Finite Group Databasebeta

← C137×#18846 #46868232

Subgroups — C137×#18846 #46868232

Order 35072.

1182 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3496×C137C211yesmaximal, characteristic
2#3254×C137C211yesmaximal, characteristic, complemented
3#3228×C137C211yesmaximal, characteristic, complemented
4#3110×C137C211yesmaximal, characteristic, complemented
5#1654×C137C212yesmaximal, complemented
6#1647×C137C212yesmaximal, complemented
7#1639×C137C212yesmaximal, complemented
8#1639×C137C212yesmaximal, complemented
9#1557×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic, complemented
11#1486×C137C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C137—21no—
13#547×C137—21no—
14#547×C137C2212yes—
15#546×C137—22no—
16#546×C137—22no—
17D42×C137—21no—
18D42×C137C2212yes—
19#537×C137—21no—
20#534×C137—22no—
21#533×C137C2211yescharacteristic, complemented
22#532×C137C2211yescharacteristic
23#532×C137C2211yescharacteristic
24C2×#121×C137—22no—
25C2×C4×D4×C137C2211yescharacteristic
26#500×C137C2211yescharacteristic, complemented
27#498×C137C2211yescharacteristic, complemented
28#486×C137C2211yescharacteristic, complemented
29#486×C137C2211yescharacteristic, complemented
30#484×C137—21no—
31#482×C137—22no—
32#478×C137—22no—
33#477×C137—21no—
34#474×C137C2212yescomplemented
35#474×C137C2212yescomplemented
36#468×C137—22no—
37#461×C137C2212yescomplemented
38#461×C137C2212yescomplemented
39#460×C137—21no—
40#450×C137—21no—
41#440×C137—22no—
42#432×C137C2211yescharacteristic, complemented
43#431×C137C2211yescharacteristic, complemented
44#423×C137C2211yescharacteristic, complemented
45#332×C137C2212yescomplemented
46#331×C137C2212yescomplemented
47D4⋊C8×C137C2212yescomplemented
48D4⋊C8×C137C2212yescomplemented
49D4⋊C8×C137C2212yescomplemented
50D4⋊C8×C137C2212yescomplemented
51H(5,𝔽2)×C137—48no—
52C2×#41×C137—22no—
53C2×#41×C137—22no—
54C2×#41×C137—21no—
55C2×#41×C137—21no—
56C22×D4×C137—24no—
57C22×D4×C137—22no—
58C22×D4×C137—22no—
59C22×D4×C137—22no—
60C22×D4×C137D411yescharacteristic
61C23×C4×C137—21no—
62C2×SD16×C137—22no—
63C2×SD16×C137—22no—
64C2×M16×C137—21no—
65C4⋊Q8×C137D411yescharacteristic
66C4⋊Q8×C137C2311yescharacteristic
67C4⋊Q8×C137C2311yescharacteristic
68C4⋊Q8×C137C2311yescharacteristic
69#127×C137D411yescharacteristic
70#127×C137C2311yescharacteristic
71#125×C137—21no—
72#124×C137—21no—
73#124×C137—21no—
74#123×C137—24no—
75#123×C137—24no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137—21no—
79C22⋊Q8×C137—21no—
80#121×C137—24no—
81#121×C137—24no—
82#121×C137—22no—
83#121×C137—22no—
84#121×C137—21no—
85#121×C137—21no—
86#121×C137—22no—
87#121×C137—22no—
88#121×C137D412yes—
89#121×C137D412yes—
90#120×C137—24no—
91#120×C137—24no—
92C4×Q8×C137—21no—
93C4×Q8×C137—21no—
94C4×D4×C137—21no—
95C4×D4×C137—21no—
96C4×D4×C137—21no—
97C4×D4×C137—21no—
98C4×D4×C137D412yes—
99C4×D4×C137D412yes—
100C4×D4×C137C2312yes—
101C4×D4×C137C2312yes—
102#117×C137—21no—
103C2×C4⋊C4×C137—22no—
104C2×C4⋊C4×C137D411yescharacteristic
105C2×C22⋊C4×C137—24no—
106C2×C22⋊C4×C137—21no—
107C2×C42×C137C2311yescharacteristic
108#107×C137—22no—
109#107×C137—22no—
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
127C24×C137—21no—
128C24×C137—24no—
129#41×C137—44no—
130#41×C137—44no—
131#41×C137—48no—
132#41×C137—44no—
133#41×C137—44no—
134C2×Q8×C137—21no—
135C2×Q8×C137—21no—
136C2×Q8×C137—21no—
137C2×Q8×C137—21no—
138C2×D4×C137—24no—
139C2×D4×C137—24no—
140C2×D4×C137—24no—
