Finite Group Databasebeta

← C37×#18422 #22639675

Subgroups — C37×#18422 #22639675

Order 9472.

960 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3538C211yesmaximal, characteristic, complemented
2C37×#3193C211yesmaximal, characteristic, complemented
3C37×#3188C211yesmaximal, characteristic
4C37×#2639C211yesmaximal, characteristic, complemented
5C37×#1617C212yesmaximal, complemented
6C37×#1615C212yesmaximal, complemented
7C37×#1599C212yesmaximal, complemented
8C37×#1597C212yesmaximal, complemented
9C37×#1504C211yesmaximal, characteristic, complemented
10C2×C37×#328C211yesmaximal, characteristic, complemented
11C37×#1472C211yesmaximal, characteristic, complemented
12C37×#563—22no—
13C37×#552—21no—
14C37×#552—21no—
15C37×#551—22no—
16D4×Q8×C37—21no—
17D4×Q8×C37—21no—
18C37×#544—21no—
19C37×#544C2212yes—
20C37×#541—21no—
21C37×#539—21no—
22C37×#539C2212yes—
23C37×#536—21no—
24C37×#525C2211yescharacteristic
25C2×C22⋊Q8×C37C2211yescharacteristic
26C2×C22⋊Q8×C37C2211yescharacteristic
27C2×#121×C37C2211yescharacteristic
28C37×#484C2212yes—
29C37×#484C2211yescharacteristic
30C37×#482—21no—
31C37×#481C2211yescharacteristic
32C37×#480—21no—
33C37×#468—22no—
34C37×#466—21no—
35C37×#465—21no—
36C37×#450C2212yes—
37C37×#450C2211yescharacteristic
38C37×#449—21no—
39C37×#448—21no—
40C37×#447C2211yescharacteristic
41C37×#407C2211yescharacteristic
42C2×C22⋊C8×C37C2211yescharacteristic
43C37×#400C2212yes—
44C37×#397—21no—
45C37×#394C2212yes—
46C37×#393—21no—
47C37×#386—22no—
48C2×#95×C37C2211yescharacteristic
49C37×#328C2212yes—
50C37×#328C2212yes—
51C37×#328C2212yes—
52C37×#327C2212yes—
53C37×#324C2212yes—
54C37×#324C2212yes—
55C2×#41×C37D411yescharacteristic
56C22×Q8×C37—21no—
57C22×Q8×C37D411yescharacteristic
58C22×Q8×C37D411yescharacteristic
59C22×D4×C37—21no—
60C22×D4×C37C2311yescharacteristic
61C23×C4×C37—21no—
62C23×C4×C37C2311yescharacteristic
63C2×SD16×C37—22no—
64C2×SD16×C37—21no—
65C2×D8×C37—21no—
66C2×M16×C37—21no—
67C22×C8×C37—21no—
68C4⋊Q8×C37—21no—
69C4⋊Q8×C37—22no—
70C4⋊Q8×C37—22no—
71C4⋊Q8×C37—22no—
72#126×C37—22no—
73#126×C37—22no—
74#126×C37—22no—
75#126×C37—22no—
76#125×C37—21no—
77#125×C37—22no—
78#125×C37—22no—
79#125×C37—22no—
80#125×C37—22no—
81#124×C37—22no—
82#123×C37—22no—
83#123×C37—22no—
84#123×C37—22no—
85#123×C37—21no—
86#123×C37—22no—
87C22⋊Q8×C37—21no—
88C22⋊Q8×C37—21no—
89C22⋊Q8×C37—22no—
90C22⋊Q8×C37—22no—
91C22⋊Q8×C37—22no—
92C22⋊Q8×C37—21no—
93C22⋊Q8×C37—21no—
94C22⋊Q8×C37—21no—
95C22⋊Q8×C37—21no—
96C22⋊Q8×C37—21no—
97C22⋊Q8×C37—21no—
98C22⋊Q8×C37—22no—
99C22⋊Q8×C37D412yes—
100C22⋊Q8×C37D411yescharacteristic
101C22⋊Q8×C37C2312yes—
102C22⋊Q8×C37C2311yescharacteristic
103#121×C37—21no—
104#121×C37—21no—
105#121×C37—21no—
106#121×C37—21no—
107#121×C37D411yescharacteristic
108#121×C37C2311yescharacteristic
109#120×C37—21no—
