Finite Group Databasebeta

← C137×#18830 #46868216

Subgroups — C137×#18830 #46868216

Order 35072.

872 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509×C137C211yesmaximal, characteristic, complemented
2#3227×C137C211yesmaximal, characteristic, complemented
3#3216×C137C211yesmaximal, characteristic, complemented
4#3115×C137C211yesmaximal, characteristic
5#1653×C137C212yesmaximal, complemented
6#1652×C137C212yesmaximal, complemented
7#1628×C137C212yesmaximal, complemented
8#1626×C137C212yesmaximal, complemented
9#1556×C137C211yesmaximal, characteristic, complemented
10#1497×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12#557×C137—21no—
13#557×C137C2212yes—
14#556×C137—21no—
15#556×C137C2212yes—
16#552×C137—22no—
17#552×C137—22no—
18#541×C137—22no—
19#536×C137—21no—
20#533×C137—21no—
21#533×C137—21no—
22#533×C137C2211yescharacteristic
23#532×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic
25#519×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#498×C137C2212yes—
28#497×C137—21no—
29#489×C137—21no—
30#486×C137C2212yes—
31#482×C137—21no—
32#481×C137—22no—
33#476×C137—22no—
34#475×C137C2211yescharacteristic
35#474×C137C2211yescharacteristic
36#462×C137C2212yes—
37#460×C137C2211yescharacteristic
38#460×C137C2212yes—
39#459×C137C2211yescharacteristic
40#452×C137—22no—
41#449×C137—21no—
42#442×C137—21no—
43#440×C137—21no—
44#432×C137C2211yescharacteristic
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#332×C137C2212yes—
48#332×C137C2212yes—
49Q8⋊C8×C137C2212yes—
50Q8⋊C8×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52D4⋊C8×C137C2212yes—
53C2×#41×C137—21no—
54C2×#41×C137D411yescharacteristic
55C22×Q8×C137—21no—
56C2×Q16×C137—22no—
57C2×SD16×C137—21no—
58C2×SD16×C137—22no—
59C2×D8×C137—21no—
60C4⋊Q8×C137—22no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137D411yescharacteristic
64C4⋊Q8×C137D412yes—
65C4⋊Q8×C137D411yescharacteristic
66C4⋊Q8×C137C2311yescharacteristic
67C4⋊Q8×C137C2312yes—
68#127×C137C2311yescharacteristic
69#126×C137—24no—
70#126×C137—24no—
71#125×C137—22no—
72#125×C137—24no—
73#125×C137—22no—
74#125×C137—24no—
75#125×C137—21no—
76#124×C137—22no—
77#124×C137—21no—
78#123×C137—24no—
79C22⋊Q8×C137—22no—
80C22⋊Q8×C137—22no—
81C22⋊Q8×C137—22no—
82C22⋊Q8×C137—22no—
83C22⋊Q8×C137—21no—
84C22⋊Q8×C137D412yes—
85#121×C137—21no—
86#121×C137—21no—
87C4×Q8×C137—22no—
88C4×Q8×C137—22no—
89C4×Q8×C137—22no—
90C4×Q8×C137—22no—
91C4×Q8×C137—22no—
92C4×Q8×C137—22no—
93C4×Q8×C137—21no—
94C4×Q8×C137D411yescharacteristic
95C4×Q8×C137C2312yes—
96C4×Q8×C137C2311yescharacteristic
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137—21no—
101C4×D4×C137D411yescharacteristic
102C4×D4×C137C2311yescharacteristic
103#117×C137—24no—
104#117×C137—22no—
105#117×C137—21no—
106#117×C137—22no—
107#117×C137—21no—
108#117×C137D411yescharacteristic
109C2×C4⋊C4×C137—21no—
110C2×C4⋊C4×C137D412yes—
111C2×C42×C137—21no—
112C2×C42×C137C2311yescharacteristic
113#107×C137—21no—
114#106×C137—21no—
115#106×C137—24no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2312yes—
119C4⋊C8×C137C2311yescharacteristic
