Finite Group Databasebeta

← C137×#18833 #46868219

Subgroups — C137×#18833 #46868219

Order 35072.

894 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511×C137C211yesmaximal, characteristic
2#3235×C137C211yesmaximal, characteristic, complemented
3#3229×C137C211yesmaximal, characteristic, complemented
4#3116×C137C211yesmaximal, characteristic, complemented
5#1654×C137C212yesmaximal, complemented
6#1652×C137C212yesmaximal, complemented
7#1638×C137C212yesmaximal, complemented
8#1635×C137C212yesmaximal, complemented
9#1556×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12#556×C137C2212yes—
13#555×C137—21no—
14#554×C137—21no—
15#553×C137—22no—
16#548×C137—22no—
17#547×C137C2212yes—
18#542×C137—22no—
19#537×C137—21no—
20#533×C137—21no—
21#533×C137C2211yescharacteristic
22#532×C137—21no—
23#532×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic
25#520×C137—21no—
26C4×#41×C137C2211yescharacteristic
27#501×C137—21no—
28#498×C137C2212yes—
29#490×C137—21no—
30#487×C137C2212yes—
31#485×C137—21no—
32#483×C137—22no—
33#482×C137—22no—
34#477×C137C2211yescharacteristic
35#476×C137C2211yescharacteristic
36#476×C137C2212yes—
37#475×C137—22no—
38#474×C137C2212yes—
39#462×C137C2211yescharacteristic
40#461×C137C2211yescharacteristic
41#452×C137—21no—
42#441×C137—21no—
43#440×C137—21no—
44#432×C137C2211yescharacteristic
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#332×C137C2212yes—
48#332×C137C2212yes—
49Q8⋊C8×C137C2212yes—
50D4⋊C8×C137C2212yes—
51D4⋊C8×C137C2212yes—
52D4⋊C8×C137C2212yes—
53C2×#41×C137—22no—
54C2×#41×C137—21no—
55C2×#41×C137D411yescharacteristic
56C22×Q8×C137—21no—
57C2×Q16×C137—21no—
58C2×SD16×C137—21no—
59C4⋊Q8×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137D412yes—
62C4⋊Q8×C137D411yescharacteristic
63C4⋊Q8×C137C2311yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65C4⋊Q8×C137C2312yes—
66#127×C137D411yescharacteristic
67#126×C137—24no—
68#125×C137—22no—
69#125×C137—21no—
70#124×C137—24no—
71#124×C137—22no—
72#124×C137—21no—
73#123×C137—24no—
74C22⋊Q8×C137—22no—
75C22⋊Q8×C137—24no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137—21no—
79C22⋊Q8×C137—21no—
80#121×C137—22no—
81#121×C137—22no—
82#121×C137—24no—
83#121×C137—22no—
84#121×C137—21no—
85#121×C137D412yes—
86C4×Q8×C137—22no—
87C4×Q8×C137—22no—
88C4×Q8×C137—21no—
89C4×Q8×C137—21no—
90C4×Q8×C137D411yescharacteristic
91C4×Q8×C137C2311yescharacteristic
92C4×D4×C137—22no—
93C4×D4×C137—24no—
94C4×D4×C137—22no—
95C4×D4×C137—21no—
96C4×D4×C137—21no—
97C4×D4×C137—21no—
98C4×D4×C137D411yescharacteristic
99C4×D4×C137C2311yescharacteristic
100C4×D4×C137C2312yes—
101#117×C137—22no—
102#117×C137—22no—
103#117×C137—21no—
104#117×C137—21no—
105#117×C137D411yescharacteristic
106C2×C4⋊C4×C137—21no—
107C2×C4⋊C4×C137—22no—
108C2×C4⋊C4×C137D412yes—
109C2×C42×C137—21no—
110C2×C42×C137C2311yescharacteristic
111#107×C137—21no—
112#107×C137—22no—
113#106×C137—21no—
114#106×C137—24no—
115#106×C137—22no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2311yescharacteristic
119C4⋊C8×C137C2312yes—
