Finite Group Databasebeta

← C137×#18842 #46868228

Subgroups — C137×#18842 #46868228

Order 35072.

960 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514×C137C211yesmaximal, characteristic
2#3239×C137C211yesmaximal, characteristic, complemented
3#3227×C137C211yesmaximal, characteristic, complemented
4#3039×C137C211yesmaximal, characteristic, complemented
5#1652×C137C212yesmaximal, complemented
6#1651×C137C212yesmaximal, complemented
7#1640×C137C212yesmaximal, complemented
8#1639×C137C212yesmaximal, complemented
9#1557×C137C211yesmaximal, characteristic, complemented
10#1495×C137C211yesmaximal, characteristic, complemented
11#1488×C137C211yesmaximal, characteristic, complemented
12C2×#137×C137—21no—
13#559×C137C2211yescharacteristic
14#556×C137C2211yescharacteristic
15#555×C137—21no—
16#551×C137—22no—
17D4×Q8×C137C2211yescharacteristic
18#548×C137—21no—
19#546×C137—21no—
20#546×C137—21no—
21D42×C137C2211yescharacteristic
22#542×C137—21no—
23#538×C137—22no—
24#532×C137C2211yescharacteristic
25C2×C4⋊Q8×C137C2211yescharacteristic, complemented
26#529×C137—21no—
27#529×C137C2211yescharacteristic
28C2×#124×C137—21no—
29#518×C137—21no—
30C4×#41×C137C2211yescharacteristic
31#501×C137—21no—
32#498×C137C2212yescomplemented
33#489×C137—21no—
34#486×C137C2212yes—
35#484×C137—21no—
36#481×C137—22no—
37#481×C137—21no—
38#476×C137—22no—
39#475×C137C2212yescomplemented
40#474×C137C2212yes—
41#462×C137—21no—
42#462×C137C2212yescomplemented
43#461×C137—21no—
44#461×C137C2212yes—
45#432×C137C2211yescharacteristic, complemented
46#431×C137C2211yescharacteristic, complemented
47#423×C137C2211yescharacteristic, complemented
48#419×C137—21no—
49#332×C137C2212yescomplemented
50#331×C137C2212yes—
51Q8⋊C8×C137C2212yescomplemented
52Q8⋊C8×C137C2212yescomplemented
53D4⋊C8×C137C2212yes—
54D4⋊C8×C137C2212yes—
55C2×#41×C137—21no—
56C2×#41×C137D411yescharacteristic
57C22×Q8×C137—21no—
58C22×Q8×C137—21no—
59C22×D4×C137—21no—
60C22×D4×C137—21no—
61#137×C137—44no—
62C2×Q16×C137—22no—
63C2×SD16×C137—22no—
64C2×M16×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137—21no—
67C4⋊Q8×C137—21no—
68C4⋊Q8×C137—21no—
69C4⋊Q8×C137C2312yes—
70C4⋊Q8×C137C2311yescharacteristic
71#127×C137C2311yescharacteristic
72#126×C137—22no—
73#126×C137—22no—
74#126×C137—21no—
75#125×C137D411yescharacteristic
76#124×C137—22no—
77#124×C137—22no—
78#124×C137—22no—
79#124×C137—21no—
80#124×C137—21no—
81#124×C137—21no—
82#124×C137—21no—
83#124×C137D411yescharacteristic
84#123×C137—22no—
85#123×C137—22no—
86#123×C137—22no—
87#123×C137—21no—
88C22⋊Q8×C137—22no—
89C22⋊Q8×C137—21no—
90C22⋊Q8×C137—21no—
91C22⋊Q8×C137—22no—
92C22⋊Q8×C137D411yescharacteristic
93#121×C137—22no—
94#121×C137—22no—
95#121×C137—21no—
96#121×C137—21no—
97#121×C137—21no—
98#121×C137D411yescharacteristic
99#121×C137D412yes—
100#120×C137—22no—
101#120×C137—22no—
102C4×Q8×C137—21no—
103C4×Q8×C137D411yescharacteristic
104C4×Q8×C137C2311yescharacteristic
105C4×Q8×C137C2311yescharacteristic
106C4×D4×C137—22no—
