Finite Group Databasebeta

← C37×#19024 #22640277

Subgroups — C37×#19024 #22640277

Order 9472.

948 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3558C211yesmaximal, characteristic
2C37×#3216C211yesmaximal, characteristic, complemented
3C37×#3172C211yesmaximal, characteristic, complemented
4C37×#3140C211yesmaximal, characteristic, complemented
5C37×#1662C212yesmaximal, complemented
6C37×#1661C212yesmaximal, complemented
7C37×#1659C212yesmaximal, complemented
8C37×#1659C212yesmaximal, complemented
9C37×#1529C211yesmaximal, characteristic, complemented
10C37×#1527C211yesmaximal, characteristic, complemented
11C37×#1525C211yesmaximal, characteristic, complemented
12C37×#562C2212yes—
13C37×H(3,𝔽4)C2212yes—
14C37×#559—21no—
15C37×#548—21no—
16C37×#546—22no—
17C37×#546—21no—
18D42×C37—21no—
19C37×#542—21no—
20C37×#538—22no—
21C37×#538—21no—
22C37×#535—21no—
23C37×#534—21no—
24C37×#533C2211yescharacteristic, complemented
25C37×#532C2211yescharacteristic
26C37×#529C2211yescharacteristic
27C2×#124×C37C2211yescharacteristic
28C37×#501C2211yescharacteristic, complemented
29C37×#497C2211yescharacteristic, complemented
30C37×#497C2212yescomplemented
31C37×#491C2211yescharacteristic, complemented
32C37×#489C2212yescomplemented
33C37×#489C2212yescomplemented
34C37×#489C2211yescharacteristic, complemented
35C37×#489C2212yescomplemented
36C37×#488—22no—
37C37×#470—21no—
38C37×#468—21no—
39C37×#462—21no—
40C37×#460—21no—
41C37×#452—22no—
42C4×Q16×C37—21no—
43C4×SD16×C37—21no—
44C37×#432C2211yescharacteristic, complemented
45C37×#419—21no—
46C37×#418—21no—
47C4×M16×C37C2211yescharacteristic, complemented
48C4×M16×C37C2211yescharacteristic, complemented
49C37×#330C2212yescomplemented
50C37×#329C2212yescomplemented
51C37×#329C2212yescomplemented
52C37×#329C2212yescomplemented
53C37×#329C2212yescomplemented
54C37×#329C2212yescomplemented
55C2×#41×C37—21no—
56C2×#41×C37D412yes—
57C22×Q8×C37D411yescharacteristic
58C22×D4×C37—22no—
59C22×D4×C37—21no—
60C22×D4×C37D411yescharacteristic
61C2×Q16×C37—21no—
62C2×Q16×C37—22no—
63C2×SD16×C37—21no—
64C2×SD16×C37—22no—
65C2×M16×C37—21no—
66C2×M16×C37—21no—
67C4⋊Q8×C37D411yescharacteristic
68C4⋊Q8×C37C2311yescharacteristic
69C4⋊Q8×C37C2311yescharacteristic
70#127×C37—21no—
71#127×C37D411yescharacteristic
72#126×C37—22no—
73#126×C37—21no—
74#125×C37—21no—
75#125×C37C2311yescharacteristic
76#124×C37—21no—
77#124×C37—22no—
78#124×C37—22no—
79#124×C37—22no—
80#124×C37—21no—
81#124×C37—21no—
82#124×C37D412yes—
83#124×C37C2312yes—
84#124×C37C2311yescharacteristic
85#124×C37C2312yes—
86#123×C37—22no—
87#123×C37—22no—
88#123×C37—21no—
89#123×C37—22no—
90#123×C37—22no—
91C22⋊Q8×C37—21no—
92C22⋊Q8×C37—22no—
93C22⋊Q8×C37—22no—
94C22⋊Q8×C37—21no—
95C22⋊Q8×C37—21no—
96C22⋊Q8×C37D411yescharacteristic
97#121×C37—22no—
98#121×C37—22no—
99#121×C37—22no—
100#121×C37—21no—
101#121×C37—22no—
102#121×C37—22no—
103#121×C37—21no—
104#121×C37—21no—
