Finite Group Databasebeta

← C137×#24055 #46873441

Subgroups — C137×#24055 #46873441

Order 35072.

1338 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×#547×C137C211yesmaximal, characteristic
2C2×#483×C137C211yesmaximal, characteristic, complemented
3C2×#417×C137C211yesmaximal, characteristic, complemented
4#2487×C137C211yesmaximal, characteristic, complemented
5#2043×C137C212yesmaximal, complemented
6#2035×C137C212yesmaximal, complemented
7#1894×C137C211yesmaximal, characteristic, complemented
8#1870×C137C211yesmaximal, characteristic, complemented
9#1864×C137C212yesmaximal, complemented
10#1788×C137C212yesmaximal, complemented
11#1788×C137C211yesmaximal, characteristic, complemented
12C22×#41×C137C2211yescharacteristic
13C22×#41×C137C2211yescharacteristic
14C22×M16×C137C2211yescharacteristic
15#547×C137—24no—
16#547×C137—24no—
17C2×C4⋊Q8×C137C2211yescharacteristic
18C2×#123×C137—21no—
19C2×#123×C137—21no—
20C2×C22⋊Q8×C137C2211yescharacteristic
21C2×C22⋊Q8×C137C2211yescharacteristic
22C2×#121×C137—21no—
23C2×C4×D4×C137C2211yescharacteristic
24C2×C4×D4×C137C2211yescharacteristic
25C22×C4⋊C4×C137—21no—
26C22×C4⋊C4×C137C2211yescharacteristic
27#483×C137—24no—
28C2×#106×C137C2212yes—
29#417×C137—24no—
30C2×#103×C137C2212yes—
31C2×#103×C137C2212yes—
32C2×#103×C137C2212yes—
33C2×#103×C137C2212yes—
34C2×#102×C137C2212yes—
35C2×#102×C137C2212yes—
36C2×C22⋊C8×C137C2211yescharacteristic
37C2×C22⋊C8×C137C2212yes—
38#400×C137—21no—
39#399×C137—21no—
40#389×C137C2212yes—
41#386×C137—21no—
42#385×C137—21no—
43#380×C137C2212yes—
44#380×C137C2211yescharacteristic
45#340×C137C2212yes—
46#336×C137C2212yes—
47C2×#41×C137—22no—
48C2×#41×C137—22no—
49C2×#41×C137—22no—
50C2×#41×C137—24no—
51C2×#41×C137—24no—
52C2×#41×C137—42no—
53C2×#41×C137—42no—
54C2×#41×C137D414yes—
55C22×Q8×C137C2311yescharacteristic
56C22×Q8×C137C2311yescharacteristic
57C22×D4×C137—21no—
58C22×D4×C137—21no—
59C22×D4×C137D411yescharacteristic
60C22×D4×C137C2311yescharacteristic
61C23×C4×C137—21no—
62C23×C4×C137—21no—
63C23×C4×C137—21no—
64C23×C4×C137—21no—
65C23×C4×C137C2311yescharacteristic
66C23×C4×C137C2311yescharacteristic
67C2×M16×C137—22no—
68C2×M16×C137—42no—
69C22×C8×C137C2312yes—
70C22×C8×C137C2312yes—
71C4⋊Q8×C137—22no—
72C4⋊Q8×C137—22no—
73#123×C137—28no—
74#123×C137—28no—
75C22⋊Q8×C137—22no—
76C22⋊Q8×C137—22no—
77C22⋊Q8×C137—22no—
78C22⋊Q8×C137D414yes—
79#121×C137—28no—
80C4×D4×C137—22no—
81C4×D4×C137—22no—
82C4×D4×C137—22no—
83C4×D4×C137—22no—
84C2×C4⋊C4×C137—28no—
85C2×C4⋊C4×C137—21no—
86C2×C4⋊C4×C137—21no—
87C2×C4⋊C4×C137—21no—
88C2×C4⋊C4×C137—21no—
89C2×C4⋊C4×C137—22no—
90C2×C4⋊C4×C137—22no—
91C2×C4⋊C4×C137—21no—
92C2×C4⋊C4×C137—22no—
93C2×C4⋊C4×C137D411yescharacteristic
94C2×C4⋊C4×C137C2312yes—
95C2×C4⋊C4×C137C2312yes—
96C2×C4⋊C4×C137C2311yescharacteristic
97C2×C22⋊C4×C137—21no—
98C2×C22⋊C4×C137—21no—
99C2×C22⋊C4×C137—21no—
100C2×C22⋊C4×C137—21no—
101C2×C22⋊C4×C137—22no—
102C2×C22⋊C4×C137D411yescharacteristic
103C2×C22⋊C4×C137D411yescharacteristic
104C2×C42×C137C2311yescharacteristic
105#106×C137—24no—
106#103×C137—24no—
107#103×C137—24no—
108#103×C137—42no—
109#103×C137—42no—
110#102×C137—24no—
111#102×C137—24no—
112C22⋊C8×C137—22no—
113C22⋊C8×C137—42no—
