Finite Group Databasebeta

← C37×#18242 #22639495

Subgroups — C37×#18242 #22639495

Order 9472.

992 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3561C211yesmaximal, characteristic, complemented
2C37×#3114C211yesmaximal, characteristic, complemented
3C37×#3101C211yesmaximal, characteristic, complemented
4C37×#2631C211yesmaximal, characteristic, complemented
5C37×#1612C212yesmaximal, complemented
6C37×#1608C212yesmaximal, complemented
7C37×#1600C212yesmaximal, complemented
8C37×#1596C212yesmaximal, complemented
9C37×#1521C211yesmaximal, characteristic, complemented
10C37×#1515C211yesmaximal, characteristic, complemented
11C37×#1469C211yesmaximal, characteristic, complemented
12C37×#563—21no—
13C37×#562—21no—
14C37×#556—21no—
15C37×#551—22no—
16D4×Q8×C37—21no—
17C37×#548—21no—
18C37×#546—21no—
19C37×#543C2212yes—
20C37×#542—21no—
21C37×#542—22no—
22C37×#538C2212yes—
23C37×#536—21no—
24C37×#533C2211yescharacteristic
25C37×#529C2211yescharacteristic
26C2×#124×C37C2211yescharacteristic
27C2×C4×D4×C37C2211yescharacteristic
28C37×#483C2211yescharacteristic
29C37×#483C2212yes—
30C37×#480C2211yescharacteristic
31C37×#478—21no—
32C37×#476—21no—
33C37×#463—21no—
34C37×#461—21no—
35C37×#450C2211yescharacteristic
36C37×#450C2212yes—
37C37×#447C2211yescharacteristic
38C37×#442—21no—
39C37×#440—21no—
40C4×SD16×C37—22no—
41C37×#432C2211yescharacteristic
42C37×#431C2211yescharacteristic
43C37×#397C2212yes—
44C37×#393—21no—
45C37×#392—21no—
46C37×#392C2212yes—
47C37×#386—22no—
48C37×#383C2211yescharacteristic
49C37×#328C2212yes—
50C37×#328C2212yes—
51C37×#328C2212yes—
52C37×#327C2212yes—
53C37×#323C2212yes—
54C37×#323C2212yes—
55C2×#41×C37—21no—
56C22×Q8×C37—21no—
57C22×Q8×C37D411yescharacteristic
58C22×D4×C37—22no—
59C22×D4×C37D411yescharacteristic
60C22×D4×C37C2311yescharacteristic
61C23×C4×C37—21no—
62C2×SD16×C37—22no—
63C2×SD16×C37—21no—
64C2×D8×C37—21no—
65C4⋊Q8×C37—21no—
66C4⋊Q8×C37—22no—
67C4⋊Q8×C37D411yescharacteristic, complemented
68C4⋊Q8×C37D411yescharacteristic, complemented
69#126×C37—22no—
70#126×C37—22no—
71#126×C37—22no—
72#126×C37—21no—
73#125×C37—21no—
74#125×C37D411yescharacteristic, complemented
75#124×C37—21no—
76#124×C37—22no—
77#124×C37—22no—
78#124×C37—21no—
79#124×C37—21no—
80#124×C37—21no—
81#124×C37—21no—
82#124×C37D411yescharacteristic, complemented
83#123×C37—22no—
84#123×C37—22no—
85#123×C37—22no—
86#123×C37—21no—
87#123×C37—22no—
88C22⋊Q8×C37—22no—
89C22⋊Q8×C37—21no—
90C22⋊Q8×C37—22no—
91C22⋊Q8×C37—22no—
92C22⋊Q8×C37—22no—
93C22⋊Q8×C37—21no—
94C22⋊Q8×C37—22no—
95C22⋊Q8×C37—22no—
96C22⋊Q8×C37—22no—
97C22⋊Q8×C37C2311yescharacteristic
98C22⋊Q8×C37C2312yes—
99#121×C37—21no—
100#121×C37—22no—
101#121×C37—22no—
102#121×C37C2311yescharacteristic
103#120×C37—22no—
104C4×Q8×C37—21no—
105C4×Q8×C37—21no—
