Finite Group Databasebeta

← C137×#18821 #46868207

Subgroups — C137×#18821 #46868207

Order 35072.

906 conjugacy classes of subgroups.

The subgroup lattice has 564 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#3209×C137C211yesmaximal, characteristic, complemented
4#3116×C137C211yesmaximal, characteristic, complemented
5#1646×C137C212yesmaximal, complemented
6#1644×C137C212yesmaximal, complemented
7#1634×C137C212yesmaximal, complemented
8#1632×C137C212yesmaximal, complemented
9#1558×C137C211yesmaximal, characteristic, complemented
10#1496×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12#556×C137—21no—
13#555×C137C2212yes—
14#554×C137C2212yes—
15#553×C137—22no—
16#548×C137—22no—
17#547×C137—21no—
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#494×C137C2212yes—
29#490×C137—21no—
30#487×C137C2212yes—
31#485×C137—21no—
32#482×C137—22no—
33#477×C137C2211yescharacteristic
34#476×C137C2211yescharacteristic
35#476×C137C2212yes—
36#474×C137C2212yes—
37#462×C137—22no—
38#462×C137C2211yescharacteristic
39#461×C137C2211yescharacteristic
40#452×C137—21no—
41#451×C137—22no—
42#441×C137—21no—
43#440×C137—21no—
44#432×C137C2211yescharacteristic
45#432×C137C2211yescharacteristic
46#431×C137C2211yescharacteristic
47#333×C137C2212yes—
48#333×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×C137D412yes—
57C2×Q16×C137—21no—
58C2×Q16×C137—24no—
59C2×SD16×C137—21no—
60C4⋊Q8×C137—21no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137D412yes—
63C4⋊Q8×C137D411yescharacteristic
64C4⋊Q8×C137C2311yescharacteristic
65C4⋊Q8×C137C2311yescharacteristic
66C4⋊Q8×C137C2312yes—
67#127×C137D411yescharacteristic
68#126×C137—24no—
69#125×C137—22no—
70#125×C137—21no—
71#124×C137—22no—
72#124×C137—24no—
73#124×C137—21no—
74#123×C137—24no—
75C22⋊Q8×C137—22no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137—24no—
79C22⋊Q8×C137—21no—
80C22⋊Q8×C137—21no—
81#121×C137—22no—
82#121×C137—24no—
83#121×C137—22no—
84#121×C137—22no—
85#121×C137—21no—
86#121×C137D412yes—
87C4×Q8×C137—22no—
88C4×Q8×C137—22no—
89C4×Q8×C137—21no—
90C4×Q8×C137—21no—
91C4×Q8×C137D411yescharacteristic
92C4×Q8×C137C2311yescharacteristic
93C4×D4×C137—22no—
94C4×D4×C137—24no—
95C4×D4×C137—22no—
96C4×D4×C137—21no—
97C4×D4×C137—21no—
98C4×D4×C137—21no—
99C4×D4×C137D411yescharacteristic
100C4×D4×C137C2311yescharacteristic
101C4×D4×C137C2312yes—
102#117×C137—22no—
103#117×C137—22no—
104#117×C137—21no—
105#117×C137—21no—
106#117×C137D411yescharacteristic
107C2×C4⋊C4×C137—21no—
108C2×C4⋊C4×C137—21no—
109C2×C4⋊C4×C137—22no—
110C2×C42×C137—21no—
111C2×C42×C137C2311yescharacteristic
112#107×C137—21no—
113#107×C137—22no—
114#106×C137—21no—
115#106×C137—22no—
116C4⋊C8×C137—21no—
117C4⋊C8×C137C2311yescharacteristic
118C4⋊C8×C137C2311yescharacteristic
119C4⋊C8×C137C2312yes—
120#103×C137—21no—
121#103×C137—24no—
122#103×C137—21no—
123#103×C137—21no—
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—21no—
141C2×Q8×C137—21no—
142C2×Q8×C137—82no—
143C2×Q8×C137C2×D411yescharacteristic
144C2×Q8×C137C2×D414yes—
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—
162Q16×C137—84no—
163SD16×C137—42no—
164C2×C8×C137—22no—
165C2×C8×C137—22no—
166C2×C8×C137—21no—
167C2×C8×C137—21no—
168C4⋊C4×C137—24no—
169C4⋊C4×C137—24no—
170C4⋊C4×C137—22no—
171C4⋊C4×C137—22no—
172C4⋊C4×C137—21no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—22no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—21no—
177C4⋊C4×C137—21no—
178C4⋊C4×C137—21no—
179C4⋊C4×C137—21no—
180C4⋊C4×C137—21no—
181C4⋊C4×C137—41no—
182C4⋊C4×C137—41no—
183C4⋊C4×C137C2×D411yescharacteristic
184C4⋊C4×C137C2×D411yescharacteristic
185C4⋊C4×C137C2×D412yes—
186C22⋊C4×C137—24no—
187C22⋊C4×C137—24no—
188C22⋊C4×C137—24no—
189C22⋊C4×C137—21no—
190C22⋊C4×C137—21no—
191C22⋊C4×C137—22no—
