Finite Group Databasebeta

← C137×#20864 #46870250

Subgroups — C137×#20864 #46870250

Order 35072.

1422 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
1C2×#520×C137C211yesmaximal, characteristic
2#2911×C137C211yesmaximal, characteristic, complemented
3C2×#421×C137C211yesmaximal, characteristic
4#2896×C137C211yesmaximal, characteristic
5#2317×C137C211yesmaximal, characteristic, complemented
6#1860×C137C214yesmaximal, complemented
7#1859×C137C214yesmaximal, complemented
8C2×#343×C137C211yesmaximal, characteristic, complemented
9#1733×C137C211yesmaximal, characteristic, complemented
10C22×#41×C137C2211yescharacteristic
11C22×M16×C137C2211yescharacteristic
12C22×M16×C137C2211yescharacteristic
13#520×C137—22no—
14#520×C137—22no—
15#520×C137—22no—
16#520×C137C2214yes—
17C2×C4×Q8×C137—21no—
18C2×C4×D4×C137—21no—
19C2×C4×D4×C137—22no—
20C2×#117×C137—21no—
21C2×#117×C137C2211yescharacteristic
22C2×#117×C137C2211yescharacteristic
23C2×#117×C137C2211yescharacteristic
24C22×C4⋊C4×C137—21no—
25C22×C4⋊C4×C137C2211yescharacteristic
26#428×C137—21no—
27#428×C137—21no—
28#428×C137C2212yescomplemented
29#428×C137C2212yescomplemented
30#427×C137—22no—
31#421×C137C418yescomplemented
32C2×#104×C137C2214yes—
33#419×C137—21no—
34#418×C137—21no—
35#417×C137C2212yes—
36#417×C137C2212yes—
37#416×C137—22no—
38#389×C137—22no—
39#385×C137—22no—
40#384×C137C2212yescomplemented
41#382×C137C2212yescomplemented
42#343×C137C2214yescomplemented
43#339×C137C2214yescomplemented
44C2×#41×C137—21no—
45C2×#41×C137—21no—
46C2×#41×C137—22no—
47C2×#41×C137—22no—
48C2×#41×C137C2×C414yes—
49C2×#41×C137D412yes—
50C2×#41×C137Q814yes—
51C2×#41×C137C2312yes—
52C22×Q8×C137D411yescharacteristic
53C22×D4×C137—21no—
54C22×D4×C137D411yescharacteristic
55C23×C4×C137—21no—
56C23×C4×C137—22no—
57C23×C4×C137D412yes—
58C23×C4×C137C2311yescharacteristic
59C2×M16×C137—21no—
60C2×M16×C137—22no—
61C2×M16×C137—21no—
62C2×M16×C137—22no—
63C2×M16×C137—22no—
64C2×M16×C137C2×C414yescomplemented
65C2×M16×C137C2312yes—
66C2×M16×C137C2312yes—
67C22×C8×C137—21no—
68C22×C8×C137—21no—
69C4×Q8×C137—24no—
70C4×Q8×C137—24no—
71C4×D4×C137—28no—
72C4×D4×C137—24no—
73C4×D4×C137—24no—
74C4×D4×C137—28no—
75#117×C137—24no—
76#117×C137—21no—
77#117×C137—21no—
78#117×C137—22no—
79#117×C137—24no—
80#117×C137—21no—
81#117×C137—21no—
82#117×C137—21no—
83#117×C137—21no—
84#117×C137C2×C414yescomplemented
85#117×C137D412yes—
86#117×C137C2312yes—
87C2×C4⋊C4×C137—22no—
88C2×C4⋊C4×C137—24no—
89C2×C4⋊C4×C137—21no—
90C2×C4⋊C4×C137—21no—
91C2×C4⋊C4×C137—21no—
92C2×C4⋊C4×C137—21no—
93C2×C4⋊C4×C137—21no—
94C2×C4⋊C4×C137—24no—
95C2×C4⋊C4×C137—24no—
96C2×C4⋊C4×C137D412yes—
97C2×C4⋊C4×C137D411yescharacteristic
98C2×C4⋊C4×C137D411yescharacteristic
99C2×C4⋊C4×C137C2311yescharacteristic
100C2×C4⋊C4×C137C2311yescharacteristic
101C2×C4⋊C4×C137C2312yes—
102C2×C22⋊C4×C137—22no—
103C2×C22⋊C4×C137—21no—
104C2×C22⋊C4×C137—21no—
105C2×C22⋊C4×C137—21no—
106C2×C42×C137—22no—
107C2×C42×C137—21no—