110C4×Q8×C37—21no—
111C4×Q8×C37—21no—
112C4×D4×C37—22no—
113C4×D4×C37—21no—
114C4×D4×C37—21no—
115C4×D4×C37—22no—
116#117×C37—21no—
117#117×C37—22no—
118#117×C37—22no—
119#117×C37D411yescharacteristic
120C2×C4⋊C4×C37—21no—
121C2×C4⋊C4×C37D411yescharacteristic
122C2×C4⋊C4×C37D411yescharacteristic
123C2×C4⋊C4×C37C2311yescharacteristic
124C2×C4⋊C4×C37C2312yes—
125C2×C22⋊C4×C37—21no—
126C2×C22⋊C4×C37—21no—
127C2×C22⋊C4×C37—22no—
128C2×C22⋊C4×C37—22no—
129C2×C22⋊C4×C37D412yes—
130#107×C37—21no—
131#107×C37—22no—
132#106×C37—21no—
133#103×C37—22no—
134#103×C37—21no—
135#102×C37—21no—
136#102×C37—21no—
137#102×C37—21no—
138#102×C37—22no—
139C22⋊C8×C37—21no—
140C22⋊C8×C37—21no—
141C22⋊C8×C37C2312yes—
142C22⋊C8×C37C2311yescharacteristic
143C22⋊C8×C37C2311yescharacteristic
144#95×C37—21no—
145#95×C37—22no—
146#95×C37—22no—
147#95×C37C2312yes—
148C24×C37—21no—
149C24×C37C2×D411yescharacteristic
150#41×C37—81no—
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×C37—81no—
158C2×Q8×C37C2×D412yes—
159C2×Q8×C37C2×D411yescharacteristic
160C2×D4×C37—21no—
161C2×D4×C37—21no—
162C2×D4×C37—21no—
163C2×D4×C37—21no—
164C2×D4×C37—21no—
165C2×D4×C37—44no—
166C2×D4×C37—41no—
167C2×D4×C37—41no—
168C2×D4×C37—81no—
169C2×D4×C37C2×D411yescharacteristic
170C22×C4×C37—21no—
171C22×C4×C37—22no—
172C22×C4×C37—21no—
173C22×C4×C37—21no—
174C22×C4×C37—21no—
175C22×C4×C37—22no—
176C22×C4×C37—42no—
177C22×C4×C37—81no—
178C22×C4×C37—81no—
179C22×C4×C37—81no—
180C22×C4×C37C2×D412yes—
181C22×C4×C37C2×D412yes—
182C22×C4×C37C2×D411yescharacteristic
183C22×C4×C37C2×D411yescharacteristic
184C22×C4×C37C2×D411yescharacteristic
185C22×C4×C37C2×D411yescharacteristic
186C22×C4×C37C2×D411yescharacteristic
187C22×C4×C37C2411yescharacteristic
188SD16×C37—81no—
189SD16×C37—82no—
190D8×C37—81no—
191M16×C37—81no—
192C2×C8×C37—21no—
193C2×C8×C37—21no—
194C2×C8×C37—22no—
195C2×C8×C37—81no—
196C4⋊C4×C37—22no—
197C4⋊C4×C37—22no—
198C4⋊C4×C37—22no—
199C4⋊C4×C37—22no—
200C4⋊C4×C37—22no—
201C4⋊C4×C37—22no—
202C4⋊C4×C37—22no—
203C4⋊C4×C37—22no—
204C4⋊C4×C37—22no—
205C4⋊C4×C37—21no—
206C4⋊C4×C37—21no—
207C4⋊C4×C37—21no—
208C4⋊C4×C37—21no—
209C4⋊C4×C37—22no—
210C4⋊C4×C37—22no—
211C4⋊C4×C37—22no—
212C4⋊C4×C37—21no—
213C4⋊C4×C37—21no—
214C4⋊C4×C37—21no—
215C4⋊C4×C37C2×D411yescharacteristic
216C4⋊C4×C37C2×D412yes—
217C4⋊C4×C37C2×D411yescharacteristic
218C22⋊C4×C37—22no—
219C22⋊C4×C37—22no—
220C22⋊C4×C37—22no—
221C22⋊C4×C37—22no—
222C22⋊C4×C37—22no—
223C22⋊C4×C37—22no—
224C22⋊C4×C37—21no—
225C22⋊C4×C37—21no—
226C22⋊C4×C37—22no—
227C22⋊C4×C37—21no—
228C22⋊C4×C37—21no—
229C22⋊C4×C37—21no—
230C22⋊C4×C37—41no—
231C22⋊C4×C37—41no—
232C22⋊C4×C37—82no—
233C22⋊C4×C37—82no—
234C42×C37—22no—
235C42×C37—22no—
236C42×C37—22no—
237C42×C37—21no—
238C23×C37—21no—
239C23×C37—21no—
240C23×C37—22no—
241C23×C37—21no—
