Finite Group Databasebeta

← C137×#23923 #46873309

Subgroups — C137×#23923 #46873309

Order 35072.

1948 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525×C137C212yesmaximal, complemented
2C2×#452×C137C212yesmaximal, complemented
3C2×#449×C137C212yesmaximal, complemented
4#2001×C137C211yesmaximal, characteristic, complemented
5#2000×C137C212yesmaximal, complemented
6#1999×C137C211yesmaximal, characteristic, complemented
7#1875×C137C212yesmaximal, complemented
8#1873×C137C212yesmaximal, complemented
9#1779×C137C211yesmaximal, characteristic, complemented
10C22×#41×C137C2212yes—
11C24×C4×C137C2211yescharacteristic
12C22×Q16×C137C2211yescharacteristic, complemented
13C22×SD16×C137C2212yescomplemented
14C22×D8×C137C2211yescharacteristic, complemented
15#525×C137—28no—
16#525×C137—28no—
17C2×#123×C137—22no—
18C2×C22⋊Q8×C137C2212yes—
19C2×C22⋊Q8×C137C2212yes—
20C2×#121×C137C2212yes—
21C2×#121×C137C2212yes—
22C2×#120×C137—22no—
23C2×C4×D4×C137—22no—
24C2×C4×D4×C137—22no—
25C2×#117×C137C2212yes—
26#452×C137—28no—
27#449×C137—28no—
28C2×#103×C137C2212yescomplemented
29C2×#103×C137C2212yescomplemented
30C2×#102×C137C2212yescomplemented
31C2×#102×C137C2212yescomplemented
32C2×C22⋊C8×C137C2212yes—
33C2×C22⋊C8×C137C2212yes—
34C2×C22⋊C8×C137C2211yescharacteristic
35C2×C22⋊C8×C137C2211yescharacteristic
36#340×C137C2211yescharacteristic, complemented
37#340×C137C2212yescomplemented
38#340×C137C2211yescharacteristic, complemented
39C25×C137D411yescharacteristic
40C2×#41×C137—24no—
41C2×#41×C137—24no—
42C2×#41×C137—24no—
43C2×#41×C137—24no—
44C2×#41×C137—42no—
45C2×#41×C137—42no—
46C22×Q8×C137C2312yes—
47C22×D4×C137—22no—
48C22×D4×C137—22no—
49C22×D4×C137—22no—
50C22×D4×C137C2312yes—
51C23×C4×C137—22no—
52C23×C4×C137—24no—
53C23×C4×C137—24no—
54C23×C4×C137—41no—
55C23×C4×C137—41no—
56C23×C4×C137D411yescharacteristic
57C23×C4×C137D412yes—
58C23×C4×C137C2311yescharacteristic
59C23×C4×C137C2312yes—
60C2×Q16×C137—24no—
61C2×Q16×C137—41no—
62C2×SD16×C137—24no—
63C2×SD16×C137—24no—
64C2×SD16×C137—42no—
65C2×D8×C137—24no—
66C2×D8×C137—41no—
67C22×C8×C137C2311yescharacteristic
68C22×C8×C137C2312yes—
69C22×C8×C137C2311yescharacteristic
70#123×C137—216no—
71C22⋊Q8×C137—24no—
72C22⋊Q8×C137—24no—
73C22⋊Q8×C137—24no—
74C22⋊Q8×C137D418yescomplemented
75#121×C137—24no—
76#121×C137—24no—
77#121×C137—24no—
78#121×C137D418yescomplemented
79#120×C137—216no—
80C4×D4×C137—216no—
81C4×D4×C137—216no—
82#117×C137—24no—
83#117×C137—24no—
84C2×C4⋊C4×C137—22no—
85C2×C4⋊C4×C137—22no—
86C2×C4⋊C4×C137C2312yes—
87C2×C4⋊C4×C137C2312yes—
88C2×C22⋊C4×C137—22no—
89C2×C22⋊C4×C137—22no—
90C2×C22⋊C4×C137—22no—
91C2×C22⋊C4×C137—22no—
92C2×C22⋊C4×C137—22no—
93C2×C42×C137—22no—
94#103×C137—24no—
95#103×C137—24no—
96#102×C137—24no—
97#102×C137—24no—
98C22⋊C8×C137—24no—
99C22⋊C8×C137—24no—
100C22⋊C8×C137—41no—
101C22⋊C8×C137—41no—
102C24×C137—24no—
103C24×C137—22no—
104C24×C137—22no—
105C24×C137—41no—
106C24×C137—42no—
107C24×C137—41no—
108C24×C137C2×D412yes—
109C24×C137C2×D411yescharacteristic
110#41×C137—48no—
111#41×C137—48no—
112#41×C137—48no—
113#41×C137—48no—
114C2×Q8×C137—22no—
115C2×Q8×C137—22no—
116C2×Q8×C137—22no—
117C2×Q8×C137—22no—
118C2×Q8×C137—22no—
119C2×Q8×C137C2×D414yes—
120C2×D4×C137—28no—
121C2×D4×C137—28no—
