Finite Group Databasebeta

← C137×#20218 #46869604

Subgroups — C137×#20218 #46869604

Order 35072.

626 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#3395×C137C211yesmaximal, characteristic, complemented
2#3386×C137C211yesmaximal, characteristic, complemented
3#3368×C137C211yesmaximal, characteristic
4#3278×C137C211yesmaximal, characteristic
5#1722×C137C211yesmaximal, characteristic, complemented
6#1702×C137C211yesmaximal, characteristic, complemented
7#1694×C137C211yesmaximal, characteristic, complemented
8#1689×C137C211yesmaximal, characteristic, complemented
9#1666×C137C211yesmaximal, characteristic, complemented
10#1649×C137C211yesmaximal, characteristic, complemented
11#1648×C137C211yesmaximal, characteristic, complemented
12#1641×C137C211yesmaximal, characteristic
13#1626×C137C211yesmaximal, characteristic, complemented
14#1593×C137C211yesmaximal, characteristic, complemented
15#1583×C137C211yesmaximal, characteristic, complemented
16C2×#137×C137—21no—
17#557×C137C2211yescharacteristic
18#557×C137C2211yescharacteristic
19#550×C137C2211yescharacteristic
20D4×Q8×C137C2211yescharacteristic
21#501×C137—21no—
22#500×C137C2211yescharacteristic
23#500×C137C2211yescharacteristic
24#494×C137—21no—
25#494×C137C2211yescharacteristic
26#492×C137C2211yescharacteristic
27#490×C137—21no—
28#486×C137C2211yescharacteristic
29#486×C137C2211yescharacteristic
30#478×C137—21no—
31#478×C137—21no—
32#477×C137—21no—
33#475×C137C2211yescharacteristic
34#474×C137C2211yescharacteristic
35#474×C137C2211yescharacteristic
36#468×C137—21no—
37#462×C137—21no—
38#462×C137—21no—
39#462×C137C2211yescharacteristic
40#462×C137C2211yescharacteristic
41#462×C137C2211yescharacteristic
42#460×C137—21no—
43#460×C137C2211yescharacteristic
44#460×C137C2211yescharacteristic
45#452×C137—21no—
46#448×C137—21no—
47#446×C137C2211yescharacteristic
48#446×C137C2211yescharacteristic
49C8×Q8×C137C2211yescharacteristic
50#440×C137—21no—
51#440×C137—21no—
52C4×Q16×C137—21no—
53C4×Q16×C137C2211yescharacteristic
54C4×SD16×C137C2211yescharacteristic
55#436×C137C2211yescharacteristic
56#334×C137C2211yescharacteristic
57#334×C137C2211yescharacteristic
58#332×C137C2211yescharacteristic
59#332×C137C2211yescharacteristic
60#332×C137C2211yescharacteristic
61#331×C137C2211yescharacteristic
62Q8⋊C8×C137C2211yescharacteristic
63Q8⋊C8×C137C2211yescharacteristic
64Q8⋊C8×C137C2211yescharacteristic
65D4⋊C8×C137C2211yescharacteristic
66#322×C137C2211yescharacteristic
67C2×#41×C137—21no—
68C22×Q8×C137—21no—
69#137×C137—44no—
70C2×Q16×C137—21no—
71C2×Q16×C137—21no—
72C2×Q16×C137—21no—
73C2×Q16×C137—21no—
74C2×Q16×C137—21no—
75C2×Q16×C137D411yescharacteristic
76C2×SD16×C137—21no—
77C2×SD16×C137—21no—
78C2×SD16×C137—21no—
79C2×SD16×C137—21no—
80C2×SD16×C137D411yescharacteristic
81C2×M16×C137—21no—
82C4⋊Q8×C137—21no—
83C4⋊Q8×C137—21no—
84C4⋊Q8×C137—21no—
85C4⋊Q8×C137C2311yescharacteristic
