Finite Group Databasebeta

← C137×#18664 #46868050

Subgroups — C137×#18664 #46868050

Order 35072.

980 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501×C137C211yesmaximal, characteristic, complemented
2#3246×C137C211yesmaximal, characteristic, complemented
3#3167×C137C211yesmaximal, characteristic, complemented
4#3036×C137C211yesmaximal, characteristic
5#1651×C137C212yesmaximal, complemented
6#1643×C137C212yesmaximal, complemented
7#1627×C137C212yesmaximal, complemented
8#1622×C137C212yesmaximal, complemented
9#1550×C137C211yesmaximal, characteristic, complemented
10#1493×C137C211yesmaximal, characteristic, complemented
11C2×D4⋊C8×C137C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C137—21no—
13#554×C137—21no—
14#554×C137—22no—
15#554×C137—22no—
16#554×C137C2212yes—
17D4×Q8×C137—21no—
18D4×Q8×C137C2212yes—
19#544×C137—21no—
20#543×C137—22no—
21#533×C137C2211yescharacteristic
22#533×C137C2211yescharacteristic
23#532×C137C2211yescharacteristic
24C2×C22⋊Q8×C137—22no—
25#518×C137—21no—
26C2×C4×D4×C137C2211yescharacteristic
27#498×C137C2211yescharacteristic, complemented
28#492×C137C2211yescharacteristic, complemented
29#487×C137C2211yescharacteristic, complemented
30#486×C137C2211yescharacteristic, complemented
31#476×C137—21no—
32#475×C137C2212yescomplemented
33#465×C137—22no—
34#465×C137—22no—
35#463×C137—22no—
36#461×C137—21no—
37#461×C137C2212yes—
38#460×C137C2212yescomplemented
39#459×C137C2212yes—
40C4×SD16×C137—22no—
41#423×C137C2211yescharacteristic
42C2×C4⋊C8×C137C2211yescharacteristic
43#417×C137—21no—
44C2×C4×C8×C137C2211yescharacteristic
45#333×C137C2212yescomplemented
46#332×C137C2212yescomplemented
47Q8⋊C8×C137C2212yescomplemented
48Q8⋊C8×C137C2212yescomplemented
49D4⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
51C2×#41×C137—21no—
52C22×Q8×C137—22no—
53C22×D4×C137D411yescharacteristic
54C23×C4×C137—21no—
55AGL(1,ℤ/8)×C137—44no—
56C2×SD16×C137—22no—
57C2×SD16×C137—22no—
58C2×SD16×C137—22no—
59C2×D8×C137—22no—
60C2×M16×C137—21no—
61C22×C8×C137—21no—
62C22×C8×C137—21no—
63C4⋊Q8×C137—22no—
64C4⋊Q8×C137—22no—
65C4⋊Q8×C137D411yescharacteristic
66C4⋊Q8×C137D411yescharacteristic
67C4⋊Q8×C137C2311yescharacteristic
68C4⋊Q8×C137C2311yescharacteristic
69C4⋊Q8×C137C2311yescharacteristic
70#127×C137C2311yescharacteristic
71#125×C137—22no—
72#125×C137—22no—
73#125×C137—21no—
74#125×C137—21no—
75#124×C137—21no—
76C22⋊Q8×C137—24no—
77C22⋊Q8×C137—24no—
78C22⋊Q8×C137—24no—
79C22⋊Q8×C137—24no—
80C22⋊Q8×C137—22no—
81C22⋊Q8×C137—24no—
82C22⋊Q8×C137—21no—
83C22⋊Q8×C137—21no—
84C22⋊Q8×C137—22no—
85C22⋊Q8×C137—24no—
86C22⋊Q8×C137D412yes—
87C22⋊Q8×C137D412yes—
88#121×C137—21no—
89#121×C137—21no—
90C4×Q8×C137—21no—
91C4×Q8×C137—21no—
92C4×Q8×C137D412yes—
93C4×Q8×C137C2312yes—
94C4×D4×C137—22no—
95C4×D4×C137—22no—
96C4×D4×C137—22no—
97C4×D4×C137—22no—
98C4×D4×C137—21no—
99C4×D4×C137—21no—
100C4×D4×C137—21no—
101C4×D4×C137—21no—
102C4×D4×C137D412yes—
103C4×D4×C137C2312yes—
104#117×C137—21no—
105#117×C137—21no—
106C2×C4⋊C4×C137—24no—
107C2×C4⋊C4×C137—22no—
108C2×C4⋊C4×C137D411yescharacteristic
109C2×C22⋊C4×C137—24no—
110C2×C22⋊C4×C137—21no—
111C2×C42×C137C2311yescharacteristic
112#106×C137—22no—
113#106×C137—22no—
114C4⋊C8×C137—21no—
115C4⋊C8×C137—21no—
116C4⋊C8×C137C2312yes—
117C4⋊C8×C137C2312yes—
118#103×C137—22no—
119#103×C137—22no—
120#103×C137—22no—
121#102×C137—22no—