192C22⋊C4×C137—21no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—41no—
195C42×C137—24no—
196C42×C137—21no—
197C42×C137—21no—
198C42×C137—21no—
199C42×C137—41no—
200C42×C137C2×D411yescharacteristic
201C42×C137C2×D411yescharacteristic
202C42×C137C2×D411yescharacteristic
203C42×C137C2411yescharacteristic
204C23×C137—21no—
205C23×C137—22no—
206C23×C137#12011yescharacteristic
207Q8×C137—21no—
208Q8×C137—21no—
209Q8×C137—44no—
210Q8×C137—44no—
211Q8×C137—42no—
212Q8×C137—42no—
213Q8×C137—41no—
214Q8×C137—41no—
215D4×C137—21no—
216D4×C137—21no—
217D4×C137—24no—
218D4×C137—42no—
219D4×C137—42no—
220D4×C137—42no—
221D4×C137—44no—
222D4×C137—84no—
223C2×C4×C137—21no—
224C2×C4×C137—21no—
225C2×C4×C137—22no—
226C2×C4×C137—21no—
227C2×C4×C137—21no—
228C2×C4×C137—22no—
229C2×C4×C137—21no—
230C2×C4×C137—24no—
231C2×C4×C137—41no—
232C2×C4×C137—42no—
233C2×C4×C137—42no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—42no—
237C2×C4×C137—41no—
238C2×C4×C137—41no—
239C2×C4×C137—82no—
240C2×C4×C137—84no—
241C2×C4×C137#12011yescharacteristic
242C2×C4×C137#12011yescharacteristic
243C2×C4×C137#12011yescharacteristic
244C2×C4×C137C22×D411yescharacteristic
245C2×C4×C137C22×D411yescharacteristic
246C2×C4×C137C22×D411yescharacteristic
247C8×C137—42no—
248C8×C137—41no—
249C8×C137—41no—
250C8×C137—42no—
251C22×C137—21no—
252C22×C137—21no—
253C22×C137—21no—
254C22×C137—41no—
255C22×C137—42no—
256C22×C137—41no—
257C22×C137—42no—
258C22×C137—41no—
259C22×C137—42no—
260C22×C137C2×#12011yescharacteristic
261C4×C137—21no—
262C4×C137—21no—
263C4×C137—21no—
264C4×C137—21no—
265C4×C137—41no—
266C4×C137—42no—
267C4×C137—41no—
268C4×C137—41no—
269C4×C137—44no—
270C4×C137—81no—
271C4×C137—81no—
272C4×C137#57511yescharacteristic
273C4×C137#57511yescharacteristic
274C4×C137#57812yes—
275C2×C137—41no—
276C2×C137—81no—
277C2×C137—82no—
278C2×C137#300511yescharacteristic
279C2×C137#301111yescharacteristic
280C2×C137#301911yescharacteristic
281#18821C13711yesmaximal, characteristic, direct product factor, Sylow
282C137#1882111yescharacteristic, direct product factor, Sylow
283#3511C2×C13711yescharacteristic
284#3235C2×C13711yescharacteristic, complemented
285#3209C2×C13711yescharacteristic, complemented
286#3116C2×C13711yescharacteristic, complemented
287#1646C2×C13712yescomplemented
288#1644C2×C13712yescomplemented
289#1634C2×C13712yescomplemented
290#1632C2×C13712yescomplemented
291#1558C2×C13711yescharacteristic, complemented
292#1496C2×C13711yescharacteristic, complemented
293#1488C2×C13711yescharacteristic, complemented
294#556—21no—
295#555C22×C13712yes—
296#554C22×C13712yes—
297#553—22no—
298#548—22no—
299#547—21no—
300#542—22no—
301#537—21no—
302#533—21no—
303#533C22×C13711yescharacteristic
304#532—21no—
305#532C22×C13711yescharacteristic
306C2×C4⋊Q8C22×C13711yescharacteristic
307#520—21no—
308C4×#41C22×C13711yescharacteristic
309#501—21no—
310#494C22×C13712yes—
311#490—21no—
312#487C22×C13712yes—
313#485—21no—
314#482—22no—
315#477C22×C13711yescharacteristic
316#476C22×C13711yescharacteristic
317#476C22×C13712yes—
318#474C22×C13712yes—
319#462—22no—
320#462C22×C13711yescharacteristic
321#461C22×C13711yescharacteristic
322#452—21no—
323#451—22no—
324#441—21no—
325#440—21no—
326#432C22×C13711yescharacteristic
327#432C22×C13711yescharacteristic
328#431C22×C13711yescharacteristic
329#333C22×C13712yes—
330#333C22×C13712yes—
331Q8⋊C8C22×C13712yes—
332D4⋊C8C22×C13712yes—
333D4⋊C8C22×C13712yes—
334D4⋊C8C22×C13712yes—
335C2×#41—22no—
336C2×#41—21no—
337C2×#41D4×C13711yescharacteristic
338C22×Q8D4×C13712yes—
339C2×Q16—21no—
340C2×Q16—24no—
341C2×SD16—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8D4×C13712yes—
345C4⋊Q8D4×C13711yescharacteristic
346C4⋊Q8C23×C13711yescharacteristic
347C4⋊Q8C23×C13711yescharacteristic
