Finite Group Databasebeta

← C137×#18616 #46868002

Subgroups — C137×#18616 #46868002

Order 35072.

964 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506×C137C211yesmaximal, characteristic, complemented
2#3214×C137C211yesmaximal, characteristic, complemented
3#3182×C137C211yesmaximal, characteristic, complemented
4#3100×C137C211yesmaximal, characteristic, complemented
5#1647×C137C212yesmaximal, complemented
6#1646×C137C212yesmaximal, complemented
7#1625×C137C212yesmaximal, complemented
8#1624×C137C212yesmaximal, complemented
9#1554×C137C211yesmaximal, characteristic, complemented
10#1490×C137C211yesmaximal, characteristic, complemented
11#1484×C137C211yesmaximal, characteristic, complemented
12#559×C137—22no—
13#556×C137—21no—
14#555×C137C2212yes—
15#553×C137—22no—
16#550×C137—21no—
17#547×C137C2212yes—
18#538×C137—22no—
19#535×C137—21no—
20#533×C137C2211yescharacteristic
21#532×C137C2211yescharacteristic
22#532×C137C2211yescharacteristic
23#525×C137—22no—
24#520×C137—21no—
25C2×C4×D4×C137C2211yescharacteristic
26#500×C137C2211yescharacteristic, complemented
27#496×C137—21no—
28#492×C137C2211yescharacteristic, complemented
29#489×C137—21no—
30#487×C137C2211yescharacteristic, complemented
31#486×C137C2211yescharacteristic, complemented
32#480×C137—21no—
33#474×C137C2212yes—
34#468×C137—22no—
35#461×C137C2212yescomplemented
36#459×C137C2212yes—
37#459×C137C2212yescomplemented
38#449×C137—22no—
39#447×C137—21no—
40#440×C137—22no—
41C4×D8×C137—22no—
42#432×C137C2211yescharacteristic
43#431×C137C2211yescharacteristic
44C4×M16×C137C2211yescharacteristic
45#333×C137C2212yescomplemented
46#332×C137C2212yescomplemented
47D4⋊C8×C137C2212yes—
48D4⋊C8×C137C2212yes—
49D4⋊C8×C137C2212yescomplemented
50D4⋊C8×C137C2212yescomplemented
51C2×#41×C137—22no—
52C2×#41×C137—21no—
53C2×#41×C137—21no—
54C22×D4×C137D411yescharacteristic
55C23×C4×C137—21no—
56C2×SD16×C137—22no—
57C2×SD16×C137—22no—
58C2×D8×C137—22no—
59C2×D8×C137—22no—
60C2×M16×C137—21no—
61C4⋊Q8×C137D411yescharacteristic
62C4⋊Q8×C137C2311yescharacteristic
63C4⋊Q8×C137C2311yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65#127×C137—22no—
66#127×C137D411yescharacteristic
67#127×C137C2311yescharacteristic
68#126×C137—24no—
69#125×C137—21no—
70#124×C137—22no—
71#124×C137—24no—
72#124×C137—21no—
73#124×C137—21no—
74#123×C137—24no—
75#123×C137—24no—
76C22⋊Q8×C137—21no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137D412yes—
79#121×C137—24no—
80#121×C137—22no—
81#121×C137—22no—
82#121×C137—22no—
83#121×C137—21no—
84#121×C137—21no—
85#121×C137—21no—
86#121×C137D412yes—
87#120×C137—24no—
88C4×Q8×C137—22no—
89C4×Q8×C137—21no—
90C4×Q8×C137D412yes—
91C4×D4×C137—24no—
92C4×D4×C137—22no—
93C4×D4×C137—24no—
94C4×D4×C137—21no—
95C4×D4×C137—21no—
96C4×D4×C137—21no—
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137D412yes—
100C4×D4×C137C2312yes—
101C4×D4×C137C2312yes—
102#117×C137—21no—
103#117×C137—22no—
104#117×C137—22no—
105C2×C4⋊C4×C137—22no—
106C2×C4⋊C4×C137D411yescharacteristic
107C2×C22⋊C4×C137—21no—
108C2×C42×C137C2311yescharacteristic
109#107×C137—22no—
110#107×C137—22no—
111C4⋊C8×C137C2312yes—
112C4⋊C8×C137C2312yes—
113#103×C137—22no—
114#103×C137—22no—
115#102×C137—22no—