106C4×Q8×C37D411yescharacteristic, complemented
107C4×Q8×C37D412yescomplemented
108C4×D4×C37—21no—
109C4×D4×C37—22no—
110C4×D4×C37—21no—
111C4×D4×C37—22no—
112C4×D4×C37—21no—
113C4×D4×C37—21no—
114C4×D4×C37—21no—
115C4×D4×C37—21no—
116C4×D4×C37D411yescharacteristic, complemented
117#117×C37—22no—
118#117×C37—21no—
119#117×C37—21no—
120C2×C4⋊C4×C37C2311yescharacteristic
121C2×C22⋊C4×C37—22no—
122C2×C22⋊C4×C37—21no—
123C2×C22⋊C4×C37—22no—
124C2×C22⋊C4×C37—22no—
125C2×C22⋊C4×C37D412yes—
126C2×C22⋊C4×C37C2312yes—
127C2×C42×C37C2311yescharacteristic
128#107×C37—21no—
129#106×C37—22no—
130#106×C37—21no—
131C4⋊C8×C37—21no—
132C4⋊C8×C37—21no—
133#103×C37—22no—
134#103×C37—21no—
135#102×C37—21no—
136#102×C37—22no—
137#102×C37—21no—
138#102×C37—21no—
139C22⋊C8×C37C2312yes—
140C22⋊C8×C37C2311yescharacteristic
141C22⋊C8×C37C2311yescharacteristic
142#97×C37—21no—
143C4×C8×C37—21no—
144#95×C37—21no—
145#95×C37—21no—
146#95×C37—22no—
147#95×C37C2312yes—
148C24×C37—21no—
149C24×C37C2×D412yes—
150#41×C37—82no—
151C2×Q8×C37—22no—
152C2×Q8×C37—21no—
153C2×Q8×C37—21no—
154C2×Q8×C37—21no—
155C2×Q8×C37—81no—
156C2×Q8×C37—82no—
157C2×Q8×C37C2×D411yescharacteristic
158C2×Q8×C37C2×D412yes—
159C2×D4×C37—21no—
160C2×D4×C37—22no—
161C2×D4×C37—22no—
162C2×D4×C37—21no—
163C2×D4×C37—21no—
164C2×D4×C37—21no—
165C2×D4×C37—21no—
166C2×D4×C37—42no—
167C2×D4×C37—44no—
168C2×D4×C37—81no—
169C2×D4×C37—82no—
170C2×D4×C37—82no—
171C2×D4×C37C2×D411yescharacteristic
172C22×C4×C37—22no—
173C22×C4×C37—22no—
174C22×C4×C37—22no—
175C22×C4×C37—21no—
176C22×C4×C37—22no—
177C22×C4×C37—21no—
178C22×C4×C37—41no—
179C22×C4×C37—41no—
180C22×C4×C37—82no—
181C22×C4×C37C2×D411yescharacteristic
182C22×C4×C37C2×D412yes—
183C22×C4×C37C2×D411yescharacteristic
184C22×C4×C37C2411yescharacteristic
185SD16×C37—42no—
186SD16×C37—44no—
187D8×C37—42no—
188C2×C8×C37—22no—
189C2×C8×C37—21no—
190C2×C8×C37—21no—
191C4⋊C4×C37—22no—
192C4⋊C4×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—22no—
202C4⋊C4×C37—21no—
203C4⋊C4×C37—22no—
204C4⋊C4×C37—21no—
205C4⋊C4×C37—21no—
206C4⋊C4×C37C2×D411yescharacteristic
207C4⋊C4×C37C2×D412yes—
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—22no—
217C22⋊C4×C37—22no—
218C22⋊C4×C37—21no—
219C22⋊C4×C37—21no—
220C22⋊C4×C37—21no—
221C22⋊C4×C37—22no—
222C22⋊C4×C37—22no—
223C22⋊C4×C37—21no—
224C22⋊C4×C37—41no—
225C22⋊C4×C37—41no—
226C22⋊C4×C37—82no—
227C22⋊C4×C37—82no—
228C42×C37—22no—
229C42×C37—22no—
230C42×C37—21no—
231C42×C37C2×D411yescharacteristic
232C42×C37C2×D411yescharacteristic
233C42×C37C2×D411yescharacteristic
234C42×C37C2×D411yescharacteristic
235C23×C37—22no—
236C23×C37—22no—
237C23×C37—21no—
238C23×C37—22no—
239C23×C37—22no—
240C23×C37—42no—
241C23×C37—41no—
242C23×C37—42no—
243C23×C37—42no—
244C23×C37—41no—
245C23×C37—42no—
246C23×C37C22×D411yescharacteristic