105#121×C37D411yescharacteristic
106#120×C37—22no—
107#120×C37—22no—
108#120×C37—22no—
109#120×C37—22no—
110#120×C37—22no—
111C4×Q8×C37—21no—
112C4×Q8×C37—21no—
113C4×Q8×C37—21no—
114C4×Q8×C37D411yescharacteristic
115C4×D4×C37—22no—
116C4×D4×C37—22no—
117C4×D4×C37—21no—
118C4×D4×C37—21no—
119C4×D4×C37—21no—
120C4×D4×C37—21no—
121C4×D4×C37D411yescharacteristic
122#117×C37—21no—
123#117×C37—21no—
124C2×C22⋊C4×C37—22no—
125C2×C22⋊C4×C37—21no—
126C2×C22⋊C4×C37—21no—
127C2×C42×C37C2311yescharacteristic
128#107×C37—21no—
129#106×C37—21no—
130C4⋊C8×C37—21no—
131#103×C37—22no—
132#103×C37—22no—
133#103×C37—21no—
134#103×C37—22no—
135#103×C37—21no—
136#103×C37—21no—
137#102×C37—22no—
138#102×C37—22no—
139#102×C37—21no—
140#102×C37—22no—
141C22⋊C8×C37—21no—
142#97×C37C2311yescharacteristic
143#97×C37C2311yescharacteristic
144#97×C37C2312yes—
145#97×C37C2312yes—
146C4×C8×C37—21no—
147C4×C8×C37—21no—
148C24×C37—22no—
149C24×C37—21no—
150#41×C37—82no—
151#41×C37—82no—
152C2×Q8×C37—21no—
153C2×Q8×C37—21no—
154C2×Q8×C37—21no—
155C2×Q8×C37—42no—
156C2×Q8×C37C2×D412yes—
157C2×Q8×C37C2×D411yescharacteristic
158C2×Q8×C37C2×D412yes—
159C2×D4×C37—22no—
160C2×D4×C37—21no—
161C2×D4×C37—22no—
162C2×D4×C37—22no—
163C2×D4×C37—22no—
164C2×D4×C37—22no—
165C2×D4×C37—22no—
166C2×D4×C37—21no—
167C2×D4×C37—21no—
168C2×D4×C37—21no—
169C2×D4×C37—42no—
170C2×D4×C37—82no—
171C2×D4×C37—84no—
172C2×D4×C37C2×D412yes—
173C2×D4×C37C2×D411yescharacteristic
174C2×D4×C37C2×D412yes—
175C22×C4×C37—22no—
176C22×C4×C37—22no—
177C22×C4×C37—21no—
178C22×C4×C37—21no—
179C22×C4×C37—22no—
180C22×C4×C37C2×D411yescharacteristic
181C22×C4×C37C2×D411yescharacteristic
182C22×C4×C37C2×D411yescharacteristic
183Q16×C37—42no—
184Q16×C37—82no—
185SD16×C37—42no—
186SD16×C37—82no—
187M16×C37—42no—
188M16×C37—42no—
189C2×C8×C37—21no—
190C2×C8×C37—22no—
191C2×C8×C37—21no—
192C2×C8×C37—22no—
193C4⋊C4×C37—22no—
194C4⋊C4×C37—22no—
195C4⋊C4×C37—21no—
196C4⋊C4×C37—21no—
197C4⋊C4×C37—21no—
198C4⋊C4×C37—21no—
199C4⋊C4×C37—21no—
200C4⋊C4×C37—21no—
201C4⋊C4×C37—21no—
202C4⋊C4×C37—21no—
203C4⋊C4×C37—21no—
204C4⋊C4×C37—21no—
205C4⋊C4×C37—41no—
206C4⋊C4×C37—41no—
207C4⋊C4×C37C2×D411yescharacteristic
208C4⋊C4×C37C2×D411yescharacteristic
209C22⋊C4×C37—22no—
210C22⋊C4×C37—22no—
211C22⋊C4×C37—22no—
212C22⋊C4×C37—22no—
213C22⋊C4×C37—22no—
214C22⋊C4×C37—21no—
215C22⋊C4×C37—22no—
216C22⋊C4×C37—21no—
217C22⋊C4×C37—22no—
218C22⋊C4×C37—21no—
219C22⋊C4×C37—22no—
220C22⋊C4×C37—22no—
221C22⋊C4×C37—21no—
222C22⋊C4×C37—21no—
223C22⋊C4×C37—22no—
224C22⋊C4×C37—22no—
225C22⋊C4×C37—41no—
226C42×C37—22no—
227C42×C37—21no—
228C42×C37—41no—
229C42×C37C2×D411yescharacteristic
230C42×C37C2×D411yescharacteristic
231C42×C37C2×D411yescharacteristic
232C42×C37C2411yescharacteristic
233C23×C37—21no—
234C23×C37—22no—
235C23×C37—22no—
236C23×C37—42no—
237C23×C37—81no—
238C23×C37—81no—
239C23×C37—82no—
240C23×C37—82no—
241C23×C37—82no—
242C23×C37#12011yescharacteristic
243Q8×C37—41no—