122C2×D4×C137—28no—
123C2×D4×C137—22no—
124C2×D4×C137—22no—
125C2×D4×C137—22no—
126C2×D4×C137—22no—
127C2×D4×C137—22no—
128C2×D4×C137—44no—
129C2×D4×C137—44no—
130C2×D4×C137—44no—
131C2×D4×C137—44no—
132C2×D4×C137—44no—
133C2×D4×C137—44no—
134C2×D4×C137C2×D414yes—
135C22×C4×C137—22no—
136C22×C4×C137—22no—
137C22×C4×C137—22no—
138C22×C4×C137—22no—
139C22×C4×C137—28no—
140C22×C4×C137—24no—
141C22×C4×C137—22no—
142C22×C4×C137—22no—
143C22×C4×C137—22no—
144C22×C4×C137—24no—
145C22×C4×C137—24no—
146C22×C4×C137—22no—
147C22×C4×C137—22no—
148C22×C4×C137—21no—
149C22×C4×C137—21no—
150C22×C4×C137—41no—
151C22×C4×C137—42no—
152C22×C4×C137—44no—
153C22×C4×C137—48no—
154C22×C4×C137—44no—
155C22×C4×C137—44no—
156C22×C4×C137—44no—
157C22×C4×C137C2×D411yescharacteristic
158C22×C4×C137C2×D412yes—
159C22×C4×C137C2×D414yes—
160C22×C4×C137C2×D414yes—
161C22×C4×C137C2411yescharacteristic
162Q16×C137—44no—
163SD16×C137—48no—
164D8×C137—44no—
165C2×C8×C137—22no—
166C2×C8×C137—22no—
167C2×C8×C137—22no—
168C2×C8×C137—22no—
169C2×C8×C137—21no—
170C2×C8×C137—22no—
171C2×C8×C137—21no—
172C4⋊C4×C137—28no—
173C4⋊C4×C137—28no—
174C4⋊C4×C137—22no—
175C4⋊C4×C137—22no—
176C4⋊C4×C137C2×D414yes—
177C4⋊C4×C137C2×D414yes—
178C22⋊C4×C137—28no—
179C22⋊C4×C137—28no—
180C22⋊C4×C137—28no—
181C22⋊C4×C137—28no—
182C22⋊C4×C137—28no—
183C42×C137—28no—
184C23×C137—22no—
185C23×C137—22no—
186C23×C137—22no—
187C23×C137—21no—
188C23×C137—22no—
189C23×C137—22no—
190C23×C137—22no—
191C23×C137—22no—
192C23×C137—21no—
193C23×C137—22no—
194C23×C137—24no—
195C23×C137—44no—
196C23×C137—48no—
197C23×C137—44no—
198C23×C137—48no—
199C23×C137—44no—
200C23×C137—42no—
201C23×C137—42no—
202C23×C137—42no—
203C23×C137—42no—
204C23×C137—42no—
205C23×C137—42no—
206C23×C137#12014yes—
207C23×C137#13512yes—
208C23×C137C22×D411yescharacteristic
209Q8×C137—24no—
210Q8×C137—24no—
211Q8×C137—42no—
212Q8×C137—42no—
213D4×C137—24no—
214D4×C137—24no—
215D4×C137—416no—
216D4×C137—416no—
217D4×C137—416no—
218D4×C137—42no—
219D4×C137—42no—
220C2×C4×C137—22no—
221C2×C4×C137—22no—
222C2×C4×C137—24no—
223C2×C4×C137—24no—
224C2×C4×C137—24no—
225C2×C4×C137—24no—
226C2×C4×C137—24no—
227C2×C4×C137—24no—
228C2×C4×C137—22no—
229C2×C4×C137—24no—
230C2×C4×C137—21no—
231C2×C4×C137—44no—
232C2×C4×C137—48no—
233C2×C4×C137—416no—
234C2×C4×C137—42no—
235C2×C4×C137—42no—
236C2×C4×C137—42no—
237C2×C4×C137—42no—
238C2×C4×C137—42no—
239C2×C4×C137—42no—
240C2×C4×C137—41no—
241C2×C4×C137—42no—
242C2×C4×C137—41no—
243C2×C4×C137#12014yes—
244C2×C4×C137C22×D412yes—
245C2×C4×C137C22×D412yes—
246C2×C4×C137H(5,𝔽2)11yescharacteristic
247C2×C4×C137H(5,𝔽2)11yescharacteristic
248C8×C137—41no—
249C8×C137—42no—
250C8×C137—41no—
251C22×C137—24no—
252C22×C137—22no—
253C22×C137—24no—
254C22×C137—24no—
255C22×C137—24no—
256C22×C137—416no—
257C22×C137—41no—
258C22×C137—42no—
259C22×C137—42no—
260C22×C137—41no—
261C22×C137—42no—
262C22×C137—42no—
263C22×C137—42no—
264C22×C137—44no—
265C22×C137—44no—
266C22×C137—44no—
267C22×C137—82no—
268C22×C137—82no—
269C22×C137—82no—
270C22×C137—82no—
271C22×C137C2×#12012yes—
272C22×C137#53411yescharacteristic
273C22×C137C2×#13511yescharacteristic
274C22×C137#57511yescharacteristic
275C22×C137#57512yes—
276C4×C137—24no—
277C4×C137—41no—
278C4×C137—44no—
279C4×C137—44no—
