Finite Group Databasebeta

← C137×#18750 #46868136

Subgroups — C137×#18750 #46868136

Order 35072.

1040 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559×C137C211yesmaximal, characteristic, complemented
2#3205×C137C211yesmaximal, characteristic
3#3103×C137C211yesmaximal, characteristic, complemented
4#3037×C137C211yesmaximal, characteristic, complemented
5#1640×C137C214yesmaximal, complemented
6#1620×C137C214yesmaximal, complemented
7#1545×C137C211yesmaximal, characteristic, complemented
8#1487×C137C211yesmaximal, characteristic, complemented
9#1479×C137C211yesmaximal, characteristic, complemented
10C2×#137×C137—21no—
11#559×C137—22no—
12#559×C137—22no—
13#559×C137—22no—
14#559×C137C2214yes—
15#532×C137C2211yescharacteristic
16C2×#127×C137C2211yescharacteristic
17C2×#124×C137—21no—
18C2×#124×C137—21no—
19C2×#123×C137—22no—
20C2×C4×Q8×C137C2211yescharacteristic
21C2×C4×Q8×C137C2211yescharacteristic
22C2×C4×D4×C137—21no—
23C2×C4×D4×C137—21no—
24#493×C137C2212yes—
25#486×C137C2212yes—
26#481×C137—21no—
27#475×C137C2212yescomplemented
28#464×C137—22no—
29#462×C137C2212yescomplemented
30#460×C137C2212yescomplemented
31#460×C137C2212yescomplemented
32#459×C137—22no—
33#448×C137—21no—
34#447×C137—22no—
35C4×SD16×C137—22no—
36#431×C137C2211yescharacteristic
37#431×C137C2211yescharacteristic
38#423×C137C2211yescharacteristic
39#417×C137—21no—
40#331×C137C2214yes—
41Q8⋊C8×C137C2214yescomplemented
42Q8⋊C8×C137C2214yescomplemented
43C2×#41×C137—21no—
44C22×Q8×C137D411yescharacteristic
45C22×Q8×C137D411yescharacteristic
46C22×D4×C137—21no—
47C22×D4×C137—21no—
48C22×D4×C137D412yes—
49C23×C4×C137—22no—
50#137×C137—44no—
51C2×Q16×C137—22no—
52C2×SD16×C137—22no—
53C2×SD16×C137—22no—
54C2×D8×C137—24no—
55C2×M16×C137—21no—
56C4⋊Q8×C137C2311yescharacteristic
57#127×C137—21no—
58#127×C137—21no—
59#127×C137D412yes—
60#127×C137C2311yescharacteristic
61#127×C137C2312yes—
62#124×C137—24no—
63#124×C137—24no—
64#124×C137—24no—
65#124×C137—24no—
66#124×C137—21no—
67#123×C137—28no—
68#123×C137—28no—
69#121×C137—21no—
70#121×C137—21no—
71C4×Q8×C137—21no—
72C4×Q8×C137—21no—
73C4×Q8×C137—21no—
74C4×Q8×C137—21no—
75C4×Q8×C137D412yes—
76C4×Q8×C137D412yes—
77C4×Q8×C137C2312yes—
78C4×Q8×C137C2312yes—
79C4×D4×C137—24no—
80C4×D4×C137—24no—
81C4×D4×C137—24no—
82C4×D4×C137—24no—
83C4×D4×C137—21no—
84C4×D4×C137—21no—
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—22no—
92C2×C42×C137—21no—
93C2×C42×C137—21no—
94C2×C42×C137C2311yescharacteristic
95#106×C137—22no—
96C4⋊C8×C137—21no—
97C4⋊C8×C137—21no—
98C4⋊C8×C137C2312yes—
99C4⋊C8×C137C2312yes—
100#103×C137—22no—
101#103×C137—22no—
102#102×C137—24no—
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—22no—
110#41×C137—42no—
111#41×C137—42no—
112C2×Q8×C137—21no—
113C2×Q8×C137—21no—
114C2×Q8×C137—22no—
115C2×Q8×C137—22no—
116C2×Q8×C137—21no—
117C2×Q8×C137—42no—
118C2×Q8×C137C2×D412yes—
119C2×Q8×C137C2×D412yes—
120C2×D4×C137—22no—
121C2×D4×C137—22no—
122C2×D4×C137—22no—
123C2×D4×C137—22no—
124C2×D4×C137—22no—
125C2×D4×C137—21no—
126C2×D4×C137—21no—
127C2×D4×C137—21no—
128C2×D4×C137—44no—
129C2×D4×C137—41no—
130C2×D4×C137—42no—
131C2×D4×C137—42no—
132C2×D4×C137—82no—
133C2×D4×C137C2×D414yes—
134C22×C4×C137—21no—
135C22×C4×C137—21no—
136C22×C4×C137—21no—
137C22×C4×C137—21no—
138C22×C4×C137—28no—
139C22×C4×C137—41no—
140C22×C4×C137—84no—
141C22×C4×C137C2×D411yescharacteristic
142C22×C4×C137C2×D411yescharacteristic
143C22×C4×C137C2×D411yescharacteristic
144Q16×C137—44no—
145SD16×C137—44no—
146SD16×C137—44no—
147D8×C137—84no—
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—28no—
165C22⋊C4×C137—41no—
166C22⋊C4×C137—41no—
167C42×C137—22no—
168C42×C137—22no—
169C42×C137—22no—
170C42×C137—22no—
171C42×C137C2×D411yescharacteristic
172C42×C137C2×D411yescharacteristic
