Finite Group Databasebeta

← C137×#25877 #46875263

Subgroups — C137×#25877 #46875263

Order 35072.

1254 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3500×C137C211yesmaximal, characteristic
2#3142×C137C211yesmaximal, characteristic, complemented
3#3080×C137C211yesmaximal, characteristic, complemented
4#2861×C137C211yesmaximal, characteristic, complemented
5#2038×C137C212yesmaximal, complemented
6#2037×C137C212yesmaximal, complemented
7#1967×C137C211yesmaximal, characteristic, complemented
8#1962×C137C211yesmaximal, characteristic, complemented
9#1793×C137C212yesmaximal, complemented
10#1792×C137C212yesmaximal, complemented
11C4×#104×C137C211yesmaximal, characteristic, complemented
12C2×2−1+4×C137C2211yescharacteristic
13C2×H(5,𝔽2)×C137C2211yescharacteristic
14#548×C137—22no—
15#546×C137—22no—
16#546×C137—24no—
17#536×C137—21no—
18#536×C137C2211yescharacteristic
19#535×C137—21no—
20#535×C137C2211yescharacteristic
21#532×C137—21no—
22#532×C137C2211yescharacteristic
23C2×#125×C137C2211yescharacteristic, complemented
24C2×#124×C137—21no—
25C2×#124×C137C2211yescharacteristic
26C4×#41×C137C2211yescharacteristic
27#491×C137—21no—
28#490×C137—21no—
29#488×C137—21no—
30#488×C137—21no—
31#479×C137—22no—
32#478×C137—22no—
33#428×C137C2212yescomplemented
34#423×C137C2211yescharacteristic, complemented
35C2×#104×C137C2212yes—
36C2×#104×C137C2212yes—
37C2×#104×C137C2212yescomplemented
38#417×C137C2212yes—
39#417×C137C2212yes—
40#417×C137C2212yescomplemented
41C2×#101×C137C2212yescomplemented
42C2×#100×C137C2212yes—
43C4×M16×C137C2211yescharacteristic, complemented
44#391×C137C2212yescomplemented
45#390×C137C2212yes—
46#388×C137—24no—
47C43×C137C2211yescharacteristic, complemented
48#339×C137C2212yescomplemented
492−1+4×C137—24no—
502−1+4×C137—42no—
51H(5,𝔽2)×C137—24no—
52H(5,𝔽2)×C137—42no—
53C2×#41×C137—21no—
54C2×#41×C137—21no—
55C2×#41×C137—21no—
56C2×#41×C137—21no—
57C2×#41×C137—22no—
58C2×#41×C137C2311yescharacteristic
59C2×#41×C137C2311yescharacteristic
60C2×#41×C137C2311yescharacteristic
61C22×Q8×C137—21no—
62C22×Q8×C137—21no—
63C22×Q8×C137C2311yescharacteristic
64C22×D4×C137—22no—
65C22×D4×C137—21no—
66C22×D4×C137—21no—
67C22×D4×C137C2311yescharacteristic
68C2×M16×C137C2312yes—
69C2×M16×C137C2312yes—
70C4⋊Q8×C137—21no—
71C4⋊Q8×C137D411yescharacteristic
72#127×C137—21no—
73#127×C137D411yescharacteristic
74#125×C137—22no—
75#125×C137—21no—
76#125×C137—21no—
77#124×C137—24no—
78#124×C137—21no—
79#124×C137—21no—
80#124×C137—21no—
81#124×C137—21no—
82#124×C137—22no—
83#124×C137D411yescharacteristic
84#124×C137D411yescharacteristic
85#123×C137—24no—
86#123×C137—22no—
87C22⋊Q8×C137—24no—
88C22⋊Q8×C137—22no—
89#121×C137—24no—
90#121×C137—24no—
91#121×C137—22no—
92#121×C137—22no—
93#120×C137—24no—
94#120×C137—22no—
95C4×Q8×C137D412yes—
96C4×D4×C137—24no—
97C4×D4×C137—22no—
98C4×D4×C137—22no—
