Finite Group Databasebeta

← C137×#18553 #46867939

Subgroups — C137×#18553 #46867939

Order 35072.

1008 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3483×C137C211yesmaximal, characteristic
2#3141×C137C211yesmaximal, characteristic, complemented
3#3079×C137C211yesmaximal, characteristic, complemented
4#3039×C137C211yesmaximal, characteristic, complemented
5#1646×C137C212yesmaximal, complemented
6#1645×C137C212yesmaximal, complemented
7#1642×C137C212yesmaximal, complemented
8#1641×C137C212yesmaximal, complemented
9#1560×C137C211yesmaximal, characteristic, complemented
10#1489×C137C211yesmaximal, characteristic, complemented
11#1489×C137C211yesmaximal, characteristic, complemented
12C2×#137×C137—21no—
13Q82×C137C2211yescharacteristic
14#555×C137—22no—
15#555×C137—21no—
16#555×C137—21no—
17#555×C137C2211yescharacteristic
18#550×C137—21no—
19#550×C137—21no—
20#550×C137C2211yescharacteristic
21#550×C137C2211yescharacteristic
22#532×C137—22no—
23#532×C137—21no—
24#532×C137—21no—
25#532×C137C2211yescharacteristic
26C2×C4⋊Q8×C137C2211yescharacteristic
27C4×#41×C137—21no—
28C4×#41×C137—21no—
29C4×#41×C137C2211yescharacteristic
30C4×#41×C137C2211yescharacteristic
31#489×C137—22no—
32#487×C137C2212yes—
33#486×C137C2212yes—
34#477×C137—21no—
35#476×C137—21no—
36#475×C137C2212yes—
37#474×C137C2212yes—
38#462×C137—21no—
39#462×C137C2212yes—
40#461×C137—21no—
41#461×C137C2212yes—
42#428×C137—21no—
43#423×C137C2211yescharacteristic
44#423×C137C2211yescharacteristic
45#419×C137—21no—
46#417×C137—21no—
47#417×C137—22no—
48C4×M16×C137C2211yescharacteristic
49#333×C137C2212yes—
50#331×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52Q8⋊C8×C137C2212yes—
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137—22no—
56C2×#41×C137—21no—
57C2×#41×C137—21no—
58C2×#41×C137D411yescharacteristic
59C2×#41×C137D411yescharacteristic
60C2×#41×C137D412yes—
61C22×Q8×C137—21no—
62#137×C137—44no—
63C2×Q16×C137—22no—
64C2×SD16×C137—22no—
65C2×M16×C137—21no—
66C2×M16×C137—21no—
67C2×M16×C137—21no—
68C4⋊Q8×C137—22no—
69C4⋊Q8×C137—21no—
70C4⋊Q8×C137—21no—
71C4⋊Q8×C137—21no—
72C4⋊Q8×C137—21no—
73C4⋊Q8×C137—21no—
74C4⋊Q8×C137C2312yes—
75C4⋊Q8×C137C2311yescharacteristic
76#127×C137—22no—
77#127×C137—21no—
78#127×C137—21no—
79#127×C137C2311yescharacteristic
80#124×C137—22no—
81#124×C137—22no—
82#124×C137—22no—
83#124×C137—22no—
84#124×C137D412yes—
85#121×C137—22no—
86#121×C137—22no—
87#121×C137—22no—
88#121×C137—22no—
89#121×C137—21no—
90#121×C137—21no—
91#121×C137—22no—
92#121×C137—22no—
93#121×C137—21no—
94#121×C137—21no—
95#121×C137—21no—
96#121×C137—21no—
97C4×Q8×C137—21no—
98C4×Q8×C137—21no—
99C4×Q8×C137D411yescharacteristic
100C4×Q8×C137D411yescharacteristic
101C4×Q8×C137C2311yescharacteristic
102C4×Q8×C137C2311yescharacteristic
103C4×D4×C137—22no—
104C4×D4×C137—22no—
105C4×D4×C137—22no—
106C4×D4×C137—22no—
107C4×D4×C137—21no—
108C4×D4×C137—21no—
109C4×D4×C137—21no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137—21no—