141C2×D4×C137—22no—
142C2×D4×C137—22no—
143C2×D4×C137—24no—
144C2×D4×C137—21no—
145C2×D4×C137—21no—
146C2×D4×C137—21no—
147C2×D4×C137—21no—
148C2×D4×C137—21no—
149C2×D4×C137—48no—
150C2×D4×C137—44no—
151C2×D4×C137—416no—
152C2×D4×C137—44no—
153C2×D4×C137—82no—
154C2×D4×C137—82no—
155C2×D4×C137C2×D412yes—
156C2×D4×C137C2×D412yes—
157C2×D4×C137C2×D412yes—
158C22×C4×C137—24no—
159C22×C4×C137—21no—
160C22×C4×C137—22no—
161C22×C4×C137—22no—
162C22×C4×C137—22no—
163C22×C4×C137—22no—
164C22×C4×C137—81no—
165C22×C4×C137—81no—
166C22×C4×C137C2×D411yescharacteristic
167C22×C4×C137C2×D411yescharacteristic
168C22×C4×C137C2×D411yescharacteristic
169SD16×C137—44no—
170SD16×C137—82no—
171M16×C137—81no—
172C2×C8×C137—21no—
173C2×C8×C137—21no—
174C2×C8×C137—22no—
175C2×C8×C137—22no—
176C4⋊C4×C137—24no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—21no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—41no—
185C4⋊C4×C137—41no—
186C4⋊C4×C137C2×D412yes—
187C4⋊C4×C137C2×D412yes—
188C4⋊C4×C137C2×D412yes—
189C22⋊C4×C137—24no—
190C22⋊C4×C137—24no—
191C22⋊C4×C137—22no—
192C22⋊C4×C137—22no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—22no—
195C22⋊C4×C137—22no—
196C22⋊C4×C137—22no—
197C22⋊C4×C137—41no—
198C42×C137—41no—
199C42×C137C2×D411yescharacteristic
200C42×C137C2×D411yescharacteristic
201C42×C137C2×D411yescharacteristic
202C42×C137C2411yescharacteristic
203C23×C137—24no—
204C23×C137—22no—
205C23×C137—22no—
206C23×C137—22no—
207C23×C137—48no—
208C23×C137—81no—
209C23×C137—81no—
210C23×C137—84no—
211C23×C137—84no—
212C23×C137#12011yescharacteristic
213Q8×C137—42no—
214Q8×C137—42no—
215Q8×C137—42no—
216Q8×C137—41no—
217Q8×C137—41no—
218D4×C137—24no—
219D4×C137—22no—
220D4×C137—22no—
221D4×C137—22no—
222D4×C137—42no—
223D4×C137—42no—
224D4×C137—44no—
225D4×C137—44no—
226D4×C137—48no—
227D4×C137—48no—
228D4×C137—42no—
229D4×C137—41no—
230D4×C137—41no—
231D4×C137—81no—
232D4×C137—84no—
233D4×C137AGL(1,ℤ/8)14yes—
234C2×C4×C137—22no—
235C2×C4×C137—21no—
236C2×C4×C137—21no—
237C2×C4×C137—22no—
238C2×C4×C137—22no—
239C2×C4×C137—22no—
240C2×C4×C137—41no—
241C2×C4×C137—44no—
242C2×C4×C137—48no—
243C2×C4×C137—42no—
244C2×C4×C137—42no—
245C2×C4×C137—42no—
246C2×C4×C137—42no—
247C2×C4×C137—41no—
248C2×C4×C137—41no—
249C2×C4×C137—82no—
250C2×C4×C137—81no—
251C2×C4×C137#12011yescharacteristic
252C2×C4×C137#12011yescharacteristic
253C2×C4×C137#12011yescharacteristic
254C2×C4×C137C22×D411yescharacteristic
255C2×C4×C137C22×D411yescharacteristic
256C2×C4×C137C22×D411yescharacteristic
257C8×C137—41no—
258C8×C137—41no—
259C8×C137—42no—
260C8×C137—42no—
261C22×C137—21no—
262C22×C137—24no—
263C22×C137—21no—
264C22×C137—21no—
265C22×C137—42no—
266C22×C137—42no—
267C22×C137—42no—
268C22×C137—42no—
269C22×C137—42no—
270C22×C137—42no—
271C22×C137—42no—
272C22×C137—42no—
273C22×C137—84no—
274C22×C137—168no—
275C22×C137C2×#12011yescharacteristic
276C4×C137—21no—
277C4×C137—21no—
278C4×C137—21no—
279C4×C137—21no—
280C4×C137—42no—
281C4×C137—42no—
282C4×C137—42no—
283C4×C137—81no—
284C4×C137—81no—
285C4×C137C2×AGL(1,ℤ/8)12yes—
286C4×C137#57611yescharacteristic
287C4×C137#57711yescharacteristic
288C2×C137—41no—
289C2×C137—44no—
290C2×C137—82no—
291C2×C137—82no—
292C2×C137#300311yescharacteristic
293C2×C137#301211yescharacteristic
294C2×C137#302211yescharacteristic
295#18846C13711yesmaximal, characteristic, direct product factor, Sylow
296C137#1884611yescharacteristic, direct product factor, Sylow