107C4×D4×C137—22no—
108C4×D4×C137—21no—
109C4×D4×C137—21no—
110C4×D4×C137—21no—
111C4×D4×C137—21no—
112C4×D4×C137—21no—
113C4×D4×C137—21no—
114C4×D4×C137—21no—
115C4×D4×C137D411yescharacteristic
116C4×D4×C137D412yes—
117C4×D4×C137C2311yescharacteristic
118C4×D4×C137C2311yescharacteristic
119#117×C137—22no—
120#117×C137—21no—
121#117×C137—21no—
122#117×C137—21no—
123#117×C137D411yescharacteristic
124C2×C4⋊C4×C137—21no—
125C2×C4⋊C4×C137—21no—
126C2×C22⋊C4×C137—22no—
127C2×C22⋊C4×C137—22no—
128C2×C22⋊C4×C137—22no—
129C2×C42×C137—21no—
130C2×C42×C137C2311yescharacteristic
131#107×C137—22no—
132#106×C137—22no—
133#106×C137—24no—
134C4⋊C8×C137—21no—
135C4⋊C8×C137—21no—
136C4⋊C8×C137C2312yes—
137C4⋊C8×C137C2312yes—
138#103×C137—22no—
139#103×C137—22no—
140#103×C137—22no—
141#102×C137—22no—
142#102×C137—24no—
143C22⋊C8×C137—21no—
144C22⋊C8×C137—21no—
145#97×C137—21no—
146C4×C8×C137C2312yes—
147C24×C137—22no—
148#41×C137—22no—
149#41×C137—41no—
150#41×C137—82no—
151C2×Q8×C137—22no—
152C2×Q8×C137—21no—
153C2×Q8×C137—22no—
154C2×Q8×C137—42no—
155C2×Q8×C137—42no—
156C2×Q8×C137—41no—
157C2×Q8×C137—82no—
158C2×Q8×C137C2×D411yescharacteristic
159C2×Q8×C137C2×D411yescharacteristic
160C2×D4×C137—22no—
161C2×D4×C137—21no—
162C2×D4×C137—22no—
163C2×D4×C137—21no—
164C2×D4×C137—22no—
165C2×D4×C137—21no—
166C2×D4×C137—21no—
167C2×D4×C137—42no—
168C2×D4×C137—42no—
169C2×D4×C137—82no—
170C2×D4×C137C2×D411yescharacteristic
171C2×D4×C137C2×D411yescharacteristic
172C2×D4×C137C2×D412yes—
173C22×C4×C137—22no—
174C22×C4×C137—22no—
175C22×C4×C137—21no—
176C22×C4×C137—21no—
177C22×C4×C137—21no—
178C22×C4×C137—21no—
179C22×C4×C137—22no—
180C22×C4×C137—41no—
181C22×C4×C137C2×D411yescharacteristic
182C22×C4×C137C2×D411yescharacteristic
183C22×C4×C137C2×D411yescharacteristic
184Q16×C137—44no—
185SD16×C137—44no—
186M16×C137—42no—
187C2×C8×C137—22no—
188C2×C8×C137—22no—
189C2×C8×C137—22no—
190C4⋊C4×C137—22no—
191C4⋊C4×C137—22no—
192C4⋊C4×C137—21no—
193C4⋊C4×C137—21no—
194C4⋊C4×C137—21no—
195C4⋊C4×C137—21no—
196C4⋊C4×C137—21no—
197C4⋊C4×C137—21no—
198C4⋊C4×C137—21no—
199C4⋊C4×C137—22no—
200C4⋊C4×C137—22no—
201C4⋊C4×C137—41no—
202C4⋊C4×C137—41no—
203C4⋊C4×C137C2×D411yescharacteristic
204C4⋊C4×C137C2×D411yescharacteristic
205C4⋊C4×C137C2×D411yescharacteristic
206C4⋊C4×C137C2×D412yes—
207C4⋊C4×C137C2×D411yescharacteristic
208C22⋊C4×C137—22no—
209C22⋊C4×C137—22no—
210C22⋊C4×C137—22no—
211C22⋊C4×C137—22no—
212C22⋊C4×C137—22no—
213C22⋊C4×C137—22no—
214C22⋊C4×C137—22no—
215C22⋊C4×C137—21no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—21no—
218C22⋊C4×C137—21no—
219C22⋊C4×C137—21no—
220C22⋊C4×C137—21no—
221C22⋊C4×C137—22no—
222C22⋊C4×C137—22no—
223C22⋊C4×C137—21no—
224C22⋊C4×C137—21no—
225C42×C137—22no—
226C42×C137—21no—
227C42×C137—21no—
228C42×C137—21no—
229C42×C137—21no—
230C42×C137C2×D411yescharacteristic
231C42×C137C2×D411yescharacteristic
232C42×C137C2×D411yescharacteristic
233C42×C137C2411yescharacteristic
234C23×C137—21no—
235C23×C137—21no—