242C23×C37—41no—
243C23×C37—42no—
244C23×C37—42no—
245C23×C37—41no—
246C23×C37—41no—
247C23×C37—41no—
248C23×C37#12011yescharacteristic
249C23×C37#12011yescharacteristic
250C23×C37C22×D411yescharacteristic
251Q8×C37—42no—
252Q8×C37—41no—
253Q8×C37—42no—
254Q8×C37—42no—
255Q8×C37—81no—
256Q8×C37—82no—
257D4×C37—41no—
258D4×C37—41no—
259D4×C37—41no—
260D4×C37—42no—
261D4×C37—41no—
262D4×C37—41no—
263D4×C37—42no—
264D4×C37—81no—
265D4×C37—82no—
266D4×C37—82no—
267C2×C4×C37—21no—
268C2×C4×C37—21no—
269C2×C4×C37—22no—
270C2×C4×C37—22no—
271C2×C4×C37—21no—
272C2×C4×C37—22no—
273C2×C4×C37—22no—
274C2×C4×C37—22no—
275C2×C4×C37—22no—
276C2×C4×C37—41no—
277C2×C4×C37—41no—
278C2×C4×C37—41no—
279C2×C4×C37—41no—
280C2×C4×C37—41no—
281C2×C4×C37—41no—
282C2×C4×C37—42no—
283C2×C4×C37—42no—
284C2×C4×C37—42no—
285C2×C4×C37—42no—
286C2×C4×C37—42no—
287C2×C4×C37—42no—
288C2×C4×C37—41no—
289C2×C4×C37—42no—
290C2×C4×C37—81no—
291C2×C4×C37—82no—
292C2×C4×C37—84no—
293C2×C4×C37#12011yescharacteristic
294C2×C4×C37#12011yescharacteristic
295C2×C4×C37C22×D411yescharacteristic
296C2×C4×C37C22×D411yescharacteristic
297C8×C37—41no—
298C8×C37—41no—
299C8×C37—42no—
300#18422C3711yesmaximal, characteristic, direct product factor, Sylow
301C22×C37—21no—
302C22×C37—21no—
303C22×C37—21no—
304C22×C37—21no—
305C22×C37—21no—
306C22×C37—21no—
307C22×C37—21no—
308C22×C37—42no—
309C22×C37—42no—
310C22×C37—41no—
311C22×C37—41no—
312C22×C37—41no—
313C22×C37—82no—
314C22×C37—82no—
315C22×C37—81no—
316C22×C37—81no—
317C22×C37—81no—
318C22×C37#45412yes—
319C22×C37C2×#12011yescharacteristic
320C22×C37#57711yescharacteristic
321C22×C37#57811yescharacteristic
322C4×C37—21no—
323C4×C37—21no—
324C4×C37—21no—
325C4×C37—21no—
326C4×C37—42no—
327C4×C37—41no—
328C4×C37—42no—
329C4×C37—41no—
330C4×C37—41no—
331C4×C37—42no—
332C4×C37—42no—
333C4×C37—82no—
334C4×C37—81no—
335#3538C2×C3711yescharacteristic, complemented
336#3193C2×C3711yescharacteristic, complemented
337#3188C2×C3711yescharacteristic
338#2639C2×C3711yescharacteristic, complemented
339#1617C2×C3712yescomplemented
340#1615C2×C3712yescomplemented
341#1599C2×C3712yescomplemented
342#1597C2×C3712yescomplemented
343#1504C2×C3711yescharacteristic, complemented
344C2×#328C2×C3711yescharacteristic, complemented
345#1472C2×C3711yescharacteristic, complemented
346C2×C37—21no—
347C2×C37—21no—
348C2×C37—22no—
349C2×C37—41no—
350C2×C37—81no—
351C2×C37—161no—
352C2×C37#301311yescharacteristic
353C2×C37C2×#45411yescharacteristic
354C2×C37#302611yescharacteristic
355#563—22no—
356#552—21no—
357#552—21no—
358#551—22no—
359D4×Q8—21no—
360D4×Q8—21no—
361#544—21no—
362#544C22×C3712yes—
363#541—21no—
364#539—21no—
365#539C22×C3712yes—
366#536—21no—
367#525C22×C3711yescharacteristic
368C2×C22⋊Q8C22×C3711yescharacteristic
369C2×C22⋊Q8C22×C3711yescharacteristic
