×C4×C137—22no—
230C2×C4×C137—21no—
231C2×C4×C137—24no—
232C2×C4×C137—41no—
233C2×C4×C137—42no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—41no—
237C2×C4×C137—41no—
238C2×C4×C137—42no—
239C2×C4×C137—41no—
240C2×C4×C137—41no—
241C2×C4×C137—82no—
242C2×C4×C137—84no—
243C2×C4×C137#12011yescharacteristic
244C2×C4×C137#12011yescharacteristic
245C2×C4×C137#12011yescharacteristic
246C2×C4×C137#13512yes—
247C2×C4×C137#13712yes—
248C2×C4×C137C22×D411yescharacteristic
249C2×C4×C137C22×D411yescharacteristic
250C2×C4×C137C22×D411yescharacteristic
251C8×C137—42no—
252C8×C137—41no—
253C8×C137—41no—
254C8×C137—42no—
255C22×C137—21no—
256C22×C137—21no—
257C22×C137—41no—
258C22×C137—42no—
259C22×C137—41no—
260C22×C137—42no—
261C22×C137—41no—
262C22×C137—42no—
263C22×C137#45212yes—
264C22×C137C2×#12011yescharacteristic
265C4×C137—21no—
266C4×C137—21no—
267C4×C137—21no—
268C4×C137—41no—
269C4×C137—42no—
270C4×C137—41no—
271C4×C137—41no—
272C4×C137—44no—
273C4×C137—81no—
274C4×C137—81no—
275C4×C137#45212yes—
276C4×C137C2×#13511yescharacteristic
277C4×C137C2×#13711yescharacteristic
278C4×C137#57711yescharacteristic
279C4×C137#57811yescharacteristic
280C2×C137—22no—
281C2×C137—81no—
282C2×C137—82no—
283C2×C137C2×#45211yescharacteristic
284C2×C137#301311yescharacteristic
285C2×C137#301911yescharacteristic
286#18877C13711yesmaximal, characteristic, direct product factor, Sylow
287C137#1887711yescharacteristic, direct product factor, Sylow
288#3511C2×C13711yescharacteristic
289#3246C2×C13711yescharacteristic, complemented
290#3209C2×C13711yescharacteristic, complemented
291#3076C2×C13711yescharacteristic, complemented
292#1646C2×C13712yescomplemented
293#1643C2×C13712yescomplemented
294#1634C2×C13712yescomplemented
295#1632C2×C13712yescomplemented
296C2×#333C2×C13711yescharacteristic, complemented
297#1480C2×C13711yescharacteristic, complemented
298#1477C2×C13711yescharacteristic, complemented
299#556—21no—
300#555C22×C13712yes—
301#554C22×C13712yes—
302#553—22no—
303#548—22no—
304#547—21no—
305#542—22no—
306#537—21no—
307#533—21no—
308#533C22×C13711yescharacteristic
309#532—21no—
310#532C22×C13711yescharacteristic
311C2×C4⋊Q8C22×C13711yescharacteristic
312#520—21no—
313C4×#41C22×C13711yescharacteristic
314#494C22×C13712yes—
315#487C22×C13712yes—
316#479—21no—
317#478—21no—
318#477C22×C13711yescharacteristic
319#476C22×C13711yescharacteristic
320#476C22×C13712yes—
321#475—21no—
322#475C22×C13711yescharacteristic
323#474C22×C13711yescharacteristic
324#465—22no—
325#462—22no—
326#461C22×C13712yes—
327#460—21no—
328#451—22no—
329#431C22×C13711yescharacteristic
330#427—21no—
331C2×C4⋊C8C22×C13711yescharacteristic
332C2×C4⋊C8C22×C13711yescharacteristic
333#416—21no—
334#333C22×C13712yes—
335#333C22×C13712yes—
336Q8⋊C8C22×C13712yes—
337D4⋊C8C22×C13712yes—
338D4⋊C8C22×C13712yes—
339D4⋊C8C22×C13712yes—
340C2×#41—22no—
341C2×#41—21no—
342C2×#41D4×C13711yescharacteristic
343C22×Q8D4×C13712yes—
344C2×Q16—24no—
345C2×SD16—22no—
346C22×C8—21no—
347C22×C8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8D4×C13712yes—
351C4⋊Q8D4×C13711yescharacteristic
352C4⋊Q8C23×C13711yescharacteristic
353C4⋊Q8C23×C13711yescharacteristic
354C4⋊Q8C23×C13712yes—
355#127D4×C13711yescharacteristic
356#126—24no—
357#125—22no—
358#125—21no—
359#124—22no—
360#124—24no—
361#124—21no—
362#123—24no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369#121—22no—
370#121—24no—
371#121—22no—
372#121—22no—
373#121—21no—
374#121D4×C13712yes—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8D4×C13711yescharacteristic
380C4×Q8C23×C13711yescharacteristic
381C4×D4—22no—
382C4×D4—24no—
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4D4×C13711yescharacteristic