297#3496C2×C13711yescharacteristic
298#3254C2×C13711yescharacteristic, complemented
299#3228C2×C13711yescharacteristic, complemented
300#3110C2×C13711yescharacteristic, complemented
301#1654C2×C13712yescomplemented
302#1647C2×C13712yescomplemented
303#1639C2×C13712yescomplemented
304#1639C2×C13712yescomplemented
305#1557C2×C13711yescharacteristic, complemented
306#1496C2×C13711yescharacteristic, complemented
307#1486C2×C13711yescharacteristic, complemented
308C2×H(5,𝔽2)—21no—
309#547—21no—
310#547C22×C13712yes—
311#546—22no—
312#546—22no—
313D42—21no—
314D42C22×C13712yes—
315#537—21no—
316#534—22no—
317#533C22×C13711yescharacteristic, complemented
318#532C22×C13711yescharacteristic
319#532C22×C13711yescharacteristic
320C2×#121—22no—
321C2×C4×D4C22×C13711yescharacteristic
322#500C22×C13711yescharacteristic, complemented
323#498C22×C13711yescharacteristic, complemented
324#486C22×C13711yescharacteristic, complemented
325#486C22×C13711yescharacteristic, complemented
326#484—21no—
327#482—22no—
328#478—22no—
329#477—21no—
330#474C22×C13712yescomplemented
331#474C22×C13712yescomplemented
332#468—22no—
333#461C22×C13712yescomplemented
334#461C22×C13712yescomplemented
335#460—21no—
336#450—21no—
337#440—22no—
338#432C22×C13711yescharacteristic, complemented
339#431C22×C13711yescharacteristic, complemented
340#423C22×C13711yescharacteristic, complemented
341#332C22×C13712yescomplemented
342#331C22×C13712yescomplemented
343D4⋊C8C22×C13712yescomplemented
344D4⋊C8C22×C13712yescomplemented
345D4⋊C8C22×C13712yescomplemented
346D4⋊C8C22×C13712yescomplemented
347H(5,𝔽2)—48no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—21no—
351C2×#41—21no—
352C22×D4—24no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4D4×C13711yescharacteristic
357C23×C4—21no—
358C2×SD16—22no—
359C2×SD16—22no—
360C2×M16—21no—
361C4⋊Q8D4×C13711yescharacteristic
362C4⋊Q8C23×C13711yescharacteristic
363C4⋊Q8C23×C13711yescharacteristic
364C4⋊Q8C23×C13711yescharacteristic
365#127D4×C13711yescharacteristic
366#127C23×C13711yescharacteristic
367#125—21no—
368#124—21no—
369#124—21no—
370#123—24no—
371#123—24no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—24no—
377#121—24no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121—21no—
382#121—22no—
383#121—22no—
384#121D4×C13712yes—
385#121D4×C13712yes—
386#120—24no—
387#120—24no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4D4×C13712yes—
395C4×D4D4×C13712yes—
396C4×D4C23×C13712yes—
397C4×D4C23×C13712yes—
398#117—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4D4×C13711yescharacteristic
401C2×C22⋊C4—24no—
402C2×C22⋊C4—21no—
403C2×C42C23×C13711yescharacteristic
404#107—22no—
405#107—22no—
406#107—22no—
407#106—22no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8C23×C13712yes—
411C4⋊C8C23×C13712yes—
412#103—22no—
413#103—22no—
414#102—22no—
415#102—22no—
416#102—22no—
417#102—22no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420#97—21no—
421C4×C8C23×C13711yescharacteristic
422C4×C8C23×C13711yescharacteristic
423C24—21no—
424C24—24no—
425#41—44no—
426#41—44no—
427#41—48no—
428#41—44no—
429#41—44no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—48no—
446C2×D4—44no—
447C2×D4—416no—
448C2×D4—44no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C2×D4×C13712yes—
452C2×D4C2×D4×C13712yes—
453C2×D4C2×D4×C13712yes—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4C2×D4×C13711yescharacteristic
463C22×C4C2×D4×C13711yescharacteristic
