Finite Group Databasebeta

← C137×#18816 #46868202

Subgroups — C137×#18816 #46868202

Order 35072.

918 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503×C137C211yesmaximal, characteristic, complemented
2#3216×C137C211yesmaximal, characteristic, complemented
3#3206×C137C211yesmaximal, characteristic, complemented
4#3115×C137C211yesmaximal, characteristic
5#1645×C137C212yesmaximal, complemented
6#1644×C137C212yesmaximal, complemented
7#1623×C137C212yesmaximal, complemented
8#1622×C137C212yesmaximal, complemented
9#1558×C137C211yesmaximal, characteristic, complemented
10#1497×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12Q82×C137C2212yes—
13#557×C137—21no—
14#554×C137—22no—
15#554×C137C2212yes—
16D4×Q8×C137—22no—
17D4×Q8×C137—21no—
18#544×C137—22no—
19#544×C137—21no—
20#533×C137—21no—
21#533×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23C2×C4⋊Q8×C137—21no—
24C2×C4⋊Q8×C137C2211yescharacteristic
25#520×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#497×C137—21no—
28#492×C137C2212yes—
29#489×C137—21no—
30#487×C137C2212yes—
31#482×C137—21no—
32#475×C137C2211yescharacteristic
33#474×C137C2211yescharacteristic
34#462×C137C2212yes—
35#461×C137—22no—
36#460×C137C2211yescharacteristic
37#460×C137C2212yes—
38#459×C137C2211yescharacteristic
39#452×C137—22no—
40#449×C137—21no—
41#448×C137—22no—
42#442×C137—21no—
43#440×C137—21no—
44#432×C137C2211yescharacteristic
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#333×C137C2212yes—
48#333×C137C2212yes—
49Q8⋊C8×C137C2212yes—
50Q8⋊C8×C137C2212yes—
51Q8⋊C8×C137C2212yes—
52D4⋊C8×C137C2212yes—
53C2×#41×C137—21no—
54C2×#41×C137D411yescharacteristic
55C22×Q8×C137—22no—
56C22×Q8×C137D412yes—
57C2×Q16×C137—22no—
58C2×SD16×C137—21no—
59C2×SD16×C137—24no—
60C2×SD16×C137—22no—
61C2×D8×C137—21no—
62C4⋊Q8×C137—22no—
63C4⋊Q8×C137—22no—
64C4⋊Q8×C137—24no—
65C4⋊Q8×C137—22no—
66C4⋊Q8×C137—22no—
67C4⋊Q8×C137—21no—
68C4⋊Q8×C137—21no—
69C4⋊Q8×C137D412yes—
70C4⋊Q8×C137D411yescharacteristic
71C4⋊Q8×C137D411yescharacteristic
72C4⋊Q8×C137C2311yescharacteristic
73C4⋊Q8×C137C2312yes—
74#127×C137C2311yescharacteristic
75#125×C137—22no—
76#125×C137—24no—
77#125×C137—22no—
78#125×C137—21no—
79#124×C137—21no—
80C22⋊Q8×C137—22no—
81C22⋊Q8×C137—24no—
82C22⋊Q8×C137—24no—
83C22⋊Q8×C137—22no—
84C22⋊Q8×C137—24no—
85C22⋊Q8×C137—21no—
86C22⋊Q8×C137D412yes—
87#121×C137—21no—
88#121×C137—21no—
89C4×Q8×C137—22no—
90C4×Q8×C137—22no—
91C4×Q8×C137—22no—
92C4×Q8×C137—21no—
93C4×Q8×C137D411yescharacteristic
94C4×Q8×C137C2312yes—
95C4×Q8×C137C2311yescharacteristic
96C4×D4×C137—24no—
97C4×D4×C137—22no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137—21no—
101C4×D4×C137—21no—
102C4×D4×C137D411yescharacteristic
103C4×D4×C137C2311yescharacteristic
104#117×C137—22no—
105#117×C137—21no—
106#117×C137—21no—
107#117×C137D411yescharacteristic
108C2×C4⋊C4×C137—22no—
109C2×C4⋊C4×C137—21no—
110C2×C4⋊C4×C137—21no—
111C2×C4⋊C4×C137—22no—
112C2×C42×C137—21no—
113C2×C42×C137C2311yescharacteristic
114#107×C137—21no—
115#106×C137—21no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2312yes—
119C4⋊C8×C137C2311yescharacteristic