122#102×C137—22no—
123#102×C137—22no—
124C4×C8×C137—21no—
125C4×C8×C137C2311yescharacteristic
126C4×C8×C137C2311yescharacteristic
127C24×C137—21no—
128#41×C137—42no—
129#41×C137—42no—
130C2×Q8×C137—24no—
131C2×Q8×C137—21no—
132C2×Q8×C137—21no—
133C2×Q8×C137—21no—
134C2×Q8×C137—82no—
135C2×Q8×C137—82no—
136C2×Q8×C137C2×D412yes—
137C2×Q8×C137C2×D412yes—
138C2×D4×C137—21no—
139C2×D4×C137—22no—
140C2×D4×C137—21no—
141C2×D4×C137—21no—
142C2×D4×C137—21no—
143C2×D4×C137—41no—
144C2×D4×C137SD1614yescomplemented
145C2×D4×C137C2×D412yes—
146C22×C4×C137—24no—
147C22×C4×C137—24no—
148C22×C4×C137—21no—
149C22×C4×C137—22no—
150C22×C4×C137—22no—
151C22×C4×C137—41no—
152C22×C4×C137—81no—
153C22×C4×C137—81no—
154C22×C4×C137C2×D411yescharacteristic
155C22×C4×C137C2×D411yescharacteristic
156C22×C4×C137C2×D411yescharacteristic
157SD16×C137—44no—
158SD16×C137—44no—
159SD16×C137—82no—
160D8×C137—44no—
161M16×C137—42no—
162C2×C8×C137—21no—
163C2×C8×C137—21no—
164C2×C8×C137—22no—
165C2×C8×C137—22no—
166C2×C8×C137—42no—
167C2×C8×C137—81no—
168C4⋊C4×C137—24no—
169C4⋊C4×C137—24no—
170C4⋊C4×C137—24no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—22no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—22no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—21no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137C2×D412yes—
185C4⋊C4×C137C2×D412yes—
186C4⋊C4×C137C2×D412yes—
187C22⋊C4×C137—24no—
188C22⋊C4×C137—24no—
189C22⋊C4×C137—22no—
190C22⋊C4×C137—22no—
191C22⋊C4×C137—22no—
192C22⋊C4×C137—22no—
193C22⋊C4×C137—41no—
194C22⋊C4×C137—41no—
195C42×C137—22no—
196C42×C137—22no—
197C42×C137C2×D411yescharacteristic
198C42×C137C2×D411yescharacteristic
199C42×C137C2×D411yescharacteristic
200C42×C137C2411yescharacteristic
201C23×C137—22no—
202C23×C137—24no—
203C23×C137—41no—
204C23×C137—81no—
205C23×C137—81no—
206C23×C137#12011yescharacteristic
207Q8×C137—22no—
208Q8×C137—22no—
209Q8×C137—42no—
210Q8×C137—41no—
211Q8×C137—41no—
212Q8×C137—41no—
213Q8×C137—41no—
214Q8×C137—81no—
215Q8×C137—84no—
216D4×C137—22no—
217D4×C137—24no—
218D4×C137—42no—
219D4×C137—41no—
220D4×C137—41no—
221D4×C137—41no—
222D4×C137—41no—
223D4×C137—82no—
224D4×C137C2×SD1614yes—
225C2×C4×C137—21no—
226C2×C4×C137—21no—
227C2×C4×C137—22no—
228C2×C4×C137—22no—
229C2×C4×C137—22no—
230C2×C4×C137—22no—
231C2×C4×C137—22no—
232C2×C4×C137—41no—
233C2×C4×C137—42no—
234C2×C4×C137—42no—
235C2×C4×C137—41no—
236C2×C4×C137—82no—
237C2×C4×C137—81no—
238C2×C4×C137—84no—
239C2×C4×C137—84no—
240C2×C4×C137—82no—
241C2×C4×C137#12011yescharacteristic
242C2×C4×C137#12011yescharacteristic
243C2×C4×C137#12011yescharacteristic
244C2×C4×C137C2×SD1612yes—
245C2×C4×C137#13712yes—
246C2×C4×C137C22×D411yescharacteristic
247C2×C4×C137C22×D411yescharacteristic
248C2×C4×C137C22×D411yescharacteristic
249C8×C137—22no—
250C8×C137—41no—
251C8×C137—42no—
252C8×C137—42no—
253C22×C137—21no—
254C22×C137—21no—
255C22×C137—24no—
256C22×C137—42no—
257C22×C137—42no—
258C22×C137—84no—
259C22×C137—81no—
260C22×C137—81no—
261C22×C137—81no—
262C22×C137#44812yes—
263C22×C137C2×#12011yescharacteristic
264C4×C137—21no—
265C4×C137—21no—
266C4×C137—21no—
267C4×C137—42no—
268C4×C137—42no—
269C4×C137—42no—
270C4×C137—42no—
271C4×C137—42no—
272C4×C137—81no—
273C4×C137#44812yes—
274C4×C137C22×SD1611yescharacteristic
275C4×C137C2×#13711yescharacteristic
276C4×C137#57611yescharacteristic