114#95×C137—22no—
115#95×C137—21no—
116#95×C137—22no—
117C24×C137—21no—
118C24×C137C2×D411yescharacteristic
119C24×C137C2×D411yescharacteristic
120#41×C137—28no—
121#41×C137—44no—
122#41×C137—48no—
123#41×C137—48no—
124#41×C137—44no—
125#41×C137—44no—
126C2×Q8×C137—21no—
127C2×Q8×C137—21no—
128C2×Q8×C137—22no—
129C2×Q8×C137—22no—
130C2×Q8×C137—22no—
131C2×Q8×C137—22no—
132C2×Q8×C137C2×D412yes—
133C2×Q8×C137C2×D412yes—
134C2×D4×C137—24no—
135C2×D4×C137—22no—
136C2×D4×C137—24no—
137C2×D4×C137—22no—
138C2×D4×C137—21no—
139C2×D4×C137—22no—
140C2×D4×C137—41no—
141C2×D4×C137—44no—
142C2×D4×C137—44no—
143C2×D4×C137C2×D412yes—
144C2×D4×C137#4114yes—
145C22×C4×C137—22no—
146C22×C4×C137—22no—
147C22×C4×C137—24no—
148C22×C4×C137—24no—
149C22×C4×C137—21no—
150C22×C4×C137—21no—
151C22×C4×C137—21no—
152C22×C4×C137—24no—
153C22×C4×C137—21no—
154C22×C4×C137—21no—
155C22×C4×C137—22no—
156C22×C4×C137—22no—
157C22×C4×C137—22no—
158C22×C4×C137—22no—
159C22×C4×C137—44no—
160C22×C4×C137—44no—
161C22×C4×C137—42no—
162C22×C4×C137—42no—
163C22×C4×C137—42no—
164C22×C4×C137—82no—
165C22×C4×C137C2×D411yescharacteristic
166C22×C4×C137C2×D411yescharacteristic
167C22×C4×C137C2×D411yescharacteristic
168C22×C4×C137C2×D412yes—
169C22×C4×C137C2×D412yes—
170C22×C4×C137C2×D411yescharacteristic
171C22×C4×C137C2411yescharacteristic
172M16×C137—44no—
173C2×C8×C137—22no—
174C2×C8×C137—22no—
175C2×C8×C137—22no—
176C2×C8×C137—22no—
177C2×C8×C137—22no—
178C2×C8×C137—22no—
179C4⋊C4×C137—24no—
180C4⋊C4×C137—24no—
181C4⋊C4×C137—24no—
182C4⋊C4×C137—22no—
183C4⋊C4×C137—24no—
184C4⋊C4×C137—21no—
185C4⋊C4×C137—22no—
186C4⋊C4×C137—22no—
187C4⋊C4×C137—22no—
188C4⋊C4×C137—22no—
189C4⋊C4×C137—42no—
190C4⋊C4×C137—82no—
191C4⋊C4×C137C2×D412yes—
192C22⋊C4×C137—24no—
193C22⋊C4×C137—24no—
194C22⋊C4×C137—24no—
195C22⋊C4×C137—22no—
196C22⋊C4×C137—22no—
197C22⋊C4×C137—42no—
198C22⋊C4×C137—82no—
199C42×C137—21no—
200C42×C137—21no—
201C23×C137—22no—
202C23×C137—22no—
203C23×C137—24no—
204C23×C137—21no—
205C23×C137—21no—
206C23×C137—22no—
207C23×C137—21no—
208C23×C137—22no—
209C23×C137—21no—
210C23×C137—21no—
211C23×C137—22no—
212C23×C137—41no—
213C23×C137—41no—
214C23×C137—42no—
215C23×C137#12012yes—
216C23×C137C22×D411yescharacteristic
217Q8×C137—22no—
218Q8×C137—22no—
219Q8×C137—24no—
220Q8×C137—41no—
221Q8×C137—41no—
222Q8×C137—42no—
223D4×C137—22no—
224D4×C137—24no—
225D4×C137—44no—
226D4×C137—48no—
227D4×C137—48no—
228D4×C137#12314yes—
229C2×C4×C137—21no—
230C2×C4×C137—22no—
231C2×C4×C137—21no—
232C2×C4×C137—22no—
233C2×C4×C137—22no—
234C2×C4×C137—24no—
235C2×C4×C137—21no—
236C2×C4×C137—22no—
237C2×C4×C137—21no—
238C2×C4×C137—22no—
239C2×C4×C137—22no—
240C2×C4×C137—22no—
241C2×C4×C137—22no—
242C2×C4×C137—48no—
243C2×C4×C137—48no—
244C2×C4×C137—48no—
245C2×C4×C137—42no—
246C2×C4×C137—42no—
247C2×C4×C137—41no—
248C2×C4×C137—41no—
249C2×C4×C137—41no—
250C2×C4×C137—41no—
251C2×C4×C137—42no—
252C2×C4×C137—42no—
253C2×C4×C137—42no—
254C2×C4×C137—42no—
255C2×C4×C137—42no—
256C2×C4×C137—42no—
257C2×C4×C137—42no—
258C2×C4×C137—84no—
259C2×C4×C137#12012yes—
260C2×C4×C137#13712yes—
261C2×C4×C137C22×D411yescharacteristic
262C2×C4×C137C22×D411yescharacteristic
263C2×C4×C137C2×#4112yes—
