Finite Group Databasebeta

← C137×#18185 #46867571

Subgroups — C137×#18185 #46867571

Order 35072.

944 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3493×C137C211yesmaximal, characteristic
2#3117×C137C212yesmaximal, complemented
3#2396×C137C211yesmaximal, characteristic, complemented
4#1618×C137C212yesmaximal, complemented
5#1617×C137C211yesmaximal, characteristic, complemented
6#1616×C137C211yesmaximal, characteristic, complemented
7#1606×C137C212yesmaximal, complemented
8#1605×C137C211yesmaximal, characteristic, complemented
9#1604×C137C211yesmaximal, characteristic, complemented
10#1509×C137C212yesmaximal, complemented
11#1464×C137C211yesmaximal, characteristic, complemented
12#556×C137—22no—
13#555×C137—22no—
14#553×C137—22no—
15#552×C137—22no—
16#544×C137C2211yescharacteristic
17#538×C137—21no—
18#538×C137C2212yes—
19#537×C137C2211yescharacteristic
20#529×C137—21no—
21#529×C137C2212yes—
22#525×C137—21no—
23#520×C137—21no—
24C2×C4×Q8×C137C2211yescharacteristic
25C2×#117×C137C2211yescharacteristic
26#484×C137C2212yes—
27#481×C137C2212yes—
28#479×C137—22no—
29#464×C137—22no—
30#451×C137C2212yes—
31#448×C137C2212yes—
32#441×C137—22no—
33#440×C137—22no—
34#432×C137C2212yes—
35#400×C137C2211yescharacteristic
36#399×C137C2211yescharacteristic
37#398×C137—21no—
38#396×C137—21no—
39#395×C137C2211yescharacteristic
40#393×C137C2211yescharacteristic
41C4×C4⋊C4×C137—21no—
42C4×C4⋊C4×C137C2211yescharacteristic
43C4×C22⋊C4×C137—21no—
44#328×C137C2212yes—
45#328×C137C2212yes—
46#327×C137C2212yes—
47#327×C137C2212yes—
48#324×C137C2211yescharacteristic
49#324×C137C2211yescharacteristic
50#323×C137C2212yes—
51C2×#41×C137—21no—
52C22×Q8×C137C2311yescharacteristic
53C23×C4×C137D411yescharacteristic
54C2×Q16×C137—22no—
55C2×SD16×C137—22no—
56C4⋊Q8×C137—22no—
57#126×C137—22no—
58#126×C137—22no—
59#126×C137—22no—
60#126×C137—22no—
61#125×C137—21no—
62#125×C137—22no—
63#125×C137D412yes—
64#124×C137—21no—
65#124×C137—22no—
66#124×C137—22no—
67#124×C137D412yes—
68#123×C137—22no—
69#123×C137—22no—
70#123×C137—22no—
71C22⋊Q8×C137—21no—
72C22⋊Q8×C137—21no—
73C22⋊Q8×C137—21no—
74C22⋊Q8×C137—22no—
75C22⋊Q8×C137C2312yes—
76#121×C137—21no—
77#121×C137—21no—
78#121×C137—22no—
79#121×C137—22no—
80#121×C137—22no—
81#121×C137—21no—
82#121×C137C2312yes—
83#120×C137—22no—
84C4×Q8×C137—22no—
85C4×Q8×C137—21no—
86C4×Q8×C137—22no—
87C4×Q8×C137—22no—
88C4×Q8×C137D412yes—
89C4×D4×C137—22no—
90C4×D4×C137—22no—
91C4×D4×C137—22no—
92C4×D4×C137—21no—
93C4×D4×C137—22no—
94C4×D4×C137—22no—
95C4×D4×C137D412yes—
96#117×C137—22no—
97#117×C137—22no—
98#117×C137—21no—
99#117×C137—22no—
100#117×C137—22no—
101#117×C137—22no—
102C2×C4⋊C4×C137—22no—
103C2×C4⋊C4×C137—21no—
104C2×C4⋊C4×C137—21no—
105C2×C4⋊C4×C137—21no—
106C2×C4⋊C4×C137—21no—
107C2×C4⋊C4×C137D411yescharacteristic
108C2×C4⋊C4×C137C2311yescharacteristic
109C2×C4⋊C4×C137C2311yescharacteristic
110C2×C22⋊C4×C137—21no—
111C2×C22⋊C4×C137—21no—
112C2×C22⋊C4×C137D411yescharacteristic
113C2×C22⋊C4×C137C2311yescharacteristic
114C2×C42×C137—21no—
115C2×C42×C137—21no—
116C2×C42×C137D411yescharacteristic
117C2×C42×C137C2311yescharacteristic
118#107×C137—22no—
