Finite Group Databasebeta

← C137×#24751 #46874137

Subgroups — C137×#24751 #46874137

Order 35072.

1076 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465×C137C212yesmaximal
2C2×#461×C137C211yesmaximal, characteristic, complemented
3C22×C4⋊C8×C137C211yesmaximal, characteristic, complemented
4#2857×C137C211yesmaximal, characteristic, complemented
5C4×#121×C137C211yesmaximal, characteristic
6#2066×C137C212yesmaximal, complemented
7#2055×C137C212yesmaximal, complemented
8#1981×C137C211yesmaximal, characteristic, complemented
9#1948×C137C211yesmaximal, characteristic, complemented
10#1892×C137C212yesmaximal, complemented
11#1802×C137C211yesmaximal, characteristic, complemented
12C22×SD16×C137C2212yes—
13C23×C8×C137C2212yes—
14C2×C4⋊Q8×C137C2211yescharacteristic
15C2×C22⋊Q8×C137C2212yes—
16C2×#121×C137C2211yescharacteristic
17C2×C4×D4×C137—21no—
18C2×C4×D4×C137C2211yescharacteristic
19C22×C42×C137C2211yescharacteristic
20#465×C137—28no—
21#461×C137—24no—
22C2×#106×C137C2212yes—
23C2×C4⋊C8×C137—24no—
24C2×C4⋊C8×C137C2212yes—
25C2×C4⋊C8×C137C2211yescharacteristic
26C2×C4⋊C8×C137C2211yescharacteristic
27C2×#103×C137C2212yes—
28C2×#103×C137C2212yes—
29C2×#103×C137C2212yes—
30C2×#102×C137C2212yes—
31#391×C137C2212yes—
32#390×C137C2211yescharacteristic
33#389×C137—21no—
34#389×C137—21no—
35#389×C137C2211yescharacteristic
36#388×C137—21no—
37#386×C137—21no—
38#384×C137C2211yescharacteristic
39#380×C137—21no—
40#380×C137—21no—
41#380×C137C2211yescharacteristic
42C4×C4⋊C4×C137C2211yescharacteristic
43C4×C22⋊C4×C137—21no—
44#340×C137C2212yes—
45#340×C137C2212yes—
46C22×Q8×C137C2312yes—
47C22×D4×C137—21no—
48C22×D4×C137C2311yescharacteristic
49C23×C4×C137—21no—
50C23×C4×C137D411yescharacteristic
51C23×C4×C137D411yescharacteristic
52C23×C4×C137C2311yescharacteristic
53C2×SD16×C137—24no—
54C2×SD16×C137—24no—
55C2×SD16×C137—42no—
56C22×C8×C137—24no—
57C22×C8×C137—24no—
58C22×C8×C137—42no—
59C22×C8×C137C2312yes—
60C22×C8×C137C2312yes—
61C4⋊Q8×C137—22no—
62C4⋊Q8×C137—22no—
63C22⋊Q8×C137—24no—
64C22⋊Q8×C137—24no—
65#121×C137—22no—
66#121×C137D414yes—
67C4×D4×C137—22no—
68C4×D4×C137—22no—
69C4×D4×C137—44no—
70C2×C4⋊C4×C137—21no—
71C2×C4⋊C4×C137—21no—
72C2×C4⋊C4×C137—21no—
73C2×C4⋊C4×C137—21no—
74C2×C4⋊C4×C137—21no—
75C2×C4⋊C4×C137—22no—
76C2×C4⋊C4×C137C2311yescharacteristic
77C2×C4⋊C4×C137C2311yescharacteristic
78C2×C4⋊C4×C137C2312yes—
79C2×C4⋊C4×C137C2311yescharacteristic
80C2×C22⋊C4×C137—21no—
81C2×C22⋊C4×C137—22no—
82C2×C22⋊C4×C137—21no—
83C2×C22⋊C4×C137—21no—
84C2×C22⋊C4×C137—21no—
85C2×C22⋊C4×C137—21no—
86C2×C42×C137—22no—
87C2×C42×C137—22no—
88C2×C42×C137—21no—
89C2×C42×C137D411yescharacteristic
90C2×C42×C137D411yescharacteristic
91C2×C42×C137C2311yescharacteristic
92C2×C42×C137C2311yescharacteristic
93#106×C137—24no—
94C4⋊C8×C137—24no—
95C4⋊C8×C137—22no—
96C4⋊C8×C137D414yescomplemented
97#103×C137—24no—
98#103×C137—24no—
99#103×C137—42no—
100#102×C137—24no—
101#95×C137—21no—
102#95×C137—21no—
103#95×C137—21no—
104#95×C137—22no—
105#95×C137—22no—
106C24×C137—21no—
107C24×C137C2×D411yescharacteristic
108C2×Q8×C137—22no—
109C2×Q8×C137—22no—
110C2×Q8×C137—22no—
111C2×Q8×C137—22no—
112C2×Q8×C137—22no—
113C2×Q8×C137—22no—
114C2×D4×C137—24no—
115C2×D4×C137—21no—