236C23×C137—22no—
237C23×C137—44no—
238C23×C137—82no—
239C23×C137—82no—
240C23×C137#12011yescharacteristic
241Q8×C137—21no—
242Q8×C137—21no—
243Q8×C137—22no—
244Q8×C137—42no—
245Q8×C137—42no—
246Q8×C137—42no—
247Q8×C137—82no—
248Q8×C137—82no—
249D4×C137—21no—
250D4×C137—22no—
251D4×C137—42no—
252D4×C137—42no—
253D4×C137—44no—
254D4×C137—41no—
255D4×C137—41no—
256D4×C137—42no—
257D4×C137—81no—
258D4×C137—82no—
259D4×C137—82no—
260D4×C137AGL(1,ℤ/8)12yes—
261C2×C4×C137—21no—
262C2×C4×C137—21no—
263C2×C4×C137—22no—
264C2×C4×C137—21no—
265C2×C4×C137—21no—
266C2×C4×C137—21no—
267C2×C4×C137—21no—
268C2×C4×C137—21no—
269C2×C4×C137—21no—
270C2×C4×C137—22no—
271C2×C4×C137—41no—
272C2×C4×C137—42no—
273C2×C4×C137—41no—
274C2×C4×C137—41no—
275C2×C4×C137—42no—
276C2×C4×C137—42no—
277C2×C4×C137—81no—
278C2×C4×C137—82no—
279C2×C4×C137—82no—
280C2×C4×C137—82no—
281C2×C4×C137—82no—
282C2×C4×C137#12011yescharacteristic
283C2×C4×C137#12011yescharacteristic
284C2×C4×C137#12011yescharacteristic
285C2×C4×C137C22×D411yescharacteristic
286C2×C4×C137C22×D411yescharacteristic
287C2×C4×C137C22×D411yescharacteristic
288C8×C137—42no—
289C8×C137—42no—
290C8×C137—42no—
291C22×C137—21no—
292C22×C137—22no—
293C22×C137—21no—
294C22×C137—21no—
295C22×C137—41no—
296C22×C137—41no—
297C22×C137—41no—
298C22×C137—42no—
299C22×C137—42no—
300C22×C137—41no—
301C22×C137—42no—
302C22×C137—41no—
303C22×C137—164no—
304C22×C137C2×#12011yescharacteristic
305C4×C137—21no—
306C4×C137—21no—
307C4×C137—21no—
308C4×C137—21no—
309C4×C137—41no—
310C4×C137—41no—
311C4×C137—41no—
312C4×C137—42no—
313C4×C137—41no—
314C4×C137—41no—
315C4×C137—41no—
316C4×C137—82no—
317C4×C137C2×AGL(1,ℤ/8)11yescharacteristic
318C4×C137#57511yescharacteristic
319C4×C137#57712yes—
320C2×C137—41no—
321C2×C137—42no—
322C2×C137—82no—
323C2×C137—81no—
324C2×C137#300611yescharacteristic
325C2×C137#300911yescharacteristic
326C2×C137#302211yescharacteristic
327#18842C13711yesmaximal, characteristic, direct product factor, Sylow
328C137#1884211yescharacteristic, direct product factor, Sylow
329#3514C2×C13711yescharacteristic
330#3239C2×C13711yescharacteristic, complemented
331#3227C2×C13711yescharacteristic, complemented
332#3039C2×C13711yescharacteristic, complemented
333#1652C2×C13712yescomplemented
334#1651C2×C13712yescomplemented
335#1640C2×C13712yescomplemented
336#1639C2×C13712yescomplemented
337#1557C2×C13711yescharacteristic, complemented
338#1495C2×C13711yescharacteristic, complemented
339#1488C2×C13711yescharacteristic, complemented
340C2×#137—21no—
341#559C22×C13711yescharacteristic
342#556C22×C13711yescharacteristic
343#555—21no—
344#551—22no—
345D4×Q8C22×C13711yescharacteristic
346#548—21no—
347#546—21no—
348#546—21no—
349D42C22×C13711yescharacteristic
350#542—21no—
351#538—22no—
352#532C22×C13711yescharacteristic
353C2×C4⋊Q8C22×C13711yescharacteristic, complemented
354#529—21no—
355#529C22×C13711yescharacteristic
356C2×#124—21no—
357#518—21no—
358C4×#41C22×C13711yescharacteristic
359#501—21no—
