Finite Group Databasebeta

← C137×#18827 #46868213

Subgroups — C137×#18827 #46868213

Order 35072.

1080 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3499×C137C211yesmaximal, characteristic
2#3235×C137C211yesmaximal, characteristic, complemented
3#3230×C137C211yesmaximal, characteristic, complemented
4#3116×C137C211yesmaximal, characteristic, complemented
5#1649×C137C212yesmaximal, complemented
6#1647×C137C212yesmaximal, complemented
7#1637×C137C212yesmaximal, complemented
8#1635×C137C212yesmaximal, complemented
9#1556×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12C2×2−1+4×C137—21no—
13D4×Q8×C137—21no—
14D4×Q8×C137C2212yes—
15#547×C137—22no—
16#547×C137—21no—
17#547×C137—22no—
18#547×C137C2212yes—
19#536×C137—22no—
20#536×C137—21no—
21#533×C137C2211yescharacteristic
22#532×C137—21no—
23#532×C137C2211yescharacteristic
24C2×C4⋊Q8×C137—21no—
25C2×C4⋊Q8×C137C2211yescharacteristic
26C4×#41×C137C2211yescharacteristic
27#501×C137—21no—
28#500×C137C2212yes—
29#490×C137—21no—
30#487×C137C2212yes—
31#485×C137—21no—
32#484×C137—22no—
33#482×C137—22no—
34#477×C137—22no—
35#477×C137C2211yescharacteristic
36#476×C137C2211yescharacteristic
37#476×C137C2212yes—
38#474×C137C2212yes—
39#462×C137C2211yescharacteristic
40#461×C137C2211yescharacteristic
41#452×C137—21no—
42#441×C137—21no—
43#440×C137—21no—
44#432×C137C2211yescharacteristic
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#332×C137C2212yes—
48#332×C137C2212yes—
49Q8⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
51D4⋊C8×C137C2212yes—
52D4⋊C8×C137C2212yes—
532−1+4×C137—44no—
542−1+4×C137—44no—
55C2×#41×C137—22no—
56C2×#41×C137—24no—
57C2×#41×C137—22no—
58C2×#41×C137—21no—
59C2×#41×C137D411yescharacteristic
60C22×Q8×C137—22no—
61C22×Q8×C137—22no—
62C22×Q8×C137—21no—
63C2×Q16×C137—21no—
64C2×SD16×C137—21no—
65C4⋊Q8×C137—22no—
66C4⋊Q8×C137—22no—
67C4⋊Q8×C137—22no—
68C4⋊Q8×C137—21no—
69C4⋊Q8×C137—21no—
70C4⋊Q8×C137D412yes—
71C4⋊Q8×C137D411yescharacteristic
72C4⋊Q8×C137C2311yescharacteristic
73C4⋊Q8×C137C2311yescharacteristic
74C4⋊Q8×C137C2312yes—
75#127×C137D411yescharacteristic
76#125×C137—21no—
77#124×C137—22no—
78#124×C137—21no—
79#123×C137—24no—
80#123×C137—24no—
81#123×C137—24no—
82C22⋊Q8×C137—24no—
83C22⋊Q8×C137—22no—
84C22⋊Q8×C137—24no—
85C22⋊Q8×C137—22no—
86C22⋊Q8×C137—21no—
87C22⋊Q8×C137—21no—
88#121×C137—22no—
89#121×C137—22no—
90#121×C137—21no—
91#121×C137D412yes—
92C4×Q8×C137—21no—
93C4×Q8×C137—21no—
94C4×Q8×C137D411yescharacteristic
95C4×Q8×C137C2311yescharacteristic
96C4×D4×C137—22no—
97C4×D4×C137—22no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137—21no—
101C4×D4×C137D411yescharacteristic
102C4×D4×C137C2311yescharacteristic
103C4×D4×C137C2312yes—
104#117×C137—21no—
105#117×C137—21no—
106#117×C137D411yescharacteristic
107C2×C4⋊C4×C137—22no—
108C2×C4⋊C4×C137—21no—
109C2×C4⋊C4×C137—22no—
110C2×C4⋊C4×C137D412yes—
111C2×C42×C137—21no—
112C2×C42×C137C2311yescharacteristic
113#107×C137—21no—
114#107×C137—22no—
115#107×C137—24no—
116#106×C137—21no—
117#106×C137—22no—
118C4⋊C8×C137—21no—
119C4⋊C8×C137C2311yescharacteristic
120C4⋊C8×C137C2311yescharacteristic
121C4⋊C8×C137C2312yes—
122#103×C137—21no—
123#103×C137—21no—
124#103×C137—21no—