120#103×C137—21no—
121#103×C137—24no—
122#103×C137—22no—
123#102×C137—21no—
124#102×C137—21no—
125#102×C137—21no—
126#102×C137—22no—
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—22no—
134#41×C137—22no—
135#41×C137—82no—
136C2×Q8×C137—24no—
137C2×Q8×C137—22no—
138C2×Q8×C137—21no—
139C2×Q8×C137—21no—
140C2×Q8×C137—84no—
141C2×Q8×C137—82no—
142C2×Q8×C137C2×D412yes—
143C2×Q8×C137C2×D411yescharacteristic
144C2×Q8×C137C2×D414yes—
145C2×D4×C137—21no—
146C2×D4×C137—21no—
147C2×D4×C137—21no—
148C2×D4×C137—21no—
149C2×D4×C137—41no—
150C2×D4×C137C2×D411yescharacteristic
151C22×C4×C137—21no—
152C22×C4×C137—24no—
153C22×C4×C137—22no—
154C22×C4×C137—21no—
155C22×C4×C137—21no—
156C22×C4×C137—41no—
157C22×C4×C137C2×D411yescharacteristic
158C22×C4×C137C2×D411yescharacteristic
159C22×C4×C137C2×D411yescharacteristic
160Q16×C137—82no—
161SD16×C137—42no—
162SD16×C137—82no—
163SD16×C137—84no—
164D8×C137—42no—
165C2×C8×C137—22no—
166C2×C8×C137—21no—
167C2×C8×C137—21no—
168C2×C8×C137—22no—
169C4⋊C4×C137—24no—
170C4⋊C4×C137—24no—
171C4⋊C4×C137—24no—
172C4⋊C4×C137—24no—
173C4⋊C4×C137—22no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—22no—
180C4⋊C4×C137—22no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—22no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137C2×D412yes—
187C4⋊C4×C137C2×D411yescharacteristic
188C4⋊C4×C137C2×D411yescharacteristic
189C22⋊C4×C137—24no—
190C22⋊C4×C137—22no—
191C22⋊C4×C137—21no—
192C22⋊C4×C137—21no—
193C22⋊C4×C137—21no—
194C22⋊C4×C137—41no—
195C22⋊C4×C137—41no—
196C42×C137—24no—
197C42×C137—21no—
198C42×C137—21no—
199C42×C137—21no—
200C42×C137—22no—
201C42×C137C2×D411yescharacteristic
202C42×C137C2×D411yescharacteristic
203C42×C137C2×D411yescharacteristic
204C42×C137C2411yescharacteristic
205C23×C137—21no—
206C23×C137—41no—
207C23×C137#12011yescharacteristic
208Q8×C137—21no—
209Q8×C137—22no—
210Q8×C137—22no—
211Q8×C137—44no—
212Q8×C137—44no—
213Q8×C137—42no—
214Q8×C137—42no—
215Q8×C137—84no—
216Q8×C137#13712yes—
217D4×C137—24no—
218D4×C137—21no—
219D4×C137—42no—
220D4×C137—41no—
221D4×C137—41no—
222D4×C137—42no—
223D4×C137—82no—
224D4×C137#13712yes—
225C2×C4×C137—24no—
226C2×C4×C137—21no—
227C2×C4×C137—21no—
228C2×C4×C137—21no—
229C2×C4×C137—22no—
230C2×C4×C137—22no—
231C2×C4×C137—21no—
232C2×C4×C137—21no—
233C2×C4×C137—24no—
234C2×C4×C137—22no—
235C2×C4×C137—41no—
236C2×C4×C137—42no—
237C2×C4×C137—42no—
238C2×C4×C137—41no—
239C2×C4×C137—82no—
240C2×C4×C137—82no—
241C2×C4×C137—84no—
242C2×C4×C137#12011yescharacteristic
243C2×C4×C137#12011yescharacteristic
244C2×C4×C137#12011yescharacteristic
245C2×C4×C137C22×D411yescharacteristic
246C2×C4×C137C22×D411yescharacteristic
247C2×C4×C137C22×D411yescharacteristic
248C8×C137—42no—
249C8×C137—41no—
250C8×C137—41no—
251C8×C137—42no—
252C22×C137—21no—
253C22×C137—22no—
254C22×C137—21no—
255C22×C137—21no—
256C22×C137—41no—
257C22×C137—41no—
258C22×C137—81no—
259C22×C137—81no—
260C22×C137—81no—
261C22×C137C2×#12011yescharacteristic
262C4×C137—21no—
263C4×C137—22no—
264C4×C137—21no—
265C4×C137—21no—
266C4×C137—21no—
267C4×C137—41no—
268C4×C137—42no—