464C22×C4C2×D4×C13711yescharacteristic
465SD16—44no—
466SD16—82no—
467M16—81no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4C2×D4×C13712yes—
483C4⋊C4C2×D4×C13712yes—
484C4⋊C4C2×D4×C13712yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—41no—
494C42—41no—
495C42C2×D4×C13711yescharacteristic
496C42C2×D4×C13711yescharacteristic
497C42C2×D4×C13711yescharacteristic
498C42C24×C13711yescharacteristic
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—48no—
504C23—81no—
505C23—81no—
506C23—84no—
507C23—84no—
508C23#120×C13711yescharacteristic
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—41no—
513Q8—41no—
514D4—24no—
515D4—22no—
516D4—22no—
517D4—22no—
518D4—42no—
519D4—42no—
520D4—44no—
521D4—44no—
522D4—48no—
523D4—48no—
524D4—42no—
525D4—41no—
526D4—41no—
527D4—81no—
528D4—84no—
529D4AGL(1,ℤ/8)×C13714yes—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4#120×C13711yescharacteristic
548C2×C4#120×C13711yescharacteristic
549C2×C4#120×C13711yescharacteristic
550C2×C4C22×D4×C13711yescharacteristic
551C2×C4C22×D4×C13711yescharacteristic
552C2×C4C22×D4×C13711yescharacteristic
553C8—41no—
554C8—41no—
555C8—42no—
556C8—42no—
557C22—21no—
558C22—24no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—84no—
570C22—168no—
571C22C2×#120×C13711yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4C2×AGL(1,ℤ/8)×C13712yes—
582C4#576×C13711yescharacteristic
583C4#577×C13711yescharacteristic
584C2—41no—
585C2—44no—
586C2—82no—
587C2—82no—
588C2#3003×C13711yescharacteristic
589C2#3012×C13711yescharacteristic
590C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3496×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. #3254×C137C2semidirect
3. #3228×C137C2semidirect
4. #3110×C137C2semidirect
5. #1654×C137C2semidirect
6. #1647×C137C2semidirect
7. #1639×C137C2semidirect
8. #1639×C137C2semidirect
9. #1557×C137C2semidirect
10. #1496×C137C2semidirect
11. #1486×C137C2semidirect
21. #533×C137C22semidirect
26. #500×C137C22semidirect
27. #498×C137C22semidirect
28. #486×C137C22semidirect
29. #486×C137C22semidirect
34. #474×C137C22semidirect
35. #474×C137C22semidirect
37. #461×C137C22semidirect
38. #461×C137C22semidirect
42. #432×C137C22semidirect
43. #431×C137C22semidirect
44. #423×C137C22semidirect
45. #332×C137C22semidirect
46. #331×C137C22semidirect
47. D4⋊C8×C137C22semidirect
48. D4⋊C8×C137C22semidirect
49. D4⋊C8×C137C22semidirect
50. D4⋊C8×C137C22semidirect
295. #18846C137direct (N × H)
296. C137#18846direct (N × H)
298. #3254C2×C137semidirect
299. #3228C2×C137semidirect
300. #3110C2×C137semidirect
301. #1654C2×C137semidirect
302. #1647C2×C137semidirect
303. #1639C2×C137semidirect
304. #1639C2×C137semidirect
305. #1557C2×C137semidirect
306. #1496C2×C137semidirect
307. #1486C2×C137semidirect
317. #533C22×C137semidirect
322. #500C22×C137semidirect
323. #498C22×C137semidirect
324. #486C22×C137semidirect
325. #486C22×C137semidirect
330. #474C22×C137semidirect
331. #474C22×C137semidirect
333. #461C22×C137semidirect
334. #461C22×C137semidirect
338. #432C22×C137semidirect
339. #431C22×C137semidirect
340. #423C22×C137semidirect
341. #332C22×C137semidirect
342. #331C22×C137semidirect
343. D4⋊C8C22×C137semidirect
344. D4⋊C8C22×C137semidirect
345. D4⋊C8C22×C137semidirect
346. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3496×C1372
#3254×C1372
#3228×C1372
#3110×C1372
#1654×C1372
#1647×C1372
#1639×C1372
#1557×C1372
#1496×C1372
#1486×C1372
#18846137