Finite Group Databasebeta

← C137×#25833 #46875219

Subgroups — C137×#25833 #46875219

Order 35072.

934 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3504×C137C211yesmaximal, characteristic
2#3165×C137C211yesmaximal, characteristic, complemented
3#3079×C137C211yesmaximal, characteristic
4#2660×C137C211yesmaximal, characteristic, complemented
5#2061×C137C212yesmaximal, complemented
6#2060×C137C212yesmaximal, complemented
7#1998×C137C211yesmaximal, characteristic, complemented
8#1995×C137C211yesmaximal, characteristic, complemented
9#1860×C137C212yesmaximal
10#1859×C137C212yesmaximal
11#1759×C137C211yesmaximal, characteristic
12#557×C137—21no—
13#557×C137—21no—
14#554×C137—21no—
15#554×C137—21no—
16#554×C137—22no—
17#554×C137—22no—
18#544×C137—21no—
19#544×C137—21no—
20#544×C137C2211yescharacteristic
21#544×C137C2211yescharacteristic
22#533×C137—21no—
23#533×C137C2211yescharacteristic
24C2×C4⋊Q8×C137C2211yescharacteristic
25C2×#125×C137—21no—
26C2×#125×C137C2211yescharacteristic
27#520×C137C2211yescharacteristic
28#520×C137C2211yescharacteristic
29C4×#41×C137C2211yescharacteristic
30#501×C137—21no—
31#501×C137—21no—
32#500×C137—21no—
33#498×C137—21no—
34#477×C137—21no—
35#476×C137—21no—
36#475×C137—21no—
37#474×C137—21no—
38#428×C137C2212yes—
39#428×C137C2212yes—
40#428×C137C2212yes—
41#423×C137C2211yescharacteristic
42C2×#104×C137C2212yes—
43#417×C137C2212yes—
44C4×M16×C137C2211yescharacteristic
45#398×C137—22no—
46#395×C137—22no—
47#384×C137C2212yes—
48#382×C137C2212yes—
49#376×C137C2211yescharacteristic
50#343×C137C2212yes—
51#343×C137C2212yes—
52#339×C137C2212yes—
53#339×C137C2212yes—
54#339×C137C2212yes—
55C2×#41×C137C2311yescharacteristic
56C22×Q8×C137—21no—
57C2×M16×C137C2312yes—
58C2×M16×C137C2312yes—
59C4⋊Q8×C137—22no—
60C4⋊Q8×C137—22no—
61C4⋊Q8×C137—21no—
62C4⋊Q8×C137—21no—
63C4⋊Q8×C137—21no—
64C4⋊Q8×C137—21no—
65C4⋊Q8×C137—21no—
66C4⋊Q8×C137—21no—
67C4⋊Q8×C137—21no—
68C4⋊Q8×C137—21no—
69C4⋊Q8×C137D411yescharacteristic
70C4⋊Q8×C137D411yescharacteristic
71#125×C137—22no—
72#125×C137—22no—
73#125×C137—22no—
74#125×C137—22no—
75#125×C137—21no—
76#125×C137—21no—
77#125×C137—21no—
78#125×C137—21no—
79#125×C137—21no—
80#125×C137—21no—
81#125×C137—21no—
82#125×C137—21no—
83#125×C137D411yescharacteristic
84#125×C137D411yescharacteristic
85C22⋊Q8×C137—22no—
86C22⋊Q8×C137—22no—
87C22⋊Q8×C137—22no—
88C22⋊Q8×C137—22no—
89C22⋊Q8×C137—22no—
90C22⋊Q8×C137—22no—
91C22⋊Q8×C137—21no—
92C22⋊Q8×C137—21no—
93C22⋊Q8×C137—22no—
94C22⋊Q8×C137—22no—
95C4×Q8×C137—22no—
96C4×Q8×C137—22no—
97C4×Q8×C137—21no—
98C4×Q8×C137—21no—
99C4×Q8×C137—21no—
100C4×Q8×C137—21no—
101C4×Q8×C137D411yescharacteristic
102C4×Q8×C137D411yescharacteristic
103C4×D4×C137—21no—
104C4×D4×C137—21no—
105C4×D4×C137—21no—
106C4×D4×C137—21no—
107C4×D4×C137—22no—
108C4×D4×C137—22no—
109C4×D4×C137D411yescharacteristic
110C4×D4×C137D411yescharacteristic
111#117×C137—21no—
112#117×C137—21no—
113#117×C137—21no—
114#117×C137C2311yescharacteristic
115#117×C137C2311yescharacteristic
116#117×C137C2311yescharacteristic
117C2×C4⋊C4×C137—22no—
118C2×C4⋊C4×C137—22no—
119C2×C4⋊C4×C137—21no—
120C2×C4⋊C4×C137—21no—
121C2×C4⋊C4×C137—21no—
122C2×C4⋊C4×C137—24no—
123C2×C4⋊C4×C137D412yes—