116#102×C137—22no—
117#102×C137—22no—
118#102×C137—22no—
119C22⋊C8×C137—21no—
120C22⋊C8×C137—21no—
121#97×C137—21no—
122#97×C137—21no—
123C4×C8×C137—21no—
124C4×C8×C137C2311yescharacteristic
125C4×C8×C137C2311yescharacteristic
126C24×C137—21no—
127#41×C137—44no—
128#41×C137—82no—
129#41×C137—82no—
130#41×C137—82no—
131C2×Q8×C137—21no—
132C2×Q8×C137—21no—
133C2×Q8×C137—21no—
134C2×Q8×C137C2×D412yes—
135C2×D4×C137—22no—
136C2×D4×C137—24no—
137C2×D4×C137—24no—
138C2×D4×C137—21no—
139C2×D4×C137—21no—
140C2×D4×C137—21no—
141C2×D4×C137—21no—
142C2×D4×C137—21no—
143C2×D4×C137—21no—
144C2×D4×C137—42no—
145C2×D4×C137—41no—
146C2×D4×C137C2×D412yes—
147C2×D4×C137C2×D412yes—
148C22×C4×C137—24no—
149C22×C4×C137—21no—
150C22×C4×C137—22no—
151C22×C4×C137—22no—
152C22×C4×C137—22no—
153C22×C4×C137—41no—
154C22×C4×C137—82no—
155C22×C4×C137C2×D411yescharacteristic
156C22×C4×C137C2×D411yescharacteristic
157C22×C4×C137C2×D411yescharacteristic
158SD16×C137—44no—
159SD16×C137—82no—
160D8×C137—44no—
161D8×C137—82no—
162M16×C137—42no—
163C2×C8×C137—21no—
164C2×C8×C137—22no—
165C2×C8×C137—21no—
166C2×C8×C137—22no—
167C4⋊C4×C137—24no—
168C4⋊C4×C137—21no—
169C4⋊C4×C137—22no—
170C4⋊C4×C137—21no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137C2×D412yes—
178C4⋊C4×C137C2×D412yes—
179C4⋊C4×C137C2×D412yes—
180C22⋊C4×C137—24no—
181C22⋊C4×C137—24no—
182C22⋊C4×C137—24no—
183C22⋊C4×C137—22no—
184C22⋊C4×C137—22no—
185C22⋊C4×C137—22no—
186C22⋊C4×C137—22no—
187C22⋊C4×C137—22no—
188C22⋊C4×C137—41no—
189C22⋊C4×C137—41no—
190C42×C137—24no—
191C42×C137—22no—
192C42×C137C2×D411yescharacteristic
193C42×C137C2×D411yescharacteristic
194C42×C137C2×D411yescharacteristic
195C42×C137C2411yescharacteristic
196C23×C137—22no—
197C23×C137—22no—
198C23×C137—42no—
199C23×C137—41no—
200C23×C137—81no—
201C23×C137—81no—
202C23×C137#12011yescharacteristic
203Q8×C137—42no—
204Q8×C137—42no—
205Q8×C137—42no—
206Q8×C137—42no—
207D4×C137—22no—
208D4×C137—22no—
209D4×C137—22no—
210D4×C137—22no—
211D4×C137—42no—
212D4×C137—42no—
213D4×C137—42no—
214D4×C137—44no—
215D4×C137—42no—
216D4×C137—42no—
217D4×C137—81no—
218D4×C137—84no—
219D4×C137—84no—
220D4×C137—82no—
221C2×C4×C137—22no—
222C2×C4×C137—22no—
223C2×C4×C137—22no—
224C2×C4×C137—22no—
225C2×C4×C137—22no—
226C2×C4×C137—22no—
227C2×C4×C137—41no—
228C2×C4×C137—44no—
229C2×C4×C137—42no—
230C2×C4×C137—42no—
231C2×C4×C137—41no—
232C2×C4×C137—82no—
233C2×C4×C137—82no—
234C2×C4×C137—81no—
235C2×C4×C137—84no—
236C2×C4×C137#12011yescharacteristic
237C2×C4×C137#12011yescharacteristic
238C2×C4×C137#12011yescharacteristic
239C2×C4×C137C22×D411yescharacteristic
240C2×C4×C137C22×D411yescharacteristic
241C2×C4×C137C22×D411yescharacteristic
242C8×C137—22no—
243C8×C137—41no—
244C8×C137—42no—
245C8×C137—42no—
246C22×C137—21no—
247C22×C137—21no—
248C22×C137—21no—
249C22×C137—42no—
250C22×C137—42no—
251C22×C137—42no—
252C22×C137—42no—
253C22×C137—42no—
254C22×C137—42no—
255C22×C137—84no—
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—42no—
265C4×C137—42no—
266C4×C137—42no—
267C4×C137—42no—
268C4×C137—81no—
269C4×C137#57512yes—
270C4×C137#57612yes—
271C2×C137—41no—
272C2×C137—82no—
273C2×C137—82no—
274C2×C137—161no—
275C2×C137#300511yescharacteristic