360#498C22×C13712yescomplemented
361#489—21no—
362#486C22×C13712yes—
363#484—21no—
364#481—22no—
365#481—21no—
366#476—22no—
367#475C22×C13712yescomplemented
368#474C22×C13712yes—
369#462—21no—
370#462C22×C13712yescomplemented
371#461—21no—
372#461C22×C13712yes—
373#432C22×C13711yescharacteristic, complemented
374#431C22×C13711yescharacteristic, complemented
375#423C22×C13711yescharacteristic, complemented
376#419—21no—
377#332C22×C13712yescomplemented
378#331C22×C13712yes—
379Q8⋊C8C22×C13712yescomplemented
380Q8⋊C8C22×C13712yescomplemented
381D4⋊C8C22×C13712yes—
382D4⋊C8C22×C13712yes—
383C2×#41—21no—
384C2×#41D4×C13711yescharacteristic
385C22×Q8—21no—
386C22×Q8—21no—
387C22×D4—21no—
388C22×D4—21no—
389#137—44no—
390C2×Q16—22no—
391C2×SD16—22no—
392C2×M16—21no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8C23×C13712yes—
398C4⋊Q8C23×C13711yescharacteristic
399#127C23×C13711yescharacteristic
400#126—22no—
401#126—22no—
402#126—21no—
403#125D4×C13711yescharacteristic
404#124—22no—
405#124—22no—
406#124—22no—
407#124—21no—
408#124—21no—
409#124—21no—
410#124—21no—
411#124D4×C13711yescharacteristic
412#123—22no—
413#123—22no—
414#123—22no—
415#123—21no—
416C22⋊Q8—22no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—22no—
420C22⋊Q8D4×C13711yescharacteristic
421#121—22no—
422#121—22no—
423#121—21no—
424#121—21no—
425#121—21no—
426#121D4×C13711yescharacteristic
427#121D4×C13712yes—
428#120—22no—
429#120—22no—
430C4×Q8—21no—
431C4×Q8D4×C13711yescharacteristic
432C4×Q8C23×C13711yescharacteristic
433C4×Q8C23×C13711yescharacteristic
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4D4×C13711yescharacteristic
444C4×D4D4×C13712yes—
445C4×D4C23×C13711yescharacteristic
446C4×D4C23×C13711yescharacteristic
447#117—22no—
448#117—21no—
449#117—21no—
450#117—21no—
451#117D4×C13711yescharacteristic
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C22⋊C4—22no—
455C2×C22⋊C4—22no—
456C2×C22⋊C4—22no—
457C2×C42—21no—
458C2×C42C23×C13711yescharacteristic
459#107—22no—
460#106—22no—
461#106—24no—
462C4⋊C8—21no—
463C4⋊C8—21no—
464C4⋊C8C23×C13712yes—
465C4⋊C8C23×C13712yes—
466#103—22no—
467#103—22no—
468#103—22no—
469#102—22no—
470#102—24no—
471C22⋊C8—21no—
472C22⋊C8—21no—
473#97—21no—
474C4×C8C23×C13712yes—
475C24—22no—
476#41—22no—
477#41—41no—
478#41—82no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—22no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—41no—
485C2×Q8—82no—
486C2×Q8C2×D4×C13711yescharacteristic
487C2×Q8C2×D4×C13711yescharacteristic
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4—82no—
498C2×D4C2×D4×C13711yescharacteristic
499C2×D4C2×D4×C13711yescharacteristic
500C2×D4C2×D4×C13712yes—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—41no—
509C22×C4C2×D4×C13711yescharacteristic
510C22×C4C2×D4×C13711yescharacteristic
511C22×C4C2×D4×C13711yescharacteristic
512Q16—44no—
513SD16—44no—
514M16—42no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4C2×D4×C13711yescharacteristic
532C4⋊C4C2×D4×C13711yescharacteristic