277C4×C137#57711yescharacteristic
278C2×C137—22no—
279C2×C137—44no—
280C2×C137—161no—
281C2×C137C2×#44811yescharacteristic
282C2×C137#301211yescharacteristic
283C2×C137#301811yescharacteristic
284#18664C13711yesmaximal, characteristic, direct product factor, Sylow
285C137#1866411yescharacteristic, direct product factor, Sylow
286#3501C2×C13711yescharacteristic, complemented
287#3246C2×C13711yescharacteristic, complemented
288#3167C2×C13711yescharacteristic, complemented
289#3036C2×C13711yescharacteristic
290#1651C2×C13712yescomplemented
291#1643C2×C13712yescomplemented
292#1627C2×C13712yescomplemented
293#1622C2×C13712yescomplemented
294#1550C2×C13711yescharacteristic, complemented
295#1493C2×C13711yescharacteristic, complemented
296C2×D4⋊C8C2×C13711yescharacteristic, complemented
297C2×AGL(1,ℤ/8)—21no—
298#554—21no—
299#554—22no—
300#554—22no—
301#554C22×C13712yes—
302D4×Q8—21no—
303D4×Q8C22×C13712yes—
304#544—21no—
305#543—22no—
306#533C22×C13711yescharacteristic
307#533C22×C13711yescharacteristic
308#532C22×C13711yescharacteristic
309C2×C22⋊Q8—22no—
310#518—21no—
311C2×C4×D4C22×C13711yescharacteristic
312#498C22×C13711yescharacteristic, complemented
313#492C22×C13711yescharacteristic, complemented
314#487C22×C13711yescharacteristic, complemented
315#486C22×C13711yescharacteristic, complemented
316#476—21no—
317#475C22×C13712yescomplemented
318#465—22no—
319#465—22no—
320#463—22no—
321#461—21no—
322#461C22×C13712yes—
323#460C22×C13712yescomplemented
324#459C22×C13712yes—
325C4×SD16—22no—
326#423C22×C13711yescharacteristic
327C2×C4⋊C8C22×C13711yescharacteristic
328#417—21no—
329C2×C4×C8C22×C13711yescharacteristic
330#333C22×C13712yescomplemented
331#332C22×C13712yescomplemented
332Q8⋊C8C22×C13712yescomplemented
333Q8⋊C8C22×C13712yescomplemented
334D4⋊C8C22×C13712yes—
335D4⋊C8C22×C13712yes—
336C2×#41—21no—
337C22×Q8—22no—
338C22×D4D4×C13711yescharacteristic
339C23×C4—21no—
340AGL(1,ℤ/8)—44no—
341C2×SD16—22no—
342C2×SD16—22no—
343C2×SD16—22no—
344C2×D8—22no—
345C2×M16—21no—
346C22×C8—21no—
347C22×C8—21no—
348C4⋊Q8—22no—
349C4⋊Q8—22no—
350C4⋊Q8D4×C13711yescharacteristic
351C4⋊Q8D4×C13711yescharacteristic
352C4⋊Q8C23×C13711yescharacteristic
353C4⋊Q8C23×C13711yescharacteristic
354C4⋊Q8C23×C13711yescharacteristic
355#127C23×C13711yescharacteristic
356#125—22no—
357#125—22no—
358#125—21no—
359#125—21no—
360#124—21no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—24no—
371C22⋊Q8D4×C13712yes—
372C22⋊Q8D4×C13712yes—
373#121—21no—
374#121—21no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8D4×C13712yes—
378C4×Q8C23×C13712yes—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4D4×C13712yes—
388C4×D4C23×C13712yes—
389#117—21no—
390#117—21no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4D4×C13711yescharacteristic
394C2×C22⋊C4—24no—
395C2×C22⋊C4—21no—
396C2×C42C23×C13711yescharacteristic
397#106—22no—
398#106—22no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8C23×C13712yes—
402C4⋊C8C23×C13712yes—
403#103—22no—
404#103—22no—
405#103—22no—
406#102—22no—
407#102—22no—
408#102—22no—
409C4×C8—21no—
410C4×C8C23×C13711yescharacteristic
411C4×C8C23×C13711yescharacteristic
412C24—21no—
413#41—42no—
414#41—42no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8C2×D4×C13712yes—
422C2×Q8C2×D4×C13712yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4SD16×C13714yescomplemented
