Finite Group Databasebeta

← C137×#18772 #46868158

Subgroups — C137×#18772 #46868158

Order 35072.

1040 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C137C211yesmaximal, characteristic, complemented
2#3206×C137C211yesmaximal, characteristic, complemented
3#3100×C137C211yesmaximal, characteristic, complemented
4#3036×C137C211yesmaximal, characteristic, complemented
5#1646×C137C214yesmaximal, complemented
6#1624×C137C214yesmaximal, complemented
7#1548×C137C211yesmaximal, characteristic, complemented
8#1486×C137C211yesmaximal, characteristic, complemented
9#1478×C137C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C137—21no—
11#555×C137—22no—
12#555×C137—22no—
13#555×C137—22no—
14#555×C137C2214yes—
15#532×C137C2211yescharacteristic
16C2×C4⋊Q8×C137C2211yescharacteristic
17C2×#124×C137—22no—
18C2×#121×C137—21no—
19C2×#121×C137—21no—
20C2×C4×D4×C137C2211yescharacteristic
21C2×C4×D4×C137C2211yescharacteristic
22C2×#117×C137—21no—
23C2×#117×C137—21no—
24#492×C137C2212yescomplemented
25#487×C137C2212yescomplemented
26#480×C137—21no—
27#474×C137C2212yescomplemented
28#463×C137—22no—
29#461×C137—22no—
30#461×C137C2212yescomplemented
31#459×C137C2212yescomplemented
32#459×C137C2212yescomplemented
33#448×C137—22no—
34#447×C137—21no—
35C4×D8×C137—22no—
36#431×C137C2211yescharacteristic
37#431×C137C2211yescharacteristic
38#423×C137C2211yescharacteristic
39#417×C137—21no—
40#333×C137C2214yescomplemented
41D4⋊C8×C137C2214yescomplemented
42D4⋊C8×C137C2214yescomplemented
43C2×#41×C137—21no—
44C22×Q8×C137D412yes—
45C22×D4×C137—22no—
46C22×D4×C137D411yescharacteristic
47C22×D4×C137D411yescharacteristic
48C23×C4×C137—21no—
49C23×C4×C137—21no—
50AGL(1,ℤ/8)×C137—44no—
51C2×SD16×C137—22no—
52C2×SD16×C137—24no—
53C2×D8×C137—22no—
54C2×D8×C137—22no—
55C2×M16×C137—21no—
56C4⋊Q8×C137—21no—
57C4⋊Q8×C137—21no—
58C4⋊Q8×C137D412yes—
59C4⋊Q8×C137C2311yescharacteristic
60C4⋊Q8×C137C2312yes—
61#127×C137C2311yescharacteristic
62#124×C137—28no—
63#124×C137—28no—
64#124×C137—21no—
65#121×C137—24no—
66#121×C137—24no—
67#121×C137—24no—
68#121×C137—24no—
69#121×C137—21no—
70#121×C137—21no—
71C4×D4×C137—21no—
72C4×D4×C137—21no—
73C4×D4×C137—21no—
74C4×D4×C137—21no—
75C4×D4×C137—21no—
76C4×D4×C137—21no—
77C4×D4×C137D412yes—
78C4×D4×C137D412yes—
79C4×D4×C137C2312yes—
80C4×D4×C137C2312yes—
81#117×C137—24no—
82#117×C137—24no—
83#117×C137—24no—
84#117×C137—24no—
85C2×C4⋊C4×C137—21no—
86C2×C4⋊C4×C137—21no—
87C2×C4⋊C4×C137D411yescharacteristic
88C2×C4⋊C4×C137D411yescharacteristic
89C2×C22⋊C4×C137—22no—
90C2×C22⋊C4×C137—22no—
91C2×C22⋊C4×C137—21no—
92C2×C22⋊C4×C137—21no—
93C2×C42×C137—22no—
94C2×C42×C137C2311yescharacteristic
95#107×C137—22no—
96C4⋊C8×C137—21no—
97C4⋊C8×C137—21no—
98C4⋊C8×C137C2312yes—
99C4⋊C8×C137C2312yes—
100#103×C137—24no—
101#103×C137—22no—
102#102×C137—22no—
103#102×C137—22no—
104#102×C137—22no—
105C22⋊C8×C137—21no—
106C22⋊C8×C137—21no—
107C4×C8×C137—21no—
108C4×C8×C137C2312yes—
109C24×C137—21no—
110C24×C137—21no—
111#41×C137—42no—
112#41×C137—42no—