120#103×C137—21no—
121#103×C137—22no—
122#102×C137—21no—
123#102×C137—21no—
124#102×C137—21no—
125#102×C137—22no—
126#102×C137—24no—
127C22⋊C8×C137—21no—
128C22⋊C8×C137—21no—
129C22⋊C8×C137—21no—
130#97×C137—21no—
131#97×C137—21no—
132C4×C8×C137C2312yes—
133#41×C137—41no—
134#41×C137—41no—
135#41×C137—82no—
136C2×Q8×C137—21no—
137C2×Q8×C137—22no—
138C2×Q8×C137—22no—
139C2×Q8×C137—21no—
140C2×Q8×C137—42no—
141C2×Q8×C137—42no—
142C2×Q8×C137C2×D411yescharacteristic
143C2×Q8×C137C2×D412yes—
144C2×D4×C137—21no—
145C2×D4×C137—21no—
146C2×D4×C137—21no—
147C2×D4×C137—21no—
148C2×D4×C137—41no—
149C2×D4×C137C2×D411yescharacteristic
150C22×C4×C137—22no—
151C22×C4×C137—21no—
152C22×C4×C137—21no—
153C22×C4×C137—21no—
154C22×C4×C137—41no—
155C22×C4×C137C2×D411yescharacteristic
156C22×C4×C137C2×D411yescharacteristic
157C22×C4×C137C2×D411yescharacteristic
158Q16×C137—82no—
159SD16×C137—42no—
160SD16×C137—82no—
161D8×C137—42no—
162C2×C8×C137—22no—
163C2×C8×C137—21no—
164C2×C8×C137—21no—
165C2×C8×C137—22no—
166C4⋊C4×C137—24no—
167C4⋊C4×C137—24no—
168C4⋊C4×C137—24no—
169C4⋊C4×C137—24no—
170C4⋊C4×C137—22no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—22no—
177C4⋊C4×C137—22no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—22no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—21no—
182C4⋊C4×C137C2×D412yes—
183C4⋊C4×C137C2×D414yes—
184C4⋊C4×C137C2×D411yescharacteristic
185C4⋊C4×C137C2×D411yescharacteristic
186C22⋊C4×C137—24no—
187C22⋊C4×C137—24no—
188C22⋊C4×C137—21no—
189C22⋊C4×C137—22no—
190C22⋊C4×C137—21no—
191C22⋊C4×C137—21no—
192C22⋊C4×C137—41no—
193C22⋊C4×C137—41no—
194C42×C137—24no—
195C42×C137—24no—
196C42×C137—21no—
197C42×C137—21no—
198C42×C137—21no—
199C42×C137—22no—
200C42×C137C2×D411yescharacteristic
201C42×C137C2×D411yescharacteristic
202C42×C137C2×D411yescharacteristic
203C42×C137C2411yescharacteristic
204C23×C137—21no—
205C23×C137—41no—
206C23×C137#12011yescharacteristic
207Q8×C137—21no—
208Q8×C137—21no—
209Q8×C137—22no—
210Q8×C137—22no—
211Q8×C137—42no—
212Q8×C137—44no—
213Q8×C137—42no—
214Q8×C137—42no—
215Q8×C137—42no—
216D4×C137—21no—
217D4×C137—21no—
218D4×C137—42no—
219D4×C137—42no—
220D4×C137—41no—
221D4×C137—41no—
222D4×C137—42no—
223D4×C137—82no—
224C2×C4×C137—21no—
225C2×C4×C137—21no—
226C2×C4×C137—22no—
227C2×C4×C137—21no—
228C2×C4×C137—21no—
229C2×C4×C137—22no—
230C2×C4×C137—22no—
231C2×C4×C137—24no—
232C2×C4×C137—21no—
233C2×C4×C137—41no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—44no—
237C2×C4×C137—42no—
238C2×C4×C137—41no—
239C2×C4×C137—82no—
240C2×C4×C137#12011yescharacteristic
241C2×C4×C137#12011yescharacteristic
242C2×C4×C137#12011yescharacteristic
243C2×C4×C137C22×D411yescharacteristic
244C2×C4×C137C22×D411yescharacteristic
245C2×C4×C137C22×D411yescharacteristic
246C8×C137—42no—
247C8×C137—41no—
248C8×C137—41no—
249C8×C137—42no—
250C22×C137—21no—
251C22×C137—21no—
252C22×C137—21no—
253C22×C137—41no—
254C22×C137—41no—
255C22×C137—41no—
256C22×C137—81no—
257C22×C137—81no—
258C22×C137—81no—
259C22×C137C2×#12011yescharacteristic
260C4×C137—21no—
261C4×C137—21no—