115C4×D4×C137D411yescharacteristic
116C4×D4×C137D411yescharacteristic
117C4×D4×C137C2311yescharacteristic
118C4×D4×C137C2311yescharacteristic
119#117×C137—22no—
120#117×C137—22no—
121#117×C137—21no—
122#117×C137—21no—
123#117×C137—21no—
124#117×C137—21no—
125#117×C137D411yescharacteristic
126#117×C137D411yescharacteristic
127C2×C4⋊C4×C137—21no—
128C2×C4⋊C4×C137—21no—
129C2×C42×C137—22no—
130C2×C42×C137—21no—
131C2×C42×C137—21no—
132C2×C42×C137C2311yescharacteristic
133#107×C137—22no—
134#106×C137—22no—
135C4⋊C8×C137—21no—
136C4⋊C8×C137—21no—
137C4⋊C8×C137C2312yes—
138C4⋊C8×C137C2312yes—
139#103×C137—22no—
140#103×C137—24no—
141#103×C137—22no—
142#103×C137—22no—
143#102×C137—22no—
144#102×C137—24no—
145#97×C137—21no—
146C4×C8×C137C2312yes—
147#41×C137—22no—
148#41×C137—22no—
149#41×C137—41no—
150#41×C137—41no—
151#41×C137—44no—
152#41×C137—42no—
153#41×C137—42no—
154#41×C137—42no—
155#41×C137—42no—
156#41×C137—84no—
157C2×Q8×C137—22no—
158C2×Q8×C137—22no—
159C2×Q8×C137—42no—
160C2×Q8×C137—42no—
161C2×Q8×C137—41no—
162C2×Q8×C137C2×D411yescharacteristic
163C2×Q8×C137C2×D411yescharacteristic
164C2×Q8×C137C2×D412yes—
165C2×D4×C137—22no—
166C2×D4×C137—22no—
167C2×D4×C137—21no—
168C2×D4×C137—21no—
169C2×D4×C137—22no—
170C2×D4×C137—22no—
171C2×D4×C137—22no—
172C2×D4×C137—22no—
173C2×D4×C137—21no—
174C2×D4×C137—21no—
175C2×D4×C137C2×D411yescharacteristic
176C2×D4×C137C2×D411yescharacteristic
177C2×D4×C137C2×D412yes—
178C22×C4×C137—22no—
179C22×C4×C137—22no—
180C22×C4×C137—21no—
181C22×C4×C137—21no—
182C22×C4×C137—22no—
183C22×C4×C137—21no—
184C22×C4×C137—21no—
185C22×C4×C137—21no—
186C22×C4×C137—21no—
187C22×C4×C137—41no—
188C22×C4×C137C2×D411yescharacteristic
189C22×C4×C137C2×D411yescharacteristic
190C22×C4×C137C2×D411yescharacteristic
191Q16×C137—44no—
192SD16×C137—44no—
193M16×C137—42no—
194M16×C137—42no—
195M16×C137—42no—
196C2×C8×C137—22no—
197C2×C8×C137—22no—
198C2×C8×C137—22no—
199C4⋊C4×C137—22no—
200C4⋊C4×C137—22no—
201C4⋊C4×C137—21no—
202C4⋊C4×C137—21no—
203C4⋊C4×C137—21no—
204C4⋊C4×C137—21no—
205C4⋊C4×C137—21no—
206C4⋊C4×C137—21no—
207C4⋊C4×C137—22no—
208C4⋊C4×C137—22no—
209C4⋊C4×C137—41no—
210C4⋊C4×C137—41no—
211C4⋊C4×C137C2×D411yescharacteristic
212C4⋊C4×C137C2×D411yescharacteristic
213C4⋊C4×C137C2×D411yescharacteristic
214C4⋊C4×C137C2×D411yescharacteristic
215C22⋊C4×C137—22no—
216C22⋊C4×C137—22no—
217C22⋊C4×C137—22no—
218C22⋊C4×C137—22no—
219C22⋊C4×C137—21no—
220C22⋊C4×C137—21no—
221C22⋊C4×C137—22no—
222C22⋊C4×C137—22no—
223C22⋊C4×C137—21no—
224C22⋊C4×C137—21no—
225C22⋊C4×C137—21no—
226C22⋊C4×C137—21no—
227C42×C137—22no—
228C42×C137—22no—
229C42×C137—21no—
230C42×C137—21no—
231C42×C137—21no—
232C42×C137—21no—
233C42×C137—21no—
234C42×C137—21no—
235C42×C137C2×D411yescharacteristic
236C42×C137C2×D411yescharacteristic
237C42×C137C2×D411yescharacteristic
238C42×C137C2411yescharacteristic
239C23×C137—21no—
240C23×C137—21no—
241C23×C137—22no—
242C23×C137#12011yescharacteristic