280C4×C137—44no—
281C4×C137D4211yescharacteristic
282C4×C137D4212yes—
283C4×C137D4211yescharacteristic
284C2×C137—44no—
285C2×C137—42no—
286C2×C137—84no—
287C2×C137#240311yescharacteristic
288C2×C137#300212yes—
289C2×C137#300512yes—
290C2×C137#319111yescharacteristic
291C2×C137#319711yescharacteristic
292#23923C13711yesmaximal, characteristic, direct product factor, Sylow
293C137#2392311yescharacteristic, direct product factor, Sylow
294C2×#525C2×C13712yescomplemented
295C2×#452C2×C13712yescomplemented
296C2×#449C2×C13712yescomplemented
297#2001C2×C13711yescharacteristic, complemented
298#2000C2×C13712yescomplemented
299#1999C2×C13711yescharacteristic, complemented
300#1875C2×C13712yescomplemented
301#1873C2×C13712yescomplemented
302#1779C2×C13711yescharacteristic, complemented
303C22×#41C22×C13712yes—
304C24×C4C22×C13711yescharacteristic
305C22×Q16C22×C13711yescharacteristic, complemented
306C22×SD16C22×C13712yescomplemented
307C22×D8C22×C13711yescharacteristic, complemented
308#525—28no—
309#525—28no—
310C2×#123—22no—
311C2×C22⋊Q8C22×C13712yes—
312C2×C22⋊Q8C22×C13712yes—
313C2×#121C22×C13712yes—
314C2×#121C22×C13712yes—
315C2×#120—22no—
316C2×C4×D4—22no—
317C2×C4×D4—22no—
318C2×#117C22×C13712yes—
319#452—28no—
320#449—28no—
321C2×#103C22×C13712yescomplemented
322C2×#103C22×C13712yescomplemented
323C2×#102C22×C13712yescomplemented
324C2×#102C22×C13712yescomplemented
325C2×C22⋊C8C22×C13712yes—
326C2×C22⋊C8C22×C13712yes—
327C2×C22⋊C8C22×C13711yescharacteristic
328C2×C22⋊C8C22×C13711yescharacteristic
329#340C22×C13711yescharacteristic, complemented
330#340C22×C13712yescomplemented
331#340C22×C13711yescharacteristic, complemented
332C25D4×C13711yescharacteristic
333C2×#41—24no—
334C2×#41—24no—
335C2×#41—24no—
336C2×#41—24no—
337C2×#41—42no—
338C2×#41—42no—
339C22×Q8C23×C13712yes—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4C23×C13712yes—
344C23×C4—22no—
345C23×C4—24no—
346C23×C4—24no—
347C23×C4—41no—
348C23×C4—41no—
349C23×C4D4×C13711yescharacteristic
350C23×C4D4×C13712yes—
351C23×C4C23×C13711yescharacteristic
352C23×C4C23×C13712yes—
353C2×Q16—24no—
354C2×Q16—41no—
355C2×SD16—24no—
356C2×SD16—24no—
357C2×SD16—42no—
358C2×D8—24no—
359C2×D8—41no—
360C22×C8C23×C13711yescharacteristic
361C22×C8C23×C13712yes—
362C22×C8C23×C13711yescharacteristic
363#123—216no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366C22⋊Q8—24no—
367C22⋊Q8D4×C13718yescomplemented
368#121—24no—
369#121—24no—
370#121—24no—
371#121D4×C13718yescomplemented
372#120—216no—
373C4×D4—216no—
374C4×D4—216no—
375#117—24no—
376#117—24no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4C23×C13712yes—
380C2×C4⋊C4C23×C13712yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C42—22no—
387#103—24no—
388#103—24no—
389#102—24no—
390#102—24no—
391C22⋊C8—24no—
392C22⋊C8—24no—
393C22⋊C8—41no—
394C22⋊C8—41no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—41no—
399C24—42no—
400C24—41no—
401C24C2×D4×C13712yes—
402C24C2×D4×C13711yescharacteristic
403#41—48no—
404#41—48no—
405#41—48no—
406#41—48no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8C2×D4×C13714yes—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4C2×D4×C13714yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4C2×D4×C13711yescharacteristic