108C2×C42×C137—21no—
109C2×C42×C137C2312yes—
110#107×C137—22no—
111#107×C137—22no—
112#106×C137—22no—
113#106×C137—22no—
114#104×C137C2×C4116yescomplemented
115#103×C137—22no—
116#103×C137—22no—
117#102×C137—22no—
118#102×C137—22no—
119#95×C137—24no—
120C24×C137—21no—
121C24×C137C2×D411yescharacteristic
122#41×C137—24no—
123#41×C137—24no—
124#41×C137—24no—
125#41×C137—24no—
126#41×C137—24no—
127#41×C137—22no—
128#41×C137—22no—
129#41×C137C4⋊C418yescomplemented
130#41×C137C22×C414yes—
131#41×C137C2×Q814yes—
132C2×Q8×C137—21no—
133C2×Q8×C137—22no—
134C2×Q8×C137C4⋊C414yescomplemented
135C2×Q8×C137C2×D412yes—
136C2×D4×C137—24no—
137C2×D4×C137—22no—
138C2×D4×C137—24no—
139C2×D4×C137—22no—
140C2×D4×C137—21no—
141C2×D4×C137—22no—
142C2×D4×C137C4⋊C414yescomplemented
143C2×D4×C137C2×D412yes—
144C22×C4×C137—21no—
145C22×C4×C137—24no—
146C22×C4×C137—22no—
147C22×C4×C137—24no—
148C22×C4×C137—22no—
149C22×C4×C137—21no—
150C22×C4×C137—22no—
151C22×C4×C137—21no—
152C22×C4×C137—22no—
153C22×C4×C137—28no—
154C22×C4×C137—22no—
155C22×C4×C137—21no—
156C22×C4×C137—21no—
157C22×C4×C137—82no—
158C22×C4×C137—84no—
159C22×C4×C137C22×C412yes—
160C22×C4×C137C2×D414yes—
161C22×C4×C137C2×D411yescharacteristic
162C22×C4×C137C2×D411yescharacteristic
163C22×C4×C137C2×D411yescharacteristic
164C22×C4×C137C2×D411yescharacteristic
165C22×C4×C137C2×D411yescharacteristic
166C22×C4×C137C2×Q812yes—
167C22×C4×C137#4112yes—
168C22×C4×C137#4112yes—
169C22×C4×C137C2411yescharacteristic
170M16×C137—24no—
171M16×C137—22no—
172M16×C137—22no—
173M16×C137—22no—
174M16×C137—42no—
175M16×C137C22×C414yescomplemented
176C2×C8×C137—24no—
177C2×C8×C137—22no—
178C2×C8×C137—22no—
179C2×C8×C137—42no—
180C4⋊C4×C137—28no—
181C4⋊C4×C137—24no—
182C4⋊C4×C137—22no—
183C4⋊C4×C137—21no—
184C4⋊C4×C137—22no—
185C4⋊C4×C137—21no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—21no—
189C4⋊C4×C137—21no—
190C4⋊C4×C137—21no—
191C4⋊C4×C137—21no—
192C4⋊C4×C137—84no—
193C4⋊C4×C137C2×D412yes—
194C4⋊C4×C137C2×D412yes—
195C22⋊C4×C137—28no—
196C22⋊C4×C137—24no—
197C22⋊C4×C137—22no—
198C22⋊C4×C137—22no—
199C22⋊C4×C137—41no—
200C22⋊C4×C137—41no—
201C42×C137—28no—
202C42×C137—22no—
203C42×C137—22no—
204C42×C137—22no—
205C42×C137—41no—
206C42×C137—41no—
207C42×C137C22×C414yescomplemented
208C23×C137—22no—
209C23×C137—24no—
210C23×C137—41no—
211C23×C137—41no—
212C23×C137—42no—
213C23×C137—42no—
214C23×C137C4×D412yes—
215C23×C137#12011yescharacteristic
216C23×C137#12011yescharacteristic
217C23×C137C22⋊Q812yes—
218C23×C137C22×D411yescharacteristic
219Q8×C137—22no—
220Q8×C137—24no—
221Q8×C137C2×C4⋊C414yes—
222D4×C137—24no—
223D4×C137—24no—
224D4×C137—28no—
225D4×C137—22no—
226D4×C137—24no—
227D4×C137C2×C4⋊C414yes—
228C2×C4×C137—22no—
229C2×C4×C137—24no—
230C2×C4×C137—24no—
231C2×C4×C137—24no—
232C2×C4×C137—28no—
233C2×C4×C137—24no—
234C2×C4×C137—22no—
235C2×C4×C137—24no—
236C2×C4×C137—22no—
237C2×C4×C137—42no—
238C2×C4×C137—42no—
239C2×C4×C137—42no—
240C2×C4×C137—44no—