86C4⋊Q8×C137C2311yescharacteristic
87C4⋊Q8×C137C2311yescharacteristic
88#127×C137—21no—
89#127×C137C2311yescharacteristic
90#125×C137—21no—
91#125×C137—21no—
92#125×C137—41no—
93#125×C137—41no—
94C22⋊Q8×C137—21no—
95C22⋊Q8×C137—41no—
96#121×C137—21no—
97#121×C137—41no—
98C4×Q8×C137—21no—
99C4×Q8×C137—21no—
100C4×Q8×C137—21no—
101C4×Q8×C137—21no—
102C4×Q8×C137C2311yescharacteristic
103C4×Q8×C137C2311yescharacteristic
104C4×Q8×C137C2311yescharacteristic
105C4×D4×C137—21no—
106C4×D4×C137—21no—
107C4×D4×C137C2311yescharacteristic
108#107×C137—21no—
109#107×C137—21no—
110#107×C137—21no—
111#107×C137—21no—
112#107×C137—41no—
113#107×C137D411yescharacteristic
114#106×C137—21no—
115#106×C137D411yescharacteristic
116C4⋊C8×C137—21no—
117C4⋊C8×C137—21no—
118C4⋊C8×C137—21no—
119C4⋊C8×C137C2311yescharacteristic
120C4⋊C8×C137C2311yescharacteristic
121C4⋊C8×C137C2311yescharacteristic
122C4⋊C8×C137C2311yescharacteristic
123#103×C137—21no—
124#103×C137—21no—
125#103×C137—21no—
126#103×C137—21no—
127#103×C137D411yescharacteristic
128#103×C137D411yescharacteristic
129#103×C137D411yescharacteristic
130#102×C137—21no—
131#102×C137—21no—
132#102×C137—21no—
133#102×C137—21no—
134#102×C137—21no—
135#102×C137—21no—
136#102×C137—41no—
137#102×C137D411yescharacteristic
138C22⋊C8×C137—21no—
139#97×C137—21no—
140#97×C137—21no—
141C4×C8×C137—21no—
142C4×C8×C137C2311yescharacteristic
143C4×C8×C137C2311yescharacteristic
144C4×C8×C137C2311yescharacteristic
145#41×C137—42no—
146#41×C137—81no—
147C2×Q8×C137—21no—
148C2×Q8×C137—21no—
149C2×Q8×C137—21no—
150C2×Q8×C137—41no—
151C2×Q8×C137—42no—
152C2×Q8×C137—81no—
153C2×Q8×C137C2×D411yescharacteristic
154C2×Q8×C137C2×D411yescharacteristic
155C2×Q8×C137C2×D411yescharacteristic
156C2×D4×C137—21no—
157C2×D4×C137—21no—
158C2×D4×C137—21no—
159C2×D4×C137—41no—
160C2×D4×C137—41no—
161C2×D4×C137C2×D411yescharacteristic
162C22×C4×C137—21no—
163C22×C4×C137—41no—
164C22×C4×C137—41no—
165Q16×C137—22no—
166Q16×C137—42no—
167Q16×C137—42no—
168Q16×C137—81no—
169Q16×C137—81no—
170Q16×C137—81no—
171SD16×C137—22no—
172SD16×C137—42no—
173SD16×C137—42no—
174SD16×C137—42no—
175SD16×C137—81no—
176M16×C137—42no—
177C2×C8×C137—21no—
178C2×C8×C137—21no—
179C2×C8×C137—21no—
180C2×C8×C137—21no—
181C2×C8×C137—21no—
182C2×C8×C137C2×D411yescharacteristic
183C2×C8×C137C2×D411yescharacteristic
184C2×C8×C137C2×D411yescharacteristic
185C2×C8×C137C2×D411yescharacteristic
186C4⋊C4×C137—21no—
187C4⋊C4×C137—21no—
188C4⋊C4×C137—21no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—21no—
191C4⋊C4×C137—21no—
192C4⋊C4×C137—21no—
193C4⋊C4×C137—21no—
194C4⋊C4×C137—21no—
195C4⋊C4×C137—41no—
196C4⋊C4×C137—41no—
197C4⋊C4×C137—41no—
198C4⋊C4×C137—41no—
199C4⋊C4×C137—41no—
200C4⋊C4×C137—41no—
201C4⋊C4×C137—41no—
202C4⋊C4×C137C2×D411yescharacteristic
203C4⋊C4×C137C2×D411yescharacteristic
204C4⋊C4×C137C2×D411yescharacteristic
205C4⋊C4×C137C2×D411yescharacteristic