451C22×C4C2×D4×C13712yes—
452C22×C4C2×D4×C13714yes—
453C22×C4C2×D4×C13714yes—
454C22×C4C24×C13711yescharacteristic
455Q16—44no—
456SD16—48no—
457D8—44no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—21no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4C2×D4×C13714yes—
470C4⋊C4C2×D4×C13714yes—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C42—28no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—21no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—22no—
487C23—24no—
488C23—44no—
489C23—48no—
490C23—44no—
491C23—48no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#120×C13714yes—
500C23#135×C13712yes—
501C23C22×D4×C13711yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506D4—24no—
507D4—24no—
508D4—416no—
509D4—416no—
510D4—416no—
511D4—42no—
512D4—42no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4#120×C13714yes—
537C2×C4C22×D4×C13712yes—
538C2×C4C22×D4×C13712yes—
539C2×C4H(5,𝔽2)×C13711yescharacteristic
540C2×C4H(5,𝔽2)×C13711yescharacteristic
541C8—41no—
542C8—42no—
543C8—41no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—416no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22C2×#120×C13712yes—
565C22#534×C13711yescharacteristic
566C22C2×#135×C13711yescharacteristic
567C22#575×C13711yescharacteristic
568C22#575×C13712yes—
569C4—24no—
570C4—41no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4D42×C13711yescharacteristic
575C4D42×C13712yes—
576C4D42×C13711yescharacteristic
577C2—44no—
578C2—42no—
579C2—84no—
580C2#2403×C13711yescharacteristic
581C2#3002×C13712yes—
582C2#3005×C13712yes—
583C2#3191×C13711yescharacteristic
584C2#3197×C13711yescharacteristic
subgroupG/H
center, socleC23×C137C22×D4
commutator, Frattini, cosocleC22×C4C24×C137
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C24×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#525×C137C2semidirect
2. C2×#452×C137C2semidirect
3. C2×#449×C137C2semidirect
4. #2001×C137C2semidirect
5. #2000×C137C2semidirect
6. #1999×C137C2semidirect
7. #1875×C137C2semidirect
8. #1873×C137C2semidirect
9. #1779×C137C2semidirect
12. C22×Q16×C137C22semidirect
13. C22×SD16×C137C22semidirect
14. C22×D8×C137C22semidirect
28. C2×#103×C137C22semidirect
29. C2×#103×C137C22semidirect
30. C2×#102×C137C22semidirect
31. C2×#102×C137C22semidirect
36. #340×C137C22semidirect
37. #340×C137C22semidirect
38. #340×C137C22semidirect
74. C22⋊Q8×C137D4semidirect
78. #121×C137D4semidirect
292. #23923C137direct (N × H)
293. C137#23923direct (N × H)
294. C2×#525C2×C137semidirect
295. C2×#452C2×C137semidirect
296. C2×#449C2×C137semidirect
297. #2001C2×C137semidirect
298. #2000C2×C137semidirect
299. #1999C2×C137semidirect
300. #1875C2×C137semidirect
301. #1873C2×C137semidirect
302. #1779C2×C137semidirect
305. C22×Q16C22×C137semidirect
306. C22×SD16C22×C137semidirect
307. C22×D8C22×C137semidirect
321. C2×#103C22×C137semidirect
322. C2×#103C22×C137semidirect
323. C2×#102C22×C137semidirect
324. C2×#102C22×C137semidirect
329. #340C22×C137semidirect
330. #340C22×C137semidirect
331. #340C22×C137semidirect
367. C22⋊Q8D4×C137semidirect
371. #121D4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#525×C1372
C2×#452×C1372
C2×#449×C1372
#2001×C1372
#2000×C1372
#1999×C1372
#1875×C1372
#1873×C1372
#1779×C1372
#23923137