Finite Group Databasebeta

← C37×#61886 #22683139

Subgroups — C37×#61886 #22683139

Order 9472.

1796 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3522C211yesmaximal, characteristic, complemented
2C37×#3480C211yesmaximal, characteristic, complemented
3C37×#3396C213yesmaximal, complemented
4C37×#3394C213yesmaximal, complemented
5C37×#3392C211yesmaximal, characteristic, complemented
6C37×#3391C211yesmaximal, characteristic, complemented
7C37×#3363C211yesmaximal, characteristic, complemented
8C37×#3362C211yesmaximal, characteristic, complemented
9C37×#3361C213yesmaximal, complemented
10C37×#3360C213yesmaximal, complemented
11C37×#3212C213yesmaximal, complemented
12C37×#3207C213yesmaximal, complemented
13C37×#3106C213yesmaximal, complemented
14C37×#3102C213yesmaximal, complemented
15C37×#2982C211yesmaximal, characteristic, complemented
16C37×#559C2213yes—
17C37×#559C2213yes—
18Q82×C37C2211yescharacteristic
19C37×#557C2211yescharacteristic
20C37×#556C2213yes—
21C37×#554C2213yes—
22C37×#553—23no—
23C37×#550C2211yescharacteristic
24C37×#550C2211yescharacteristic
25C37×#548—21no—
26C37×#548—21no—
27C37×#547—23no—
28C37×#542—23no—
29C37×#532C2213yes—
30C37×#532C2213yes—
31C37×#529—23no—
32C37×#529C2213yes—
33C37×#520—21no—
34C4×#41×C37—21no—
35C4×#41×C37C2213yes—
36C2×C4×Q8×C37C2211yescharacteristic
37C37×#495C2216yes—
38C37×#493C2213yes—
39C37×#492C2213yes—
40C37×#488C2216yes—
41C37×#487C2213yes—
42C37×#486C2213yes—
43C37×#485C2211yescharacteristic
44C37×#482C2213yes—
45C37×#479C2213yes—
46C37×#478C2213yes—
47C37×#477C2213yes—
48C37×#476C2213yes—
49C37×#464C2213yes—
50C37×#463C2216yes—
51C37×#463C2213yes—
52C37×#463C2216yes—
53C37×#462C2213yes—
54C37×#461C2213yes—
55C37×#460C2213yes—
56C37×#460C2213yes—
57C37×#459C2213yes—
58C37×#459C2213yes—
59C37×#452C2213yes—
60C37×#449C2213yes—
61C37×#449C2216yes—
62C8×Q8×C37C2211yescharacteristic
63C8×Q8×C37C2211yescharacteristic
64C8×Q8×C37C2213yes—
65C8×Q8×C37C2213yes—
66C4×Q16×C37C2213yes—
67C4×SD16×C37C2213yes—
68C4×SD16×C37C2213yes—
69C4×SD16×C37C2216yes—
70C4×D8×C37C2213yes—
71C4×D8×C37C2216yes—
72C37×#431C2213yes—
73C37×#431C2213yes—
74C2×#41×C37—21no—
75C2×#41×C37—21no—
76C2×#41×C37—23no—
77C2×#41×C37C2313yes—
78C22×Q8×C37D411yescharacteristic
79C2×Q16×C37C2313yes—
80C2×SD16×C37C2316yes—
81C2×SD16×C37C2313yes—
82C2×SD16×C37C2313yes—
83C2×D8×C37C2316yes—
84C2×D8×C37C2313yes—
85C4⋊Q8×C37—23no—
86C4⋊Q8×C37C2313yes—
87C4⋊Q8×C37C2313yes—
88#127×C37C2313yes—
89#127×C37C2313yes—
90#126×C37—26no—
91#126×C37—23no—
92#125×C37—23no—
93#125×C37C2313yes—
94#124×C37—23no—
95#124×C37—23no—
96#124×C37—23no—
97#124×C37C2313yes—
98#124×C37C2316yes—
99#123×C37—23no—
100#123×C37—26no—
101#123×C37—26no—
102C22⋊Q8×C37—23no—
103C22⋊Q8×C37—23no—
104C22⋊Q8×C37—26no—
105C22⋊Q8×C37C2311yescharacteristic
106#121×C37—23no—
107#121×C37—23no—
108#121×C37—23no—
109#121×C37—23no—
110#121×C37—23no—
111#121×C37C2313yes—
112C4×Q8×C37—21no—
113C4×Q8×C37—21no—
114C4×Q8×C37D411yescharacteristic