120#103×C137—21no—
121#103×C137—21no—
122#103×C137—21no—
123#103×C137—24no—
124#102×C137—21no—
125#102×C137—22no—
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—41no—
134#41×C137—41no—
135#41×C137—42no—
136#41×C137—42no—
137#41×C137—84no—
138C2×Q8×C137—21no—
139C2×Q8×C137—22no—
140C2×Q8×C137—22no—
141C2×Q8×C137—21no—
142C2×Q8×C137—21no—
143C2×Q8×C137—82no—
144C2×Q8×C137C2×D411yescharacteristic
145C2×D4×C137—21no—
146C2×D4×C137—22no—
147C2×D4×C137—24no—
148C2×D4×C137—21no—
149C2×D4×C137—21no—
150C2×D4×C137C2×D412yes—
151C2×D4×C137C2×D411yescharacteristic
152C22×C4×C137—24no—
153C22×C4×C137—22no—
154C22×C4×C137—21no—
155C22×C4×C137—21no—
156C22×C4×C137—21no—
157C22×C4×C137—22no—
158C22×C4×C137C2×D411yescharacteristic
159C22×C4×C137C2×D411yescharacteristic
160C22×C4×C137C2×D411yescharacteristic
161Q16×C137—42no—
162SD16×C137—42no—
163C2×C8×C137—22no—
164C2×C8×C137—22no—
165C2×C8×C137—21no—
166C2×C8×C137—21no—
167C4⋊C4×C137—24no—
168C4⋊C4×C137—24no—
169C4⋊C4×C137—22no—
170C4⋊C4×C137—22no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—22no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—21no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—41no—
180C4⋊C4×C137—41no—
181C4⋊C4×C137C2×D411yescharacteristic
182C4⋊C4×C137C2×D414yes—
183C4⋊C4×C137C2×D411yescharacteristic
184C4⋊C4×C137C2×D412yes—
185C22⋊C4×C137—24no—
186C22⋊C4×C137—24no—
187C22⋊C4×C137—24no—
188C22⋊C4×C137—21no—
189C22⋊C4×C137—21no—
190C22⋊C4×C137—22no—
191C22⋊C4×C137—21no—
192C22⋊C4×C137—22no—
193C22⋊C4×C137—41no—
194C42×C137—24no—
195C42×C137—21no—
196C42×C137—21no—
197C42×C137—21no—
198C42×C137—41no—
199C42×C137C2×D411yescharacteristic
200C42×C137C2×D411yescharacteristic
201C42×C137C2×D411yescharacteristic
202C42×C137C2411yescharacteristic
203C23×C137—21no—
204C23×C137—22no—
205C23×C137#12011yescharacteristic
206Q8×C137—21no—
207Q8×C137—21no—
208Q8×C137—42no—
209Q8×C137—42no—
210Q8×C137—44no—
211Q8×C137—41no—
212Q8×C137—41no—
213Q8×C137—82no—
214D4×C137—21no—
215D4×C137—21no—
216D4×C137—24no—
217D4×C137—42no—
218D4×C137—42no—
219D4×C137—42no—
220D4×C137—44no—
221D4×C137—84no—
222C2×C4×C137—21no—
223C2×C4×C137—21no—
224C2×C4×C137—22no—
225C2×C4×C137—21no—
226C2×C4×C137—21no—
227C2×C4×C137—22no—
228C2×C4×C137—21no—
229C2×C4×C137—24no—
230C2×C4×C137—41no—
231C2×C4×C137—42no—
232C2×C4×C137—42no—
233C2×C4×C137—42no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—41no—
237C2×C4×C137—41no—
238C2×C4×C137—82no—
239C2×C4×C137—84no—
240C2×C4×C137#12011yescharacteristic
241C2×C4×C137#12011yescharacteristic
242C2×C4×C137#12011yescharacteristic
243C2×C4×C137C22×D411yescharacteristic
244C2×C4×C137C22×D411yescharacteristic
245C2×C4×C137C22×D411yescharacteristic
246C8×C137—42no—
247C8×C137—41no—
248C8×C137—41no—
249C8×C137—42no—
250C22×C137—21no—
251C22×C137—21no—
252C22×C137—21no—
253C22×C137—41no—
254C22×C137—42no—
255C22×C137—41no—
256C22×C137—42no—
257C22×C137—41no—
258C22×C137—42no—
259C22×C137C2×#12011yescharacteristic
260C4×C137—21no—
261C4×C137—21no—
262C4×C137—21no—
263C4×C137—21no—