276C2×C137#300811yescharacteristic
277C2×C137#301811yescharacteristic
278#18616C13711yesmaximal, characteristic, direct product factor, Sylow
279C137#1861611yescharacteristic, direct product factor, Sylow
280#3506C2×C13711yescharacteristic, complemented
281#3214C2×C13711yescharacteristic, complemented
282#3182C2×C13711yescharacteristic, complemented
283#3100C2×C13711yescharacteristic, complemented
284#1647C2×C13712yescomplemented
285#1646C2×C13712yescomplemented
286#1625C2×C13712yescomplemented
287#1624C2×C13712yescomplemented
288#1554C2×C13711yescharacteristic, complemented
289#1490C2×C13711yescharacteristic, complemented
290#1484C2×C13711yescharacteristic, complemented
291#559—22no—
292#556—21no—
293#555C22×C13712yes—
294#553—22no—
295#550—21no—
296#547C22×C13712yes—
297#538—22no—
298#535—21no—
299#533C22×C13711yescharacteristic
300#532C22×C13711yescharacteristic
301#532C22×C13711yescharacteristic
302#525—22no—
303#520—21no—
304C2×C4×D4C22×C13711yescharacteristic
305#500C22×C13711yescharacteristic, complemented
306#496—21no—
307#492C22×C13711yescharacteristic, complemented
308#489—21no—
309#487C22×C13711yescharacteristic, complemented
310#486C22×C13711yescharacteristic, complemented
311#480—21no—
312#474C22×C13712yes—
313#468—22no—
314#461C22×C13712yescomplemented
315#459C22×C13712yes—
316#459C22×C13712yescomplemented
317#449—22no—
318#447—21no—
319#440—22no—
320C4×D8—22no—
321#432C22×C13711yescharacteristic
322#431C22×C13711yescharacteristic
323C4×M16C22×C13711yescharacteristic
324#333C22×C13712yescomplemented
325#332C22×C13712yescomplemented
326D4⋊C8C22×C13712yes—
327D4⋊C8C22×C13712yes—
328D4⋊C8C22×C13712yescomplemented
329D4⋊C8C22×C13712yescomplemented
330C2×#41—22no—
331C2×#41—21no—
332C2×#41—21no—
333C22×D4D4×C13711yescharacteristic
334C23×C4—21no—
335C2×SD16—22no—
336C2×SD16—22no—
337C2×D8—22no—
338C2×D8—22no—
339C2×M16—21no—
340C4⋊Q8D4×C13711yescharacteristic
341C4⋊Q8C23×C13711yescharacteristic
342C4⋊Q8C23×C13711yescharacteristic
343C4⋊Q8C23×C13711yescharacteristic
344#127—22no—
345#127D4×C13711yescharacteristic
346#127C23×C13711yescharacteristic
347#126—24no—
348#125—21no—
349#124—22no—
350#124—24no—
351#124—21no—
352#124—21no—
353#123—24no—
354#123—24no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8D4×C13712yes—
358#121—24no—
359#121—22no—
360#121—22no—
361#121—22no—
362#121—21no—
363#121—21no—
364#121—21no—
365#121D4×C13712yes—
366#120—24no—
367C4×Q8—22no—
368C4×Q8—21no—
369C4×Q8D4×C13712yes—
370C4×D4—24no—
371C4×D4—22no—
372C4×D4—24no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4D4×C13712yes—
379C4×D4C23×C13712yes—
380C4×D4C23×C13712yes—
381#117—21no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4D4×C13711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C42C23×C13711yescharacteristic
388#107—22no—
389#107—22no—
390C4⋊C8C23×C13712yes—
391C4⋊C8C23×C13712yes—
392#103—22no—
393#103—22no—
394#102—22no—
395#102—22no—
396#102—22no—
397#102—22no—
398C22⋊C8—21no—
399C22⋊C8—21no—
400#97—21no—
401#97—21no—
402C4×C8—21no—
403C4×C8C23×C13711yescharacteristic
404C4×C8C23×C13711yescharacteristic
405C24—21no—
406#41—44no—
407#41—82no—
408#41—82no—