243Q8×C137—21no—
244Q8×C137—21no—
245Q8×C137—42no—
246Q8×C137—42no—
247Q8×C137—42no—
248Q8×C137—82no—
249Q8×C137—82no—
250Q8×C137AGL(1,ℤ/8)12yes—
251Q8×C137#13712yescomplemented
252D4×C137—21no—
253D4×C137—21no—
254D4×C137—21no—
255D4×C137—21no—
256D4×C137—42no—
257D4×C137—42no—
258D4×C137—42no—
259D4×C137—44no—
260D4×C137—44no—
261D4×C137—41no—
262D4×C137—41no—
263D4×C137—41no—
264D4×C137—41no—
265D4×C137—82no—
266D4×C137—82no—
267D4×C137—82no—
268D4×C137—82no—
269C2×C4×C137—21no—
270C2×C4×C137—21no—
271C2×C4×C137—21no—
272C2×C4×C137—21no—
273C2×C4×C137—21no—
274C2×C4×C137—21no—
275C2×C4×C137—22no—
276C2×C4×C137—21no—
277C2×C4×C137—21no—
278C2×C4×C137—22no—
279C2×C4×C137—21no—
280C2×C4×C137—21no—
281C2×C4×C137—42no—
282C2×C4×C137—42no—
283C2×C4×C137—44no—
284C2×C4×C137—44no—
285C2×C4×C137—41no—
286C2×C4×C137—41no—
287C2×C4×C137—41no—
288C2×C4×C137—41no—
289C2×C4×C137—44no—
290C2×C4×C137—42no—
291C2×C4×C137—42no—
292C2×C4×C137—42no—
293C2×C4×C137—82no—
294C2×C4×C137#12011yescharacteristic
295C2×C4×C137#12011yescharacteristic
296C2×C4×C137#12011yescharacteristic
297C2×C4×C137C22×D411yescharacteristic
298C2×C4×C137C22×D411yescharacteristic
299C2×C4×C137C22×D411yescharacteristic
300C8×C137—42no—
301C8×C137—42no—
302C8×C137—42no—
303C22×C137—21no—
304C22×C137—21no—
305C22×C137—21no—
306C22×C137—41no—
307C22×C137—42no—
308C22×C137—41no—
309C22×C137—41no—
310C22×C137—42no—
311C22×C137—41no—
312C22×C137—42no—
313C22×C137—41no—
314C22×C137—41no—
315C22×C137C2×#12011yescharacteristic
316C4×C137—21no—
317C4×C137—22no—
318C4×C137—21no—
319C4×C137—22no—
320C4×C137—21no—
321C4×C137—41no—
322C4×C137—41no—
323C4×C137—42no—
324C4×C137—41no—
325C4×C137—41no—
326C4×C137—82no—
327C4×C137#45512yes—
328C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
329C4×C137C2×#13711yescharacteristic
330C4×C137#57511yescharacteristic
331C4×C137#57511yescharacteristic
332C2×C137—41no—
333C2×C137—81no—
334C2×C137—81no—
335C2×C137—82no—
336C2×C137#300611yescharacteristic
337C2×C137#300711yescharacteristic
338C2×C137C2×#45511yescharacteristic
339#18553C13711yesmaximal, characteristic, direct product factor, Sylow
340C137#1855311yescharacteristic, direct product factor, Sylow
341#3483C2×C13711yescharacteristic
342#3141C2×C13711yescharacteristic, complemented
343#3079C2×C13711yescharacteristic, complemented
344#3039C2×C13711yescharacteristic, complemented
345#1646C2×C13712yescomplemented
346#1645C2×C13712yescomplemented
347#1642C2×C13712yescomplemented
348#1641C2×C13712yescomplemented
349#1560C2×C13711yescharacteristic, complemented
350#1489C2×C13711yescharacteristic, complemented
351#1489C2×C13711yescharacteristic, complemented
352C2×#137—21no—
353Q82C22×C13711yescharacteristic
354#555—22no—
355#555—21no—
356#555—21no—
357#555C22×C13711yescharacteristic
358#550—21no—
359#550—21no—
360#550C22×C13711yescharacteristic
361#550C22×C13711yescharacteristic
362#532—22no—
363#532—21no—
364#532—21no—
365#532C22×C13711yescharacteristic