244Q8×C37—42no—
245Q8×C37—42no—
246Q8×C37—41no—
247Q8×C37—41no—
248Q8×C37—41no—
249Q8×C37—41no—
250Q8×C37—42no—
251D4×C37—41no—
252D4×C37—42no—
253D4×C37—42no—
254D4×C37—42no—
255D4×C37—44no—
256D4×C37—42no—
257D4×C37—42no—
258D4×C37—81no—
259D4×C37—82no—
260D4×C37—82no—
261D4×C37—82no—
262D4×C37—82no—
263D4×C37—82no—
264C2×C4×C37—22no—
265C2×C4×C37—21no—
266C2×C4×C37—22no—
267C2×C4×C37—22no—
268C2×C4×C37—21no—
269C2×C4×C37—21no—
270C2×C4×C37—41no—
271C2×C4×C37—41no—
272C2×C4×C37—42no—
273C2×C4×C37—41no—
274C2×C4×C37—41no—
275C2×C4×C37—82no—
276C2×C4×C37—81no—
277C2×C4×C37—82no—
278C2×C4×C37—82no—
279C2×C4×C37#12011yescharacteristic
280C2×C4×C37#12011yescharacteristic
281C2×C4×C37#12011yescharacteristic
282C2×C4×C37C22×D411yescharacteristic
283C2×C4×C37C22×D411yescharacteristic
284C2×C4×C37C22×D411yescharacteristic
285C8×C37—22no—
286C8×C37—42no—
287C8×C37—41no—
288C8×C37—42no—
289#19024C3711yesmaximal, characteristic, direct product factor, Sylow
290C22×C37—21no—
291C22×C37—21no—
292C22×C37—21no—
293C22×C37—41no—
294C22×C37—42no—
295C22×C37—42no—
296C22×C37—41no—
297C22×C37—42no—
298C22×C37—42no—
299C22×C37—42no—
300C22×C37—42no—
301C22×C37—41no—
302C22×C37—162no—
303C22×C37—164no—
304C22×C37C2×#12011yescharacteristic
305C4×C37—21no—
306C4×C37—21no—
307C4×C37—21no—
308C4×C37—21no—
309C4×C37—21no—
310C4×C37—41no—
311C4×C37—42no—
312C4×C37—41no—
313C4×C37—41no—
314C4×C37—42no—
315C4×C37—81no—
316C4×C37—81no—
317C4×C37#45512yes—
318#3558C2×C3711yescharacteristic
319#3216C2×C3711yescharacteristic, complemented
320#3172C2×C3711yescharacteristic, complemented
321#3140C2×C3711yescharacteristic, complemented
322#1662C2×C3712yescomplemented
323#1661C2×C3712yescomplemented
324#1659C2×C3712yescomplemented
325#1659C2×C3712yescomplemented
326#1529C2×C3711yescharacteristic, complemented
327#1527C2×C3711yescharacteristic, complemented
328#1525C2×C3711yescharacteristic, complemented
329C2×C37—41no—
330C2×C37—82no—
331C2×C37—81no—
332C2×C37—82no—
333C2×C37C2×#45511yescharacteristic
334C2×C37#301811yescharacteristic
335C2×C37#301911yescharacteristic
336#562C22×C3712yes—
337H(3,𝔽4)C22×C3712yes—
338#559—21no—
339#548—21no—
340#546—22no—
341#546—21no—
342D42—21no—
343#542—21no—
344#538—22no—
345#538—21no—
346#535—21no—
347#534—21no—
348#533C22×C3711yescharacteristic, complemented
349#532C22×C3711yescharacteristic
350#529C22×C3711yescharacteristic
351C2×#124C22×C3711yescharacteristic
352#501C22×C3711yescharacteristic, complemented
353#497C22×C3711yescharacteristic, complemented
354#497C22×C3712yescomplemented
355#491C22×C3711yescharacteristic, complemented
356#489C22×C3712yescomplemented
357#489C22×C3712yescomplemented
358#489C22×C3711yescharacteristic, complemented
359#489C22×C3712yescomplemented
360#488—22no—
361#470—21no—
362#468—21no—
363#462—21no—
364#460—21no—
365#452—22no—
366C4×Q16—21no—
367C4×SD16—21no—
368#432C22×C3711yescharacteristic, complemented
369#419—21no—