124C2×C4⋊C4×C137D412yes—
125C2×C4⋊C4×C137C2312yes—
126C2×C4⋊C4×C137C2311yescharacteristic
127C2×C4⋊C4×C137C2311yescharacteristic
128C2×C42×C137—21no—
129C2×C42×C137C2312yes—
130C2×C42×C137C2311yescharacteristic
131#107×C137—22no—
132#107×C137—22no—
133#107×C137—22no—
134#106×C137—22no—
135#106×C137—22no—
136#106×C137—22no—
137C4⋊C8×C137—21no—
138C4⋊C8×C137—21no—
139#104×C137Q818yescomplemented
140#103×C137—22no—
141#102×C137—22no—
142#97×C137—21no—
143C4×C8×C137—21no—
144#95×C137—21no—
145#95×C137—21no—
146#95×C137—24no—
147#41×C137—22no—
148#41×C137C2×Q814yes—
149C2×Q8×C137—22no—
150C2×Q8×C137—22no—
151C2×Q8×C137—21no—
152C2×Q8×C137—21no—
153C2×Q8×C137—41no—
154C2×Q8×C137—41no—
155C2×Q8×C137—82no—
156C2×Q8×C137C2×D411yescharacteristic
157C2×D4×C137—21no—
158C2×D4×C137C2×D411yescharacteristic
159C22×C4×C137—21no—
160C22×C4×C137—22no—
161C22×C4×C137—22no—
162C22×C4×C137—22no—
163C22×C4×C137—21no—
164C22×C4×C137—21no—
165C22×C4×C137—22no—
166C22×C4×C137C2×D412yes—
167C22×C4×C137C2×D412yes—
168C22×C4×C137C2×D411yescharacteristic
169C22×C4×C137C2×D411yescharacteristic
170C22×C4×C137C2411yescharacteristic
171M16×C137—22no—
172M16×C137—24no—
173M16×C137C2×Q814yes—
174C2×C8×C137—22no—
175C2×C8×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—22no—
183C4⋊C4×C137—22no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—22no—
189C4⋊C4×C137—22no—
190C4⋊C4×C137—21no—
191C4⋊C4×C137—21no—
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—21no—
200C4⋊C4×C137—22no—
201C4⋊C4×C137—22no—
202C4⋊C4×C137—41no—
203C4⋊C4×C137—41no—
204C4⋊C4×C137—84no—
205C4⋊C4×C137C2×D411yescharacteristic
206C4⋊C4×C137C2×D411yescharacteristic
207C4⋊C4×C137C2×D411yescharacteristic
208C4⋊C4×C137C2×D411yescharacteristic
209C4⋊C4×C137C2×D411yescharacteristic
210C4⋊C4×C137C2×D411yescharacteristic
211C22⋊C4×C137—22no—
212C22⋊C4×C137—22no—
213C22⋊C4×C137—21no—
214C22⋊C4×C137—21no—
215C22⋊C4×C137—21no—
216C22⋊C4×C137—21no—
217C22⋊C4×C137—21no—
218C42×C137—22no—
219C42×C137—22no—
220C42×C137—21no—
221C42×C137—21no—
222C42×C137—21no—
223C42×C137—21no—
224C42×C137—21no—
225C42×C137—22no—
226C42×C137C2×D411yescharacteristic
227C42×C137C2×D411yescharacteristic
228C42×C137C2×D411yescharacteristic
229C42×C137C2×D411yescharacteristic
230C42×C137C2×Q814yes—
231C23×C137—21no—
232C23×C137C22×D411yescharacteristic
233Q8×C137—21no—
234Q8×C137—81no—
235Q8×C137—81no—
236Q8×C137—82no—
237Q8×C137—82no—
238Q8×C137—82no—
239Q8×C137—82no—
240Q8×C137C4⋊Q812yes—
241D4×C137—24no—
242D4×C137—21no—
243D4×C137C4⋊Q812yes—
244C2×C4×C137—21no—
245C2×C4×C137—24no—
246C2×C4×C137—21no—
247C2×C4×C137—21no—
248C2×C4×C137—21no—
249C2×C4×C137—21no—
250C2×C4×C137—22no—
251C2×C4×C137—22no—
252C2×C4×C137—21no—
253C2×C4×C137—21no—
254C2×C4×C137—24no—
255C2×C4×C137—41no—
256C2×C4×C137—41no—
257C2×C4×C137—42no—
258C2×C4×C137—42no—
259C2×C4×C137—42no—
260C2×C4×C137—42no—
261C2×C4×C137—44no—
262C2×C4×C137—42no—
263C2×C4×C137—81no—
264C2×C4×C137—81no—
265C2×C4×C137—82no—
266C2×C4×C137—82no—
267C2×C4×C137#12011yescharacteristic
268C2×C4×C137#12011yescharacteristic
269C2×C4×C137#12011yescharacteristic
270C2×C4×C137#12011yescharacteristic
271C2×C4×C137C22×D411yescharacteristic
272C2×C4×C137C22×D411yescharacteristic
