Finite Group Databasebeta

← C137×#20318 #46869704

Subgroups — C137×#20318 #46869704

Order 35072.

556 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3387×C137C211yesmaximal, characteristic, complemented
2#3372×C137C211yesmaximal, characteristic
3#3369×C137C211yesmaximal, characteristic
4#3356×C137C211yesmaximal, characteristic
5#1707×C137C211yesmaximal, characteristic, complemented
6#1704×C137C211yesmaximal, characteristic, complemented
7#1704×C137C211yesmaximal, characteristic, complemented
8#1692×C137C211yesmaximal, characteristic, complemented
9#1691×C137C211yesmaximal, characteristic
10#1650×C137C211yesmaximal, characteristic
11#1648×C137C211yesmaximal, characteristic, complemented
12#1633×C137C211yesmaximal, characteristic, complemented
13#1632×C137C211yesmaximal, characteristic
14#1591×C137C211yesmaximal, characteristic, complemented
15#1590×C137C211yesmaximal, characteristic
16Q82×C137C2211yescharacteristic
17#557×C137C2211yescharacteristic
18#556×C137C2211yescharacteristic
19#554×C137C2211yescharacteristic
20#501×C137—21no—
21#500×C137C2211yescharacteristic
22#498×C137C2211yescharacteristic
23#498×C137C2211yescharacteristic
24#494×C137C2211yescharacteristic
25#491×C137—21no—
26#489×C137—21no—
27#487×C137C2211yescharacteristic
28#487×C137C2211yescharacteristic
29#485×C137—21no—
30#483×C137—21no—
31#479×C137—21no—
32#479×C137—21no—
33#477×C137C2211yescharacteristic
34#477×C137C2211yescharacteristic
35#477×C137C2211yescharacteristic
36#476×C137C2211yescharacteristic
37#475×C137—21no—
38#475×C137—21no—
39#475×C137C2211yescharacteristic
40#475×C137C2211yescharacteristic
41#470×C137—21no—
42#464×C137—21no—
43#462×C137C2211yescharacteristic
44#461×C137—21no—
45#461×C137C2211yescharacteristic
46#446×C137C2211yescharacteristic
47#446×C137C2211yescharacteristic
48#446×C137C2211yescharacteristic
49#441×C137—21no—
50#441×C137—21no—
51#440×C137—21no—
52C4×Q16×C137C2211yescharacteristic
53C4×SD16×C137C2211yescharacteristic
54#436×C137C2211yescharacteristic
55#428×C137—21no—
56#335×C137C2211yescharacteristic
57#334×C137C2211yescharacteristic
58#333×C137C2211yescharacteristic
59#333×C137C2211yescharacteristic
60#332×C137C2211yescharacteristic
61#332×C137C2211yescharacteristic
62Q8⋊C8×C137C2211yescharacteristic
63Q8⋊C8×C137C2211yescharacteristic
64Q8⋊C8×C137C2211yescharacteristic
65D4⋊C8×C137C2211yescharacteristic
66C82×C137C2211yescharacteristic
67C2×Q16×C137—21no—
68C2×Q16×C137—41no—
69C2×Q16×C137D411yescharacteristic
70C2×SD16×C137—21no—
71C2×SD16×C137D411yescharacteristic
72C2×M16×C137—21no—
73C4⋊Q8×C137—21no—
74C4⋊Q8×C137—21no—
75C4⋊Q8×C137—21no—
76C4⋊Q8×C137—41no—
77C4⋊Q8×C137C2311yescharacteristic
78C4⋊Q8×C137C2311yescharacteristic
79C4⋊Q8×C137C2311yescharacteristic
80C4⋊Q8×C137C2311yescharacteristic
81#126×C137—41no—
82#125×C137—21no—
83#125×C137—21no—
84#125×C137—41no—
85#124×C137—21no—
86C22⋊Q8×C137—21no—
87C22⋊Q8×C137—21no—
88C22⋊Q8×C137—41no—
89C4×Q8×C137—21no—
90C4×Q8×C137—21no—
91C4×Q8×C137—21no—
92C4×Q8×C137—21no—
93C4×Q8×C137—21no—
94C4×Q8×C137C2311yescharacteristic
95C4×Q8×C137C2311yescharacteristic
96C4×Q8×C137C2311yescharacteristic
97C4×D4×C137—21no—
98C4×D4×C137C2311yescharacteristic
99#117×C137—21no—
100C2×C4⋊C4×C137—21no—
101#107×C137—21no—
102#107×C137—21no—
103#107×C137—21no—
104#107×C137—41no—