370#418—21no—
371C4×M16C22×C3711yescharacteristic, complemented
372C4×M16C22×C3711yescharacteristic, complemented
373#330C22×C3712yescomplemented
374#329C22×C3712yescomplemented
375#329C22×C3712yescomplemented
376#329C22×C3712yescomplemented
377#329C22×C3712yescomplemented
378#329C22×C3712yescomplemented
379C37#1902411yescharacteristic, direct product factor, Sylow
380C2×#41—21no—
381C2×#41D4×C3712yes—
382C22×Q8D4×C3711yescharacteristic
383C22×D4—22no—
384C22×D4—21no—
385C22×D4D4×C3711yescharacteristic
386C2×Q16—21no—
387C2×Q16—22no—
388C2×SD16—21no—
389C2×SD16—22no—
390C2×M16—21no—
391C2×M16—21no—
392C4⋊Q8D4×C3711yescharacteristic
393C4⋊Q8C23×C3711yescharacteristic
394C4⋊Q8C23×C3711yescharacteristic
395#127—21no—
396#127D4×C3711yescharacteristic
397#126—22no—
398#126—21no—
399#125—21no—
400#125C23×C3711yescharacteristic
401#124—21no—
402#124—22no—
403#124—22no—
404#124—22no—
405#124—21no—
406#124—21no—
407#124D4×C3712yes—
408#124C23×C3712yes—
409#124C23×C3711yescharacteristic
410#124C23×C3712yes—
411#123—22no—
412#123—22no—
413#123—21no—
414#123—22no—
415#123—22no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8D4×C3711yescharacteristic
422#121—22no—
423#121—22no—
424#121—22no—
425#121—21no—
426#121—22no—
427#121—22no—
428#121—21no—
429#121—21no—
430#121D4×C3711yescharacteristic
431#120—22no—
432#120—22no—
433#120—22no—
434#120—22no—
435#120—22no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8D4×C3711yescharacteristic
440C4×D4—22no—
441C4×D4—22no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4D4×C3711yescharacteristic
447#117—21no—
448#117—21no—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C42C23×C3711yescharacteristic
453#107—21no—
454#106—21no—
455C4⋊C8—21no—
456#103—22no—
457#103—22no—
458#103—21no—
459#103—22no—
460#103—21no—
461#103—21no—
462#102—22no—
463#102—22no—
464#102—21no—
465#102—22no—
466C22⋊C8—21no—
467#97C23×C3711yescharacteristic
468#97C23×C3711yescharacteristic
469#97C23×C3712yes—
470#97C23×C3712yes—
471C4×C8—21no—
472C4×C8—21no—
473C24—22no—
474C24—21no—
475#41—82no—
476#41—82no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—42no—
481C2×Q8C2×D4×C3712yes—
482C2×Q8C2×D4×C3711yescharacteristic
483C2×Q8C2×D4×C3712yes—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—82no—
496C2×D4—84no—
497C2×D4C2×D4×C3712yes—
498C2×D4C2×D4×C3711yescharacteristic
499C2×D4C2×D4×C3712yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4C2×D4×C3711yescharacteristic
506C22×C4C2×D4×C3711yescharacteristic
507C22×C4C2×D4×C3711yescharacteristic
508Q16—42no—
509Q16—82no—
510SD16—42no—
511SD16—82no—
512M16—42no—
513M16—42no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—21no—
517C2×C8—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4C2×D4×C3711yescharacteristic
533C4⋊C4C2×D4×C3711yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—41no—
551C42—22no—
552C42—21no—
553C42—41no—
554C42C2×D4×C3711yescharacteristic
555C42C2×D4×C3711yescharacteristic