125#103×C137—24no—
126#102×C137—21no—
127#102×C137—22no—
128#102×C137—22no—
129C22⋊C8×C137—21no—
130C22⋊C8×C137—21no—
131C22⋊C8×C137—21no—
132#97×C137—21no—
133#97×C137—21no—
134C4×C8×C137C2312yes—
135#41×C137—22no—
136#41×C137—22no—
137#41×C137—48no—
138#41×C137—42no—
139#41×C137—42no—
140#41×C137—416no—
141#41×C137—44no—
142#41×C137—44no—
143C2×Q8×C137—24no—
144C2×Q8×C137—22no—
145C2×Q8×C137—21no—
146C2×Q8×C137—22no—
147C2×Q8×C137—21no—
148C2×Q8×C137—21no—
149C2×Q8×C137—48no—
150C2×Q8×C137—42no—
151C2×Q8×C137—42no—
152C2×Q8×C137—44no—
153C2×Q8×C137—44no—
154C2×Q8×C137C2×D411yescharacteristic
155C2×D4×C137—21no—
156C2×D4×C137—24no—
157C2×D4×C137—22no—
158C2×D4×C137—21no—
159C2×D4×C137—21no—
160C2×D4×C137C2×D412yes—
161C2×D4×C137C2×D411yescharacteristic
162C22×C4×C137—21no—
163C22×C4×C137—24no—
164C22×C4×C137—24no—
165C22×C4×C137—22no—
166C22×C4×C137—21no—
167C22×C4×C137—21no—
168C22×C4×C137—22no—
169C22×C4×C137C2×D411yescharacteristic
170C22×C4×C137C2×D411yescharacteristic
171C22×C4×C137C2×D411yescharacteristic
172Q16×C137—42no—
173SD16×C137—42no—
174C2×C8×C137—22no—
175C2×C8×C137—22no—
176C2×C8×C137—21no—
177C2×C8×C137—21no—
178C4⋊C4×C137—24no—
179C4⋊C4×C137—24no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—22no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—22no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—21no—
187C4⋊C4×C137—21no—
188C4⋊C4×C137—21no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—41no—
191C4⋊C4×C137—41no—
192C4⋊C4×C137C2×D411yescharacteristic
193C4⋊C4×C137C2×D414yes—
194C4⋊C4×C137C2×D411yescharacteristic
195C4⋊C4×C137C2×D412yes—
196C22⋊C4×C137—24no—
197C22⋊C4×C137—24no—
198C22⋊C4×C137—21no—
199C22⋊C4×C137—21no—
200C22⋊C4×C137—22no—
201C22⋊C4×C137—21no—
202C22⋊C4×C137—22no—
203C22⋊C4×C137—41no—
204C42×C137—21no—
205C42×C137—21no—
206C42×C137—21no—
207C42×C137—41no—
208C42×C137C2×D411yescharacteristic
209C42×C137C2×D411yescharacteristic
210C42×C137C2×D411yescharacteristic
211C42×C137C2411yescharacteristic
212C23×C137—21no—
213C23×C137—22no—
214C23×C137#12011yescharacteristic
215Q8×C137—21no—
216Q8×C137—44no—
217Q8×C137—42no—
218Q8×C137—48no—
219Q8×C137—42no—
220Q8×C137—42no—
221Q8×C137—42no—
222Q8×C137—41no—
223Q8×C137—41no—
224Q8×C137AGL(1,ℤ/8)12yes—
225D4×C137—24no—
226D4×C137—21no—
227D4×C137—24no—
228D4×C137—42no—
229D4×C137—42no—
230D4×C137—44no—
231D4×C137—48no—
232D4×C137AGL(1,ℤ/8)12yes—
233C2×C4×C137—24no—
234C2×C4×C137—21no—
235C2×C4×C137—21no—
236C2×C4×C137—21no—
237C2×C4×C137—22no—
238C2×C4×C137—21no—
239C2×C4×C137—21no—
240C2×C4×C137—24no—
241C2×C4×C137—22no—
242C2×C4×C137—41no—
243C2×C4×C137—42no—
244C2×C4×C137—48no—
245C2×C4×C137—44no—
246C2×C4×C137—48no—
247C2×C4×C137—42no—
248C2×C4×C137—42no—
249C2×C4×C137—42no—
250C2×C4×C137—41no—
251C2×C4×C137—41no—
252C2×C4×C137#12011yescharacteristic
253C2×C4×C137#12011yescharacteristic
254C2×C4×C137#12011yescharacteristic
255C2×C4×C137C22×D411yescharacteristic
256C2×C4×C137C22×D411yescharacteristic
257C2×C4×C137C22×D411yescharacteristic
258C8×C137—42no—
259C8×C137—41no—
260C8×C137—41no—
261C8×C137—42no—
262C22×C137—21no—
263C22×C137—22no—
264C22×C137—21no—
265C22×C137—21no—
266C22×C137—41no—
267C22×C137—41no—
268C22×C137—42no—