105#107×C137D411yescharacteristic
106#106×C137—21no—
107#106×C137—21no—
108#106×C137—21no—
109#106×C137—21no—
110#106×C137—21no—
111#106×C137—21no—
112#106×C137—21no—
113#106×C137D411yescharacteristic
114C4⋊C8×C137—21no—
115C4⋊C8×C137—21no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2311yescharacteristic
119C4⋊C8×C137C2311yescharacteristic
120C4⋊C8×C137C2311yescharacteristic
121#103×C137—21no—
122#103×C137—21no—
123#103×C137—21no—
124#103×C137—21no—
125#103×C137—21no—
126#103×C137—21no—
127#103×C137—21no—
128#103×C137—21no—
129#103×C137—21no—
130#103×C137—21no—
131#103×C137—21no—
132#103×C137D411yescharacteristic
133#103×C137D411yescharacteristic
134#103×C137D411yescharacteristic
135#102×C137—21no—
136#102×C137D411yescharacteristic
137C22⋊C8×C137—21no—
138#97×C137—21no—
139#97×C137—21no—
140#97×C137—21no—
141C4×C8×C137C2311yescharacteristic
142C4×C8×C137C2311yescharacteristic
143C4×C8×C137C2311yescharacteristic
144C2×Q8×C137—21no—
145C2×Q8×C137—21no—
146C2×Q8×C137—21no—
147C2×Q8×C137—21no—
148C2×Q8×C137—41no—
149C2×Q8×C137C2×D411yescharacteristic
150C2×Q8×C137C2×D411yescharacteristic
151C2×Q8×C137C2×D411yescharacteristic
152C2×D4×C137C2×D411yescharacteristic
153C22×C4×C137—21no—
154C22×C4×C137—41no—
155Q16×C137—22no—
156Q16×C137—42no—
157Q16×C137—82no—
158SD16×C137—41no—
159SD16×C137—42no—
160M16×C137—42no—
161C2×C8×C137—21no—
162C2×C8×C137—21no—
163C2×C8×C137—21no—
164C2×C8×C137—21no—
165C2×C8×C137—21no—
166C2×C8×C137C2×D411yescharacteristic
167C2×C8×C137C2×D411yescharacteristic
168C2×C8×C137C2×D411yescharacteristic
169C2×C8×C137C2×D411yescharacteristic
170C4⋊C4×C137—21no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—41no—
182C4⋊C4×C137—41no—
183C4⋊C4×C137—41no—
184C4⋊C4×C137—41no—
185C4⋊C4×C137—41no—
186C4⋊C4×C137—41no—
187C4⋊C4×C137—41no—
188C4⋊C4×C137—41no—
189C4⋊C4×C137C2×D411yescharacteristic
190C4⋊C4×C137C2×D411yescharacteristic
191C4⋊C4×C137C2×D411yescharacteristic
192C4⋊C4×C137C2×D411yescharacteristic
193C22⋊C4×C137—21no—
194C22⋊C4×C137—41no—
195C22⋊C4×C137—41no—
196C22⋊C4×C137—41no—
197C42×C137—21no—
198C42×C137—21no—
199C42×C137—21no—
200C42×C137—41no—
201C42×C137—41no—
202C42×C137—41no—
203C42×C137C2411yescharacteristic
204C23×C137—21no—
205Q8×C137—21no—
206Q8×C137—21no—
207Q8×C137—21no—
208Q8×C137—21no—
209Q8×C137—21no—
210Q8×C137—21no—
211Q8×C137—41no—
212Q8×C137—41no—
213Q8×C137—41no—
214Q8×C137—41no—
215Q8×C137—81no—
216Q8×C137—81no—
217Q8×C137—161no—
218D4×C137—21no—
219D4×C137—21no—
220C2×C4×C137—21no—
221C2×C4×C137—21no—
222C2×C4×C137—21no—
223C2×C4×C137—21no—
224C2×C4×C137—21no—
225C2×C4×C137—21no—
226C2×C4×C137—21no—
227C2×C4×C137—41no—
228C2×C4×C137—41no—
229C2×C4×C137—41no—
230C2×C4×C137—41no—
231C2×C4×C137—41no—
232C2×C4×C137—41no—
233C2×C4×C137—161no—
234C2×C4×C137C22×D411yescharacteristic
235C2×C4×C137C22×D411yescharacteristic
236C2×C4×C137H(5,𝔽2)11yescharacteristic
237C8×C137—22no—
238C8×C137—21no—
239C8×C137—21no—
240C8×C137—21no—
241C8×C137—41no—
242C8×C137—41no—
243C8×C137—41no—
244C8×C137—41no—
245C8×C137#13712yes—
246C22×C137—41no—
247C22×C137—41no—
248C22×C137—41no—
249C22×C137D4211yescharacteristic
250C4×C137—21no—
251C4×C137—41no—
252C4×C137—41no—
253C4×C137—41no—
254C4×C137—41no—
255C4×C137—41no—
256C4×C137—41no—
257C4×C137—41no—
258C4×C137—81no—
259C4×C137—81no—