348C4⋊Q8C23×C13712yes—
349#127D4×C13711yescharacteristic
350#126—24no—
351#125—22no—
352#125—21no—
353#124—22no—
354#124—24no—
355#124—21no—
356#123—24no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363#121—22no—
364#121—24no—
365#121—22no—
366#121—22no—
367#121—21no—
368#121D4×C13712yes—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8D4×C13711yescharacteristic
374C4×Q8C23×C13711yescharacteristic
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4D4×C13711yescharacteristic
382C4×D4C23×C13711yescharacteristic
383C4×D4C23×C13712yes—
384#117—22no—
385#117—22no—
386#117—21no—
387#117—21no—
388#117D4×C13711yescharacteristic
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—22no—
392C2×C42—21no—
393C2×C42C23×C13711yescharacteristic
394#107—21no—
395#107—22no—
396#106—21no—
397#106—22no—
398C4⋊C8—21no—
399C4⋊C8C23×C13711yescharacteristic
400C4⋊C8C23×C13711yescharacteristic
401C4⋊C8C23×C13712yes—
402#103—21no—
403#103—24no—
404#103—21no—
405#103—21no—
406#102—21no—
407#102—22no—
408#102—22no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412#97—21no—
413#97—21no—
414C4×C8C23×C13712yes—
415#41—41no—
416#41—41no—
417#41—42no—
418#41—42no—
419#41—84no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—82no—
425C2×Q8C2×D4×C13711yescharacteristic
426C2×Q8C2×D4×C13714yes—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×D4×C13712yes—
433C2×D4C2×D4×C13711yescharacteristic
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442C22×C4C2×D4×C13711yescharacteristic
443Q16—42no—
444Q16—84no—
445SD16—42no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—21no—
449C2×C8—21no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4C2×D4×C13711yescharacteristic
466C4⋊C4C2×D4×C13711yescharacteristic
467C4⋊C4C2×D4×C13712yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—21no—
475C22⋊C4—22no—
476C22⋊C4—41no—
477C42—24no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42C2×D4×C13711yescharacteristic
483C42C2×D4×C13711yescharacteristic
484C42C2×D4×C13711yescharacteristic
485C42C24×C13711yescharacteristic
486C23—21no—
487C23—22no—
488C23#120×C13711yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497D4—21no—
498D4—21no—
499D4—24no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—84no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4#120×C13711yescharacteristic
524C2×C4#120×C13711yescharacteristic
525C2×C4#120×C13711yescharacteristic
526C2×C4C22×D4×C13711yescharacteristic
527C2×C4C22×D4×C13711yescharacteristic
528C2×C4C22×D4×C13711yescharacteristic
529C8—42no—
530C8—41no—
531C8—41no—
532C8—42no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22C2×#120×C13711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—42no—
549C4—41no—
550C4—41no—
551C4—44no—
552C4—81no—
553C4—81no—
554C4#575×C13711yescharacteristic
555C4#575×C13711yescharacteristic
556C4#578×C13712yes—
557C2—41no—
558C2—81no—
559C2—82no—
560C2#3005×C13711yescharacteristic
561C2#3011×C13711yescharacteristic
562C2#3019×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. #3209×C137C2semidirect
4. #3116×C137C2semidirect
5. #1646×C137C2semidirect
6. #1644×C137C2semidirect
7. #1634×C137C2semidirect
8. #1632×C137C2semidirect
9. #1558×C137C2semidirect
10. #1496×C137C2semidirect
11. #1488×C137C2semidirect
281. #18821C137direct (N × H)
282. C137#18821direct (N × H)
284. #3235C2×C137semidirect
285. #3209C2×C137semidirect
286. #3116C2×C137semidirect
287. #1646C2×C137semidirect
288. #1644C2×C137semidirect
289. #1634C2×C137semidirect
290. #1632C2×C137semidirect
291. #1558C2×C137semidirect
292. #1496C2×C137semidirect
293. #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
#3209×C1372
#3116×C1372
#1646×C1372
#1644×C1372
#1634×C1372
#1632×C1372
#1558×C1372
#1496×C1372
#1488×C1372
#18821137