241C2×C4×C137—42no—
242C2×C4×C137—44no—
243C2×C4×C137—42no—
244C2×C4×C137—42no—
245C2×C4×C137—41no—
246C2×C4×C137—41no—
247C2×C4×C137—82no—
248C2×C4×C137—88no—
249C2×C4×C137C2×C4⋊C412yes—
250C2×C4×C137C2×C4⋊C412yes—
251C2×C4×C137C4×D412yes—
252C2×C4×C137C4×D412yes—
253C2×C4×C137C4×D412yes—
254C2×C4×C137#12011yescharacteristic
255C2×C4×C137#12011yescharacteristic
256C2×C4×C137C22⋊Q812yes—
257C2×C4×C137C22⋊Q812yes—
258C2×C4×C137C22⋊Q812yes—
259C2×C4×C137#12312yes—
260C2×C4×C137#12312yes—
261C2×C4×C137C23×C411yescharacteristic
262C2×C4×C137C22×D411yescharacteristic
263C2×C4×C137C22×D411yescharacteristic
264C2×C4×C137C22×Q811yescharacteristic
265C2×C4×C137C2×#4111yescharacteristic
266C2×C4×C137C2×#4111yescharacteristic
267C8×C137—24no—
268C8×C137—42no—
269C22×C137—21no—
270C22×C137—21no—
271C22×C137—21no—
272C22×C137—21no—
273C22×C137—21no—
274C22×C137—21no—
275C22×C137—24no—
276C22×C137—42no—
277C22×C137—42no—
278C22×C137—48no—
279C22×C137—42no—
280C22×C137—42no—
281C22×C137#38512yes—
282C22×C137#38512yes—
283C22×C137C2×C4×D411yescharacteristic
284C22×C137C2×#12011yescharacteristic
285C22×C137C2×C22⋊Q811yescharacteristic
286C4×C137—24no—
287C4×C137—44no—
288C4×C137—44no—
289C4×C137—42no—
290C4×C137—42no—
291C4×C137—81no—
292C4×C137—81no—
293C4×C137#38512yes—
294C4×C137#38512yes—
295C4×C137C22×C4⋊C411yescharacteristic
296C4×C137C2×C4×D411yescharacteristic
297C4×C137C2×C22⋊Q811yescharacteristic
298C4×C137C2×#12311yescharacteristic
299C2×C137—21no—
300C2×C137—22no—
301C2×C137—21no—
302C2×C137—22no—
303C2×C137—44no—
304C2×C137C2×#38511yescharacteristic
305C2×C137#301911yescharacteristic
306C2×C137#302011yescharacteristic
307#20864C13711yesmaximal, characteristic, direct product factor, Sylow
308C137#2086411yescharacteristic, direct product factor, Sylow
309C2×#520C2×C13711yescharacteristic
310#2911C2×C13711yescharacteristic, complemented
311C2×#421C2×C13711yescharacteristic
312#2896C2×C13711yescharacteristic
313#2317C2×C13711yescharacteristic, complemented
314#1860C2×C13714yescomplemented
315#1859C2×C13714yescomplemented
316C2×#343C2×C13711yescharacteristic, complemented
317#1733C2×C13711yescharacteristic, complemented
318C22×#41C22×C13711yescharacteristic
319C22×M16C22×C13711yescharacteristic
320C22×M16C22×C13711yescharacteristic
321#520—22no—
322#520—22no—
323#520—22no—
324#520C22×C13714yes—
325C2×C4×Q8—21no—
326C2×C4×D4—21no—
327C2×C4×D4—22no—
328C2×#117—21no—
329C2×#117C22×C13711yescharacteristic
330C2×#117C22×C13711yescharacteristic
331C2×#117C22×C13711yescharacteristic
332C22×C4⋊C4—21no—
333C22×C4⋊C4C22×C13711yescharacteristic
334#428—21no—
335#428—21no—
336#428C22×C13712yescomplemented
337#428C22×C13712yescomplemented
338#427—22no—
339#421C4×C13718yescomplemented
340C2×#104C22×C13714yes—
341#419—21no—
342#418—21no—
343#417C22×C13712yes—
344#417C22×C13712yes—
345#416—22no—
346#389—22no—
347#385—22no—
348#384C22×C13712yescomplemented
349#382C22×C13712yescomplemented
350#343C22×C13714yescomplemented