260C4×C137—81no—
261C4×C137—81no—
262C4×C137C2×#13711yescharacteristic
263C4×C137#57511yescharacteristic
264C4×C137#57811yescharacteristic
265C4×C137#57811yescharacteristic
266C2×C137—81no—
267C2×C137#327611yescharacteristic
268C2×C137#327711yescharacteristic
269C2×C137#329111yescharacteristic
270#20318C13711yesmaximal, characteristic, direct product factor, Sylow
271C137#2031811yescharacteristic, direct product factor, Sylow
272#3387C2×C13711yescharacteristic, complemented
273#3372C2×C13711yescharacteristic
274#3369C2×C13711yescharacteristic
275#3356C2×C13711yescharacteristic
276#1707C2×C13711yescharacteristic, complemented
277#1704C2×C13711yescharacteristic, complemented
278#1704C2×C13711yescharacteristic, complemented
279#1692C2×C13711yescharacteristic, complemented
280#1691C2×C13711yescharacteristic
281#1650C2×C13711yescharacteristic
282#1648C2×C13711yescharacteristic, complemented
283#1633C2×C13711yescharacteristic, complemented
284#1632C2×C13711yescharacteristic
285#1591C2×C13711yescharacteristic, complemented
286#1590C2×C13711yescharacteristic
287Q82C22×C13711yescharacteristic
288#557C22×C13711yescharacteristic
289#556C22×C13711yescharacteristic
290#554C22×C13711yescharacteristic
291#501—21no—
292#500C22×C13711yescharacteristic
293#498C22×C13711yescharacteristic
294#498C22×C13711yescharacteristic
295#494C22×C13711yescharacteristic
296#491—21no—
297#489—21no—
298#487C22×C13711yescharacteristic
299#487C22×C13711yescharacteristic
300#485—21no—
301#483—21no—
302#479—21no—
303#479—21no—
304#477C22×C13711yescharacteristic
305#477C22×C13711yescharacteristic
306#477C22×C13711yescharacteristic
307#476C22×C13711yescharacteristic
308#475—21no—
309#475—21no—
310#475C22×C13711yescharacteristic
311#475C22×C13711yescharacteristic
312#470—21no—
313#464—21no—
314#462C22×C13711yescharacteristic
315#461—21no—
316#461C22×C13711yescharacteristic
317#446C22×C13711yescharacteristic
318#446C22×C13711yescharacteristic
319#446C22×C13711yescharacteristic
320#441—21no—
321#441—21no—
322#440—21no—
323C4×Q16C22×C13711yescharacteristic
324C4×SD16C22×C13711yescharacteristic
325#436C22×C13711yescharacteristic
326#428—21no—
327#335C22×C13711yescharacteristic
328#334C22×C13711yescharacteristic
329#333C22×C13711yescharacteristic
330#333C22×C13711yescharacteristic
331#332C22×C13711yescharacteristic
332#332C22×C13711yescharacteristic
333Q8⋊C8C22×C13711yescharacteristic
334Q8⋊C8C22×C13711yescharacteristic
335Q8⋊C8C22×C13711yescharacteristic
336D4⋊C8C22×C13711yescharacteristic
337C82C22×C13711yescharacteristic
338C2×Q16—21no—
339C2×Q16—41no—
340C2×Q16D4×C13711yescharacteristic
341C2×SD16—21no—
342C2×SD16D4×C13711yescharacteristic
343C2×M16—21no—
344C4⋊Q8—21no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—41no—
348C4⋊Q8C23×C13711yescharacteristic
349C4⋊Q8C23×C13711yescharacteristic
350C4⋊Q8C23×C13711yescharacteristic
351C4⋊Q8C23×C13711yescharacteristic
352#126—41no—
353#125—21no—
354#125—21no—
355#125—41no—
356#124—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—41no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8C23×C13711yescharacteristic
366C4×Q8C23×C13711yescharacteristic
367C4×Q8C23×C13711yescharacteristic
368C4×D4—21no—
369C4×D4C23×C13711yescharacteristic
370#117—21no—
371C2×C4⋊C4—21no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—41no—