366C2×C4⋊Q8C22×C13711yescharacteristic
367C4×#41—21no—
368C4×#41—21no—
369C4×#41C22×C13711yescharacteristic
370C4×#41C22×C13711yescharacteristic
371#489—22no—
372#487C22×C13712yes—
373#486C22×C13712yes—
374#477—21no—
375#476—21no—
376#475C22×C13712yes—
377#474C22×C13712yes—
378#462—21no—
379#462C22×C13712yes—
380#461—21no—
381#461C22×C13712yes—
382#428—21no—
383#423C22×C13711yescharacteristic
384#423C22×C13711yescharacteristic
385#419—21no—
386#417—21no—
387#417—22no—
388C4×M16C22×C13711yescharacteristic
389#333C22×C13712yes—
390#331C22×C13712yes—
391Q8⋊C8C22×C13712yes—
392Q8⋊C8C22×C13712yes—
393D4⋊C8C22×C13712yes—
394D4⋊C8C22×C13712yes—
395C2×#41—22no—
396C2×#41—21no—
397C2×#41—21no—
398C2×#41D4×C13711yescharacteristic
399C2×#41D4×C13711yescharacteristic
400C2×#41D4×C13712yes—
401C22×Q8—21no—
402#137—44no—
403C2×Q16—22no—
404C2×SD16—22no—
405C2×M16—21no—
406C2×M16—21no—
407C2×M16—21no—
408C4⋊Q8—22no—
409C4⋊Q8—21no—
410C4⋊Q8—21no—
411C4⋊Q8—21no—
412C4⋊Q8—21no—
413C4⋊Q8—21no—
414C4⋊Q8C23×C13712yes—
415C4⋊Q8C23×C13711yescharacteristic
416#127—22no—
417#127—21no—
418#127—21no—
419#127C23×C13711yescharacteristic
420#124—22no—
421#124—22no—
422#124—22no—
423#124—22no—
424#124D4×C13712yes—
425#121—22no—
426#121—22no—
427#121—22no—
428#121—22no—
429#121—21no—
430#121—21no—
431#121—22no—
432#121—22no—
433#121—21no—
434#121—21no—
435#121—21no—
436#121—21no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8D4×C13711yescharacteristic
440C4×Q8D4×C13711yescharacteristic
441C4×Q8C23×C13711yescharacteristic
442C4×Q8C23×C13711yescharacteristic
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—22no—
446C4×D4—22no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—21no—
452C4×D4—21no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4D4×C13711yescharacteristic
456C4×D4D4×C13711yescharacteristic
457C4×D4C23×C13711yescharacteristic
458C4×D4C23×C13711yescharacteristic
459#117—22no—
460#117—22no—
461#117—21no—
462#117—21no—
463#117—21no—
464#117—21no—
465#117D4×C13711yescharacteristic
466#117D4×C13711yescharacteristic
467C2×C4⋊C4—21no—
468C2×C4⋊C4—21no—
469C2×C42—22no—
470C2×C42—21no—
471C2×C42—21no—
472C2×C42C23×C13711yescharacteristic
473#107—22no—
474#106—22no—
475C4⋊C8—21no—
476C4⋊C8—21no—
477C4⋊C8C23×C13712yes—
478C4⋊C8C23×C13712yes—
479#103—22no—
480#103—24no—
481#103—22no—
482#103—22no—
483#102—22no—
484#102—24no—
485#97—21no—
486C4×C8C23×C13712yes—
487#41—22no—
488#41—22no—
489#41—41no—
490#41—41no—
491#41—44no—
492#41—42no—
493#41—42no—
494#41—42no—
495#41—42no—
496#41—84no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8—41no—
502C2×Q8C2×D4×C13711yescharacteristic
503C2×Q8C2×D4×C13711yescharacteristic
504C2×Q8C2×D4×C13712yes—
505C2×D4—22no—
506C2×D4—22no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4C2×D4×C13711yescharacteristic
516C2×D4C2×D4×C13711yescharacteristic
517C2×D4C2×D4×C13712yes—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—41no—
528C22×C4C2×D4×C13711yescharacteristic