264C2×C4×C137H(5,𝔽2)12yes—
265C8×C137—42no—
266C8×C137—42no—
267C22×C137—22no—
268C22×C137—24no—
269C22×C137—22no—
270C22×C137—22no—
271C22×C137—42no—
272C22×C137—44no—
273C22×C137—42no—
274C22×C137—42no—
275C22×C137—42no—
276C22×C137—44no—
277C22×C137—42no—
278C22×C137—41no—
279C22×C137—41no—
280C22×C137—41no—
281C22×C137—42no—
282C22×C137C2×#12011yescharacteristic
283C22×C137#54612yes—
284C22×C137C2×#13711yescharacteristic
285C22×C137#57511yescharacteristic
286C22×C137#57712yes—
287C4×C137—22no—
288C4×C137—22no—
289C4×C137—41no—
290C4×C137—42no—
291C4×C137—42no—
292C4×C137—42no—
293C4×C137—41no—
294C4×C137—82no—
295C4×C137—82no—
296C4×C137C2×#12311yescharacteristic
297C4×C137#53411yescharacteristic
298C4×C137#54612yes—
299C2×C137—42no—
300C2×C137—44no—
301C2×C137—81no—
302C2×C137#240511yescharacteristic
303C2×C137#300711yescharacteristic
304C2×C137#301011yescharacteristic
305C2×C137#330712yes—
306C2×C137#333112yes—
307#24055C13711yesmaximal, characteristic, direct product factor, Sylow
308C137#2405511yescharacteristic, direct product factor, Sylow
309C2×#547C2×C13711yescharacteristic
310C2×#483C2×C13711yescharacteristic, complemented
311C2×#417C2×C13711yescharacteristic, complemented
312#2487C2×C13711yescharacteristic, complemented
313#2043C2×C13712yescomplemented
314#2035C2×C13712yescomplemented
315#1894C2×C13711yescharacteristic, complemented
316#1870C2×C13711yescharacteristic, complemented
317#1864C2×C13712yescomplemented
318#1788C2×C13712yescomplemented
319#1788C2×C13711yescharacteristic, complemented
320C22×#41C22×C13711yescharacteristic
321C22×#41C22×C13711yescharacteristic
322C22×M16C22×C13711yescharacteristic
323#547—24no—
324#547—24no—
325C2×C4⋊Q8C22×C13711yescharacteristic
326C2×#123—21no—
327C2×#123—21no—
328C2×C22⋊Q8C22×C13711yescharacteristic
329C2×C22⋊Q8C22×C13711yescharacteristic
330C2×#121—21no—
331C2×C4×D4C22×C13711yescharacteristic
332C2×C4×D4C22×C13711yescharacteristic
333C22×C4⋊C4—21no—
334C22×C4⋊C4C22×C13711yescharacteristic
335#483—24no—
336C2×#106C22×C13712yes—
337#417—24no—
338C2×#103C22×C13712yes—
339C2×#103C22×C13712yes—
340C2×#103C22×C13712yes—
341C2×#103C22×C13712yes—
342C2×#102C22×C13712yes—
343C2×#102C22×C13712yes—
344C2×C22⋊C8C22×C13711yescharacteristic
345C2×C22⋊C8C22×C13712yes—
346#400—21no—
347#399—21no—
348#389C22×C13712yes—
349#386—21no—
350#385—21no—
351#380C22×C13711yescharacteristic
352#380C22×C13712yes—
353#340C22×C13712yes—
354#336C22×C13712yes—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—24no—
359C2×#41—24no—
360C2×#41—42no—
361C2×#41—42no—
362C2×#41D4×C13714yes—
363C22×Q8C23×C13711yescharacteristic
364C22×Q8C23×C13711yescharacteristic
365C22×D4—21no—
366C22×D4—21no—
367C22×D4D4×C13711yescharacteristic
368C22×D4C23×C13711yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—21no—
373C23×C4C23×C13711yescharacteristic
374C23×C4C23×C13711yescharacteristic
375C2×M16—22no—
376C2×M16—42no—
377C22×C8C23×C13712yes—
378C22×C8C23×C13712yes—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381#123—28no—
382#123—28no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C13714yes—
387#121—28no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4D4×C13711yescharacteristic