115C4×Q8×C37D413yes—
116C4×Q8×C37C2313yes—
117C4×Q8×C37C2313yes—
118C4×Q8×C37C2313yes—
119C4×Q8×C37C2311yescharacteristic
120C4×Q8×C37C2313yes—
121C4×D4×C37—23no—
122C4×D4×C37—23no—
123C4×D4×C37—23no—
124C4×D4×C37—23no—
125C4×D4×C37—23no—
126C4×D4×C37—23no—
127C4×D4×C37—23no—
128C4×D4×C37—23no—
129C4×D4×C37C2313yes—
130C4×D4×C37C2316yes—
131C4×D4×C37C2313yes—
132C4×D4×C37C2313yes—
133#117×C37—23no—
134#117×C37—23no—
135#117×C37—23no—
136#117×C37C2311yescharacteristic
137C2×C4⋊C4×C37—23no—
138C2×C4⋊C4×C37D413yes—
139C2×C42×C37—23no—
140C2×C42×C37C2313yes—
141#107×C37C2311yescharacteristic
142#107×C37C2313yes—
143#106×C37C2313yes—
144#106×C37C2311yescharacteristic
145C4⋊C8×C37C2313yes—
146C4⋊C8×C37C2313yes—
147C4⋊C8×C37C2313yes—
148C4⋊C8×C37C2313yes—
149#103×C37C2311yescharacteristic
150#103×C37C2316yes—
151#103×C37C2313yes—
152#103×C37C2311yescharacteristic
153#103×C37C2313yes—
154#102×C37C2313yes—
155#102×C37C2313yes—
156#102×C37C2316yes—
157#102×C37C2313yes—
158#102×C37C2313yes—
159C22⋊C8×C37C2311yescharacteristic
160C22⋊C8×C37C2313yes—
161C4×C8×C37C2316yes—
162C4×C8×C37C2313yes—
163C4×C8×C37C2313yes—
164#41×C37—43no—
165#41×C37—43no—
166#41×C37—44no—
167#41×C37—44no—
168#41×C37—86no—
169C2×Q8×C37—24no—
170C2×Q8×C37—23no—
171C2×Q8×C37C2×D413yes—
172C2×Q8×C37C2×D411yescharacteristic
173C2×Q8×C37C2413yes—
174C2×Q8×C37C2411yescharacteristic
175C2×D4×C37—23no—
176C2×D4×C37—23no—
177C2×D4×C37—23no—
178C2×D4×C37—23no—
179C2×D4×C37—23no—
180C2×D4×C37—23no—
181C2×D4×C37C2413yes—
182C2×D4×C37C2413yes—
183C22×C4×C37—23no—
184C22×C4×C37—23no—
185C22×C4×C37—23no—
186C22×C4×C37—23no—
187C22×C4×C37—26no—
188C22×C4×C37C2×D413yes—
189C22×C4×C37C2×D413yes—
190C22×C4×C37C2411yescharacteristic
191Q16×C37—43no—
192SD16×C37—43no—
193SD16×C37—46no—
194SD16×C37—43no—
195D8×C37—43no—
196D8×C37—46no—
197C2×C8×C37C2413yes—
198C2×C8×C37C2411yescharacteristic
199C2×C8×C37C2413yes—
200C2×C8×C37C2411yescharacteristic
201C4⋊C4×C37—23no—
202C4⋊C4×C37—23no—
203C4⋊C4×C37—21no—
204C4⋊C4×C37—23no—
205C4⋊C4×C37—23no—
206C4⋊C4×C37—23no—
207C4⋊C4×C37—23no—
208C4⋊C4×C37—23no—
209C4⋊C4×C37—23no—
210C4⋊C4×C37C2×D413yes—
211C4⋊C4×C37C2×D413yes—
212C4⋊C4×C37C2×D416yes—
213C4⋊C4×C37C2411yescharacteristic
214C4⋊C4×C37C2413yes—
215C4⋊C4×C37C2411yescharacteristic
216C4⋊C4×C37C2411yescharacteristic
217C22⋊C4×C37—21no—
218C22⋊C4×C37—21no—
219C22⋊C4×C37—26no—
220C22⋊C4×C37—23no—
221C22⋊C4×C37—23no—
222C22⋊C4×C37—23no—
223C22⋊C4×C37—23no—
224C22⋊C4×C37—26no—
225C22⋊C4×C37—23no—
226C22⋊C4×C37—23no—
227C22⋊C4×C37—26no—
228C42×C37—21no—
229C42×C37—23no—
230C42×C37—23no—
231C42×C37—23no—
232C42×C37C2×D413yes—
233C42×C37C2×D413yes—
234C42×C37C2413yes—
235C42×C37C2413yes—
236C23×C37—23no—
237C23×C37—23no—
238C23×C37C22×D411yescharacteristic
239Q8×C37—26no—
240Q8×C37—21no—
241Q8×C37—23no—
242Q8×C37—23no—
243Q8×C37—21no—
244Q8×C37—83no—
245Q8×C37#13514yes—
246D4×C37—23no—
247D4×C37—23no—
248D4×C37—23no—
249D4×C37—23no—
250D4×C37—46no—