99C4×D4×C137D412yes—
100#117×C137—21no—
101#117×C137C2311yescharacteristic
102C2×C4⋊C4×C137—22no—
103C2×C4⋊C4×C137C2312yes—
104C2×C22⋊C4×C137—24no—
105C2×C22⋊C4×C137—24no—
106C2×C22⋊C4×C137D414yes—
107C2×C42×C137—21no—
108C2×C42×C137—22no—
109C2×C42×C137C2312yes—
110C2×C42×C137C2311yescharacteristic
111#107×C137—22no—
112#106×C137—22no—
113C4⋊C8×C137—22no—
114#104×C137—24no—
115#104×C137—42no—
116#104×C137—42no—
117#103×C137—22no—
118#103×C137—22no—
119#103×C137—22no—
120#102×C137—22no—
121#102×C137—22no—
122#102×C137—22no—
123#101×C137—42no—
124#100×C137—42no—
125#97×C137—21no—
126C4×C8×C137—21no—
127#95×C137—24no—
128C24×C137—22no—
129C24×C137C2×D412yes—
130#41×C137—22no—
131#41×C137—22no—
132#41×C137—22no—
133#41×C137—22no—
134#41×C137—22no—
135#41×C137—44no—
136#41×C137—48no—
137#41×C137—44no—
138#41×C137—42no—
139#41×C137—42no—
140#41×C137—42no—
141#41×C137—42no—
142#41×C137#4114yes—
143C2×Q8×C137—22no—
144C2×Q8×C137—21no—
145C2×Q8×C137—21no—
146C2×Q8×C137—22no—
147C2×Q8×C137—22no—
148C2×Q8×C137—44no—
149C2×Q8×C137—42no—
150C2×Q8×C137—42no—
151C2×Q8×C137C2×D411yescharacteristic
152C2×Q8×C137C2×D411yescharacteristic
153C2×Q8×C137C2×D411yescharacteristic
154C2×D4×C137—21no—
155C2×D4×C137—22no—
156C2×D4×C137—22no—
157C2×D4×C137—22no—
158C2×D4×C137—21no—
159C2×D4×C137—21no—
160C2×D4×C137—24no—
161C2×D4×C137—22no—
162C2×D4×C137—21no—
163C2×D4×C137—21no—
164C2×D4×C137—22no—
165C2×D4×C137—22no—
166C2×D4×C137—44no—
167C2×D4×C137—48no—
168C2×D4×C137—42no—
169C2×D4×C137—42no—
170C2×D4×C137C2×D411yescharacteristic
171C2×D4×C137C2×D411yescharacteristic
172C2×D4×C137C2×D411yescharacteristic
173C22×C4×C137—21no—
174C22×C4×C137—22no—
175C22×C4×C137—22no—
176C22×C4×C137—22no—
177C22×C4×C137—22no—
178C22×C4×C137—21no—
179C22×C4×C137—21no—
180C22×C4×C137—22no—
181C22×C4×C137—22no—
182C22×C4×C137C2×D412yes—
183C22×C4×C137C2×D412yes—
184C22×C4×C137C2411yescharacteristic
185M16×C137—22no—
186M16×C137—22no—
187M16×C137—22no—
188M16×C137—22no—
189C2×C8×C137—22no—
190C2×C8×C137—22no—
191C4⋊C4×C137—24no—
192C4⋊C4×C137—22no—
193C4⋊C4×C137—22no—
194C4⋊C4×C137—22no—
195C4⋊C4×C137—22no—
196C4⋊C4×C137—42no—
197C4⋊C4×C137—42no—
198C4⋊C4×C137C2×D412yes—
199C22⋊C4×C137—24no—
200C22⋊C4×C137—24no—
201C22⋊C4×C137—21no—
202C22⋊C4×C137—22no—
203C22⋊C4×C137—24no—
204C22⋊C4×C137—24no—
205C22⋊C4×C137—22no—
206C22⋊C4×C137—22no—
207C22⋊C4×C137—48no—
208C42×C137—22no—
209C42×C137—21no—
210C42×C137—22no—
211C42×C137—22no—
212C42×C137—44no—
213C42×C137C2×D411yescharacteristic
214C42×C137C2×D411yescharacteristic
215C42×C137C2×D412yes—
216C23×C137—21no—
217C23×C137—22no—
218C23×C137—21no—
219C23×C137—21no—
220C23×C137—24no—
221C23×C137—44no—
222C23×C137—44no—
223C23×C137—84no—
224C23×C137C22×D411yescharacteristic
225Q8×C137—21no—
226Q8×C137—22no—
227Q8×C137—22no—
228Q8×C137—44no—
229Q8×C137—41no—
230Q8×C137—41no—
231Q8×C137—42no—