116C2×D4×C137—21no—
117C2×D4×C137—21no—
118C2×D4×C137—21no—
119C2×D4×C137—21no—
120C2×D4×C137—42no—
121C2×D4×C137—42no—
122C2×D4×C137C2×D412yes—
123C22×C4×C137—22no—
124C22×C4×C137—22no—
125C22×C4×C137—21no—
126C22×C4×C137—21no—
127C22×C4×C137—21no—
128C22×C4×C137—21no—
129C22×C4×C137—22no—
130C22×C4×C137—21no—
131C22×C4×C137—22no—
132C22×C4×C137—21no—
133C22×C4×C137—21no—
134C22×C4×C137—22no—
135C22×C4×C137—22no—
136C22×C4×C137—41no—
137C22×C4×C137—42no—
138C22×C4×C137—42no—
139C22×C4×C137—42no—
140C22×C4×C137SD1614yescomplemented
141C22×C4×C137C2×D411yescharacteristic
142C22×C4×C137C2×D411yescharacteristic
143C22×C4×C137C2×D411yescharacteristic
144C22×C4×C137C2×D412yes—
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×C137—48no—
156C2×C8×C137C2×D414yes—
157C4⋊C4×C137—21no—
158C4⋊C4×C137—22no—
159C4⋊C4×C137—21no—
160C4⋊C4×C137—21no—
161C4⋊C4×C137—22no—
162C4⋊C4×C137—21no—
163C4⋊C4×C137—21no—
164C4⋊C4×C137—42no—
165C4⋊C4×C137—44no—
166C4⋊C4×C137—42no—
167C4⋊C4×C137—42no—
168C4⋊C4×C137—42no—
169C4⋊C4×C137—42no—
170C4⋊C4×C137C2×D412yes—
171C22⋊C4×C137—24no—
172C22⋊C4×C137—42no—
173C22⋊C4×C137—44no—
174C22⋊C4×C137—42no—
175C22⋊C4×C137—42no—
176C22⋊C4×C137—42no—
177C42×C137—22no—
178C42×C137—22no—
179C42×C137—21no—
180C42×C137—21no—
181C42×C137—42no—
182C42×C137C2×D412yes—
183C42×C137#4112yes—
184C23×C137—21no—
185C23×C137—21no—
186C23×C137—21no—
187C23×C137—21no—
188C23×C137—22no—
189C23×C137—41no—
190C23×C137—41no—
191C23×C137—41no—
192C23×C137—41no—
193C23×C137—41no—
194C23×C137—41no—
195C23×C137#12112yes—
196C23×C137C2×SD1612yes—
197C23×C137#13712yes—
198C23×C137C22×D411yescharacteristic
199Q8×C137—42no—
200Q8×C137—42no—
201Q8×C137—42no—
202Q8×C137—42no—
203D4×C137—22no—
204D4×C137—22no—
205D4×C137—48no—
206D4×C137—41no—
207D4×C137—41no—
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—22no—
215C2×C4×C137—22no—
216C2×C4×C137—22no—
217C2×C4×C137—22no—
218C2×C4×C137—28no—
219C2×C4×C137—21no—
220C2×C4×C137—22no—
221C2×C4×C137—21no—
222C2×C4×C137—21no—
223C2×C4×C137—44no—
224C2×C4×C137—41no—
225C2×C4×C137—41no—
226C2×C4×C137—42no—
227C2×C4×C137—41no—
228C2×C4×C137—42no—
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—42no—
238C2×C4×C137—84no—
239C2×C4×C137#12112yes—
240C2×C4×C137#12112yes—
241C2×C4×C137#12112yes—
242C2×C4×C137C2×SD1612yes—
243C2×C4×C137C2×SD1612yes—
244C2×C4×C137#13512yes—
245C2×C4×C137C22×D411yescharacteristic
246C2×C4×C137C22×D411yescharacteristic
247C2×C4×C137C2×#4111yescharacteristic
248C2×C4×C137C2×#4111yescharacteristic
249C2×C4×C137H(5,𝔽2)11yescharacteristic
250C2×C4×C1372−1+411yescharacteristic
251C8×C137—24no—
252C8×C137—42no—
253C22×C137—22no—
254C22×C137—22no—
255C22×C137—22no—
256C22×C137—22no—
257C22×C137—22no—
258C22×C137—22no—
259C22×C137—42no—
260C22×C137—42no—
261C22×C137—42no—
262C22×C137—81no—
263C22×C137—81no—
264C22×C137—81no—
265C22×C137—81no—
266C22×C137#46114yescomplemented
267C22×C137C2×#12111yescharacteristic
268C22×C137C2×#12111yescharacteristic
269C22×C137#53711yescharacteristic
270C22×C137C22×SD1611yescharacteristic
271C22×C137C2×#13511yescharacteristic
272C22×C137C2×#13711yescharacteristic
273C22×C137#57511yescharacteristic
274C4×C137—22no—
275C4×C137—22no—
276C4×C137—22no—
277C4×C137—42no—