262C4×C137—21no—
263C4×C137—21no—
264C4×C137—41no—
265C4×C137—44no—
266C4×C137—41no—
267C4×C137—42no—
268C4×C137—41no—
269C4×C137—42no—
270C4×C137—81no—
271C4×C137#57511yescharacteristic
272C4×C137#57511yescharacteristic
273C4×C137#57712yes—
274C2×C137—41no—
275C2×C137—81no—
276C2×C137—161no—
277C2×C137#300511yescharacteristic
278C2×C137#301011yescharacteristic
279C2×C137#302011yescharacteristic
280#18830C13711yesmaximal, characteristic, direct product factor, Sylow
281C137#1883011yescharacteristic, direct product factor, Sylow
282#3509C2×C13711yescharacteristic, complemented
283#3227C2×C13711yescharacteristic, complemented
284#3216C2×C13711yescharacteristic, complemented
285#3115C2×C13711yescharacteristic
286#1653C2×C13712yescomplemented
287#1652C2×C13712yescomplemented
288#1628C2×C13712yescomplemented
289#1626C2×C13712yescomplemented
290#1556C2×C13711yescharacteristic, complemented
291#1497C2×C13711yescharacteristic, complemented
292#1488C2×C13711yescharacteristic, complemented
293#557—21no—
294#557C22×C13712yes—
295#556—21no—
296#556C22×C13712yes—
297#552—22no—
298#552—22no—
299#541—22no—
300#536—21no—
301#533—21no—
302#533—21no—
303#533C22×C13711yescharacteristic
304#532C22×C13711yescharacteristic
305C2×C4⋊Q8C22×C13711yescharacteristic
306#519—21no—
307C4×#41C22×C13711yescharacteristic
308#498C22×C13712yes—
309#497—21no—
310#489—21no—
311#486C22×C13712yes—
312#482—21no—
313#481—22no—
314#476—22no—
315#475C22×C13711yescharacteristic
316#474C22×C13711yescharacteristic
317#462C22×C13712yes—
318#460C22×C13711yescharacteristic
319#460C22×C13712yes—
320#459C22×C13711yescharacteristic
321#452—22no—
322#449—21no—
323#442—21no—
324#440—21no—
325#432C22×C13711yescharacteristic
326#432C22×C13711yescharacteristic
327#431C22×C13711yescharacteristic
328#332C22×C13712yes—
329#332C22×C13712yes—
330Q8⋊C8C22×C13712yes—
331Q8⋊C8C22×C13712yes—
332Q8⋊C8C22×C13712yes—
333D4⋊C8C22×C13712yes—
334C2×#41—21no—
335C2×#41D4×C13711yescharacteristic
336C22×Q8—21no—
337C2×Q16—22no—
338C2×SD16—21no—
339C2×SD16—22no—
340C2×D8—21no—
341C4⋊Q8—22no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8D4×C13711yescharacteristic
345C4⋊Q8D4×C13712yes—
346C4⋊Q8D4×C13711yescharacteristic
347C4⋊Q8C23×C13711yescharacteristic
348C4⋊Q8C23×C13712yes—
349#127C23×C13711yescharacteristic
350#126—24no—
351#126—24no—
352#125—22no—
353#125—24no—
354#125—22no—
355#125—24no—
356#125—21no—
357#124—22no—
358#124—21no—
359#123—24no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—21no—
365C22⋊Q8D4×C13712yes—
366#121—21no—
367#121—21no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—21no—
375C4×Q8D4×C13711yescharacteristic
376C4×Q8C23×C13711yescharacteristic
377C4×Q8C23×C13712yes—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4D4×C13711yescharacteristic
383C4×D4C23×C13711yescharacteristic
384#117—24no—
385#117—22no—
386#117—21no—
387#117—22no—
388#117—21no—
389#117D4×C13711yescharacteristic
390C2×C4⋊C4—21no—
391C2×C4⋊C4D4×C13712yes—
392C2×C42—21no—
393C2×C42C23×C13711yescharacteristic
394#107—21no—
395#106—21no—
396#106—24no—
397C4⋊C8—21no—