247Q8×C37—41no—
248Q8×C37—42no—
249Q8×C37—42no—
250Q8×C37—42no—
251Q8×C37—81no—
252Q8×C37—82no—
253D4×C37—22no—
254D4×C37—22no—
255D4×C37—41no—
256D4×C37—41no—
257D4×C37—41no—
258D4×C37—42no—
259D4×C37—42no—
260D4×C37—81no—
261D4×C37—82no—
262D4×C37—82no—
263D4×C37—88no—
264C2×C4×C37—22no—
265C2×C4×C37—21no—
266C2×C4×C37—22no—
267C2×C4×C37—21no—
268C2×C4×C37—22no—
269C2×C4×C37—22no—
270C2×C4×C37—21no—
271C2×C4×C37—41no—
272C2×C4×C37—41no—
273C2×C4×C37—42no—
274C2×C4×C37—42no—
275C2×C4×C37—42no—
276C2×C4×C37—42no—
277C2×C4×C37—41no—
278C2×C4×C37—42no—
279C2×C4×C37—42no—
280C2×C4×C37—82no—
281C2×C4×C37—82no—
282C2×C4×C37—82no—
283C2×C4×C37—82no—
284C2×C4×C37—82no—
285C2×C4×C37#12011yescharacteristic
286C2×C4×C37#12011yescharacteristic
287C2×C4×C37#12011yescharacteristic
288C2×C4×C37#12011yescharacteristic
289C2×C4×C37C22×D411yescharacteristic
290C2×C4×C37C22×D411yescharacteristic
291C8×C37—41no—
292C8×C37—41no—
293C8×C37—42no—
294#18242C3711yesmaximal, characteristic, direct product factor, Sylow
295C22×C37—21no—
296C22×C37—21no—
297C22×C37—44no—
298C22×C37—41no—
299C22×C37—42no—
300C22×C37—41no—
301C22×C37—42no—
302C22×C37—42no—
303C22×C37—42no—
304C22×C37—41no—
305C22×C37—82no—
306C22×C37—82no—
307C22×C37—82no—
308C22×C37—81no—
309C22×C37—81no—
310C22×C37C2×#12011yescharacteristic
311C22×C37#57712yes—
312C4×C37—21no—
313C4×C37—21no—
314C4×C37—21no—
315C4×C37—21no—
316C4×C37—21no—
317C4×C37—21no—
318C4×C37—41no—
319C4×C37—41no—
320C4×C37—42no—
321C4×C37—42no—
322C4×C37—41no—
323C4×C37—42no—
324C4×C37—82no—
325C4×C37—81no—
326#3561C2×C3711yescharacteristic, complemented
327#3114C2×C3711yescharacteristic, complemented
328#3101C2×C3711yescharacteristic, complemented
329#2631C2×C3711yescharacteristic, complemented
330#1612C2×C3712yescomplemented
331#1608C2×C3712yescomplemented
332#1600C2×C3712yescomplemented
333#1596C2×C3712yescomplemented
334#1521C2×C3711yescharacteristic, complemented
335#1515C2×C3711yescharacteristic, complemented
336#1469C2×C3711yescharacteristic, complemented
337C2×C37—22no—
338C2×C37—81no—
339C2×C37—82no—
340C2×C37—82no—
341C2×C37#301011yescharacteristic
342C2×C37#301911yescharacteristic
343C2×C37#302511yescharacteristic
344#563—21no—
345#562—21no—
346#556—21no—
347#551—22no—
348D4×Q8—21no—
349#548—21no—
350#546—21no—
351#543C22×C3712yes—
352#542—21no—
353#542—22no—
354#538C22×C3712yes—
355#536—21no—
356#533C22×C3711yescharacteristic
357#529C22×C3711yescharacteristic
358C2×#124C22×C3711yescharacteristic
359C2×C4×D4C22×C3711yescharacteristic
360#483C22×C3712yes—
361#483C22×C3711yescharacteristic
362#480C22×C3711yescharacteristic
363#478—21no—
364#476—21no—
365#463—21no—
366#461—21no—
367#450C22×C3712yes—
368#450C22×C3711yescharacteristic
369#447C22×C3711yescharacteristic
370#442—21no—
371#440—21no—