278C4×C137—41no—
279C4×C137—82no—
280C4×C137—81no—
281C4×C137—82no—
282C4×C137—81no—
283C4×C137#46514yescomplemented
284C4×C137D4211yescharacteristic
285C4×C137#54711yescharacteristic
286C4×C137#54811yescharacteristic
287C4×C137#55311yescharacteristic
288C2×C137—24no—
289C2×C137—82no—
290C2×C137#241611yescharacteristic
291C2×C137C2×#46111yescharacteristic
292C2×C137#303511yescharacteristic
293C2×C137C2×#46511yescharacteristic
294C2×C137#305611yescharacteristic
295C2×C137#324611yescharacteristic
296C2×C137#324811yescharacteristic
297#24751C13711yesmaximal, characteristic, direct product factor, Sylow
298C137#2475111yescharacteristic, direct product factor, Sylow
299C2×#465C2×C13712yes—
300C2×#461C2×C13711yescharacteristic, complemented
301C22×C4⋊C8C2×C13711yescharacteristic, complemented
302#2857C2×C13711yescharacteristic, complemented
303C4×#121C2×C13711yescharacteristic
304#2066C2×C13712yescomplemented
305#2055C2×C13712yescomplemented
306#1981C2×C13711yescharacteristic, complemented
307#1948C2×C13711yescharacteristic, complemented
308#1892C2×C13712yescomplemented
309#1802C2×C13711yescharacteristic, complemented
310C22×SD16C22×C13712yes—
311C23×C8C22×C13712yes—
312C2×C4⋊Q8C22×C13711yescharacteristic
313C2×C22⋊Q8C22×C13712yes—
314C2×#121C22×C13711yescharacteristic
315C2×C4×D4—21no—
316C2×C4×D4C22×C13711yescharacteristic
317C22×C42C22×C13711yescharacteristic
318#465—28no—
319#461—24no—
320C2×#106C22×C13712yes—
321C2×C4⋊C8—24no—
322C2×C4⋊C8C22×C13712yes—
323C2×C4⋊C8C22×C13711yescharacteristic
324C2×C4⋊C8C22×C13711yescharacteristic
325C2×#103C22×C13712yes—
326C2×#103C22×C13712yes—
327C2×#103C22×C13712yes—
328C2×#102C22×C13712yes—
329#391C22×C13712yes—
330#390C22×C13711yescharacteristic
331#389—21no—
332#389—21no—
333#389C22×C13711yescharacteristic
334#388—21no—
335#386—21no—
336#384C22×C13711yescharacteristic
337#380—21no—
338#380—21no—
339#380C22×C13711yescharacteristic
340C4×C4⋊C4C22×C13711yescharacteristic
341C4×C22⋊C4—21no—
342#340C22×C13712yes—
343#340C22×C13712yes—
344C22×Q8C23×C13712yes—
345C22×D4—21no—
346C22×D4C23×C13711yescharacteristic
347C23×C4—21no—
348C23×C4D4×C13711yescharacteristic
349C23×C4D4×C13711yescharacteristic
350C23×C4C23×C13711yescharacteristic
351C2×SD16—24no—
352C2×SD16—24no—
353C2×SD16—42no—
354C22×C8—24no—
355C22×C8—24no—
356C22×C8—42no—
357C22×C8C23×C13712yes—
358C22×C8C23×C13712yes—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363#121—22no—
364#121D4×C13714yes—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—44no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C23×C13711yescharacteristic
375C2×C4⋊C4C23×C13712yes—
376C2×C4⋊C4C23×C13711yescharacteristic
377C2×C4⋊C4C23×C13711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—21no—
387C2×C42D4×C13711yescharacteristic
388C2×C42D4×C13711yescharacteristic
389C2×C42C23×C13711yescharacteristic
390C2×C42C23×C13711yescharacteristic
391#106—24no—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8D4×C13714yescomplemented
395#103—24no—
396#103—24no—
397#103—42no—
398#102—24no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404C24—21no—
405C24C2×D4×C13711yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4C2×D4×C13712yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4SD16×C13714yescomplemented