398C4⋊C8C23×C13712yes—
399C4⋊C8C23×C13711yescharacteristic
400C4⋊C8C23×C13711yescharacteristic
401#103—21no—
402#103—22no—
403#102—21no—
404#102—21no—
405#102—21no—
406#102—22no—
407#102—24no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411#97—21no—
412#97—21no—
413C4×C8C23×C13712yes—
414#41—41no—
415#41—41no—
416#41—82no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8C2×D4×C13711yescharacteristic
424C2×Q8C2×D4×C13712yes—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—41no—
430C2×D4C2×D4×C13711yescharacteristic
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—41no—
436C22×C4C2×D4×C13711yescharacteristic
437C22×C4C2×D4×C13711yescharacteristic
438C22×C4C2×D4×C13711yescharacteristic
439Q16—82no—
440SD16—42no—
441SD16—82no—
442D8—42no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4C2×D4×C13712yes—
464C4⋊C4C2×D4×C13714yes—
465C4⋊C4C2×D4×C13711yescharacteristic
466C4⋊C4C2×D4×C13711yescharacteristic
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C42—24no—
476C42—24no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—22no—
481C42C2×D4×C13711yescharacteristic
482C42C2×D4×C13711yescharacteristic
483C42C2×D4×C13711yescharacteristic
484C42C24×C13711yescharacteristic
485C23—21no—
486C23—41no—
487C23#120×C13711yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—22no—
491Q8—22no—
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497D4—21no—
498D4—21no—
499D4—42no—
500D4—42no—
501D4—41no—
502D4—41no—
503D4—42no—
504D4—82no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4#120×C13711yescharacteristic
522C2×C4#120×C13711yescharacteristic
523C2×C4#120×C13711yescharacteristic
524C2×C4C22×D4×C13711yescharacteristic
525C2×C4C22×D4×C13711yescharacteristic
526C2×C4C22×D4×C13711yescharacteristic
527C8—42no—
528C8—41no—
529C8—41no—
530C8—42no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C2×#120×C13711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—44no—
547C4—41no—
548C4—42no—
549C4—41no—
550C4—42no—
551C4—81no—
552C4#575×C13711yescharacteristic
553C4#575×C13711yescharacteristic
554C4#577×C13712yes—
555C2—41no—
556C2—81no—
557C2—161no—
558C2#3005×C13711yescharacteristic
559C2#3010×C13711yescharacteristic
560C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3115×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
1. #3509×C137C2semidirect
2. #3227×C137C2semidirect
3. #3216×C137C2semidirect
5. #1653×C137C2semidirect
6. #1652×C137C2semidirect
7. #1628×C137C2semidirect
8. #1626×C137C2semidirect
9. #1556×C137C2semidirect
10. #1497×C137C2semidirect
11. #1488×C137C2semidirect
280. #18830C137direct (N × H)
281. C137#18830direct (N × H)
282. #3509C2×C137semidirect
283. #3227C2×C137semidirect
284. #3216C2×C137semidirect
286. #1653C2×C137semidirect
287. #1652C2×C137semidirect
288. #1628C2×C137semidirect
289. #1626C2×C137semidirect
290. #1556C2×C137semidirect
291. #1497C2×C137semidirect
292. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3509×C1372
#3227×C1372
#3216×C1372
#3115×C1372
#1653×C1372
#1652×C1372
#1628×C1372
#1626×C1372
#1556×C1372
#1497×C1372
#1488×C1372
#18830137