556C42C2×D4×C3711yescharacteristic
557C42C24×C3711yescharacteristic
558C23—21no—
559C23—22no—
560C23—22no—
561C23—42no—
562C23—81no—
563C23—81no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23#120×C3711yescharacteristic
568Q8—41no—
569Q8—42no—
570Q8—42no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—42no—
576D4—41no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—42no—
582D4—42no—
583D4—81no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—82no—
588D4—82no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4#120×C3711yescharacteristic
605C2×C4#120×C3711yescharacteristic
606C2×C4#120×C3711yescharacteristic
607C2×C4C22×D4×C3711yescharacteristic
608C2×C4C22×D4×C3711yescharacteristic
609C2×C4C22×D4×C3711yescharacteristic
610C8—22no—
611C8—42no—
612C8—41no—
613C8—42no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—162no—
627C22—164no—
628C22C2×#120×C3711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4C37×#45512yes—
642C2—41no—
643C2—81no—
644C2—82no—
645C2—82no—
646C2C2×C37×#45511yescharacteristic
647C2C37×#301811yescharacteristic
648C2C37×#301911yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#3558C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3216C2semidirect
3. C37×#3172C2semidirect
4. C37×#3140C2semidirect
5. C37×#1662C2semidirect
6. C37×#1661C2semidirect
7. C37×#1659C2semidirect
8. C37×#1659C2semidirect
9. C37×#1529C2semidirect
10. C37×#1527C2semidirect
11. C37×#1525C2semidirect
24. C37×#533C22semidirect
28. C37×#501C22semidirect
29. C37×#497C22semidirect
30. C37×#497C22semidirect
31. C37×#491C22semidirect
32. C37×#489C22semidirect
33. C37×#489C22semidirect
34. C37×#489C22semidirect
35. C37×#489C22semidirect
44. C37×#432C22semidirect
47. C4×M16×C37C22semidirect
48. C4×M16×C37C22semidirect
49. C37×#330C22semidirect
50. C37×#329C22semidirect
51. C37×#329C22semidirect
52. C37×#329C22semidirect
53. C37×#329C22semidirect
54. C37×#329C22semidirect
289. #19024C37direct (N × H)
319. #3216C2×C37semidirect
320. #3172C2×C37semidirect
321. #3140C2×C37semidirect
322. #1662C2×C37semidirect
323. #1661C2×C37semidirect
324. #1659C2×C37semidirect
325. #1659C2×C37semidirect
326. #1529C2×C37semidirect
327. #1527C2×C37semidirect
328. #1525C2×C37semidirect
348. #533C22×C37semidirect
352. #501C22×C37semidirect
353. #497C22×C37semidirect
354. #497C22×C37semidirect
355. #491C22×C37semidirect
356. #489C22×C37semidirect
357. #489C22×C37semidirect
358. #489C22×C37semidirect
359. #489C22×C37semidirect
368. #432C22×C37semidirect
371. C4×M16C22×C37semidirect
372. C4×M16C22×C37semidirect
373. #330C22×C37semidirect
374. #329C22×C37semidirect
375. #329C22×C37semidirect
376. #329C22×C37semidirect
377. #329C22×C37semidirect
378. #329C22×C37semidirect
379. C37#19024direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35582
C37×#32162
C37×#31722
C37×#31402
C37×#16622
C37×#16612
C37×#16592
C37×#15292
C37×#15272
C37×#15252
#1902437