232Q8×C137#12412yes—
233D4×C137—24no—
234D4×C137—21no—
235D4×C137—22no—
236D4×C137—22no—
237D4×C137—44no—
238D4×C137—44no—
239D4×C137—44no—
240D4×C137—42no—
241D4×C137—42no—
242D4×C137—41no—
243D4×C137—41no—
244D4×C137—42no—
245D4×C137—82no—
246D4×C137—84no—
247D4×C137#12412yes—
248C2×C4×C137—21no—
249C2×C4×C137—24no—
250C2×C4×C137—24no—
251C2×C4×C137—22no—
252C2×C4×C137—21no—
253C2×C4×C137—22no—
254C2×C4×C137—21no—
255C2×C4×C137—24no—
256C2×C4×C137—42no—
257C2×C4×C137—44no—
258C2×C4×C137—44no—
259C2×C4×C137—42no—
260C2×C4×C137—44no—
261C2×C4×C137—42no—
262C2×C4×C137—41no—
263C2×C4×C137—41no—
264C2×C4×C137—41no—
265C2×C4×C137—41no—
266C2×C4×C137—44no—
267C2×C4×C137—42no—
268C2×C4×C137—82no—
269C2×C4×C137#12012yes—
270C2×C4×C137#12012yes—
271C2×C4×C137C22×D411yescharacteristic
272C2×C4×C137C22×D411yescharacteristic
273C2×C4×C137C2×#4112yes—
274C2×C4×C137H(5,𝔽2)12yes—
275C8×C137—42no—
276C8×C137—42no—
277C22×C137—22no—
278C22×C137—22no—
279C22×C137—42no—
280C22×C137—41no—
281C22×C137—41no—
282C22×C137—42no—
283C22×C137—42no—
284C22×C137—44no—
285C22×C137—42no—
286C22×C137—84no—
287C22×C137—164no—
288C22×C137C2×#12011yescharacteristic
289C22×C137#54612yes—
290C4×C137—22no—
291C4×C137—22no—
292C4×C137—22no—
293C4×C137—44no—
294C4×C137—42no—
295C4×C137—41no—
296C4×C137—41no—
297C4×C137—42no—
298C4×C137—82no—
299C4×C137C2×#12411yescharacteristic
300C4×C137#53511yescharacteristic
301C4×C137#54612yes—
302C2×C137—22no—
303C2×C137—42no—
304C2×C137—81no—
305C2×C137—82no—
306C2×C137—81no—
307C2×C137#243511yescharacteristic
308C2×C137#301812yes—
309#25877C13711yesmaximal, characteristic, direct product factor, Sylow
310C137#2587711yescharacteristic, direct product factor, Sylow
311#3500C2×C13711yescharacteristic
312#3142C2×C13711yescharacteristic, complemented
313#3080C2×C13711yescharacteristic, complemented
314#2861C2×C13711yescharacteristic, complemented
315#2038C2×C13712yescomplemented
316#2037C2×C13712yescomplemented
317#1967C2×C13711yescharacteristic, complemented
318#1962C2×C13711yescharacteristic, complemented
319#1793C2×C13712yescomplemented
320#1792C2×C13712yescomplemented
321C4×#104C2×C13711yescharacteristic, complemented
322C2×2−1+4C22×C13711yescharacteristic
323C2×H(5,𝔽2)C22×C13711yescharacteristic
324#548—22no—
325#546—22no—
326#546—24no—
327#536—21no—
328#536C22×C13711yescharacteristic
329#535—21no—
330#535C22×C13711yescharacteristic
331#532—21no—
332#532C22×C13711yescharacteristic
333C2×#125C22×C13711yescharacteristic, complemented
334C2×#124—21no—
335C2×#124C22×C13711yescharacteristic
336C4×#41C22×C13711yescharacteristic
337#491—21no—
338#490—21no—
339#488—21no—
340#488—21no—
341#479—22no—
342#478—22no—
343#428C22×C13712yescomplemented
344#423C22×C13711yescharacteristic, complemented
345C2×#104C22×C13712yes—
346C2×#104C22×C13712yes—
347C2×#104C22×C13712yescomplemented