269C22×C137—42no—
270C22×C137—42no—
271C22×C137C2×#12011yescharacteristic
272C4×C137—21no—
273C4×C137—22no—
274C4×C137—21no—
275C4×C137—21no—
276C4×C137—21no—
277C4×C137—44no—
278C4×C137—42no—
279C4×C137—41no—
280C4×C137—41no—
281C4×C137—81no—
282C4×C137—81no—
283C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C137#57812yes—
286C2×C137—41no—
287C2×C137—42no—
288C2×C137—82no—
289C2×C137#300311yescharacteristic
290C2×C137#301111yescharacteristic
291C2×C137#302011yescharacteristic
292#18827C13711yesmaximal, characteristic, direct product factor, Sylow
293C137#1882711yescharacteristic, direct product factor, Sylow
294#3499C2×C13711yescharacteristic
295#3235C2×C13711yescharacteristic, complemented
296#3230C2×C13711yescharacteristic, complemented
297#3116C2×C13711yescharacteristic, complemented
298#1649C2×C13712yescomplemented
299#1647C2×C13712yescomplemented
300#1637C2×C13712yescomplemented
301#1635C2×C13712yescomplemented
302#1556C2×C13711yescharacteristic, complemented
303#1496C2×C13711yescharacteristic, complemented
304#1488C2×C13711yescharacteristic, complemented
305C2×2−1+4—21no—
306D4×Q8—21no—
307D4×Q8C22×C13712yes—
308#547—22no—
309#547—21no—
310#547—22no—
311#547C22×C13712yes—
312#536—22no—
313#536—21no—
314#533C22×C13711yescharacteristic
315#532—21no—
316#532C22×C13711yescharacteristic
317C2×C4⋊Q8—21no—
318C2×C4⋊Q8C22×C13711yescharacteristic
319C4×#41C22×C13711yescharacteristic
320#501—21no—
321#500C22×C13712yes—
322#490—21no—
323#487C22×C13712yes—
324#485—21no—
325#484—22no—
326#482—22no—
327#477—22no—
328#477C22×C13711yescharacteristic
329#476C22×C13711yescharacteristic
330#476C22×C13712yes—
331#474C22×C13712yes—
332#462C22×C13711yescharacteristic
333#461C22×C13711yescharacteristic
334#452—21no—
335#441—21no—
336#440—21no—
337#432C22×C13711yescharacteristic
338#432C22×C13711yescharacteristic
339#431C22×C13711yescharacteristic
340#332C22×C13712yes—
341#332C22×C13712yes—
342Q8⋊C8C22×C13712yes—
343D4⋊C8C22×C13712yes—
344D4⋊C8C22×C13712yes—
345D4⋊C8C22×C13712yes—
3462−1+4—44no—
3472−1+4—44no—
348C2×#41—22no—
349C2×#41—24no—
350C2×#41—22no—
351C2×#41—21no—
352C2×#41D4×C13711yescharacteristic
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—21no—
356C2×Q16—21no—
357C2×SD16—21no—
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8D4×C13711yescharacteristic
364C4⋊Q8D4×C13712yes—
365C4⋊Q8C23×C13711yescharacteristic
366C4⋊Q8C23×C13711yescharacteristic
367C4⋊Q8C23×C13712yes—
368#127D4×C13711yescharacteristic
369#125—21no—
370#124—22no—
371#124—21no—
372#123—24no—
373#123—24no—
374#123—24no—
375C22⋊Q8—24no—
376C22⋊Q8—22no—
377C22⋊Q8—24no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381#121—22no—
382#121—22no—
383#121—21no—
384#121D4×C13712yes—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8D4×C13711yescharacteristic
388C4×Q8C23×C13711yescharacteristic
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4D4×C13711yescharacteristic
395C4×D4C23×C13711yescharacteristic
396C4×D4C23×C13712yes—
397#117—21no—
398#117—21no—
399#117D4×C13711yescharacteristic
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4D4×C13712yes—
404C2×C42—21no—
405C2×C42C23×C13711yescharacteristic
406#107—21no—
407#107—22no—