113C2×Q8×C137—22no—
114C2×Q8×C137—21no—
115C2×Q8×C137—82no—
116C2×Q8×C137C2×D414yes—
117C2×D4×C137—28no—
118C2×D4×C137—21no—
119C2×D4×C137—21no—
120C2×D4×C137—22no—
121C2×D4×C137—22no—
122C2×D4×C137—21no—
123C2×D4×C137—21no—
124C2×D4×C137—21no—
125C2×D4×C137—42no—
126C2×D4×C137—41no—
127C2×D4×C137—84no—
128C2×D4×C137C2×D412yes—
129C2×D4×C137C2×D412yes—
130C22×C4×C137—21no—
131C22×C4×C137—21no—
132C22×C4×C137—22no—
133C22×C4×C137—22no—
134C22×C4×C137—22no—
135C22×C4×C137—22no—
136C22×C4×C137—22no—
137C22×C4×C137—44no—
138C22×C4×C137—42no—
139C22×C4×C137—42no—
140C22×C4×C137—41no—
141C22×C4×C137C2×D411yescharacteristic
142C22×C4×C137C2×D411yescharacteristic
143C22×C4×C137C2×D411yescharacteristic
144SD16×C137—44no—
145SD16×C137—84no—
146D8×C137—44no—
147D8×C137—44no—
148M16×C137—42no—
149C2×C8×C137—22no—
150C2×C8×C137—22no—
151C2×C8×C137—22no—
152C4⋊C4×C137—22no—
153C4⋊C4×C137—22no—
154C4⋊C4×C137—21no—
155C4⋊C4×C137—21no—
156C4⋊C4×C137—22no—
157C4⋊C4×C137—22no—
158C4⋊C4×C137—21no—
159C4⋊C4×C137—21no—
160C4⋊C4×C137C2×D412yes—
161C4⋊C4×C137C2×D412yes—
162C22⋊C4×C137—28no—
163C22⋊C4×C137—28no—
164C22⋊C4×C137—22no—
165C22⋊C4×C137—22no—
166C22⋊C4×C137—22no—
167C22⋊C4×C137—22no—
168C22⋊C4×C137—41no—
169C22⋊C4×C137—41no—
170C42×C137—28no—
171C42×C137C2×D411yescharacteristic
172C42×C137C2×D411yescharacteristic
173C42×C137C2×D411yescharacteristic
174C42×C137C2411yescharacteristic
175C23×C137—22no—
176C23×C137—22no—
177C23×C137—42no—
178C23×C137—42no—
179C23×C137—42no—
180C23×C137—42no—
181C23×C137—42no—
182C23×C137—42no—
183C23×C137—41no—
184C23×C137#12011yescharacteristic
185Q8×C137—44no—
186Q8×C137—41no—
187Q8×C137—41no—
188Q8×C137—82no—
189D4×C137—22no—
190D4×C137—22no—
191D4×C137—22no—
192D4×C137—22no—
193D4×C137—42no—
194D4×C137—42no—
195D4×C137—41no—
196D4×C137—41no—
197D4×C137—42no—
198D4×C137—42no—
199D4×C137—88no—
200D4×C137—82no—
201C2×C4×C137—22no—
202C2×C4×C137—22no—
203C2×C4×C137—22no—
204C2×C4×C137—24no—
205C2×C4×C137—41no—
206C2×C4×C137—44no—
207C2×C4×C137—42no—
208C2×C4×C137—44no—
209C2×C4×C137—42no—
210C2×C4×C137—42no—
211C2×C4×C137—42no—
212C2×C4×C137—44no—
213C2×C4×C137—44no—
214C2×C4×C137—41no—
215C2×C4×C137—82no—
216C2×C4×C137#12011yescharacteristic
217C2×C4×C137#12011yescharacteristic
218C2×C4×C137#12011yescharacteristic
219C2×C4×C137#13512yes—
220C2×C4×C137AGL(1,ℤ/8)12yes—
221C2×C4×C137C22×D411yescharacteristic
222C2×C4×C137C22×D411yescharacteristic
223C2×C4×C137C22×D411yescharacteristic
224C8×C137—42no—
225C8×C137—42no—
226C8×C137—42no—
227C22×C137—21no—
228C22×C137—21no—
229C22×C137—42no—
230C22×C137—42no—
231C22×C137—42no—
232C22×C137—42no—
233C22×C137—42no—
234C22×C137—42no—
235C22×C137—84no—
236C22×C137—84no—
237C22×C137—81no—
238C22×C137—81no—
239C22×C137—81no—
240C22×C137#44912yes—
241C22×C137C2×#12011yescharacteristic
242C4×C137—21no—
243C4×C137—21no—
244C4×C137—21no—
245C4×C137—42no—
246C4×C137—42no—
247C4×C137—44no—
248C4×C137—81no—
249C4×C137#44912yes—
250C4×C137C2×#13511yescharacteristic