206C22⋊C4×C137—21no—
207C22⋊C4×C137—41no—
208C22⋊C4×C137—41no—
209C42×C137—21no—
210C42×C137—21no—
211C42×C137—21no—
212C42×C137—41no—
213C42×C137—41no—
214C42×C137C2411yescharacteristic
215C23×C137—21no—
216C23×C137—41no—
217Q8×C137—21no—
218Q8×C137—21no—
219Q8×C137—21no—
220Q8×C137—21no—
221Q8×C137—21no—
222Q8×C137—41no—
223Q8×C137—41no—
224Q8×C137—41no—
225Q8×C137—41no—
226Q8×C137—41no—
227Q8×C137—41no—
228Q8×C137—82no—
229Q8×C137#13712yescomplemented
230D4×C137—21no—
231D4×C137—21no—
232D4×C137—41no—
233D4×C137—41no—
234D4×C137—41no—
235D4×C137—41no—
236D4×C137—81no—
237D4×C137—82no—
238D4×C137—161no—
239C2×C4×C137—21no—
240C2×C4×C137—21no—
241C2×C4×C137—21no—
242C2×C4×C137—21no—
243C2×C4×C137—21no—
244C2×C4×C137—21no—
245C2×C4×C137—21no—
246C2×C4×C137—41no—
247C2×C4×C137—41no—
248C2×C4×C137—41no—
249C2×C4×C137—41no—
250C2×C4×C137—41no—
251C2×C4×C137—82no—
252C2×C4×C137—82no—
253C2×C4×C137C22×D411yescharacteristic
254C2×C4×C137C22×D411yescharacteristic
255C2×C4×C137H(5,𝔽2)11yescharacteristic
256C8×C137—21no—
257C8×C137—21no—
258C8×C137—21no—
259C8×C137—41no—
260C8×C137—41no—
261C8×C137—41no—
262C8×C137—41no—
263C8×C137—41no—
264C8×C137AGL(1,ℤ/8)12yes—
265C22×C137—41no—
266C22×C137—41no—
267C22×C137—41no—
268C22×C137—81no—
269C22×C137—81no—
270C22×C137—81no—
271C22×C137D4211yescharacteristic
272C4×C137—22no—
273C4×C137—21no—
274C4×C137—41no—
275C4×C137—41no—
276C4×C137—41no—
277C4×C137—41no—
278C4×C137—41no—
279C4×C137—41no—
280C4×C137—81no—
281C4×C137—81no—
282C4×C137—81no—
283C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C137C2×#13711yescharacteristic
285C4×C137#57611yescharacteristic
286C4×C137#57811yescharacteristic
287C2×C137—81no—
288C2×C137—161no—
289C2×C137#327011yescharacteristic
290C2×C137#327411yescharacteristic
291C2×C137#328911yescharacteristic
292#20218C13711yesmaximal, characteristic, direct product factor, Sylow
293C137#2021811yescharacteristic, direct product factor, Sylow
294#3395C2×C13711yescharacteristic, complemented
295#3386C2×C13711yescharacteristic, complemented
296#3368C2×C13711yescharacteristic
297#3278C2×C13711yescharacteristic
298#1722C2×C13711yescharacteristic, complemented
299#1702C2×C13711yescharacteristic, complemented
300#1694C2×C13711yescharacteristic, complemented
301#1689C2×C13711yescharacteristic, complemented
302#1666C2×C13711yescharacteristic, complemented
303#1649C2×C13711yescharacteristic, complemented
304#1648C2×C13711yescharacteristic, complemented
305#1641C2×C13711yescharacteristic
306#1626C2×C13711yescharacteristic, complemented
307#1593C2×C13711yescharacteristic, complemented
308#1583C2×C13711yescharacteristic, complemented
309C2×#137—21no—
310#557C22×C13711yescharacteristic
311#557C22×C13711yescharacteristic
312#550C22×C13711yescharacteristic
313D4×Q8C22×C13711yescharacteristic
314#501—21no—
315#500C22×C13711yescharacteristic
316#500C22×C13711yescharacteristic
317#494—21no—
