Finite Group Databasebeta

← C137×#22043 #46871429

Subgroups — C137×#22043 #46871429

Order 35072.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450×C137C211yesmaximal, characteristic
2C4×C22⋊Q8×C137C211yesmaximal, characteristic, complemented
3C2×#390×C137C211yesmaximal, characteristic
4#1968×C137C212yesmaximal, complemented
5#1934×C137C212yesmaximal, complemented
6#1926×C137C212yesmaximal, complemented
7#1880×C137C212yesmaximal, complemented
8#1872×C137C211yesmaximal, characteristic, complemented
9#1853×C137C211yesmaximal, characteristic, complemented
10#1786×C137C211yesmaximal, characteristic
11C4×C22⋊C8×C137C211yesmaximal, characteristic, complemented
12C23×D4×C137C2211yescharacteristic
13C22×SD16×C137C2212yes—
14C2×C22⋊Q8×C137C2211yescharacteristic
15C2×C4×Q8×C137C2211yescharacteristic, complemented
16C22×C4⋊C4×C137C2211yescharacteristic
17C22×C22⋊C4×C137—21no—
18C22×C22⋊C4×C137—21no—
19C22×C42×C137C2211yescharacteristic
20#450×C137C418yescomplemented
21C2×#106×C137C2212yescomplemented
22C2×#103×C137C2212yescomplemented
23C2×#102×C137C2212yes—
24C2×#102×C137C2212yes—
25C2×#102×C137C2212yes—
26C2×C22⋊C8×C137C2211yescharacteristic
27C2×C22⋊C8×C137C2211yescharacteristic
28C2×C4×C8×C137C2212yescomplemented
29#391×C137C2211yescharacteristic, complemented
30#390×C137—21no—
31#390×C137—21no—
32#390×C137C2212yes—
33#390×C137C2212yes—
34#389×C137C2211yescharacteristic, complemented
35#387×C137—21no—
36#385×C137—21no—
37#380×C137C2211yescharacteristic
38C4×C4⋊C4×C137—21no—
39C4×C4⋊C4×C137C2211yescharacteristic, complemented
40C4×C22⋊C4×C137—21no—
41#340×C137C2212yescomplemented
42#340×C137C2212yescomplemented
43#336×C137C2212yescomplemented
44C25×C137—21no—
45C22×Q8×C137C2311yescharacteristic
46C22×D4×C137—21no—
47C22×D4×C137—22no—
48C22×D4×C137—42no—
49C22×D4×C137—42no—
50C22×D4×C137C2×C414yes—
51C22×D4×C137C2312yes—
52C23×C4×C137—21no—
53C23×C4×C137D411yescharacteristic
54C23×C4×C137D411yescharacteristic
55C23×C4×C137C2311yescharacteristic
56C2×SD16×C137—42no—
57C2×SD16×C137—42no—
58C2×SD16×C137C2×C418yescomplemented
59C22×C8×C137C2312yes—
60C22×C8×C137C2312yes—
61C22⋊Q8×C137—22no—
62C22⋊Q8×C137C2×C414yes—
63C4×Q8×C137—42no—
64C2×C4⋊C4×C137—21no—
65C2×C4⋊C4×C137—21no—
66C2×C4⋊C4×C137—21no—
67C2×C4⋊C4×C137—21no—
68C2×C4⋊C4×C137—42no—
69C2×C4⋊C4×C137C2311yescharacteristic
70C2×C4⋊C4×C137C2311yescharacteristic
71C2×C4⋊C4×C137C2312yes—
72C2×C4⋊C4×C137C2311yescharacteristic
73C2×C22⋊C4×C137—21no—
74C2×C22⋊C4×C137—21no—
75C2×C22⋊C4×C137—22no—
76C2×C22⋊C4×C137—22no—
77C2×C22⋊C4×C137—22no—
78C2×C22⋊C4×C137—22no—
79C2×C22⋊C4×C137—44no—
80C2×C22⋊C4×C137—44no—
81C2×C42×C137—21no—
82C2×C42×C137—21no—
83C2×C42×C137—42no—
84C2×C42×C137D411yescharacteristic
85C2×C42×C137D411yescharacteristic
86C2×C42×C137C2311yescharacteristic
87C2×C42×C137C2311yescharacteristic
88#106×C137—42no—
89#103×C137—42no—
90#102×C137—42no—
91#102×C137—42no—
92#102×C137C2×C418yescomplemented
93C22⋊C8×C137—22no—
94C22⋊C8×C137C2×C414yes—
95C4×C8×C137—42no—
96#95×C137—21no—
97#95×C137—21no—
98#95×C137—21no—
99#95×C137—21no—
100#95×C137—21no—
101C24×C137—24no—
102C24×C137—22no—
103C24×C137—42no—
104C24×C137—42no—
105C24×C137—42no—
106C24×C137—42no—
107C24×C137—42no—
108C24×C137—42no—
109C24×C137C2×D411yescharacteristic
110C2×Q8×C137—21no—
111C2×Q8×C137—22no—
112C2×Q8×C137—22no—
113C2×Q8×C137C22×C412yes—
114C2×D4×C137—24no—
115C2×D4×C137—22no—
116C2×D4×C137—22no—