251C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C137#57512yes—
253C2×C137—22no—
254C2×C137—82no—
255C2×C137—82no—
256C2×C137—161no—
257C2×C137C2×#44911yescharacteristic
258C2×C137#300511yescharacteristic
259C2×C137#301911yescharacteristic
260#18772C13711yesmaximal, characteristic, direct product factor, Sylow
261C137#1877211yescharacteristic, direct product factor, Sylow
262C2×#555C2×C13711yescharacteristic, complemented
263#3206C2×C13711yescharacteristic, complemented
264#3100C2×C13711yescharacteristic, complemented
265#3036C2×C13711yescharacteristic, complemented
266#1646C2×C13714yescomplemented
267#1624C2×C13714yescomplemented
268#1548C2×C13711yescharacteristic, complemented
269#1486C2×C13711yescharacteristic, complemented
270#1478C2×C13711yescharacteristic, complemented
271C2×AGL(1,ℤ/8)—21no—
272#555—22no—
273#555—22no—
274#555—22no—
275#555C22×C13714yes—
276#532C22×C13711yescharacteristic
277C2×C4⋊Q8C22×C13711yescharacteristic
278C2×#124—22no—
279C2×#121—21no—
280C2×#121—21no—
281C2×C4×D4C22×C13711yescharacteristic
282C2×C4×D4C22×C13711yescharacteristic
283C2×#117—21no—
284C2×#117—21no—
285#492C22×C13712yescomplemented
286#487C22×C13712yescomplemented
287#480—21no—
288#474C22×C13712yescomplemented
289#463—22no—
290#461—22no—
291#461C22×C13712yescomplemented
292#459C22×C13712yescomplemented
293#459C22×C13712yescomplemented
294#448—22no—
295#447—21no—
296C4×D8—22no—
297#431C22×C13711yescharacteristic
298#431C22×C13711yescharacteristic
299#423C22×C13711yescharacteristic
300#417—21no—
301#333C22×C13714yescomplemented
302D4⋊C8C22×C13714yescomplemented
303D4⋊C8C22×C13714yescomplemented
304C2×#41—21no—
305C22×Q8D4×C13712yes—
306C22×D4—22no—
307C22×D4D4×C13711yescharacteristic
308C22×D4D4×C13711yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311AGL(1,ℤ/8)—44no—
312C2×SD16—22no—
313C2×SD16—24no—
314C2×D8—22no—
315C2×D8—22no—
316C2×M16—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8D4×C13712yes—
320C4⋊Q8C23×C13711yescharacteristic
321C4⋊Q8C23×C13712yes—
322#127C23×C13711yescharacteristic
323#124—28no—
324#124—28no—
325#124—21no—
326#121—24no—
327#121—24no—
328#121—24no—
329#121—24no—
330#121—21no—
331#121—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4D4×C13712yes—
339C4×D4D4×C13712yes—
340C4×D4C23×C13712yes—
341C4×D4C23×C13712yes—
342#117—24no—
343#117—24no—
344#117—24no—
345#117—24no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4D4×C13711yescharacteristic
349C2×C4⋊C4D4×C13711yescharacteristic
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C42—22no—
355C2×C42C23×C13711yescharacteristic
356#107—22no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8C23×C13712yes—
360C4⋊C8C23×C13712yes—
361#103—24no—
362#103—22no—
363#102—22no—
364#102—22no—
365#102—22no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368C4×C8—21no—
369C4×C8C23×C13712yes—
370C24—21no—
371C24—21no—
372#41—42no—
373#41—42no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—82no—
377C2×Q8C2×D4×C13714yes—
378C2×D4—28no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—84no—