173C42×C137C2×D411yescharacteristic
174C42×C137C2411yescharacteristic
175C23×C137—24no—
176C23×C137—44no—
177C23×C137—44no—
178C23×C137—44no—
179C23×C137—41no—
180C23×C137#12011yescharacteristic
181Q8×C137—22no—
182Q8×C137—22no—
183Q8×C137—22no—
184Q8×C137—22no—
185Q8×C137—42no—
186Q8×C137—42no—
187Q8×C137—41no—
188Q8×C137—41no—
189D4×C137—44no—
190D4×C137—44no—
191D4×C137—42no—
192D4×C137—42no—
193D4×C137—41no—
194D4×C137—42no—
195D4×C137—41no—
196D4×C137—44no—
197D4×C137—42no—
198D4×C137—44no—
199D4×C137—82no—
200D4×C137—82no—
201C2×C4×C137—22no—
202C2×C4×C137—22no—
203C2×C4×C137—22no—
204C2×C4×C137—22no—
205C2×C4×C137—22no—
206C2×C4×C137—41no—
207C2×C4×C137—42no—
208C2×C4×C137—42no—
209C2×C4×C137—42no—
210C2×C4×C137—42no—
211C2×C4×C137—41no—
212C2×C4×C137—88no—
213C2×C4×C137—82no—
214C2×C4×C137#12011yescharacteristic
215C2×C4×C137#12011yescharacteristic
216C2×C4×C137#12011yescharacteristic
217C2×C4×C137#13512yes—
218C2×C4×C137AGL(1,ℤ/8)12yes—
219C2×C4×C137C22×D411yescharacteristic
220C2×C4×C137C22×D411yescharacteristic
221C2×C4×C137C22×D411yescharacteristic
222C8×C137—42no—
223C8×C137—42no—
224C8×C137—42no—
225C22×C137—21no—
226C22×C137—21no—
227C22×C137—44no—
228C22×C137—44no—
229C22×C137—44no—
230C22×C137—88no—
231C22×C137—81no—
232C22×C137—81no—
233C22×C137—81no—
234C22×C137#44912yes—
235C22×C137C2×#12011yescharacteristic
236C4×C137—21no—
237C4×C137—21no—
238C4×C137—21no—
239C4×C137—42no—
240C4×C137—42no—
241C4×C137—42no—
242C4×C137—42no—
243C4×C137—81no—
244C4×C137#44912yes—
245C4×C137C2×#13511yescharacteristic
246C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C137#57512yes—
248C2×C137—22no—
249C2×C137—84no—
250C2×C137—161no—
251C2×C137C2×#44911yescharacteristic
252C2×C137#300511yescharacteristic
253C2×C137#301911yescharacteristic
254#18750C13711yesmaximal, characteristic, direct product factor, Sylow
255C137#1875011yescharacteristic, direct product factor, Sylow
256C2×#559C2×C13711yescharacteristic, complemented
257#3205C2×C13711yescharacteristic
258#3103C2×C13711yescharacteristic, complemented
259#3037C2×C13711yescharacteristic, complemented
260#1640C2×C13714yescomplemented
261#1620C2×C13714yescomplemented
262#1545C2×C13711yescharacteristic, complemented
263#1487C2×C13711yescharacteristic, complemented
264#1479C2×C13711yescharacteristic, complemented
265C2×#137—21no—
266#559—22no—
267#559—22no—
268#559—22no—
269#559C22×C13714yes—
270#532C22×C13711yescharacteristic
271C2×#127C22×C13711yescharacteristic
272C2×#124—21no—
273C2×#124—21no—
274C2×#123—22no—
275C2×C4×Q8C22×C13711yescharacteristic
276C2×C4×Q8C22×C13711yescharacteristic
277C2×C4×D4—21no—
278C2×C4×D4—21no—
279#493C22×C13712yes—
280#486C22×C13712yes—
281#481—21no—
282#475C22×C13712yescomplemented
283#464—22no—
284#462C22×C13712yescomplemented
285#460C22×C13712yescomplemented
286#460C22×C13712yescomplemented
287#459—22no—
288#448—21no—
289#447—22no—
290C4×SD16—22no—
291#431C22×C13711yescharacteristic
292#431C22×C13711yescharacteristic
293#423C22×C13711yescharacteristic
294#417—21no—
295#331C22×C13714yes—
296Q8⋊C8C22×C13714yescomplemented
297Q8⋊C8C22×C13714yescomplemented
298C2×#41—21no—
299C22×Q8D4×C13711yescharacteristic
300C22×Q8D4×C13711yescharacteristic
301C22×D4—21no—
302C22×D4—21no—
303C22×D4D4×C13712yes—
304C23×C4—22no—
305#137—44no—
306C2×Q16—22no—
307C2×SD16—22no—
308C2×SD16—22no—
309C2×D8—24no—
310C2×M16—21no—
311C4⋊Q8C23×C13711yescharacteristic
312#127—21no—
313#127—21no—
314#127D4×C13712yes—
315#127C23×C13711yescharacteristic
316#127C23×C13712yes—
317#124—24no—
318#124—24no—
319#124—24no—
320#124—24no—
321#124—21no—
322#123—28no—
323#123—28no—
324#121—21no—
325#121—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8D4×C13712yes—
331C4×Q8D4×C13712yes—
332C4×Q8C23×C13712yes—