117C2×D4×C137—22no—
118C2×D4×C137—22no—
119C2×D4×C137—22no—
120C2×D4×C137—48no—
121C2×D4×C137—42no—
122C2×D4×C137—42no—
123C2×D4×C137—48no—
124C2×D4×C137C22×C414yes—
125C22×C4×C137—22no—
126C22×C4×C137—22no—
127C22×C4×C137—22no—
128C22×C4×C137—21no—
129C22×C4×C137—21no—
130C22×C4×C137—22no—
131C22×C4×C137—21no—
132C22×C4×C137—21no—
133C22×C4×C137—22no—
134C22×C4×C137—41no—
135C22×C4×C137—41no—
136C22×C4×C137—41no—
137C22×C4×C137—42no—
138C22×C4×C137—42no—
139C22×C4×C137—42no—
140C22×C4×C137SD1614yescomplemented
141C22×C4×C137C22×C412yes—
142C22×C4×C137C2×D411yescharacteristic
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D411yescharacteristic
145C22×C4×C137C2×D411yescharacteristic
146C22×C4×C137C2×D411yescharacteristic
147C22×C4×C137#4112yes—
148C22×C4×C137C2411yescharacteristic
149SD16×C137—48no—
150C2×C8×C137—22no—
151C2×C8×C137—22no—
152C2×C8×C137—22no—
153C2×C8×C137—22no—
154C2×C8×C137—22no—
155C2×C8×C137C22×C414yes—
156C4⋊C4×C137—24no—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—22no—
160C4⋊C4×C137—22no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—42no—
163C4⋊C4×C137—42no—
164C4⋊C4×C137—42no—
165C4⋊C4×C137C22×C412yes—
166C22⋊C4×C137—24no—
167C22⋊C4×C137—44no—
168C22⋊C4×C137—42no—
169C22⋊C4×C137—44no—
170C22⋊C4×C137—44no—
171C22⋊C4×C137—44no—
172C42×C137—22no—
173C42×C137—22no—
174C42×C137—41no—
175C42×C137—41no—
176C42×C137—42no—
177C42×C137—42no—
178C23×C137—21no—
179C23×C137—21no—
180C23×C137—21no—
181C23×C137—21no—
182C23×C137—24no—
183C23×C137—48no—
184C23×C137—48no—
185C23×C137—42no—
186C23×C137—42no—
187C23×C137—42no—
188C23×C137—42no—
189C23×C137—42no—
190C23×C137—42no—
191C23×C137—48no—
192C23×C137—84no—
193C23×C137—84no—
194C23×C137—84no—
195C23×C137—84no—
196C23×C137C4×D412yes—
197C23×C137C2×SD1612yes—
198C23×C137#13512yes—
199C23×C137C22×D411yescharacteristic
200Q8×C137—24no—
201Q8×C137—42no—
202D4×C137—24no—
203D4×C137—24no—
204D4×C137—48no—
205D4×C137—42no—
206D4×C137—42no—
207C2×C4×C137—21no—
208C2×C4×C137—21no—
209C2×C4×C137—21no—
210C2×C4×C137—21no—
211C2×C4×C137—21no—
212C2×C4×C137—21no—
213C2×C4×C137—21no—
214C2×C4×C137—21no—
215C2×C4×C137—24no—
216C2×C4×C137—24no—
217C2×C4×C137—22no—
218C2×C4×C137—22no—
219C2×C4×C137—21no—
220C2×C4×C137—21no—
221C2×C4×C137—21no—
222C2×C4×C137—44no—
223C2×C4×C137—44no—
224C2×C4×C137—41no—
225C2×C4×C137—41no—
226C2×C4×C137—41no—
227C2×C4×C137—41no—
228C2×C4×C137—41no—
229C2×C4×C137—42no—
230C2×C4×C137—41no—
231C2×C4×C137—42no—
232C2×C4×C137—41no—
233C2×C4×C137—41no—
234C2×C4×C137—42no—
235C2×C4×C137—41no—
236C2×C4×C137—41no—
237C2×C4×C137—84no—
238C2×C4×C137C4×D412yes—
239C2×C4×C137C2×SD1612yes—
240C2×C4×C137C2×SD1612yes—
241C2×C4×C137AGL(1,ℤ/8)12yes—
242C2×C4×C137C23×C411yescharacteristic
243C2×C4×C137C2×#4111yescharacteristic
244C2×C4×C137H(5,𝔽2)12yes—
245C2×C4×C137H(5,𝔽2)12yes—
246C8×C137—24no—
247C8×C137—42no—
248C22×C137—22no—
249C22×C137—22no—
250C22×C137—22no—
251C22×C137—22no—
252C22×C137—22no—
253C22×C137—22no—
254C22×C137—44no—
255C22×C137—44no—
256C22×C137—44no—
257C22×C137—816no—
258C22×C137—82no—
259C22×C137—82no—
260C22×C137—82no—
261C22×C137—82no—
262C22×C137C4×SD1614yes—
263C22×C137C2×C4×D411yescharacteristic
264C22×C137#53411yescharacteristic
265C22×C137#53511yescharacteristic
266C22×C137C22×SD1611yescharacteristic
267C22×C137C2×#13511yescharacteristic
268C22×C137C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C137#57511yescharacteristic