408#107—24no—
409#106—21no—
410#106—22no—
411C4⋊C8—21no—
412C4⋊C8C23×C13711yescharacteristic
413C4⋊C8C23×C13711yescharacteristic
414C4⋊C8C23×C13712yes—
415#103—21no—
416#103—21no—
417#103—21no—
418#103—24no—
419#102—21no—
420#102—22no—
421#102—22no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425#97—21no—
426#97—21no—
427C4×C8C23×C13712yes—
428#41—22no—
429#41—22no—
430#41—48no—
431#41—42no—
432#41—42no—
433#41—416no—
434#41—44no—
435#41—44no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—48no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8C2×D4×C13711yescharacteristic
448C2×D4—21no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4C2×D4×C13712yes—
454C2×D4C2×D4×C13711yescharacteristic
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4C2×D4×C13711yescharacteristic
463C22×C4C2×D4×C13711yescharacteristic
464C22×C4C2×D4×C13711yescharacteristic
465Q16—42no—
466SD16—42no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4C2×D4×C13711yescharacteristic
486C4⋊C4C2×D4×C13714yes—
487C4⋊C4C2×D4×C13711yescharacteristic
488C4⋊C4C2×D4×C13712yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—41no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—41no—
501C42C2×D4×C13711yescharacteristic
502C42C2×D4×C13711yescharacteristic
503C42C2×D4×C13711yescharacteristic
504C42C24×C13711yescharacteristic
505C23—21no—
506C23—22no—
507C23#120×C13711yescharacteristic
508Q8—21no—
509Q8—44no—
510Q8—42no—
511Q8—48no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—41no—
516Q8—41no—
517Q8AGL(1,ℤ/8)×C13712yes—
518D4—24no—
519D4—21no—
520D4—24no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—48no—
525D4AGL(1,ℤ/8)×C13712yes—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4#120×C13711yescharacteristic
546C2×C4#120×C13711yescharacteristic
547C2×C4#120×C13711yescharacteristic
548C2×C4C22×D4×C13711yescharacteristic
549C2×C4C22×D4×C13711yescharacteristic
550C2×C4C22×D4×C13711yescharacteristic
551C8—42no—
552C8—41no—
553C8—41no—
554C8—42no—
555C22—21no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22C2×#120×C13711yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
577C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
578C4#578×C13712yes—
579C2—41no—
580C2—42no—
581C2—82no—
582C2#3003×C13711yescharacteristic
583C2#3011×C13711yescharacteristic
584C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3499×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. #3235×C137C2semidirect
3. #3230×C137C2semidirect
4. #3116×C137C2semidirect
5. #1649×C137C2semidirect
6. #1647×C137C2semidirect
7. #1637×C137C2semidirect
8. #1635×C137C2semidirect
9. #1556×C137C2semidirect
10. #1496×C137C2semidirect
11. #1488×C137C2semidirect
292. #18827C137direct (N × H)
293. C137#18827direct (N × H)
295. #3235C2×C137semidirect
296. #3230C2×C137semidirect
297. #3116C2×C137semidirect
298. #1649C2×C137semidirect
299. #1647C2×C137semidirect
300. #1637C2×C137semidirect
301. #1635C2×C137semidirect
302. #1556C2×C137semidirect
303. #1496C2×C137semidirect
304. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3499×C1372
#3235×C1372
#3230×C1372
#3116×C1372
#1649×C1372
#1647×C1372
#1637×C1372
#1635×C1372
#1556×C1372
#1496×C1372
#1488×C1372
#18827137