439C22×C4C2×D4×C13711yescharacteristic
440C22×C4C2×D4×C13711yescharacteristic
441C22×C4C2×D4×C13711yescharacteristic
442C22×C4C2×D4×C13712yes—
443C22×C4C2×D4×C13711yescharacteristic
444C22×C4C2×D4×C13711yescharacteristic
445C22×C4#41×C13712yes—
446C22×C4C24×C13711yescharacteristic
447SD16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—48no—
454C2×C8C2×D4×C13714yes—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4C2×D4×C13712yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—22no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42C2×D4×C13712yes—
481C42#41×C13712yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23#121×C13712yes—
494C23C2×SD16×C13712yes—
495C23#137×C13712yes—
496C23C22×D4×C13711yescharacteristic
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4#121×C13712yes—
538C2×C4#121×C13712yes—
539C2×C4#121×C13712yes—
540C2×C4C2×SD16×C13712yes—
541C2×C4C2×SD16×C13712yes—
542C2×C4#135×C13712yes—
543C2×C4C22×D4×C13711yescharacteristic
544C2×C4C22×D4×C13711yescharacteristic
545C2×C4C2×#41×C13711yescharacteristic
546C2×C4C2×#41×C13711yescharacteristic
547C2×C4H(5,𝔽2)×C13711yescharacteristic
548C2×C42−1+4×C13711yescharacteristic
549C8—24no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22#461×C13714yescomplemented
565C22C2×#121×C13711yescharacteristic
566C22C2×#121×C13711yescharacteristic
567C22#537×C13711yescharacteristic
568C22C22×SD16×C13711yescharacteristic
569C22C2×#135×C13711yescharacteristic
570C22C2×#137×C13711yescharacteristic
571C22#575×C13711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—82no—
579C4—81no—
580C4—82no—
581C4#465×C13714yescomplemented
582C4D42×C13711yescharacteristic
583C4#547×C13711yescharacteristic
584C4#548×C13711yescharacteristic
585C4#553×C13711yescharacteristic
586C2—24no—
587C2—82no—
588C2#2416×C13711yescharacteristic
589C2C2×#461×C13711yescharacteristic
590C2#3035×C13711yescharacteristic
591C2C2×#465×C13711yescharacteristic
592C2#3056×C13711yescharacteristic
593C2#3246×C13711yescharacteristic
594C2#3248×C13711yescharacteristic
subgroupG/H
center, socleC23×C137C22×D4
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×#121×C137C22
LCSG ▷ C22×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. C2×#461×C137C2semidirect
3. C22×C4⋊C8×C137C2semidirect
4. #2857×C137C2semidirect
6. #2066×C137C2semidirect
7. #2055×C137C2semidirect
8. #1981×C137C2semidirect
9. #1948×C137C2semidirect
10. #1892×C137C2semidirect
11. #1802×C137C2semidirect
96. C4⋊C8×C137D4semidirect
140. C22×C4×C137SD16semidirect
266. C22×C137#461semidirect
283. C4×C137#465semidirect
297. #24751C137direct (N × H)
298. C137#24751direct (N × H)
300. C2×#461C2×C137semidirect
301. C22×C4⋊C8C2×C137semidirect
302. #2857C2×C137semidirect
304. #2066C2×C137semidirect
305. #2055C2×C137semidirect
306. #1981C2×C137semidirect
307. #1948C2×C137semidirect
308. #1892C2×C137semidirect
309. #1802C2×C137semidirect
394. C4⋊C8D4×C137semidirect
438. C22×C4SD16×C137semidirect
564. C22#461×C137semidirect
581. C4#465×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#465×C1372
C2×#461×C1372
C22×C4⋊C8×C1372
#2857×C1372
C4×#121×C1372
#2066×C1372
#2055×C1372
#1981×C1372
#1948×C1372
#1892×C1372
#1802×C1372
#24751137