269C4×C137—41no—
270C4×C137—44no—
271C4×C137—42no—
272C4×C137—81no—
273C4×C137C2×#13711yescharacteristic
274C4×C137C2×#13711yescharacteristic
275C4×C137#57712yes—
276C2×C137—41no—
277C2×C137—42no—
278C2×C137—161no—
279C2×C137#300411yescharacteristic
280C2×C137#301011yescharacteristic
281C2×C137#301911yescharacteristic
282#18816C13711yesmaximal, characteristic, direct product factor, Sylow
283C137#1881611yescharacteristic, direct product factor, Sylow
284#3503C2×C13711yescharacteristic, complemented
285#3216C2×C13711yescharacteristic, complemented
286#3206C2×C13711yescharacteristic, complemented
287#3115C2×C13711yescharacteristic
288#1645C2×C13712yescomplemented
289#1644C2×C13712yescomplemented
290#1623C2×C13712yescomplemented
291#1622C2×C13712yescomplemented
292#1558C2×C13711yescharacteristic, complemented
293#1497C2×C13711yescharacteristic, complemented
294#1488C2×C13711yescharacteristic, complemented
295Q82C22×C13712yes—
296#557—21no—
297#554—22no—
298#554C22×C13712yes—
299D4×Q8—22no—
300D4×Q8—21no—
301#544—22no—
302#544—21no—
303#533—21no—
304#533C22×C13711yescharacteristic
305#532C22×C13711yescharacteristic
306C2×C4⋊Q8—21no—
307C2×C4⋊Q8C22×C13711yescharacteristic
308#520—21no—
309C4×#41C22×C13711yescharacteristic
310#497—21no—
311#492C22×C13712yes—
312#489—21no—
313#487C22×C13712yes—
314#482—21no—
315#475C22×C13711yescharacteristic
316#474C22×C13711yescharacteristic
317#462C22×C13712yes—
318#461—22no—
319#460C22×C13711yescharacteristic
320#460C22×C13712yes—
321#459C22×C13711yescharacteristic
322#452—22no—
323#449—21no—
324#448—22no—
325#442—21no—
326#440—21no—
327#432C22×C13711yescharacteristic
328#432C22×C13711yescharacteristic
329#431C22×C13711yescharacteristic
330#333C22×C13712yes—
331#333C22×C13712yes—
332Q8⋊C8C22×C13712yes—
333Q8⋊C8C22×C13712yes—
334Q8⋊C8C22×C13712yes—
335D4⋊C8C22×C13712yes—
336C2×#41—21no—
337C2×#41D4×C13711yescharacteristic
338C22×Q8—22no—
339C22×Q8D4×C13712yes—
340C2×Q16—22no—
341C2×SD16—21no—
342C2×SD16—24no—
343C2×SD16—22no—
344C2×D8—21no—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347C4⋊Q8—24no—
348C4⋊Q8—22no—
349C4⋊Q8—22no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8D4×C13712yes—
353C4⋊Q8D4×C13711yescharacteristic
354C4⋊Q8D4×C13711yescharacteristic
355C4⋊Q8C23×C13711yescharacteristic
356C4⋊Q8C23×C13712yes—
357#127C23×C13711yescharacteristic
358#125—22no—
359#125—24no—
360#125—22no—
361#125—21no—
362#124—21no—
363C22⋊Q8—22no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366C22⋊Q8—22no—
367C22⋊Q8—24no—
368C22⋊Q8—21no—
369C22⋊Q8D4×C13712yes—
370#121—21no—
371#121—21no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×Q8—21no—
376C4×Q8D4×C13711yescharacteristic
377C4×Q8C23×C13711yescharacteristic
378C4×Q8C23×C13712yes—
379C4×D4—24no—
380C4×D4—22no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4D4×C13711yescharacteristic
386C4×D4C23×C13711yescharacteristic
387#117—22no—
388#117—21no—
389#117—21no—
390#117D4×C13711yescharacteristic
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C42—21no—
396C2×C42C23×C13711yescharacteristic
397#107—21no—
398#106—21no—
399C4⋊C8—21no—