388C4×D4C23×C13711yescharacteristic
389C4×D4C23×C13712yes—
390#117—22no—
391#117—22no—
392#117—21no—
393#117—21no—
394#117D4×C13711yescharacteristic
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C42—21no—
399C2×C42C23×C13711yescharacteristic
400#107—21no—
401#107—21no—
402#106—21no—
403#106—21no—
404#106—22no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8C23×C13711yescharacteristic
409C4⋊C8C23×C13711yescharacteristic
410C4⋊C8C23×C13712yes—
411#103—21no—
412#103—21no—
413#103—24no—
414#103—22no—
415#102—21no—
416#102—21no—
417#102—22no—
418C22⋊C8—21no—
419C4×C8C23×C13712yes—
420#41—22no—
421#41—22no—
422#41—42no—
423#41—42no—
424#41—84no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—82no—
430C2×Q8C2×D4×C13711yescharacteristic
431C2×Q8C2×D4×C13714yes—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4C2×D4×C13711yescharacteristic
438C2×D4C2×D4×C13712yes—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4C2×D4×C13711yescharacteristic
446C22×C4C2×D4×C13711yescharacteristic
447C22×C4C2×D4×C13711yescharacteristic
448Q16—84no—
449SD16—82no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—42no—
455C2×C8—81no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4C2×D4×C13712yes—
472C4⋊C4C2×D4×C13711yescharacteristic
473C4⋊C4C2×D4×C13711yescharacteristic
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C42—24no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—41no—
488C42C2×D4×C13711yescharacteristic
489C42C2×D4×C13711yescharacteristic
490C42C2×D4×C13711yescharacteristic
491C42C24×C13711yescharacteristic
492C23—21no—
493C23—22no—
494C23#120×C13711yescharacteristic
495Q8—21no—
496Q8—21no—
497Q8—44no—
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—41no—
502Q8—41no—
503D4—21no—
504D4—21no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—84no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—84no—
530C2×C4#120×C13711yescharacteristic
531C2×C4#120×C13711yescharacteristic
532C2×C4#120×C13711yescharacteristic
533C2×C4#135×C13712yes—
534C2×C4#137×C13712yes—
535C2×C4C22×D4×C13711yescharacteristic
536C2×C4C22×D4×C13711yescharacteristic
537C2×C4C22×D4×C13711yescharacteristic
538C8—42no—
539C8—41no—
540C8—41no—
541C8—42no—
542C22—21no—
543C22—21no—
544C22—41no—
545C22—42no—
546C22—41no—
547C22—42no—
548C22—41no—
549C22—42no—
550C22#452×C13712yes—
551C22C2×#120×C13711yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—41no—
559C4—44no—
560C4—81no—
561C4—81no—
562C4#452×C13712yes—
563C4C2×#135×C13711yescharacteristic
564C4C2×#137×C13711yescharacteristic
565C4#577×C13711yescharacteristic
566C4#578×C13711yescharacteristic
567C2—22no—
568C2—81no—
569C2—82no—
570C2C2×#452×C13711yescharacteristic
571C2#3013×C13711yescharacteristic
572C2#3019×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3511×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. #3246×C137C2semidirect
3. #3209×C137C2semidirect
4. #3076×C137C2semidirect
5. #1646×C137C2semidirect
6. #1643×C137C2semidirect
7. #1634×C137C2semidirect
8. #1632×C137C2semidirect
9. C2×#333×C137C2semidirect
10. #1480×C137C2semidirect
11. #1477×C137C2semidirect
286. #18877C137direct (N × H)
287. C137#18877direct (N × H)
289. #3246C2×C137semidirect
290. #3209C2×C137semidirect
291. #3076C2×C137semidirect
292. #1646C2×C137semidirect
293. #1643C2×C137semidirect
294. #1634C2×C137semidirect
295. #1632C2×C137semidirect
296. C2×#333C2×C137semidirect
297. #1480C2×C137semidirect
298. #1477C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3511×C1372
#3246×C1372
#3209×C1372
#3076×C1372
#1646×C1372
#1643×C1372
#1634×C1372
#1632×C1372
C2×#333×C1372
#1480×C1372
#1477×C1372
#18877137