529C22×C4C2×D4×C13711yescharacteristic
530C22×C4C2×D4×C13711yescharacteristic
531Q16—44no—
532SD16—44no—
533M16—42no—
534M16—42no—
535M16—42no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4C2×D4×C13711yescharacteristic
552C4⋊C4C2×D4×C13711yescharacteristic
553C4⋊C4C2×D4×C13711yescharacteristic
554C4⋊C4C2×D4×C13711yescharacteristic
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—21no—
566C22⋊C4—21no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42C2×D4×C13711yescharacteristic
576C42C2×D4×C13711yescharacteristic
577C42C2×D4×C13711yescharacteristic
578C42C24×C13711yescharacteristic
579C23—21no—
580C23—21no—
581C23—22no—
582C23#120×C13711yescharacteristic
583Q8—21no—
584Q8—21no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—82no—
589Q8—82no—
590Q8AGL(1,ℤ/8)×C13712yes—
591Q8#137×C13712yescomplemented
592D4—21no—
593D4—21no—
594D4—21no—
595D4—21no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—44no—
600D4—44no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—82no—
606D4—82no—
607D4—82no—
608D4—82no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—82no—
634C2×C4#120×C13711yescharacteristic
635C2×C4#120×C13711yescharacteristic
636C2×C4#120×C13711yescharacteristic
637C2×C4C22×D4×C13711yescharacteristic
638C2×C4C22×D4×C13711yescharacteristic
639C2×C4C22×D4×C13711yescharacteristic
640C8—42no—
641C8—42no—
642C8—42no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—41no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22C2×#120×C13711yescharacteristic
656C4—21no—
657C4—22no—
658C4—21no—
659C4—22no—
660C4—21no—
661C4—41no—
662C4—41no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—82no—
667C4#455×C13712yes—
668C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
669C4C2×#137×C13711yescharacteristic
670C4#575×C13711yescharacteristic
671C4#575×C13711yescharacteristic
672C2—41no—
673C2—81no—
674C2—81no—
675C2—82no—
676C2#3006×C13711yescharacteristic
677C2#3007×C13711yescharacteristic
678C2C2×#455×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3483×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. #3141×C137C2semidirect
3. #3079×C137C2semidirect
4. #3039×C137C2semidirect
5. #1646×C137C2semidirect
6. #1645×C137C2semidirect
7. #1642×C137C2semidirect
8. #1641×C137C2semidirect
9. #1560×C137C2semidirect
10. #1489×C137C2semidirect
11. #1489×C137C2semidirect
251. Q8×C137#137semidirect
339. #18553C137direct (N × H)
340. C137#18553direct (N × H)
342. #3141C2×C137semidirect
343. #3079C2×C137semidirect
344. #3039C2×C137semidirect
345. #1646C2×C137semidirect
346. #1645C2×C137semidirect
347. #1642C2×C137semidirect
348. #1641C2×C137semidirect
349. #1560C2×C137semidirect
350. #1489C2×C137semidirect
351. #1489C2×C137semidirect
591. Q8#137×C137semidirect

Maximal subgroups

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

subgroupindex
#3483×C1372
#3141×C1372
#3079×C1372
#3039×C1372
#1646×C1372
#1645×C1372
#1642×C1372
#1641×C1372
#1560×C1372
#1489×C1372
#18553137