402C2×C4⋊C4C23×C13712yes—
403C2×C4⋊C4C23×C13712yes—
404C2×C4⋊C4C23×C13711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4D4×C13711yescharacteristic
411C2×C22⋊C4D4×C13711yescharacteristic
412C2×C42C23×C13711yescharacteristic
413#106—24no—
414#103—24no—
415#103—24no—
416#103—42no—
417#103—42no—
418#102—24no—
419#102—24no—
420C22⋊C8—22no—
421C22⋊C8—42no—
422#95—22no—
423#95—21no—
424#95—22no—
425C24—21no—
426C24C2×D4×C13711yescharacteristic
427C24C2×D4×C13711yescharacteristic
428#41—28no—
429#41—44no—
430#41—48no—
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8C2×D4×C13712yes—
441C2×Q8C2×D4×C13712yes—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4C2×D4×C13712yes—
452C2×D4#41×C13714yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—82no—
473C22×C4C2×D4×C13711yescharacteristic
474C22×C4C2×D4×C13711yescharacteristic
475C22×C4C2×D4×C13711yescharacteristic
476C22×C4C2×D4×C13712yes—
477C22×C4C2×D4×C13712yes—
478C22×C4C2×D4×C13711yescharacteristic
479C22×C4C24×C13711yescharacteristic
480M16—44no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—82no—
499C4⋊C4C2×D4×C13712yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—82no—
507C42—21no—
508C42—21no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23#120×C13712yes—
524C23C22×D4×C13711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—24no—
528Q8—41no—
529Q8—41no—
530Q8—42no—
531D4—22no—
532D4—24no—
533D4—44no—
534D4—48no—
535D4—48no—
536D4#123×C13714yes—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4#120×C13712yes—
568C2×C4#137×C13712yes—
569C2×C4C22×D4×C13711yescharacteristic
570C2×C4C22×D4×C13711yescharacteristic
571C2×C4C2×#41×C13712yes—
572C2×C4H(5,𝔽2)×C13712yes—
573C8—42no—
574C8—42no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22C2×#120×C13711yescharacteristic
591C22#546×C13712yes—
592C22C2×#137×C13711yescharacteristic
593C22#575×C13711yescharacteristic
594C22#577×C13712yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—82no—
603C4—82no—
604C4C2×#123×C13711yescharacteristic
605C4#534×C13711yescharacteristic
606C4#546×C13712yes—
607C2—42no—
608C2—44no—
609C2—81no—
610C2#2405×C13711yescharacteristic
611C2#3007×C13711yescharacteristic
612C2#3010×C13711yescharacteristic
613C2#3307×C13712yes—
614C2#3331×C13712yes—
subgroupG/H
center, socleC23×C137C22×D4
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×#547×C137C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#483×C137C2semidirect
3. C2×#417×C137C2semidirect
4. #2487×C137C2semidirect
5. #2043×C137C2semidirect
6. #2035×C137C2semidirect
7. #1894×C137C2semidirect
8. #1870×C137C2semidirect
9. #1864×C137C2semidirect
10. #1788×C137C2semidirect
11. #1788×C137C2semidirect
307. #24055C137direct (N × H)
308. C137#24055direct (N × H)
310. C2×#483C2×C137semidirect
311. C2×#417C2×C137semidirect
312. #2487C2×C137semidirect
313. #2043C2×C137semidirect
314. #2035C2×C137semidirect
315. #1894C2×C137semidirect
316. #1870C2×C137semidirect
317. #1864C2×C137semidirect
318. #1788C2×C137semidirect
319. #1788C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#547×C1372
C2×#483×C1372
C2×#417×C1372
#2487×C1372
#2043×C1372
#2035×C1372
#1894×C1372
#1870×C1372
#1864×C1372
#1788×C1372
#24055137