270C4×C137—22no—
271C4×C137—22no—
272C4×C137—22no—
273C4×C137—22no—
274C4×C137—42no—
275C4×C137—41no—
276C4×C137—42no—
277C4×C137—81no—
278C4×C137—82no—
279C4×C137—81no—
280C4×C137#51812yes—
281C4×C137#53812yes—
282C2×C137—24no—
283C2×C137—84no—
284C2×C137#232111yescharacteristic
285C2×C137C2×C4×SD1611yescharacteristic
286C2×C137C4×AGL(1,ℤ/8)11yescharacteristic
287C2×C137#318711yescharacteristic
288C2×C137#318811yescharacteristic
289C2×C137#320811yescharacteristic
290C2×C137#321111yescharacteristic
291#22043C13711yesmaximal, characteristic, direct product factor, Sylow
292C137#2204311yescharacteristic, direct product factor, Sylow
293C2×#450C2×C13711yescharacteristic
294C4×C22⋊Q8C2×C13711yescharacteristic, complemented
295C2×#390C2×C13711yescharacteristic
296#1968C2×C13712yescomplemented
297#1934C2×C13712yescomplemented
298#1926C2×C13712yescomplemented
299#1880C2×C13712yescomplemented
300#1872C2×C13711yescharacteristic, complemented
301#1853C2×C13711yescharacteristic, complemented
302#1786C2×C13711yescharacteristic
303C4×C22⋊C8C2×C13711yescharacteristic, complemented
304C23×D4C22×C13711yescharacteristic
305C22×SD16C22×C13712yes—
306C2×C22⋊Q8C22×C13711yescharacteristic
307C2×C4×Q8C22×C13711yescharacteristic, complemented
308C22×C4⋊C4C22×C13711yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4—21no—
311C22×C42C22×C13711yescharacteristic
312#450C4×C13718yescomplemented
313C2×#106C22×C13712yescomplemented
314C2×#103C22×C13712yescomplemented
315C2×#102C22×C13712yes—
316C2×#102C22×C13712yes—
317C2×#102C22×C13712yes—
318C2×C22⋊C8C22×C13711yescharacteristic
319C2×C22⋊C8C22×C13711yescharacteristic
320C2×C4×C8C22×C13712yescomplemented
321#391C22×C13711yescharacteristic, complemented
322#390—21no—
323#390—21no—
324#390C22×C13712yes—
325#390C22×C13712yes—
326#389C22×C13711yescharacteristic, complemented
327#387—21no—
328#385—21no—
329#380C22×C13711yescharacteristic
330C4×C4⋊C4—21no—
331C4×C4⋊C4C22×C13711yescharacteristic, complemented
332C4×C22⋊C4—21no—
333#340C22×C13712yescomplemented
334#340C22×C13712yescomplemented
335#336C22×C13712yescomplemented
336C25—21no—
337C22×Q8C23×C13711yescharacteristic
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C2×C4×C13714yes—
343C22×D4C23×C13712yes—
344C23×C4—21no—
345C23×C4D4×C13711yescharacteristic
346C23×C4D4×C13711yescharacteristic
347C23×C4C23×C13711yescharacteristic
348C2×SD16—42no—
349C2×SD16—42no—
350C2×SD16C2×C4×C13718yescomplemented
351C22×C8C23×C13712yes—
352C22×C8C23×C13712yes—
353C22⋊Q8—22no—
354C22⋊Q8C2×C4×C13714yes—
355C4×Q8—42no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4C23×C13712yes—
362C2×C4⋊C4C23×C13711yescharacteristic
363C2×C4⋊C4C23×C13711yescharacteristic
364C2×C4⋊C4C23×C13711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—42no—
376C2×C42D4×C13711yescharacteristic
377C2×C42D4×C13711yescharacteristic
378C2×C42C23×C13711yescharacteristic
379C2×C42C23×C13711yescharacteristic
380#106—42no—
381#103—42no—
382#102—42no—
383#102—42no—
384#102C2×C4×C13718yescomplemented
385C22⋊C8—22no—
386C22⋊C8C2×C4×C13714yes—
387C4×C8—42no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C2×D4×C13711yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22×C4×C13712yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C22×C4×C13714yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4SD16×C13714yescomplemented
433C22×C4C22×C4×C13712yes—
434C22×C4C2×D4×C13711yescharacteristic