119#106×C137—22no—
120C4⋊C8×C137—22no—
121#103×C137—22no—
122#103×C137—22no—
123#103×C137—22no—
124#102×C137—22no—
125C22⋊C8×C137C2312yes—
126C22⋊C8×C137C2312yes—
127#97×C137—22no—
128#95×C137—21no—
129#95×C137—21no—
130#95×C137—21no—
131#95×C137—21no—
132#95×C137C2311yescharacteristic
133#95×C137C2311yescharacteristic
134C24×C137C2×D411yescharacteristic
135#41×C137—42no—
136#41×C137—42no—
137C2×Q8×C137—21no—
138C2×Q8×C137—21no—
139C2×Q8×C137—22no—
140C2×Q8×C137—42no—
141C2×Q8×C137C2×D412yes—
142C2×D4×C137—21no—
143C2×D4×C137—22no—
144C2×D4×C137—22no—
145C2×D4×C137—22no—
146C2×D4×C137—21no—
147C2×D4×C137C2×D412yes—
148C22×C4×C137—22no—
149C22×C4×C137—21no—
150C22×C4×C137—21no—
151C22×C4×C137—22no—
152C22×C4×C137—21no—
153C22×C4×C137—21no—
154C22×C4×C137—22no—
155C22×C4×C137—22no—
156C22×C4×C137—21no—
157C22×C4×C137—21no—
158C22×C4×C137—41no—
159C22×C4×C137C2×D411yescharacteristic
160C22×C4×C137C2×D411yescharacteristic
161C22×C4×C137C2×D411yescharacteristic
162C22×C4×C137C2×D411yescharacteristic
163C22×C4×C137C2×D411yescharacteristic
164C22×C4×C137C2411yescharacteristic
165Q16×C137—44no—
166SD16×C137—44no—
167C2×C8×C137—22no—
168C2×C8×C137—22no—
169C4⋊C4×C137—21no—
170C4⋊C4×C137—22no—
171C4⋊C4×C137—21no—
172C4⋊C4×C137—22no—
173C4⋊C4×C137—21no—
174C4⋊C4×C137—21no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137—22no—
177C4⋊C4×C137—22no—
178C4⋊C4×C137—22no—
179C4⋊C4×C137—22no—
180C4⋊C4×C137—22no—
181C4⋊C4×C137—22no—
182C4⋊C4×C137—42no—
183C4⋊C4×C137—42no—
184C4⋊C4×C137—42no—
185C4⋊C4×C137—42no—
186C4⋊C4×C137C2×D412yes—
187C4⋊C4×C137C2×D412yes—
188C22⋊C4×C137—22no—
189C22⋊C4×C137—22no—
190C22⋊C4×C137—22no—
191C22⋊C4×C137—22no—
192C22⋊C4×C137—22no—
193C22⋊C4×C137—22no—
194C22⋊C4×C137—22no—
195C22⋊C4×C137—22no—
196C22⋊C4×C137—22no—
197C22⋊C4×C137—22no—
198C22⋊C4×C137—22no—
199C22⋊C4×C137—42no—
200C22⋊C4×C137—42no—
201C42×C137—22no—
202C42×C137—22no—
203C42×C137—21no—
204C42×C137—21no—
205C42×C137—22no—
206C42×C137—42no—
207C42×C137—41no—
208C42×C137—41no—
209C42×C137C2×D412yes—
210C42×C137C2×D412yes—
211C23×C137—21no—
212C23×C137—22no—
213C23×C137—21no—
214C23×C137—21no—
215C23×C137—41no—
216C23×C137—41no—
217C23×C137C22×D411yescharacteristic
218Q8×C137—24no—
219Q8×C137—42no—
220Q8×C137—42no—
221Q8×C137—42no—
222Q8×C137—81no—
223D4×C137—42no—
224D4×C137—42no—
225D4×C137—42no—
226D4×C137—42no—
227D4×C137—82no—
228D4×C137—81no—
229D4×C137—82no—
230C2×C4×C137—21no—
231C2×C4×C137—21no—
232C2×C4×C137—21no—
233C2×C4×C137—21no—
234C2×C4×C137—22no—
235C2×C4×C137—22no—
236C2×C4×C137—21no—
237C2×C4×C137—22no—
238C2×C4×C137—21no—
239C2×C4×C137—22no—
240C2×C4×C137—22no—
241C2×C4×C137—41no—
242C2×C4×C137—41no—
243C2×C4×C137—42no—
244C2×C4×C137—42no—
245C2×C4×C137—44no—
246C2×C4×C137—41no—
247C2×C4×C137—41no—
248C2×C4×C137—41no—
249C2×C4×C137—41no—
250C2×C4×C137—41no—
251C2×C4×C137—41no—
252C2×C4×C137—41no—
253C2×C4×C137—41no—
254C2×C4×C137—41no—
255C2×C4×C137—41no—
256C2×C4×C137—41no—
257C2×C4×C137—41no—
258C2×C4×C137—82no—
259C2×C4×C137—82no—
260C2×C4×C137#12011yescharacteristic
261C2×C4×C137#12011yescharacteristic
262C2×C4×C137#12012yes—
263C2×C4×C137#13712yes—