430C2×D4C2×D4×C13712yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4C2×D4×C13711yescharacteristic
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442SD16—44no—
443SD16—44no—
444SD16—82no—
445D8—44no—
446M16—42no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—42no—
452C2×C8—81no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4C2×D4×C13712yes—
470C4⋊C4C2×D4×C13712yes—
471C4⋊C4C2×D4×C13712yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C42—22no—
481C42—22no—
482C42C2×D4×C13711yescharacteristic
483C42C2×D4×C13711yescharacteristic
484C42C2×D4×C13711yescharacteristic
485C42C24×C13711yescharacteristic
486C23—22no—
487C23—24no—
488C23—41no—
489C23—81no—
490C23—81no—
491C23#120×C13711yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—81no—
500Q8—84no—
501D4—24no—
502D4—22no—
503D4—42no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—82no—
509D4C2×SD16×C13714yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4#120×C13711yescharacteristic
527C2×C4#120×C13711yescharacteristic
528C2×C4#120×C13711yescharacteristic
529C2×C4C2×SD16×C13712yes—
530C2×C4#137×C13712yes—
531C2×C4C22×D4×C13711yescharacteristic
532C2×C4C22×D4×C13711yescharacteristic
533C2×C4C22×D4×C13711yescharacteristic
534C8—22no—
535C8—41no—
536C8—42no—
537C8—42no—
538C22—21no—
539C22—21no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—84no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#448×C13712yes—
548C22C2×#120×C13711yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4#448×C13712yes—
559C4C22×SD16×C13711yescharacteristic
560C4C2×#137×C13711yescharacteristic
561C4#576×C13711yescharacteristic
562C4#577×C13711yescharacteristic
563C2—22no—
564C2—44no—
565C2—161no—
566C2C2×#448×C13711yescharacteristic
567C2#3012×C13711yescharacteristic
568C2#3018×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3036×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. #3501×C137C2semidirect
2. #3246×C137C2semidirect
3. #3167×C137C2semidirect
5. #1651×C137C2semidirect
6. #1643×C137C2semidirect
7. #1627×C137C2semidirect
8. #1622×C137C2semidirect
9. #1550×C137C2semidirect
10. #1493×C137C2semidirect
11. C2×D4⋊C8×C137C2semidirect
27. #498×C137C22semidirect
28. #492×C137C22semidirect
29. #487×C137C22semidirect
30. #486×C137C22semidirect
32. #475×C137C22semidirect
38. #460×C137C22semidirect
45. #333×C137C22semidirect
46. #332×C137C22semidirect
47. Q8⋊C8×C137C22semidirect
48. Q8⋊C8×C137C22semidirect
144. C2×D4×C137SD16semidirect
284. #18664C137direct (N × H)
285. C137#18664direct (N × H)
286. #3501C2×C137semidirect
287. #3246C2×C137semidirect
288. #3167C2×C137semidirect
290. #1651C2×C137semidirect
291. #1643C2×C137semidirect
292. #1627C2×C137semidirect
293. #1622C2×C137semidirect
294. #1550C2×C137semidirect
295. #1493C2×C137semidirect
296. C2×D4⋊C8C2×C137semidirect
312. #498C22×C137semidirect
313. #492C22×C137semidirect
314. #487C22×C137semidirect
315. #486C22×C137semidirect
317. #475C22×C137semidirect
323. #460C22×C137semidirect
330. #333C22×C137semidirect
331. #332C22×C137semidirect
332. Q8⋊C8C22×C137semidirect
333. Q8⋊C8C22×C137semidirect
429. C2×D4SD16×C137semidirect

Maximal subgroups

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

subgroupindex
#3501×C1372
#3246×C1372
#3167×C1372
#3036×C1372
#1651×C1372
#1643×C1372
#1627×C1372
#1622×C1372
#1550×C1372
#1493×C1372
C2×D4⋊C8×C1372
#18664137