435C22×C4C2×D4×C13711yescharacteristic
436C22×C4C2×D4×C13711yescharacteristic
437C22×C4C2×D4×C13711yescharacteristic
438C22×C4C2×D4×C13711yescharacteristic
439C22×C4#41×C13712yes—
440C22×C4C24×C13711yescharacteristic
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8C22×C4×C13714yes—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4C22×C4×C13712yes—
458C22⋊C4—24no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—42no—
469C42—42no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23C4×D4×C13712yes—
489C23C2×SD16×C13712yes—
490C23#135×C13712yes—
491C23C22×D4×C13711yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4C4×D4×C13712yes—
531C2×C4C2×SD16×C13712yes—
532C2×C4C2×SD16×C13712yes—
533C2×C4AGL(1,ℤ/8)×C13712yes—
534C2×C4C23×C4×C13711yescharacteristic
535C2×C4C2×#41×C13711yescharacteristic
536C2×C4H(5,𝔽2)×C13712yes—
537C2×C4H(5,𝔽2)×C13712yes—
538C8—24no—
539C8—42no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C4×SD16×C13714yes—
555C22C2×C4×D4×C13711yescharacteristic
556C22#534×C13711yescharacteristic
557C22#535×C13711yescharacteristic
558C22C22×SD16×C13711yescharacteristic
559C22C2×#135×C13711yescharacteristic
560C22C2×AGL(1,ℤ/8)×C13711yescharacteristic
561C22#575×C13711yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—81no—
570C4—82no—
571C4—81no—
572C4#518×C13712yes—
573C4#538×C13712yes—
574C2—24no—
575C2—84no—
576C2#2321×C13711yescharacteristic
577C2C2×C4×SD16×C13711yescharacteristic
578C2C4×AGL(1,ℤ/8)×C13711yescharacteristic
579C2#3187×C13711yescharacteristic
580C2#3188×C13711yescharacteristic
581C2#3208×C13711yescharacteristic
582C2#3211×C13711yescharacteristic
subgroupG/H
center, socleC23×C137C22×D4
commutatorC2×C4C23×C4×C137
Frattini, cosocleC22×C4C24×C137
prime radicalC23×D4×C137C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C22×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Q8×C137C2semidirect
4. #1968×C137C2semidirect
5. #1934×C137C2semidirect
6. #1926×C137C2semidirect
7. #1880×C137C2semidirect
8. #1872×C137C2semidirect
9. #1853×C137C2semidirect
11. C4×C22⋊C8×C137C2semidirect
15. C2×C4×Q8×C137C22semidirect
20. #450×C137C4semidirect
21. C2×#106×C137C22semidirect
22. C2×#103×C137C22semidirect
28. C2×C4×C8×C137C22semidirect
29. #391×C137C22semidirect
34. #389×C137C22semidirect
39. C4×C4⋊C4×C137C22semidirect
41. #340×C137C22semidirect
42. #340×C137C22semidirect
43. #336×C137C22semidirect
58. C2×SD16×C137C2×C4semidirect
92. #102×C137C2×C4semidirect
140. C22×C4×C137SD16semidirect
291. #22043C137direct (N × H)
292. C137#22043direct (N × H)
294. C4×C22⋊Q8C2×C137semidirect
296. #1968C2×C137semidirect
297. #1934C2×C137semidirect
298. #1926C2×C137semidirect
299. #1880C2×C137semidirect
300. #1872C2×C137semidirect
301. #1853C2×C137semidirect
303. C4×C22⋊C8C2×C137semidirect
307. C2×C4×Q8C22×C137semidirect
312. #450C4×C137semidirect
313. C2×#106C22×C137semidirect
314. C2×#103C22×C137semidirect
320. C2×C4×C8C22×C137semidirect
321. #391C22×C137semidirect
326. #389C22×C137semidirect
331. C4×C4⋊C4C22×C137semidirect
333. #340C22×C137semidirect
334. #340C22×C137semidirect
335. #336C22×C137semidirect
350. C2×SD16C2×C4×C137semidirect
384. #102C2×C4×C137semidirect
432. C22×C4SD16×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#450×C1372
C4×C22⋊Q8×C1372
C2×#390×C1372
#1968×C1372
#1934×C1372
#1926×C1372
#1880×C1372
#1872×C1372
#1853×C1372
#1786×C1372
C4×C22⋊C8×C1372
#22043137