251D4×C37—46no—
252D4×C37—46no—
253D4×C37—46no—
254D4×C37—83no—
255D4×C37—83no—
256C2×C4×C37—26no—
257C2×C4×C37—21no—
258C2×C4×C37—21no—
259C2×C4×C37—21no—
260C2×C4×C37—21no—
261C2×C4×C37—21no—
262C2×C4×C37—23no—
263C2×C4×C37—23no—
264C2×C4×C37—21no—
265C2×C4×C37—43no—
266C2×C4×C37—46no—
267C2×C4×C37—46no—
268C2×C4×C37—46no—
269C2×C4×C37—83no—
270C2×C4×C37—86no—
271C2×C4×C37C22×D411yescharacteristic
272C2×C4×C37C22×D413yes—
273C2×C4×C37C22×D413yes—
274C2×C4×C37C22×D413yes—
275C2×C4×C37C22×D413yes—
276C2×C4×C37C2511yescharacteristic
277C8×C37—21no—
278C8×C37—23no—
279C8×C37—23no—
280C8×C37—21no—
281#61886C3711yesmaximal, characteristic, direct product factor, Sylow
282C22×C37—21no—
283C22×C37—21no—
284C22×C37—21no—
285C22×C37—43no—
286C22×C37—43no—
287C22×C37—43no—
288C22×C37—43no—
289C22×C37—43no—
290C22×C37—43no—
291C22×C37C23×D411yescharacteristic
292C4×C37—23no—
293C4×C37—23no—
294C4×C37—21no—
295C4×C37—23no—
296C4×C37—41no—
297C4×C37—43no—
298C4×C37—43no—
299C4×C37—41no—
300C4×C37—41no—
301C4×C37—41no—
302C4×C37C2×#13516yes—
303C4×C37#58511yescharacteristic
304C4×C37#58511yescharacteristic
305#3522C2×C3711yescharacteristic, complemented
306#3480C2×C3711yescharacteristic, complemented
307#3396C2×C3713yescomplemented
308#3394C2×C3713yescomplemented
309#3392C2×C3711yescharacteristic, complemented
310#3391C2×C3711yescharacteristic, complemented
311#3363C2×C3711yescharacteristic, complemented
312#3362C2×C3711yescharacteristic, complemented
313#3361C2×C3713yescomplemented
314#3360C2×C3713yescomplemented
315#3212C2×C3713yescomplemented
316#3207C2×C3713yescomplemented
317#3106C2×C3713yescomplemented
318#3102C2×C3713yescomplemented
319#2982C2×C3711yescharacteristic, complemented
320C2×C37—41no—
321C2×C37—83no—
322C2×C37—83no—
323C2×C37#348711yescharacteristic
324C2×C37C22×#13511yescharacteristic
325C2×C37#358511yescharacteristic
326#559C22×C3713yes—
327#559C22×C3713yes—
328Q82C22×C3711yescharacteristic
329#557C22×C3711yescharacteristic
330#556C22×C3713yes—
331#554C22×C3713yes—
332#553—23no—
333#550C22×C3711yescharacteristic
334#550C22×C3711yescharacteristic
335#548—21no—
336#548—21no—
337#547—23no—
338#542—23no—
339#532C22×C3713yes—
340#532C22×C3713yes—
341#529—23no—
342#529C22×C3713yes—
343#520—21no—
344C4×#41—21no—
345C4×#41C22×C3713yes—
346C2×C4×Q8C22×C3711yescharacteristic
347#495C22×C3716yes—
348#493C22×C3713yes—
349#492C22×C3713yes—
350#488C22×C3716yes—
351#487C22×C3713yes—
352#486C22×C3713yes—
353#485C22×C3711yescharacteristic
354#482C22×C3713yes—
355#479C22×C3713yes—
356#478C22×C3713yes—
357#477C22×C3713yes—
358#476C22×C3713yes—
359#464C22×C3713yes—
360#463C22×C3713yes—
361#463C22×C3716yes—
362#463C22×C3716yes—
363#462C22×C3713yes—
364#461C22×C3713yes—
365#460C22×C3713yes—
366#460C22×C3713yes—
367#459C22×C3713yes—
368#459C22×C3713yes—
369#452C22×C3713yes—
370#449C22×C3713yes—
371#449C22×C3716yes—
372C8×Q8C22×C3713yes—
373C8×Q8C22×C3713yes—
374C8×Q8C22×C3711yescharacteristic
375C8×Q8C22×C3711yescharacteristic