533C4⋊C4C2×D4×C13711yescharacteristic
534C4⋊C4C2×D4×C13712yes—
535C4⋊C4C2×D4×C13711yescharacteristic
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C42—22no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42C2×D4×C13711yescharacteristic
559C42C2×D4×C13711yescharacteristic
560C42C2×D4×C13711yescharacteristic
561C42C24×C13711yescharacteristic
562C23—21no—
563C23—21no—
564C23—22no—
565C23—44no—
566C23—82no—
567C23—82no—
568C23#120×C13711yescharacteristic
569Q8—21no—
570Q8—21no—
571Q8—22no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—82no—
576Q8—82no—
577D4—21no—
578D4—22no—
579D4—42no—
580D4—42no—
581D4—44no—
582D4—41no—
583D4—41no—
584D4—42no—
585D4—82no—
586D4—81no—
587D4—82no—
588D4AGL(1,ℤ/8)×C13712yes—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4#120×C13711yescharacteristic
611C2×C4#120×C13711yescharacteristic
612C2×C4#120×C13711yescharacteristic
613C2×C4C22×D4×C13711yescharacteristic
614C2×C4C22×D4×C13711yescharacteristic
615C2×C4C22×D4×C13711yescharacteristic
616C8—42no—
617C8—42no—
618C8—42no—
619C22—21no—
620C22—22no—
621C22—21no—
622C22—21no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—42no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—164no—
632C22C2×#120×C13711yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4C2×AGL(1,ℤ/8)×C13711yescharacteristic
646C4#575×C13711yescharacteristic
647C4#577×C13712yes—
648C2—41no—
649C2—42no—
650C2—81no—
651C2—82no—
652C2#3006×C13711yescharacteristic
653C2#3009×C13711yescharacteristic
654C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radical#3514×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. #3239×C137C2semidirect
3. #3227×C137C2semidirect
4. #3039×C137C2semidirect
5. #1652×C137C2semidirect
6. #1651×C137C2semidirect
7. #1640×C137C2semidirect
8. #1639×C137C2semidirect
9. #1557×C137C2semidirect
10. #1495×C137C2semidirect
11. #1488×C137C2semidirect
25. C2×C4⋊Q8×C137C22semidirect
32. #498×C137C22semidirect
39. #475×C137C22semidirect
42. #462×C137C22semidirect
45. #432×C137C22semidirect
46. #431×C137C22semidirect
47. #423×C137C22semidirect
49. #332×C137C22semidirect
51. Q8⋊C8×C137C22semidirect
52. Q8⋊C8×C137C22semidirect
327. #18842C137direct (N × H)
328. C137#18842direct (N × H)
330. #3239C2×C137semidirect
331. #3227C2×C137semidirect
332. #3039C2×C137semidirect
333. #1652C2×C137semidirect
334. #1651C2×C137semidirect
335. #1640C2×C137semidirect
336. #1639C2×C137semidirect
337. #1557C2×C137semidirect
338. #1495C2×C137semidirect
339. #1488C2×C137semidirect
353. C2×C4⋊Q8C22×C137semidirect
360. #498C22×C137semidirect
367. #475C22×C137semidirect
370. #462C22×C137semidirect
373. #432C22×C137semidirect
374. #431C22×C137semidirect
375. #423C22×C137semidirect
377. #332C22×C137semidirect
379. Q8⋊C8C22×C137semidirect
380. Q8⋊C8C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3514×C1372
#3239×C1372
#3227×C1372
#3039×C1372
#1652×C1372
#1651×C1372
#1640×C1372
#1639×C1372
#1557×C1372
#1495×C1372
#1488×C1372
#18842137