273C2×C4×C137C22×Q812yes—
274C2×C4×C1372−1+412yes—
275C8×C137—24no—
276C8×C137—42no—
277C22×C137—21no—
278C22×C137—21no—
279C22×C137—22no—
280C22×C137—41no—
281C22×C137—41no—
282C22×C137C2×#12011yescharacteristic
283C22×C137D4×Q812yes—
284C4×C137—22no—
285C4×C137—21no—
286C4×C137—21no—
287C4×C137—21no—
288C4×C137—21no—
289C4×C137—44no—
290C4×C137—42no—
291C4×C137—41no—
292C4×C137—41no—
293C4×C137—41no—
294C4×C137—42no—
295C4×C137—41no—
296C4×C137—41no—
297C4×C137—41no—
298C4×C137—82no—
299C4×C137C2×C4⋊Q811yescharacteristic
300C4×C137#53611yescharacteristic
301C4×C137D4×Q812yes—
302C2×C137—22no—
303C2×C137—42no—
304C2×C137#243411yescharacteristic
305C2×C137#302111yescharacteristic
306C2×C137#302211yescharacteristic
307#25833C13711yesmaximal, characteristic, direct product factor, Sylow
308C137#2583311yescharacteristic, direct product factor, Sylow
309#3504C2×C13711yescharacteristic
310#3165C2×C13711yescharacteristic, complemented
311#3079C2×C13711yescharacteristic
312#2660C2×C13711yescharacteristic, complemented
313#2061C2×C13712yescomplemented
314#2060C2×C13712yescomplemented
315#1998C2×C13711yescharacteristic, complemented
316#1995C2×C13711yescharacteristic, complemented
317#1860C2×C13712yes—
318#1859C2×C13712yes—
319#1759C2×C13711yescharacteristic
320#557—21no—
321#557—21no—
322#554—21no—
323#554—21no—
324#554—22no—
325#554—22no—
326#544—21no—
327#544—21no—
328#544C22×C13711yescharacteristic
329#544C22×C13711yescharacteristic
330#533—21no—
331#533C22×C13711yescharacteristic
332C2×C4⋊Q8C22×C13711yescharacteristic
333C2×#125—21no—
334C2×#125C22×C13711yescharacteristic
335#520C22×C13711yescharacteristic
336#520C22×C13711yescharacteristic
337C4×#41C22×C13711yescharacteristic
338#501—21no—
339#501—21no—
340#500—21no—
341#498—21no—
342#477—21no—
343#476—21no—
344#475—21no—
345#474—21no—
346#428C22×C13712yes—
347#428C22×C13712yes—
348#428C22×C13712yes—
349#423C22×C13711yescharacteristic
350C2×#104C22×C13712yes—
351#417C22×C13712yes—
352C4×M16C22×C13711yescharacteristic
353#398—22no—
354#395—22no—
355#384C22×C13712yes—
356#382C22×C13712yes—
357#376C22×C13711yescharacteristic
358#343C22×C13712yes—
359#343C22×C13712yes—
360#339C22×C13712yes—
361#339C22×C13712yes—
362#339C22×C13712yes—
363C2×#41C23×C13711yescharacteristic
364C22×Q8—21no—
365C2×M16C23×C13712yes—
366C2×M16C23×C13712yes—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8D4×C13711yescharacteristic
378C4⋊Q8D4×C13711yescharacteristic
379#125—22no—
380#125—22no—
381#125—22no—
382#125—22no—
383#125—21no—
384#125—21no—
385#125—21no—
386#125—21no—
387#125—21no—
388#125—21no—
389#125—21no—
390#125—21no—
391#125D4×C13711yescharacteristic
392#125D4×C13711yescharacteristic
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8D4×C13711yescharacteristic
410C4×Q8D4×C13711yescharacteristic
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4D4×C13711yescharacteristic
418C4×D4D4×C13711yescharacteristic
419#117—21no—
420#117—21no—
421#117—21no—
422#117C23×C13711yescharacteristic
423#117C23×C13711yescharacteristic
424#117C23×C13711yescharacteristic
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—24no—
431C2×C4⋊C4D4×C13712yes—
432C2×C4⋊C4D4×C13712yes—
433C2×C4⋊C4C23×C13712yes—