376C4×Q16C22×C3713yes—
377C4×SD16C22×C3713yes—
378C4×SD16C22×C3713yes—
379C4×SD16C22×C3716yes—
380C4×D8C22×C3713yes—
381C4×D8C22×C3716yes—
382#431C22×C3713yes—
383#431C22×C3713yes—
384C37#6188611yescharacteristic, direct product factor, Sylow
385C2×#41—21no—
386C2×#41—21no—
387C2×#41—23no—
388C2×#41C23×C3713yes—
389C22×Q8D4×C3711yescharacteristic
390C2×Q16C23×C3713yes—
391C2×SD16C23×C3716yes—
392C2×SD16C23×C3713yes—
393C2×SD16C23×C3713yes—
394C2×D8C23×C3716yes—
395C2×D8C23×C3713yes—
396C4⋊Q8—23no—
397C4⋊Q8C23×C3713yes—
398C4⋊Q8C23×C3713yes—
399#127C23×C3713yes—
400#127C23×C3713yes—
401#126—26no—
402#126—23no—
403#125—23no—
404#125C23×C3713yes—
405#124—23no—
406#124—23no—
407#124—23no—
408#124C23×C3713yes—
409#124C23×C3716yes—
410#123—23no—
411#123—26no—
412#123—26no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—26no—
416C22⋊Q8C23×C3711yescharacteristic
417#121—23no—
418#121—23no—
419#121—23no—
420#121—23no—
421#121—23no—
422#121C23×C3713yes—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C3711yescharacteristic
426C4×Q8D4×C3713yes—
427C4×Q8C23×C3713yes—
428C4×Q8C23×C3713yes—
429C4×Q8C23×C3713yes—
430C4×Q8C23×C3711yescharacteristic
431C4×Q8C23×C3713yes—
432C4×D4—23no—
433C4×D4—23no—
434C4×D4—23no—
435C4×D4—23no—
436C4×D4—23no—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—23no—
440C4×D4C23×C3713yes—
441C4×D4C23×C3713yes—
442C4×D4C23×C3713yes—
443C4×D4C23×C3716yes—
444#117—23no—
445#117—23no—
446#117—23no—
447#117C23×C3711yescharacteristic
448C2×C4⋊C4—23no—
449C2×C4⋊C4D4×C3713yes—
450C2×C42—23no—
451C2×C42C23×C3713yes—
452#107C23×C3711yescharacteristic
453#107C23×C3713yes—
454#106C23×C3713yes—
455#106C23×C3711yescharacteristic
456C4⋊C8C23×C3713yes—
457C4⋊C8C23×C3713yes—
458C4⋊C8C23×C3713yes—
459C4⋊C8C23×C3713yes—
460#103C23×C3711yescharacteristic
461#103C23×C3716yes—
462#103C23×C3713yes—
463#103C23×C3711yescharacteristic
464#103C23×C3713yes—
465#102C23×C3713yes—
466#102C23×C3713yes—
467#102C23×C3716yes—
468#102C23×C3713yes—
469#102C23×C3713yes—
470C22⋊C8C23×C3711yescharacteristic
471C22⋊C8C23×C3713yes—
472C4×C8C23×C3716yes—
473C4×C8C23×C3713yes—
474C4×C8C23×C3713yes—
475#41—43no—
476#41—43no—
477#41—44no—
478#41—44no—
479#41—86no—
480C2×Q8—24no—
481C2×Q8—23no—
482C2×Q8C2×D4×C3713yes—
483C2×Q8C2×D4×C3711yescharacteristic
484C2×Q8C24×C3713yes—
485C2×Q8C24×C3711yescharacteristic
486C2×D4—23no—
487C2×D4—23no—
488C2×D4—23no—
489C2×D4—23no—
490C2×D4—23no—
491C2×D4—23no—
492C2×D4C24×C3713yes—
493C2×D4C24×C3713yes—
494C22×C4—23no—
495C22×C4—23no—
496C22×C4—23no—
497C22×C4—23no—
498C22×C4—26no—
499C22×C4C2×D4×C3713yes—
500C22×C4C2×D4×C3713yes—
501C22×C4C24×C3711yescharacteristic
502Q16—43no—
503SD16—43no—
504SD16—46no—
505SD16—43no—
506D8—43no—
507D8—46no—
508C2×C8C24×C3713yes—
509C2×C8C24×C3711yescharacteristic
510C2×C8C24×C3713yes—
511C2×C8C24×C3711yescharacteristic
512C4⋊C4—23no—
513C4⋊C4—23no—
514C4⋊C4—21no—
515C4⋊C4—23no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4C2×D4×C3713yes—