264C2×C4×C137C22×D412yes—
265C8×C137—42no—
266C8×C137—42no—
267C22×C137—21no—
268C22×C137—21no—
269C22×C137—42no—
270C22×C137—41no—
271C22×C137—42no—
272C22×C137—42no—
273C22×C137—41no—
274C22×C137—42no—
275C22×C137—41no—
276C22×C137—81no—
277C22×C137C2×#12011yescharacteristic
278C22×C137C2×#13711yescharacteristic
279C22×C137#57511yescharacteristic
280C4×C137—22no—
281C4×C137—21no—
282C4×C137—21no—
283C4×C137—22no—
284C4×C137—22no—
285C4×C137—42no—
286C4×C137—42no—
287C4×C137—41no—
288C4×C137—42no—
289C4×C137—42no—
290C4×C137—41no—
291C4×C137—42no—
292C4×C137—81no—
293C4×C137—81no—
294C2×C137—21no—
295C2×C137—21no—
296C2×C137—82no—
297C2×C137—81no—
298C2×C137#300711yescharacteristic
299C2×C137#301811yescharacteristic
300C2×C137#302511yescharacteristic
301#18185C13711yesmaximal, characteristic, direct product factor, Sylow
302C137#1818511yescharacteristic, direct product factor, Sylow
303#3493C2×C13711yescharacteristic
304#3117C2×C13712yescomplemented
305#2396C2×C13711yescharacteristic, complemented
306#1618C2×C13712yescomplemented
307#1617C2×C13711yescharacteristic, complemented
308#1616C2×C13711yescharacteristic, complemented
309#1606C2×C13712yescomplemented
310#1605C2×C13711yescharacteristic, complemented
311#1604C2×C13711yescharacteristic, complemented
312#1509C2×C13712yescomplemented
313#1464C2×C13711yescharacteristic, complemented
314#556—22no—
315#555—22no—
316#553—22no—
317#552—22no—
318#544C22×C13711yescharacteristic
319#538—21no—
320#538C22×C13712yes—
321#537C22×C13711yescharacteristic
322#529—21no—
323#529C22×C13712yes—
324#525—21no—
325#520—21no—
326C2×C4×Q8C22×C13711yescharacteristic
327C2×#117C22×C13711yescharacteristic
328#484C22×C13712yes—
329#481C22×C13712yes—
330#479—22no—
331#464—22no—
332#451C22×C13712yes—
333#448C22×C13712yes—
334#441—22no—
335#440—22no—
336#432C22×C13712yes—
337#400C22×C13711yescharacteristic
338#399C22×C13711yescharacteristic
339#398—21no—
340#396—21no—
341#395C22×C13711yescharacteristic
342#393C22×C13711yescharacteristic
343C4×C4⋊C4—21no—
344C4×C4⋊C4C22×C13711yescharacteristic
345C4×C22⋊C4—21no—
346#328C22×C13712yes—
347#328C22×C13712yes—
348#327C22×C13712yes—
349#327C22×C13712yes—
350#324C22×C13711yescharacteristic
351#324C22×C13711yescharacteristic
352#323C22×C13712yes—
353C2×#41—21no—
354C22×Q8C23×C13711yescharacteristic
355C23×C4D4×C13711yescharacteristic
356C2×Q16—22no—
357C2×SD16—22no—
358C4⋊Q8—22no—
359#126—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#125—21no—
364#125—22no—
365#125D4×C13712yes—
366#124—21no—
367#124—22no—
368#124—22no—
369#124D4×C13712yes—
370#123—22no—
371#123—22no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8C23×C13712yes—
378#121—21no—
379#121—21no—
380#121—22no—
381#121—22no—
382#121—22no—
383#121—21no—
384#121C23×C13712yes—
385#120—22no—
386C4×Q8—22no—
387C4×Q8—21no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8D4×C13712yes—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4D4×C13712yes—
398#117—22no—
399#117—22no—
400#117—21no—
401#117—22no—
402#117—22no—
403#117—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4D4×C13711yescharacteristic