372C4×SD16—22no—
373#432C22×C3711yescharacteristic
374#431C22×C3711yescharacteristic
375#397C22×C3712yes—
376#393—21no—
377#392—21no—
378#392C22×C3712yes—
379#386—22no—
380#383C22×C3711yescharacteristic
381#328C22×C3712yes—
382#328C22×C3712yes—
383#328C22×C3712yes—
384#327C22×C3712yes—
385#323C22×C3712yes—
386#323C22×C3712yes—
387C37#1824211yescharacteristic, direct product factor, Sylow
388C2×#41—21no—
389C22×Q8—21no—
390C22×Q8D4×C3711yescharacteristic
391C22×D4—22no—
392C22×D4D4×C3711yescharacteristic
393C22×D4C23×C3711yescharacteristic
394C23×C4—21no—
395C2×SD16—22no—
396C2×SD16—21no—
397C2×D8—21no—
398C4⋊Q8—21no—
399C4⋊Q8—22no—
400C4⋊Q8D4×C3711yescharacteristic, complemented
401C4⋊Q8D4×C3711yescharacteristic, complemented
402#126—22no—
403#126—22no—
404#126—22no—
405#126—21no—
406#125—21no—
407#125D4×C3711yescharacteristic, complemented
408#124—21no—
409#124—22no—
410#124—22no—
411#124—21no—
412#124—21no—
413#124—21no—
414#124—21no—
415#124D4×C3711yescharacteristic, complemented
416#123—22no—
417#123—22no—
418#123—22no—
419#123—21no—
420#123—22no—
421C22⋊Q8—22no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—22no—
428C22⋊Q8—22no—
429C22⋊Q8—22no—
430C22⋊Q8C23×C3711yescharacteristic
431C22⋊Q8C23×C3712yes—
432#121—21no—
433#121—22no—
434#121—22no—
435#121C23×C3711yescharacteristic
436#120—22no—
437C4×Q8—21no—
438C4×Q8—21no—
439C4×Q8D4×C3711yescharacteristic, complemented
440C4×Q8D4×C3712yescomplemented
441C4×D4—21no—
442C4×D4—22no—
443C4×D4—21no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4D4×C3711yescharacteristic, complemented
450#117—22no—
451#117—21no—
452#117—21no—
453C2×C4⋊C4C23×C3711yescharacteristic
454C2×C22⋊C4—22no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—22no—
457C2×C22⋊C4—22no—
458C2×C22⋊C4D4×C3712yes—
459C2×C22⋊C4C23×C3712yes—
460C2×C42C23×C3711yescharacteristic
461#107—21no—
462#106—22no—
463#106—21no—
464C4⋊C8—21no—
465C4⋊C8—21no—
466#103—22no—
467#103—21no—
468#102—21no—
469#102—22no—
470#102—21no—
471#102—21no—
472C22⋊C8C23×C3712yes—
473C22⋊C8C23×C3711yescharacteristic
474C22⋊C8C23×C3711yescharacteristic
475#97—21no—
476C4×C8—21no—
477#95—21no—
478#95—21no—
479#95—22no—
480#95C23×C3712yes—
481C24—21no—
482C24C2×D4×C3712yes—
483#41—82no—
484C2×Q8—22no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—81no—
489C2×Q8—82no—
490C2×Q8C2×D4×C3711yescharacteristic
491C2×Q8C2×D4×C3712yes—
492C2×D4—21no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—42no—
500C2×D4—44no—
501C2×D4—81no—
502C2×D4—82no—
503C2×D4—82no—
504C2×D4C2×D4×C3711yescharacteristic
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—41no—
512C22×C4—41no—
513C22×C4—82no—
514C22×C4C2×D4×C3711yescharacteristic
515C22×C4C2×D4×C3712yes—
516C22×C4C2×D4×C3711yescharacteristic
517C22×C4C24×C3711yescharacteristic