264C4×C137—41no—
265C4×C137—42no—
266C4×C137—41no—
267C4×C137—44no—
268C4×C137—41no—
269C4×C137—81no—
270C4×C137—81no—
271C4×C137#57511yescharacteristic
272C4×C137#57511yescharacteristic
273C4×C137#57712yes—
274C2×C137—41no—
275C2×C137—81no—
276C2×C137—82no—
277C2×C137#300511yescharacteristic
278C2×C137#300911yescharacteristic
279C2×C137#302011yescharacteristic
280#18833C13711yesmaximal, characteristic, direct product factor, Sylow
281C137#1883311yescharacteristic, direct product factor, Sylow
282#3511C2×C13711yescharacteristic
283#3235C2×C13711yescharacteristic, complemented
284#3229C2×C13711yescharacteristic, complemented
285#3116C2×C13711yescharacteristic, complemented
286#1654C2×C13712yescomplemented
287#1652C2×C13712yescomplemented
288#1638C2×C13712yescomplemented
289#1635C2×C13712yescomplemented
290#1556C2×C13711yescharacteristic, complemented
291#1496C2×C13711yescharacteristic, complemented
292#1488C2×C13711yescharacteristic, complemented
293#556C22×C13712yes—
294#555—21no—
295#554—21no—
296#553—22no—
297#548—22no—
298#547C22×C13712yes—
299#542—22no—
300#537—21no—
301#533—21no—
302#533C22×C13711yescharacteristic
303#532—21no—
304#532C22×C13711yescharacteristic
305C2×C4⋊Q8C22×C13711yescharacteristic
306#520—21no—
307C4×#41C22×C13711yescharacteristic
308#501—21no—
309#498C22×C13712yes—
310#490—21no—
311#487C22×C13712yes—
312#485—21no—
313#483—22no—
314#482—22no—
315#477C22×C13711yescharacteristic
316#476C22×C13711yescharacteristic
317#476C22×C13712yes—
318#475—22no—
319#474C22×C13712yes—
320#462C22×C13711yescharacteristic
321#461C22×C13711yescharacteristic
322#452—21no—
323#441—21no—
324#440—21no—
325#432C22×C13711yescharacteristic
326#432C22×C13711yescharacteristic
327#431C22×C13711yescharacteristic
328#332C22×C13712yes—
329#332C22×C13712yes—
330Q8⋊C8C22×C13712yes—
331D4⋊C8C22×C13712yes—
332D4⋊C8C22×C13712yes—
333D4⋊C8C22×C13712yes—
334C2×#41—22no—
335C2×#41—21no—
336C2×#41D4×C13711yescharacteristic
337C22×Q8—21no—
338C2×Q16—21no—
339C2×SD16—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8D4×C13712yes—
343C4⋊Q8D4×C13711yescharacteristic
344C4⋊Q8C23×C13711yescharacteristic
345C4⋊Q8C23×C13711yescharacteristic
346C4⋊Q8C23×C13712yes—
347#127D4×C13711yescharacteristic
348#126—24no—
349#125—22no—
350#125—21no—
351#124—24no—
352#124—22no—
353#124—21no—
354#123—24no—
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361#121—22no—
362#121—22no—
363#121—24no—
364#121—22no—
365#121—21no—
366#121D4×C13712yes—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8D4×C13711yescharacteristic
372C4×Q8C23×C13711yescharacteristic
373C4×D4—22no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4D4×C13711yescharacteristic
380C4×D4C23×C13711yescharacteristic
381C4×D4C23×C13712yes—
382#117—22no—
383#117—22no—
384#117—21no—
385#117—21no—
386#117D4×C13711yescharacteristic
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4D4×C13712yes—
390C2×C42—21no—
391C2×C42C23×C13711yescharacteristic
392#107—21no—
393#107—22no—
394#106—21no—
395#106—24no—
396#106—22no—
397C4⋊C8—21no—