351#339C22×C13714yescomplemented
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41C2×C4×C13714yes—
357C2×#41D4×C13712yes—
358C2×#41Q8×C13714yes—
359C2×#41C23×C13712yes—
360C22×Q8D4×C13711yescharacteristic
361C22×D4—21no—
362C22×D4D4×C13711yescharacteristic
363C23×C4—21no—
364C23×C4—22no—
365C23×C4D4×C13712yes—
366C23×C4C23×C13711yescharacteristic
367C2×M16—21no—
368C2×M16—22no—
369C2×M16—21no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16C2×C4×C13714yescomplemented
373C2×M16C23×C13712yes—
374C2×M16C23×C13712yes—
375C22×C8—21no—
376C22×C8—21no—
377C4×Q8—24no—
378C4×Q8—24no—
379C4×D4—28no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—28no—
383#117—24no—
384#117—21no—
385#117—21no—
386#117—22no—
387#117—24no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117C2×C4×C13714yescomplemented
393#117D4×C13712yes—
394#117C23×C13712yes—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4D4×C13712yes—
405C2×C4⋊C4D4×C13711yescharacteristic
406C2×C4⋊C4D4×C13711yescharacteristic
407C2×C4⋊C4C23×C13711yescharacteristic
408C2×C4⋊C4C23×C13711yescharacteristic
409C2×C4⋊C4C23×C13712yes—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C42—22no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C23×C13712yes—
418#107—22no—
419#107—22no—
420#106—22no—
421#106—22no—
422#104C2×C4×C137116yescomplemented
423#103—22no—
424#103—22no—
425#102—22no—
426#102—22no—
427#95—24no—
428C24—21no—
429C24C2×D4×C13711yescharacteristic
430#41—24no—
431#41—24no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—22no—
436#41—22no—
437#41C4⋊C4×C13718yescomplemented
438#41C22×C4×C13714yes—
439#41C2×Q8×C13714yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8C4⋊C4×C13714yescomplemented
443C2×Q8C2×D4×C13712yes—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4C4⋊C4×C13714yescomplemented
451C2×D4C2×D4×C13712yes—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4C22×C4×C13712yes—
468C22×C4C2×D4×C13714yes—
469C22×C4C2×D4×C13711yescharacteristic
470C22×C4C2×D4×C13711yescharacteristic
471C22×C4C2×D4×C13711yescharacteristic
472C22×C4C2×D4×C13711yescharacteristic
473C22×C4C2×D4×C13711yescharacteristic
474C22×C4C2×Q8×C13712yes—
475C22×C4#41×C13712yes—
476C22×C4#41×C13712yes—
477C22×C4C24×C13711yescharacteristic
478M16—24no—
479M16—22no—
480M16—22no—
481M16—22no—
482M16—42no—
483M16C22×C4×C13714yescomplemented
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—42no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—84no—
501C4⋊C4C2×D4×C13712yes—
502C4⋊C4C2×D4×C13712yes—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C42—28no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—41no—
514C42—41no—
515C42C22×C4×C13714yescomplemented
516C23—24no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23C4×D4×C13712yes—
523C23#120×C13711yescharacteristic
524C23#120×C13711yescharacteristic
525C23C22⋊Q8×C13712yes—
526C23C22×D4×C13711yescharacteristic
527Q8—24no—
528Q8—22no—
529Q8C2×C4⋊C4×C13714yes—