370C2×#121C22×C3711yescharacteristic
371#484C22×C3712yes—
372#484C22×C3711yescharacteristic
373#482—21no—
374#481C22×C3711yescharacteristic
375#480—21no—
376#468—22no—
377#466—21no—
378#465—21no—
379#450C22×C3712yes—
380#450C22×C3711yescharacteristic
381#449—21no—
382#448—21no—
383#447C22×C3711yescharacteristic
384#407C22×C3711yescharacteristic
385C2×C22⋊C8C22×C3711yescharacteristic
386#400C22×C3712yes—
387#397—21no—
388#394C22×C3712yes—
389#393—21no—
390#386—22no—
391C2×#95C22×C3711yescharacteristic
392#328C22×C3712yes—
393#328C22×C3712yes—
394#328C22×C3712yes—
395#327C22×C3712yes—
396#324C22×C3712yes—
397#324C22×C3712yes—
398C37#1842211yescharacteristic, direct product factor, Sylow
399C2×#41D4×C3711yescharacteristic
400C22×Q8—21no—
401C22×Q8D4×C3711yescharacteristic
402C22×Q8D4×C3711yescharacteristic
403C22×D4—21no—
404C22×D4C23×C3711yescharacteristic
405C23×C4—21no—
406C23×C4C23×C3711yescharacteristic
407C2×SD16—22no—
408C2×SD16—21no—
409C2×D8—21no—
410C2×M16—21no—
411C22×C8—21no—
412C4⋊Q8—21no—
413C4⋊Q8—22no—
414C4⋊Q8—22no—
415C4⋊Q8—22no—
416#126—22no—
417#126—22no—
418#126—22no—
419#126—22no—
420#125—21no—
421#125—22no—
422#125—22no—
423#125—22no—
424#125—22no—
425#124—22no—
426#123—22no—
427#123—22no—
428#123—22no—
429#123—21no—
430#123—22no—
431C22⋊Q8—21no—
432C22⋊Q8—21no—
433C22⋊Q8—22no—
434C22⋊Q8—22no—
435C22⋊Q8—22no—
436C22⋊Q8—21no—
437C22⋊Q8—21no—
438C22⋊Q8—21no—
439C22⋊Q8—21no—
440C22⋊Q8—21no—
441C22⋊Q8—21no—
442C22⋊Q8—22no—
443C22⋊Q8D4×C3712yes—
444C22⋊Q8D4×C3711yescharacteristic
445C22⋊Q8C23×C3712yes—
446C22⋊Q8C23×C3711yescharacteristic
447#121—21no—
448#121—21no—
449#121—21no—
450#121—21no—
451#121D4×C3711yescharacteristic
452#121C23×C3711yescharacteristic
453#120—21no—
454C4×Q8—21no—
455C4×Q8—21no—
456C4×D4—22no—
457C4×D4—21no—
458C4×D4—21no—
459C4×D4—22no—
460#117—21no—
461#117—22no—
462#117—22no—
463#117D4×C3711yescharacteristic
464C2×C4⋊C4—21no—
465C2×C4⋊C4D4×C3711yescharacteristic
466C2×C4⋊C4D4×C3711yescharacteristic
467C2×C4⋊C4C23×C3711yescharacteristic
468C2×C4⋊C4C23×C3712yes—
469C2×C22⋊C4—21no—
470C2×C22⋊C4—21no—
471C2×C22⋊C4—22no—
472C2×C22⋊C4—22no—
473C2×C22⋊C4D4×C3712yes—
474#107—21no—
475#107—22no—
476#106—21no—
477#103—22no—
478#103—21no—
479#102—21no—
480#102—21no—
481#102—21no—
482#102—22no—
483C22⋊C8—21no—
484C22⋊C8—21no—
485C22⋊C8C23×C3712yes—
486C22⋊C8C23×C3711yescharacteristic
487C22⋊C8C23×C3711yescharacteristic
488#95—21no—
489#95—22no—
490#95—22no—
491#95C23×C3712yes—
492C24—21no—
493C24C2×D4×C3711yescharacteristic
494#41—81no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—81no—
500C2×Q8—82no—
501C2×Q8—81no—
502C2×Q8C2×D4×C3712yes—
503C2×Q8C2×D4×C3711yescharacteristic
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—44no—
510C2×D4—41no—
511C2×D4—41no—
512C2×D4—81no—
513C2×D4C2×D4×C3711yescharacteristic