398C4⋊C8C23×C13711yescharacteristic
399C4⋊C8C23×C13711yescharacteristic
400C4⋊C8C23×C13712yes—
401#103—21no—
402#103—21no—
403#103—21no—
404#103—24no—
405#102—21no—
406#102—22no—
407#102—22no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411#97—21no—
412#97—21no—
413C4×C8C23×C13712yes—
414#41—41no—
415#41—41no—
416#41—42no—
417#41—42no—
418#41—84no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—82no—
425C2×Q8C2×D4×C13711yescharacteristic
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4C2×D4×C13712yes—
432C2×D4C2×D4×C13711yescharacteristic
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4C2×D4×C13711yescharacteristic
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442Q16—42no—
443SD16—42no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—21no—
447C2×C8—21no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4C2×D4×C13711yescharacteristic
463C4⋊C4C2×D4×C13714yes—
464C4⋊C4C2×D4×C13711yescharacteristic
465C4⋊C4C2×D4×C13712yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—41no—
480C42C2×D4×C13711yescharacteristic
481C42C2×D4×C13711yescharacteristic
482C42C2×D4×C13711yescharacteristic
483C42C24×C13711yescharacteristic
484C23—21no—
485C23—22no—
486C23#120×C13711yescharacteristic
487Q8—21no—
488Q8—21no—
489Q8—42no—
490Q8—42no—
491Q8—44no—
492Q8—41no—
493Q8—41no—
494Q8—82no—
495D4—21no—
496D4—21no—
497D4—24no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—84no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4#120×C13711yescharacteristic
522C2×C4#120×C13711yescharacteristic
523C2×C4#120×C13711yescharacteristic
524C2×C4C22×D4×C13711yescharacteristic
525C2×C4C22×D4×C13711yescharacteristic
526C2×C4C22×D4×C13711yescharacteristic
527C8—42no—
528C8—41no—
529C8—41no—
530C8—42no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22C2×#120×C13711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—42no—
547C4—41no—
548C4—44no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4#575×C13711yescharacteristic
553C4#575×C13711yescharacteristic
554C4#577×C13712yes—
555C2—41no—
556C2—81no—
557C2—82no—
558C2#3005×C13711yescharacteristic
559C2#3009×C13711yescharacteristic
560C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3511×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
2. #3235×C137C2semidirect
3. #3229×C137C2semidirect
4. #3116×C137C2semidirect
5. #1654×C137C2semidirect
6. #1652×C137C2semidirect
7. #1638×C137C2semidirect
8. #1635×C137C2semidirect
9. #1556×C137C2semidirect
10. #1496×C137C2semidirect
11. #1488×C137C2semidirect
280. #18833C137direct (N × H)
281. C137#18833direct (N × H)
283. #3235C2×C137semidirect
284. #3229C2×C137semidirect
285. #3116C2×C137semidirect
286. #1654C2×C137semidirect
287. #1652C2×C137semidirect
288. #1638C2×C137semidirect
289. #1635C2×C137semidirect
290. #1556C2×C137semidirect
291. #1496C2×C137semidirect
292. #1488C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3511×C1372
#3235×C1372
#3229×C1372
#3116×C1372
#1654×C1372
#1652×C1372
#1638×C1372
#1635×C1372
#1556×C1372
#1496×C1372
#1488×C1372
#18833137