318#494C22×C13711yescharacteristic
319#492C22×C13711yescharacteristic
320#490—21no—
321#486C22×C13711yescharacteristic
322#486C22×C13711yescharacteristic
323#478—21no—
324#478—21no—
325#477—21no—
326#475C22×C13711yescharacteristic
327#474C22×C13711yescharacteristic
328#474C22×C13711yescharacteristic
329#468—21no—
330#462—21no—
331#462—21no—
332#462C22×C13711yescharacteristic
333#462C22×C13711yescharacteristic
334#462C22×C13711yescharacteristic
335#460—21no—
336#460C22×C13711yescharacteristic
337#460C22×C13711yescharacteristic
338#452—21no—
339#448—21no—
340#446C22×C13711yescharacteristic
341#446C22×C13711yescharacteristic
342C8×Q8C22×C13711yescharacteristic
343#440—21no—
344#440—21no—
345C4×Q16—21no—
346C4×Q16C22×C13711yescharacteristic
347C4×SD16C22×C13711yescharacteristic
348#436C22×C13711yescharacteristic
349#334C22×C13711yescharacteristic
350#334C22×C13711yescharacteristic
351#332C22×C13711yescharacteristic
352#332C22×C13711yescharacteristic
353#332C22×C13711yescharacteristic
354#331C22×C13711yescharacteristic
355Q8⋊C8C22×C13711yescharacteristic
356Q8⋊C8C22×C13711yescharacteristic
357Q8⋊C8C22×C13711yescharacteristic
358D4⋊C8C22×C13711yescharacteristic
359#322C22×C13711yescharacteristic
360C2×#41—21no—
361C22×Q8—21no—
362#137—44no—
363C2×Q16—21no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×Q16—21no—
367C2×Q16—21no—
368C2×Q16D4×C13711yescharacteristic
369C2×SD16—21no—
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16D4×C13711yescharacteristic
374C2×M16—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8C23×C13711yescharacteristic
379C4⋊Q8C23×C13711yescharacteristic
380C4⋊Q8C23×C13711yescharacteristic
381#127—21no—
382#127C23×C13711yescharacteristic
383#125—21no—
384#125—21no—
385#125—41no—
386#125—41no—
387C22⋊Q8—21no—
388C22⋊Q8—41no—
389#121—21no—
390#121—41no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8C23×C13711yescharacteristic
396C4×Q8C23×C13711yescharacteristic
397C4×Q8C23×C13711yescharacteristic
398C4×D4—21no—
399C4×D4—21no—
400C4×D4C23×C13711yescharacteristic
401#107—21no—
402#107—21no—
403#107—21no—
404#107—21no—
405#107—41no—
406#107D4×C13711yescharacteristic
407#106—21no—
408#106D4×C13711yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8—21no—
412C4⋊C8C23×C13711yescharacteristic
413C4⋊C8C23×C13711yescharacteristic
414C4⋊C8C23×C13711yescharacteristic
415C4⋊C8C23×C13711yescharacteristic
416#103—21no—
417#103—21no—
418#103—21no—
419#103—21no—
420#103D4×C13711yescharacteristic
421#103D4×C13711yescharacteristic
422#103D4×C13711yescharacteristic
423#102—21no—
424#102—21no—
425#102—21no—
426#102—21no—
427#102—21no—
428#102—21no—
429#102—41no—
430#102D4×C13711yescharacteristic
431C22⋊C8—21no—
432#97—21no—
433#97—21no—
434C4×C8—21no—
435C4×C8C23×C13711yescharacteristic
436C4×C8C23×C13711yescharacteristic
437C4×C8C23×C13711yescharacteristic
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—41no—