518SD16—42no—
519SD16—44no—
520D8—42no—
521C2×C8—22no—
522C2×C8—21no—
523C2×C8—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4C2×D4×C3711yescharacteristic
540C4⋊C4C2×D4×C3712yes—
541C4⋊C4C2×D4×C3711yescharacteristic
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—21no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C42—22no—
562C42—22no—
563C42—21no—
564C42C2×D4×C3711yescharacteristic
565C42C2×D4×C3711yescharacteristic
566C42C2×D4×C3711yescharacteristic
567C42C2×D4×C3711yescharacteristic
568C23—22no—
569C23—22no—
570C23—21no—
571C23—22no—
572C23—22no—
573C23—42no—
574C23—41no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—42no—
579C23C22×D4×C3711yescharacteristic
580Q8—41no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—81no—
585Q8—82no—
586D4—22no—
587D4—22no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—42no—
592D4—42no—
593D4—81no—
594D4—82no—
595D4—82no—
596D4—88no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4#120×C3711yescharacteristic
619C2×C4#120×C3711yescharacteristic
620C2×C4#120×C3711yescharacteristic
621C2×C4#120×C3711yescharacteristic
622C2×C4C22×D4×C3711yescharacteristic
623C2×C4C22×D4×C3711yescharacteristic
624C8—41no—
625C8—41no—
626C8—42no—
627C22—21no—
628C22—21no—
629C22—44no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22C2×#120×C3711yescharacteristic
643C22C37×#57712yes—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—82no—
657C4—81no—
658C2—22no—
659C2—82no—
660C2—82no—
661C2—81no—
662C2C37×#301011yescharacteristic
663C2C37×#301911yescharacteristic
664C2C37×#302511yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3561C2semidirect
2. C37×#3114C2semidirect
3. C37×#3101C2semidirect
4. C37×#2631C2semidirect
5. C37×#1612C2semidirect
6. C37×#1608C2semidirect
7. C37×#1600C2semidirect
8. C37×#1596C2semidirect
9. C37×#1521C2semidirect
10. C37×#1515C2semidirect
11. C37×#1469C2semidirect
67. C4⋊Q8×C37D4semidirect
68. C4⋊Q8×C37D4semidirect
74. #125×C37D4semidirect
82. #124×C37D4semidirect
106. C4×Q8×C37D4semidirect
107. C4×Q8×C37D4semidirect
116. C4×D4×C37D4semidirect
294. #18242C37direct (N × H)
326. #3561C2×C37semidirect
327. #3114C2×C37semidirect
328. #3101C2×C37semidirect
329. #2631C2×C37semidirect
330. #1612C2×C37semidirect
331. #1608C2×C37semidirect
332. #1600C2×C37semidirect
333. #1596C2×C37semidirect
334. #1521C2×C37semidirect
335. #1515C2×C37semidirect
336. #1469C2×C37semidirect
387. C37#18242direct (N × H)
400. C4⋊Q8D4×C37semidirect
401. C4⋊Q8D4×C37semidirect
407. #125D4×C37semidirect
415. #124D4×C37semidirect
439. C4×Q8D4×C37semidirect
440. C4×Q8D4×C37semidirect
449. C4×D4D4×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#35612
C37×#31142
C37×#31012
C37×#26312
C37×#16122
C37×#16082
C37×#16002
C37×#15962
C37×#15212
C37×#15152
C37×#14692
#1824237