409#41—82no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8C2×D4×C13712yes—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—41no—
425C2×D4C2×D4×C13712yes—
426C2×D4C2×D4×C13712yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—82no—
434C22×C4C2×D4×C13711yescharacteristic
435C22×C4C2×D4×C13711yescharacteristic
436C22×C4C2×D4×C13711yescharacteristic
437SD16—44no—
438SD16—82no—
439D8—44no—
440D8—82no—
441M16—42no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4C2×D4×C13712yes—
457C4⋊C4C2×D4×C13712yes—
458C4⋊C4C2×D4×C13712yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C42—24no—
470C42—22no—
471C42C2×D4×C13711yescharacteristic
472C42C2×D4×C13711yescharacteristic
473C42C2×D4×C13711yescharacteristic
474C42C24×C13711yescharacteristic
475C23—22no—
476C23—22no—
477C23—42no—
478C23—41no—
479C23—81no—
480C23—81no—
481C23#120×C13711yescharacteristic
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—42no—
486D4—22no—
487D4—22no—
488D4—22no—
489D4—22no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—81no—
497D4—84no—
498D4—84no—
499D4—82no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—84no—
515C2×C4#120×C13711yescharacteristic
516C2×C4#120×C13711yescharacteristic
517C2×C4#120×C13711yescharacteristic
518C2×C4C22×D4×C13711yescharacteristic
519C2×C4C22×D4×C13711yescharacteristic
520C2×C4C22×D4×C13711yescharacteristic
521C8—22no—
522C8—41no—
523C8—42no—
524C8—42no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—84no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22C2×#120×C13711yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4#575×C13712yes—
549C4#576×C13712yes—
550C2—41no—
551C2—82no—
552C2—82no—
553C2—161no—
554C2#3005×C13711yescharacteristic
555C2#3008×C13711yescharacteristic
556C2#3018×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
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. #3506×C137C2semidirect
2. #3214×C137C2semidirect
3. #3182×C137C2semidirect
4. #3100×C137C2semidirect
5. #1647×C137C2semidirect
6. #1646×C137C2semidirect
7. #1625×C137C2semidirect
8. #1624×C137C2semidirect
9. #1554×C137C2semidirect
10. #1490×C137C2semidirect
11. #1484×C137C2semidirect
26. #500×C137C22semidirect
28. #492×C137C22semidirect
30. #487×C137C22semidirect
31. #486×C137C22semidirect
35. #461×C137C22semidirect
37. #459×C137C22semidirect
45. #333×C137C22semidirect
46. #332×C137C22semidirect
49. D4⋊C8×C137C22semidirect
50. D4⋊C8×C137C22semidirect
278. #18616C137direct (N × H)
279. C137#18616direct (N × H)
280. #3506C2×C137semidirect
281. #3214C2×C137semidirect
282. #3182C2×C137semidirect
283. #3100C2×C137semidirect
284. #1647C2×C137semidirect
285. #1646C2×C137semidirect
286. #1625C2×C137semidirect
287. #1624C2×C137semidirect
288. #1554C2×C137semidirect
289. #1490C2×C137semidirect
290. #1484C2×C137semidirect
305. #500C22×C137semidirect
307. #492C22×C137semidirect
309. #487C22×C137semidirect
310. #486C22×C137semidirect
314. #461C22×C137semidirect
316. #459C22×C137semidirect
324. #333C22×C137semidirect
325. #332C22×C137semidirect
328. D4⋊C8C22×C137semidirect
329. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3506×C1372
#3214×C1372
#3182×C1372
#3100×C1372
#1647×C1372
#1646×C1372
#1625×C1372
#1624×C1372
#1554×C1372
#1490×C1372
#1484×C1372
#18616137