348#417C22×C13712yes—
349#417C22×C13712yes—
350#417C22×C13712yescomplemented
351C2×#101C22×C13712yescomplemented
352C2×#100C22×C13712yes—
353C4×M16C22×C13711yescharacteristic, complemented
354#391C22×C13712yescomplemented
355#390C22×C13712yes—
356#388—24no—
357C43C22×C13711yescharacteristic, complemented
358#339C22×C13712yescomplemented
3592−1+4—24no—
3602−1+4—42no—
361H(5,𝔽2)—24no—
362H(5,𝔽2)—42no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—22no—
368C2×#41C23×C13711yescharacteristic
369C2×#41C23×C13711yescharacteristic
370C2×#41C23×C13711yescharacteristic
371C22×Q8—21no—
372C22×Q8—21no—
373C22×Q8C23×C13711yescharacteristic
374C22×D4—22no—
375C22×D4—21no—
376C22×D4—21no—
377C22×D4C23×C13711yescharacteristic
378C2×M16C23×C13712yes—
379C2×M16C23×C13712yes—
380C4⋊Q8—21no—
381C4⋊Q8D4×C13711yescharacteristic
382#127—21no—
383#127D4×C13711yescharacteristic
384#125—22no—
385#125—21no—
386#125—21no—
387#124—24no—
388#124—21no—
389#124—21no—
390#124—21no—
391#124—21no—
392#124—22no—
393#124D4×C13711yescharacteristic
394#124D4×C13711yescharacteristic
395#123—24no—
396#123—22no—
397C22⋊Q8—24no—
398C22⋊Q8—22no—
399#121—24no—
400#121—24no—
401#121—22no—
402#121—22no—
403#120—24no—
404#120—22no—
405C4×Q8D4×C13712yes—
406C4×D4—24no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4D4×C13712yes—
410#117—21no—
411#117C23×C13711yescharacteristic
412C2×C4⋊C4—22no—
413C2×C4⋊C4C23×C13712yes—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4D4×C13714yes—
417C2×C42—21no—
418C2×C42—22no—
419C2×C42C23×C13712yes—
420C2×C42C23×C13711yescharacteristic
421#107—22no—
422#106—22no—
423C4⋊C8—22no—
424#104—24no—
425#104—42no—
426#104—42no—
427#103—22no—
428#103—22no—
429#103—22no—
430#102—22no—
431#102—22no—
432#102—22no—
433#101—42no—
434#100—42no—
435#97—21no—
436C4×C8—21no—
437#95—24no—
438C24—22no—
439C24C2×D4×C13712yes—
440#41—22no—
441#41—22no—
442#41—22no—
443#41—22no—
444#41—22no—
445#41—44no—
446#41—48no—
447#41—44no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—42no—
452#41#41×C13714yes—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—44no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8C2×D4×C13711yescharacteristic
462C2×Q8C2×D4×C13711yescharacteristic
463C2×Q8C2×D4×C13711yescharacteristic
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—24no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—44no—
477C2×D4—48no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4C2×D4×C13711yescharacteristic
481C2×D4C2×D4×C13711yescharacteristic
482C2×D4C2×D4×C13711yescharacteristic
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4C2×D4×C13712yes—
493C22×C4C2×D4×C13712yes—
494C22×C4C24×C13711yescharacteristic
495M16—22no—
496M16—22no—
497M16—22no—
498M16—22no—
499C2×C8—22no—
500C2×C8—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4C2×D4×C13712yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—48no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—44no—