434C2×C4⋊C4C23×C13711yescharacteristic
435C2×C4⋊C4C23×C13711yescharacteristic
436C2×C42—21no—
437C2×C42C23×C13712yes—
438C2×C42C23×C13711yescharacteristic
439#107—22no—
440#107—22no—
441#107—22no—
442#106—22no—
443#106—22no—
444#106—22no—
445C4⋊C8—21no—
446C4⋊C8—21no—
447#104Q8×C13718yescomplemented
448#103—22no—
449#102—22no—
450#97—21no—
451C4×C8—21no—
452#95—21no—
453#95—21no—
454#95—24no—
455#41—22no—
456#41C2×Q8×C13714yes—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—41no—
462C2×Q8—41no—
463C2×Q8—82no—
464C2×Q8C2×D4×C13711yescharacteristic
465C2×D4—21no—
466C2×D4C2×D4×C13711yescharacteristic
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4C2×D4×C13712yes—
475C22×C4C2×D4×C13712yes—
476C22×C4C2×D4×C13711yescharacteristic
477C22×C4C2×D4×C13711yescharacteristic
478C22×C4C24×C13711yescharacteristic
479M16—22no—
480M16—24no—
481M16C2×Q8×C13714yes—
482C2×C8—22no—
483C2×C8—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—84no—
513C4⋊C4C2×D4×C13711yescharacteristic
514C4⋊C4C2×D4×C13711yescharacteristic
515C4⋊C4C2×D4×C13711yescharacteristic
516C4⋊C4C2×D4×C13711yescharacteristic
517C4⋊C4C2×D4×C13711yescharacteristic
518C4⋊C4C2×D4×C13711yescharacteristic
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42C2×D4×C13711yescharacteristic
535C42C2×D4×C13711yescharacteristic
536C42C2×D4×C13711yescharacteristic
537C42C2×D4×C13711yescharacteristic
538C42C2×Q8×C13714yes—
539C23—21no—
540C23C22×D4×C13711yescharacteristic
541Q8—21no—
542Q8—81no—
543Q8—81no—
544Q8—82no—
545Q8—82no—
546Q8—82no—
547Q8—82no—
548Q8C4⋊Q8×C13712yes—
549D4—24no—
550D4—21no—
551D4C4⋊Q8×C13712yes—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4#120×C13711yescharacteristic
576C2×C4#120×C13711yescharacteristic
577C2×C4#120×C13711yescharacteristic
578C2×C4#120×C13711yescharacteristic
579C2×C4C22×D4×C13711yescharacteristic
580C2×C4C22×D4×C13711yescharacteristic
581C2×C4C22×Q8×C13712yes—
582C2×C42−1+4×C13712yes—
583C8—24no—
584C8—42no—
585C22—21no—
586C22—21no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22C2×#120×C13711yescharacteristic
591C22D4×Q8×C13712yes—
592C4—22no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—44no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—82no—
607C4C2×C4⋊Q8×C13711yescharacteristic
608C4#536×C13711yescharacteristic
609C4D4×Q8×C13712yes—
610C2—22no—
611C2—42no—
612C2#2434×C13711yescharacteristic
613C2#3021×C13711yescharacteristic
614C2#3022×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×#41×C137C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3165×C137C2semidirect
4. #2660×C137C2semidirect
5. #2061×C137C2semidirect
6. #2060×C137C2semidirect
7. #1998×C137C2semidirect
8. #1995×C137C2semidirect
139. #104×C137Q8semidirect
307. #25833C137direct (N × H)
308. C137#25833direct (N × H)
310. #3165C2×C137semidirect
312. #2660C2×C137semidirect
313. #2061C2×C137semidirect
314. #2060C2×C137semidirect
315. #1998C2×C137semidirect
316. #1995C2×C137semidirect
447. #104Q8×C137semidirect

Maximal subgroups

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

subgroupindex
#3504×C1372
#3165×C1372
#3079×C1372
#2660×C1372
#2061×C1372
#2060×C1372
#1998×C1372
#1995×C1372
#1860×C1372
#1859×C1372
#1759×C1372
#25833137