333C4×Q8C23×C13712yes—
334C4×D4—24no—
335C4×D4—24no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—21no—
339C4×D4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4D4×C13711yescharacteristic
343C2×C4⋊C4D4×C13711yescharacteristic
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C23×C13711yescharacteristic
350#106—22no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8C23×C13712yes—
354C4⋊C8C23×C13712yes—
355#103—22no—
356#103—22no—
357#102—24no—
358#102—22no—
359#102—22no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C4×C8—21no—
363C4×C8C23×C13712yes—
364C24—22no—
365#41—42no—
366#41—42no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—42no—
373C2×Q8C2×D4×C13712yes—
374C2×Q8C2×D4×C13712yes—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—44no—
384C2×D4—41no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—82no—
388C2×D4C2×D4×C13714yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—28no—
394C22×C4—41no—
395C22×C4—84no—
396C22×C4C2×D4×C13711yescharacteristic
397C22×C4C2×D4×C13711yescharacteristic
398C22×C4C2×D4×C13711yescharacteristic
399Q16—44no—
400SD16—44no—
401SD16—44no—
402D8—84no—
403M16—42no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4C2×D4×C13712yes—
416C4⋊C4C2×D4×C13712yes—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42C2×D4×C13711yescharacteristic
427C42C2×D4×C13711yescharacteristic
428C42C2×D4×C13711yescharacteristic
429C42C24×C13711yescharacteristic
430C23—24no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—41no—
435C23#120×C13711yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—22no—
439Q8—22no—
440Q8—42no—
441Q8—42no—
442Q8—41no—
443Q8—41no—
444D4—44no—
445D4—44no—
446D4—42no—
447D4—44no—
448D4—42no—
449D4—41no—
450D4—42no—
451D4—41no—
452D4—44no—
453D4—42no—
454D4—82no—
455D4—82no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—88no—
468C2×C4—82no—
469C2×C4#120×C13711yescharacteristic
470C2×C4#120×C13711yescharacteristic
471C2×C4#120×C13711yescharacteristic
472C2×C4#135×C13712yes—
473C2×C4AGL(1,ℤ/8)×C13712yes—
474C2×C4C22×D4×C13711yescharacteristic
475C2×C4C22×D4×C13711yescharacteristic
476C2×C4C22×D4×C13711yescharacteristic
477C8—42no—
478C8—42no—
479C8—42no—
480C22—21no—
481C22—21no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—88no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#449×C13712yes—
490C22C2×#120×C13711yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4#449×C13712yes—
500C4C2×#135×C13711yescharacteristic
501C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
502C4#575×C13712yes—
503C2—22no—
504C2—84no—
505C2—161no—
506C2C2×#449×C13711yescharacteristic
507C2#3005×C13711yescharacteristic
508C2#3019×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3205×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. C2×#559×C137C2semidirect
3. #3103×C137C2semidirect
4. #3037×C137C2semidirect
5. #1640×C137C2semidirect
6. #1620×C137C2semidirect
7. #1545×C137C2semidirect
8. #1487×C137C2semidirect
9. #1479×C137C2semidirect
27. #475×C137C22semidirect
29. #462×C137C22semidirect
30. #460×C137C22semidirect
31. #460×C137C22semidirect
41. Q8⋊C8×C137C22semidirect
42. Q8⋊C8×C137C22semidirect
254. #18750C137direct (N × H)
255. C137#18750direct (N × H)
256. C2×#559C2×C137semidirect
258. #3103C2×C137semidirect
259. #3037C2×C137semidirect
260. #1640C2×C137semidirect
261. #1620C2×C137semidirect
262. #1545C2×C137semidirect
263. #1487C2×C137semidirect
264. #1479C2×C137semidirect
282. #475C22×C137semidirect
284. #462C22×C137semidirect
285. #460C22×C137semidirect
286. #460C22×C137semidirect
296. Q8⋊C8C22×C137semidirect
297. Q8⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#559×C1372
#3205×C1372
#3103×C1372
#3037×C1372
#1640×C1372
#1620×C1372
#1545×C1372
#1487×C1372
#1479×C1372
#18750137