522C4⋊C4C2×D4×C3713yes—
523C4⋊C4C2×D4×C3716yes—
524C4⋊C4C24×C3711yescharacteristic
525C4⋊C4C24×C3713yes—
526C4⋊C4C24×C3711yescharacteristic
527C4⋊C4C24×C3711yescharacteristic
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—26no—
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—23no—
534C22⋊C4—23no—
535C22⋊C4—26no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—26no—
539C42—21no—
540C42—23no—
541C42—23no—
542C42—23no—
543C42C2×D4×C3713yes—
544C42C2×D4×C3713yes—
545C42C24×C3713yes—
546C42C24×C3713yes—
547C23—23no—
548C23—23no—
549C23C22×D4×C3711yescharacteristic
550Q8—26no—
551Q8—21no—
552Q8—23no—
553Q8—23no—
554Q8—21no—
555Q8—83no—
556Q8#135×C3714yes—
557D4—23no—
558D4—23no—
559D4—23no—
560D4—23no—
561D4—46no—
562D4—46no—
563D4—46no—
564D4—46no—
565D4—83no—
566D4—83no—
567C2×C4—26no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—21no—
576C2×C4—43no—
577C2×C4—46no—
578C2×C4—46no—
579C2×C4—46no—
580C2×C4—86no—
581C2×C4—83no—
582C2×C4C22×D4×C3711yescharacteristic
583C2×C4C22×D4×C3713yes—
584C2×C4C22×D4×C3713yes—
585C2×C4C22×D4×C3713yes—
586C2×C4C22×D4×C3713yes—
587C2×C4C25×C3711yescharacteristic
588C8—21no—
589C8—23no—
590C8—23no—
591C8—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—43no—
596C22—43no—
597C22—43no—
598C22—43no—
599C22—43no—
600C22—43no—
601C22C23×D4×C3711yescharacteristic
602C4—21no—
603C4—23no—
604C4—23no—
605C4—23no—
606C4—43no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—43no—
612C4C2×#135×C3716yes—
613C4C37×#58511yescharacteristic
614C4C37×#58511yescharacteristic
615C2—41no—
616C2—83no—
617C2—83no—
618C2C37×#348711yescharacteristic
619C2C22×#135×C3711yescharacteristic
620C2C37×#358511yescharacteristic
subgroupG/H
center, socleC22×C37C23×D4
commutator, Frattini, cosocleC2×C4C25×C37
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C4×Q8×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3522C2semidirect
2. C37×#3480C2semidirect
3. C37×#3396C2semidirect
4. C37×#3394C2semidirect
5. C37×#3392C2semidirect
6. C37×#3391C2semidirect
7. C37×#3363C2semidirect
8. C37×#3362C2semidirect
9. C37×#3361C2semidirect
10. C37×#3360C2semidirect
11. C37×#3212C2semidirect
12. C37×#3207C2semidirect
13. C37×#3106C2semidirect
14. C37×#3102C2semidirect
15. C37×#2982C2semidirect
281. #61886C37direct (N × H)
305. #3522C2×C37semidirect
306. #3480C2×C37semidirect
307. #3396C2×C37semidirect
308. #3394C2×C37semidirect
309. #3392C2×C37semidirect
310. #3391C2×C37semidirect
311. #3363C2×C37semidirect
312. #3362C2×C37semidirect
313. #3361C2×C37semidirect
314. #3360C2×C37semidirect
315. #3212C2×C37semidirect
316. #3207C2×C37semidirect
317. #3106C2×C37semidirect
318. #3102C2×C37semidirect
319. #2982C2×C37semidirect
384. C37#61886direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35222
C37×#34802
C37×#33962
C37×#33942
C37×#33922
C37×#33912
C37×#33632
C37×#33622
C37×#33612
C37×#33602
C37×#32122
C37×#32072
C37×#31062
C37×#31022
C37×#29822
#6188637