444C2×Q8—42no—
445C2×Q8—81no—
446C2×Q8C2×D4×C13711yescharacteristic
447C2×Q8C2×D4×C13711yescharacteristic
448C2×Q8C2×D4×C13711yescharacteristic
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4C2×D4×C13711yescharacteristic
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—41no—
458Q16—22no—
459Q16—42no—
460Q16—42no—
461Q16—81no—
462Q16—81no—
463Q16—81no—
464SD16—22no—
465SD16—42no—
466SD16—42no—
467SD16—42no—
468SD16—81no—
469M16—42no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8C2×D4×C13711yescharacteristic
476C2×C8C2×D4×C13711yescharacteristic
477C2×C8C2×D4×C13711yescharacteristic
478C2×C8C2×D4×C13711yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4C2×D4×C13711yescharacteristic
496C4⋊C4C2×D4×C13711yescharacteristic
497C4⋊C4C2×D4×C13711yescharacteristic
498C4⋊C4C2×D4×C13711yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42—41no—
507C42C24×C13711yescharacteristic
508C23—21no—
509C23—41no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—82no—
522Q8#137×C13712yescomplemented
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—81no—
530D4—82no—
531D4—161no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C22×D4×C13711yescharacteristic
547C2×C4C22×D4×C13711yescharacteristic
548C2×C4H(5,𝔽2)×C13711yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8AGL(1,ℤ/8)×C13712yes—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22D42×C13711yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
577C4C2×#137×C13711yescharacteristic
578C4#576×C13711yescharacteristic
579C4#578×C13711yescharacteristic
580C2—81no—
581C2—161no—
582C2#3270×C13711yescharacteristic
583C2#3274×C13711yescharacteristic
584C2#3289×C13711yescharacteristic
subgroupG/H
center, socleC22×C137D42
commutator, Frattini, cosocleC42C24×C137
prime radical#550×C137C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3395×C137C2semidirect
2. #3386×C137C2semidirect
5. #1722×C137C2semidirect
6. #1702×C137C2semidirect
7. #1694×C137C2semidirect
8. #1689×C137C2semidirect
9. #1666×C137C2semidirect
10. #1649×C137C2semidirect
11. #1648×C137C2semidirect
13. #1626×C137C2semidirect
14. #1593×C137C2semidirect
15. #1583×C137C2semidirect
229. Q8×C137#137semidirect
292. #20218C137direct (N × H)
293. C137#20218direct (N × H)
294. #3395C2×C137semidirect
295. #3386C2×C137semidirect
298. #1722C2×C137semidirect
299. #1702C2×C137semidirect
300. #1694C2×C137semidirect
301. #1689C2×C137semidirect
302. #1666C2×C137semidirect
303. #1649C2×C137semidirect
304. #1648C2×C137semidirect
306. #1626C2×C137semidirect
307. #1593C2×C137semidirect
308. #1583C2×C137semidirect
522. Q8#137×C137semidirect

Maximal subgroups

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

subgroupindex
#3395×C1372
#3386×C1372
#3368×C1372
#3278×C1372
#1722×C1372
#1702×C1372
#1694×C1372
#1689×C1372
#1666×C1372
#1649×C1372
#1648×C1372
#1641×C1372
#1626×C1372
#1593×C1372
#1583×C1372
#20218137