530D4—28no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4C2×C4⋊C4×C13714yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—88no—
557C2×C4C2×C4⋊C4×C13712yes—
558C2×C4C2×C4⋊C4×C13712yes—
559C2×C4C4×D4×C13712yes—
560C2×C4C4×D4×C13712yes—
561C2×C4C4×D4×C13712yes—
562C2×C4#120×C13711yescharacteristic
563C2×C4#120×C13711yescharacteristic
564C2×C4C22⋊Q8×C13712yes—
565C2×C4C22⋊Q8×C13712yes—
566C2×C4C22⋊Q8×C13712yes—
567C2×C4#123×C13712yes—
568C2×C4#123×C13712yes—
569C2×C4C23×C4×C13711yescharacteristic
570C2×C4C22×D4×C13711yescharacteristic
571C2×C4C22×D4×C13711yescharacteristic
572C2×C4C22×Q8×C13711yescharacteristic
573C2×C4C2×#41×C13711yescharacteristic
574C2×C4C2×#41×C13711yescharacteristic
575C8—24no—
576C8—42no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#385×C13712yes—
590C22#385×C13712yes—
591C22C2×C4×D4×C13711yescharacteristic
592C22C2×#120×C13711yescharacteristic
593C22C2×C22⋊Q8×C13711yescharacteristic
594C4—24no—
595C4—44no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4#385×C13712yes—
602C4#385×C13712yes—
603C4C22×C4⋊C4×C13711yescharacteristic
604C4C2×C4×D4×C13711yescharacteristic
605C4C2×C22⋊Q8×C13711yescharacteristic
606C4C2×#123×C13711yescharacteristic
607C2—21no—
608C2—22no—
609C2—22no—
610C2—21no—
611C2—44no—
612C2C2×#385×C13711yescharacteristic
613C2#3019×C13711yescharacteristic
614C2#3020×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutatorC2×C4C23×C4×C137
Frattini, cosocleC22×C4C24×C137
prime radicalC22×#41×C137C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2911×C137C2semidirect
5. #2317×C137C2semidirect
6. #1860×C137C2semidirect
7. #1859×C137C2semidirect
8. C2×#343×C137C2semidirect
9. #1733×C137C2semidirect
28. #428×C137C22semidirect
29. #428×C137C22semidirect
31. #421×C137C4semidirect
40. #384×C137C22semidirect
41. #382×C137C22semidirect
42. #343×C137C22semidirect
43. #339×C137C22semidirect
64. C2×M16×C137C2×C4semidirect
84. #117×C137C2×C4semidirect
114. #104×C137C2×C4semidirect
129. #41×C137C4⋊C4semidirect
134. C2×Q8×C137C4⋊C4semidirect
142. C2×D4×C137C4⋊C4semidirect
175. M16×C137C22×C4semidirect
207. C42×C137C22×C4semidirect
307. #20864C137direct (N × H)
308. C137#20864direct (N × H)
310. #2911C2×C137semidirect
313. #2317C2×C137semidirect
314. #1860C2×C137semidirect
315. #1859C2×C137semidirect
316. C2×#343C2×C137semidirect
317. #1733C2×C137semidirect
336. #428C22×C137semidirect
337. #428C22×C137semidirect
339. #421C4×C137semidirect
348. #384C22×C137semidirect
349. #382C22×C137semidirect
350. #343C22×C137semidirect
351. #339C22×C137semidirect
372. C2×M16C2×C4×C137semidirect
392. #117C2×C4×C137semidirect
422. #104C2×C4×C137semidirect
437. #41C4⋊C4×C137semidirect
442. C2×Q8C4⋊C4×C137semidirect
450. C2×D4C4⋊C4×C137semidirect
483. M16C22×C4×C137semidirect
515. C42C22×C4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#520×C1372
#2911×C1372
C2×#421×C1372
#2896×C1372
#2317×C1372
#1860×C1372
#1859×C1372
C2×#343×C1372
#1733×C1372
#20864137