376#107D4×C13711yescharacteristic
377#106—21no—
378#106—21no—
379#106—21no—
380#106—21no—
381#106—21no—
382#106—21no—
383#106—21no—
384#106D4×C13711yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8C23×C13711yescharacteristic
389C4⋊C8C23×C13711yescharacteristic
390C4⋊C8C23×C13711yescharacteristic
391C4⋊C8C23×C13711yescharacteristic
392#103—21no—
393#103—21no—
394#103—21no—
395#103—21no—
396#103—21no—
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103D4×C13711yescharacteristic
404#103D4×C13711yescharacteristic
405#103D4×C13711yescharacteristic
406#102—21no—
407#102D4×C13711yescharacteristic
408C22⋊C8—21no—
409#97—21no—
410#97—21no—
411#97—21no—
412C4×C8C23×C13711yescharacteristic
413C4×C8C23×C13711yescharacteristic
414C4×C8C23×C13711yescharacteristic
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—41no—
420C2×Q8C2×D4×C13711yescharacteristic
421C2×Q8C2×D4×C13711yescharacteristic
422C2×Q8C2×D4×C13711yescharacteristic
423C2×D4C2×D4×C13711yescharacteristic
424C22×C4—21no—
425C22×C4—41no—
426Q16—22no—
427Q16—42no—
428Q16—82no—
429SD16—41no—
430SD16—42no—
431M16—42no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8C2×D4×C13711yescharacteristic
438C2×C8C2×D4×C13711yescharacteristic
439C2×C8C2×D4×C13711yescharacteristic
440C2×C8C2×D4×C13711yescharacteristic
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×D4×C13711yescharacteristic
461C4⋊C4C2×D4×C13711yescharacteristic
462C4⋊C4C2×D4×C13711yescharacteristic
463C4⋊C4C2×D4×C13711yescharacteristic
464C22⋊C4—21no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42C24×C13711yescharacteristic
475C23—21no—
476Q8—21no—
477Q8—21no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—81no—
488Q8—161no—
489D4—21no—
490D4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—161no—
505C2×C4C22×D4×C13711yescharacteristic
506C2×C4C22×D4×C13711yescharacteristic
507C2×C4H(5,𝔽2)×C13711yescharacteristic
508C8—22no—
509C8—21no—
510C8—21no—
511C8—21no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8#137×C13712yes—
517C22—41no—
518C22—41no—
519C22—41no—
520C22D42×C13711yescharacteristic
521C4—21no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4C2×#137×C13711yescharacteristic
534C4#575×C13711yescharacteristic
535C4#578×C13711yescharacteristic
536C4#578×C13711yescharacteristic
537C2—81no—
538C2#3276×C13711yescharacteristic
539C2#3277×C13711yescharacteristic
540C2#3291×C13711yescharacteristic
subgroupG/H
center, socleC22×C137D42
commutator, Frattini, cosocleC42C24×C137
prime radicalC2×D4×C137C2×D4
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. #3387×C137C2semidirect
5. #1707×C137C2semidirect
6. #1704×C137C2semidirect
7. #1704×C137C2semidirect
8. #1692×C137C2semidirect
11. #1648×C137C2semidirect
12. #1633×C137C2semidirect
14. #1591×C137C2semidirect
270. #20318C137direct (N × H)
271. C137#20318direct (N × H)
272. #3387C2×C137semidirect
276. #1707C2×C137semidirect
277. #1704C2×C137semidirect
278. #1704C2×C137semidirect
279. #1692C2×C137semidirect
282. #1648C2×C137semidirect
283. #1633C2×C137semidirect
285. #1591C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3387×C1372
#3372×C1372
#3369×C1372
#3356×C1372
#1707×C1372
#1704×C1372
#1692×C1372
#1691×C1372
#1650×C1372
#1648×C1372
#1633×C1372
#1632×C1372
#1591×C1372
#1590×C1372
#20318137