410C2×C4⋊C4C23×C13711yescharacteristic
411C2×C4⋊C4C23×C13711yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4D4×C13711yescharacteristic
415C2×C22⋊C4C23×C13711yescharacteristic
416C2×C42—21no—
417C2×C42—21no—
418C2×C42D4×C13711yescharacteristic
419C2×C42C23×C13711yescharacteristic
420#107—22no—
421#106—22no—
422C4⋊C8—22no—
423#103—22no—
424#103—22no—
425#103—22no—
426#102—22no—
427C22⋊C8C23×C13712yes—
428C22⋊C8C23×C13712yes—
429#97—22no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95C23×C13711yescharacteristic
435#95C23×C13711yescharacteristic
436C24C2×D4×C13711yescharacteristic
437#41—42no—
438#41—42no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—42no—
443C2×Q8C2×D4×C13712yes—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4C2×D4×C13712yes—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4C2×D4×C13711yescharacteristic
462C22×C4C2×D4×C13711yescharacteristic
463C22×C4C2×D4×C13711yescharacteristic
464C22×C4C2×D4×C13711yescharacteristic
465C22×C4C2×D4×C13711yescharacteristic
466C22×C4C24×C13711yescharacteristic
467Q16—44no—
468SD16—44no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4C2×D4×C13712yes—
489C4⋊C4C2×D4×C13712yes—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—22no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42C2×D4×C13712yes—
512C42C2×D4×C13712yes—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—41no—
518C23—41no—
519C23C22×D4×C13711yescharacteristic
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—81no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—82no—
530D4—81no—
531D4—82no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4#120×C13711yescharacteristic
563C2×C4#120×C13711yescharacteristic
564C2×C4#120×C13712yes—
565C2×C4#137×C13712yes—
566C2×C4C22×D4×C13712yes—
567C8—42no—
568C8—42no—
569C22—21no—
570C22—21no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—81no—
579C22C2×#120×C13711yescharacteristic
580C22C2×#137×C13711yescharacteristic
581C22#575×C13711yescharacteristic
582C4—22no—
583C4—21no—
584C4—21no—
585C4—22no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—81no—
595C4—81no—
596C2—21no—
597C2—21no—
598C2—81no—
599C2—82no—
600C2#3007×C13711yescharacteristic
601C2#3018×C13711yescharacteristic
602C2#3025×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radical#3493×C137C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3117×C137C2semidirect
3. #2396×C137C2semidirect
4. #1618×C137C2semidirect
5. #1617×C137C2semidirect
6. #1616×C137C2semidirect
7. #1606×C137C2semidirect
8. #1605×C137C2semidirect
9. #1604×C137C2semidirect
10. #1509×C137C2semidirect
11. #1464×C137C2semidirect
301. #18185C137direct (N × H)
302. C137#18185direct (N × H)
304. #3117C2×C137semidirect
305. #2396C2×C137semidirect
306. #1618C2×C137semidirect
307. #1617C2×C137semidirect
308. #1616C2×C137semidirect
309. #1606C2×C137semidirect
310. #1605C2×C137semidirect
311. #1604C2×C137semidirect
312. #1509C2×C137semidirect
313. #1464C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3493×C1372
#3117×C1372
#2396×C1372
#1618×C1372
#1617×C1372
#1616×C1372
#1606×C1372
#1605×C1372
#1604×C1372
#1509×C1372
#1464×C1372
#18185137