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—42no—
521C22×C4—81no—
522C22×C4—81no—
523C22×C4—81no—
524C22×C4C2×D4×C3712yes—
525C22×C4C2×D4×C3712yes—
526C22×C4C2×D4×C3711yescharacteristic
527C22×C4C2×D4×C3711yescharacteristic
528C22×C4C2×D4×C3711yescharacteristic
529C22×C4C2×D4×C3711yescharacteristic
530C22×C4C2×D4×C3711yescharacteristic
531C22×C4C24×C3711yescharacteristic
532SD16—81no—
533SD16—82no—
534D8—81no—
535M16—81no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—81no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4C2×D4×C3711yescharacteristic
560C4⋊C4C2×D4×C3712yes—
561C4⋊C4C2×D4×C3711yescharacteristic
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—22no—
571C22⋊C4—21no—
572C22⋊C4—21no—
573C22⋊C4—21no—
574C22⋊C4—41no—
575C22⋊C4—41no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—21no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23#120×C3711yescharacteristic
593C23#120×C3711yescharacteristic
594C23C22×D4×C3711yescharacteristic
595Q8—42no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—81no—
600Q8—82no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—42no—
605D4—41no—
606D4—41no—
607D4—42no—
608D4—81no—
609D4—82no—
610D4—82no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—81no—
635C2×C4—82no—
636C2×C4—84no—
637C2×C4#120×C3711yescharacteristic
638C2×C4#120×C3711yescharacteristic
639C2×C4C22×D4×C3711yescharacteristic
640C2×C4C22×D4×C3711yescharacteristic
641C8—41no—
642C8—41no—
643C8—42no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—82no—
657C22—82no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C37×#45412yes—
662C22C2×#120×C3711yescharacteristic
663C22C37×#57711yescharacteristic
664C22C37×#57811yescharacteristic
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—81no—
678C2—21no—
679C2—21no—
680C2—22no—
681C2—41no—
682C2—81no—
683C2—161no—
684C2C37×#301311yescharacteristic
685C2C2×C37×#45411yescharacteristic
686C2C37×#302611yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#3188C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3538C2semidirect
2. C37×#3193C2semidirect
4. C37×#2639C2semidirect
5. C37×#1617C2semidirect
6. C37×#1615C2semidirect
7. C37×#1599C2semidirect
8. C37×#1597C2semidirect
9. C37×#1504C2semidirect
10. C2×C37×#328C2semidirect
11. C37×#1472C2semidirect
300. #18422C37direct (N × H)
335. #3538C2×C37semidirect
336. #3193C2×C37semidirect
338. #2639C2×C37semidirect
339. #1617C2×C37semidirect
340. #1615C2×C37semidirect
341. #1599C2×C37semidirect
342. #1597C2×C37semidirect
343. #1504C2×C37semidirect
344. C2×#328C2×C37semidirect
345. #1472C2×C37semidirect
398. C37#18422direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35382
C37×#31932
C37×#31882
C37×#26392
C37×#16172
C37×#16152
C37×#15992
C37×#15972
C37×#15042
C2×C37×#3282
C37×#14722
#1842237