400C4⋊C8C23×C13712yes—
401C4⋊C8C23×C13711yescharacteristic
402C4⋊C8C23×C13711yescharacteristic
403#103—21no—
404#103—24no—
405#103—22no—
406#102—21no—
407#102—21no—
408#102—21no—
409#102—22no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413#97—21no—
414#97—21no—
415C4×C8C23×C13712yes—
416#41—22no—
417#41—22no—
418#41—82no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—84no—
424C2×Q8—82no—
425C2×Q8C2×D4×C13712yes—
426C2×Q8C2×D4×C13711yescharacteristic
427C2×Q8C2×D4×C13714yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—41no—
433C2×D4C2×D4×C13711yescharacteristic
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—41no—
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442C22×C4C2×D4×C13711yescharacteristic
443Q16—82no—
444SD16—42no—
445SD16—82no—
446SD16—84no—
447D8—42no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4C2×D4×C13712yes—
470C4⋊C4C2×D4×C13711yescharacteristic
471C4⋊C4C2×D4×C13711yescharacteristic
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—24no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—22no—
484C42C2×D4×C13711yescharacteristic
485C42C2×D4×C13711yescharacteristic
486C42C2×D4×C13711yescharacteristic
487C42C24×C13711yescharacteristic
488C23—21no—
489C23—41no—
490C23#120×C13711yescharacteristic
491Q8—21no—
492Q8—22no—
493Q8—22no—
494Q8—44no—
495Q8—44no—
496Q8—42no—
497Q8—42no—
498Q8—84no—
499Q8#137×C13712yes—
500D4—24no—
501D4—21no—
502D4—42no—
503D4—41no—
504D4—41no—
505D4—42no—
506D4—82no—
507D4#137×C13712yes—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4#120×C13711yescharacteristic
526C2×C4#120×C13711yescharacteristic
527C2×C4#120×C13711yescharacteristic
528C2×C4C22×D4×C13711yescharacteristic
529C2×C4C22×D4×C13711yescharacteristic
530C2×C4C22×D4×C13711yescharacteristic
531C8—42no—
532C8—41no—
533C8—41no—
534C8—42no—
535C22—21no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C2×#120×C13711yescharacteristic
545C4—21no—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—41no—
551C4—42no—
552C4—41no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4C2×#137×C13711yescharacteristic
557C4C2×#137×C13711yescharacteristic
558C4#577×C13712yes—
559C2—41no—
560C2—42no—
561C2—161no—
562C2#3004×C13711yescharacteristic
563C2#3010×C13711yescharacteristic
564C2#3019×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. #3503×C137C2semidirect
2. #3216×C137C2semidirect
3. #3206×C137C2semidirect
5. #1645×C137C2semidirect
6. #1644×C137C2semidirect
7. #1623×C137C2semidirect
8. #1622×C137C2semidirect
9. #1558×C137C2semidirect
10. #1497×C137C2semidirect
11. #1488×C137C2semidirect
282. #18816C137direct (N × H)
283. C137#18816direct (N × H)
284. #3503C2×C137semidirect
285. #3216C2×C137semidirect
286. #3206C2×C137semidirect
288. #1645C2×C137semidirect
289. #1644C2×C137semidirect
290. #1623C2×C137semidirect
291. #1622C2×C137semidirect
292. #1558C2×C137semidirect
293. #1497C2×C137semidirect
294. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3503×C1372
#3216×C1372
#3206×C1372
#3115×C1372
#1645×C1372
#1644×C1372
#1623×C1372
#1622×C1372
#1558×C1372
#1497×C1372
#1488×C1372
#18816137