389C2×D4C2×D4×C13712yes—
390C2×D4C2×D4×C13712yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4C2×D4×C13711yescharacteristic
403C22×C4C2×D4×C13711yescharacteristic
404C22×C4C2×D4×C13711yescharacteristic
405SD16—44no—
406SD16—84no—
407D8—44no—
408D8—44no—
409M16—42no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4C2×D4×C13712yes—
422C4⋊C4C2×D4×C13712yes—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C42—28no—
432C42C2×D4×C13711yescharacteristic
433C42C2×D4×C13711yescharacteristic
434C42C2×D4×C13711yescharacteristic
435C42C24×C13711yescharacteristic
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23#120×C13711yescharacteristic
446Q8—44no—
447Q8—41no—
448Q8—41no—
449Q8—82no—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—41no—
458D4—41no—
459D4—42no—
460D4—88no—
461D4—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4#120×C13711yescharacteristic
478C2×C4#120×C13711yescharacteristic
479C2×C4#120×C13711yescharacteristic
480C2×C4#135×C13712yes—
481C2×C4AGL(1,ℤ/8)×C13712yes—
482C2×C4C22×D4×C13711yescharacteristic
483C2×C4C22×D4×C13711yescharacteristic
484C2×C4C22×D4×C13711yescharacteristic
485C8—42no—
486C8—42no—
487C8—42no—
488C22—21no—
489C22—21no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—84no—
497C22—84no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22#449×C13712yes—
502C22C2×#120×C13711yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—42no—
508C4—44no—
509C4—81no—
510C4#449×C13712yes—
511C4C2×#135×C13711yescharacteristic
512C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
513C4#575×C13712yes—
514C2—22no—
515C2—82no—
516C2—82no—
517C2—161no—
518C2C2×#449×C13711yescharacteristic
519C2#3005×C13711yescharacteristic
520C2#3019×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. C2×#555×C137C2semidirect
2. #3206×C137C2semidirect
3. #3100×C137C2semidirect
4. #3036×C137C2semidirect
5. #1646×C137C2semidirect
6. #1624×C137C2semidirect
7. #1548×C137C2semidirect
8. #1486×C137C2semidirect
9. #1478×C137C2semidirect
24. #492×C137C22semidirect
25. #487×C137C22semidirect
27. #474×C137C22semidirect
30. #461×C137C22semidirect
31. #459×C137C22semidirect
32. #459×C137C22semidirect
40. #333×C137C22semidirect
41. D4⋊C8×C137C22semidirect
42. D4⋊C8×C137C22semidirect
260. #18772C137direct (N × H)
261. C137#18772direct (N × H)
262. C2×#555C2×C137semidirect
263. #3206C2×C137semidirect
264. #3100C2×C137semidirect
265. #3036C2×C137semidirect
266. #1646C2×C137semidirect
267. #1624C2×C137semidirect
268. #1548C2×C137semidirect
269. #1486C2×C137semidirect
270. #1478C2×C137semidirect
285. #492C22×C137semidirect
286. #487C22×C137semidirect
288. #474C22×C137semidirect
291. #461C22×C137semidirect
292. #459C22×C137semidirect
293. #459C22×C137semidirect
301. #333C22×C137semidirect
302. D4⋊C8C22×C137semidirect
303. D4⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#555×C1372
#3206×C1372
#3100×C1372
#3036×C1372
#1646×C1372
#1624×C1372
#1548×C1372
#1486×C1372
#1478×C1372
#18772137