523C42C2×D4×C13711yescharacteristic
524C42C2×D4×C13711yescharacteristic
525C42C2×D4×C13712yes—
526C23—21no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—44no—
532C23—44no—
533C23—84no—
534C23C22×D4×C13711yescharacteristic
535Q8—21no—
536Q8—22no—
537Q8—22no—
538Q8—44no—
539Q8—41no—
540Q8—41no—
541Q8—42no—
542Q8#124×C13712yes—
543D4—24no—
544D4—21no—
545D4—22no—
546D4—22no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—42no—
551D4—42no—
552D4—41no—
553D4—41no—
554D4—42no—
555D4—82no—
556D4—84no—
557D4#124×C13712yes—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—82no—
579C2×C4#120×C13712yes—
580C2×C4#120×C13712yes—
581C2×C4C22×D4×C13711yescharacteristic
582C2×C4C22×D4×C13711yescharacteristic
583C2×C4C2×#41×C13712yes—
584C2×C4H(5,𝔽2)×C13712yes—
585C8—42no—
586C8—42no—
587C22—22no—
588C22—22no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—84no—
597C22—164no—
598C22C2×#120×C13711yescharacteristic
599C22#546×C13712yes—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—44no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—82no—
609C4C2×#124×C13711yescharacteristic
610C4#535×C13711yescharacteristic
611C4#546×C13712yes—
612C2—22no—
613C2—42no—
614C2—82no—
615C2—81no—
616C2—81no—
617C2#2435×C13711yescharacteristic
618C2#3018×C13712yes—
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3500×C137C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3142×C137C2semidirect
3. #3080×C137C2semidirect
4. #2861×C137C2semidirect
5. #2038×C137C2semidirect
6. #2037×C137C2semidirect
7. #1967×C137C2semidirect
8. #1962×C137C2semidirect
9. #1793×C137C2semidirect
10. #1792×C137C2semidirect
11. C4×#104×C137C2semidirect
23. C2×#125×C137C22semidirect
33. #428×C137C22semidirect
34. #423×C137C22semidirect
37. C2×#104×C137C22semidirect
40. #417×C137C22semidirect
41. C2×#101×C137C22semidirect
43. C4×M16×C137C22semidirect
44. #391×C137C22semidirect
47. C43×C137C22semidirect
48. #339×C137C22semidirect
309. #25877C137direct (N × H)
310. C137#25877direct (N × H)
312. #3142C2×C137semidirect
313. #3080C2×C137semidirect
314. #2861C2×C137semidirect
315. #2038C2×C137semidirect
316. #2037C2×C137semidirect
317. #1967C2×C137semidirect
318. #1962C2×C137semidirect
319. #1793C2×C137semidirect
320. #1792C2×C137semidirect
321. C4×#104C2×C137semidirect
333. C2×#125C22×C137semidirect
343. #428C22×C137semidirect
344. #423C22×C137semidirect
347. C2×#104C22×C137semidirect
350. #417C22×C137semidirect
351. C2×#101C22×C137semidirect
353. C4×M16C22×C137semidirect
354. #391C22×C137semidirect
357. C43C22×C137semidirect
358. #339C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3500×C1372
#3142×C1372
#3080×C1372
#2861×C1372
#2038×C1372
#2037×C1372
#1967×C1372
#1962×C1372
#1793×C1372
#1792×C1372
C4×#104×C1372
#25877137