Finite Group Databasebeta

← C37×#29665 #22650918

Subgroups — C37×#29665 #22650918

Order 9472.

1164 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#529C211yesmaximal, characteristic
2C2×C37×#462C211yesmaximal, characteristic, complemented
3C2×C37×#461C211yesmaximal, characteristic, complemented
4C37×#2857C211yesmaximal, characteristic, complemented
5C37×#2026C212yesmaximal, complemented
6C37×#2015C212yesmaximal, complemented
7C37×#2001C212yesmaximal, complemented
8C37×#2000C212yesmaximal
9C37×#1803C211yesmaximal, characteristic, complemented
10C37×#1802C211yesmaximal, characteristic, complemented
11C37×#1797C211yesmaximal, characteristic, complemented
12C22×Q16×C37C2212yes—
13C22×SD16×C37C2212yes—
14C2×C4⋊Q8×C37C2211yescharacteristic
15C37×#529—24no—
16C37×#529—24no—
17C2×#126×C37—21no—
18C2×#125×C37C2211yescharacteristic
19C2×#124×C37C2211yescharacteristic
20C2×#123×C37—21no—
21C2×C22⋊Q8×C37C2211yescharacteristic
22C2×C22⋊Q8×C37C2212yes—
23C2×#121×C37C2211yescharacteristic
24C2×C4×Q8×C37C2211yescharacteristic
25C2×C4×D4×C37—21no—
26C2×C4×D4×C37C2211yescharacteristic
27C2×#117×C37—21no—
28C22×C42×C37C2211yescharacteristic
29C37×#462—24no—
30C37×#461—24no—
31C2×C4⋊C8×C37C2211yescharacteristic
32C2×C4⋊C8×C37C2211yescharacteristic
33C2×#103×C37C2212yes—
34C2×#103×C37C2212yes—
35C2×#103×C37C2212yes—
36C2×#102×C37C2212yes—
37C2×C22⋊C8×C37C2212yes—
38C37×#391C2212yes—
39C37×#389—21no—
40C37×#389—21no—
41C37×#384C2211yescharacteristic
42C37×#380—21no—
43C37×#380—21no—
44C37×#340C2212yes—
45C37×#340C2212yes—
46C37×#336C2212yes—
47C22×Q8×C37C2312yes—
48C22×Q8×C37C2311yescharacteristic
49C22×D4×C37—21no—
50C22×D4×C37C2311yescharacteristic
51C23×C4×C37—21no—
52C23×C4×C37D411yescharacteristic
53C23×C4×C37D411yescharacteristic
54C23×C4×C37C2311yescharacteristic
55C2×Q16×C37—24no—
56C2×Q16×C37—42no—
57C2×Q16×C37—42no—
58C2×SD16×C37—24no—
59C2×SD16×C37—42no—
60C2×SD16×C37—42no—
61C22×C8×C37C2312yes—
62C22×C8×C37C2312yes—
63C4⋊Q8×C37—22no—
64C4⋊Q8×C37—22no—
65#126×C37—28no—
66#125×C37—22no—
67#125×C37—22no—
68#124×C37—22no—
69#124×C37—22no—
70#123×C37—28no—
71C22⋊Q8×C37—22no—
72C22⋊Q8×C37—22no—
73C22⋊Q8×C37—42no—
74C22⋊Q8×C37—42no—
75#121×C37—22no—
76#121×C37—22no—
77C4×Q8×C37—22no—
78C4×Q8×C37D414yes—
79C4×D4×C37—28no—
80C4×D4×C37—22no—
81C4×D4×C37D414yes—
82#117×C37—28no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37—21no—
85C2×C4⋊C4×C37—21no—
86C2×C4⋊C4×C37—21no—
87C2×C4⋊C4×C37—22no—
88C2×C4⋊C4×C37—21no—
89C2×C4⋊C4×C37—21no—
90C2×C4⋊C4×C37—21no—
91C2×C4⋊C4×C37—21no—
92C2×C4⋊C4×C37C2312yes—
93C2×C4⋊C4×C37C2311yescharacteristic
94C2×C4⋊C4×C37C2311yescharacteristic
95C2×C22⋊C4×C37—21no—
96C2×C22⋊C4×C37—21no—
97C2×C22⋊C4×C37—21no—
98C2×C22⋊C4×C37—21no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37—22no—
101C2×C42×C37—21no—
102C2×C42×C37—22no—
103C2×C42×C37—22no—
104C2×C42×C37D411yescharacteristic
105C2×C42×C37D411yescharacteristic
106C2×C42×C37C2311yescharacteristic
107C2×C42×C37C2311yescharacteristic
108C4⋊C8×C37—22no—
109C4⋊C8×C37—22no—
110#103×C37—24no—
111#103×C37—24no—
112#103×C37—42no—
113#102×C37—42no—
114C22⋊C8×C37—42no—
115#95×C37—22no—
116#95×C37—22no—
117C24×C37—21no—
118C24×C37C2×D411yescharacteristic
119C2×Q8×C37—21no—
120C2×Q8×C37—22no—
121C2×Q8×C37—22no—
122C2×Q8×C37—21no—
123C2×Q8×C37—21no—
124C2×Q8×C37—21no—
125C2×Q8×C37—22no—
126C2×Q8×C37—22no—
127C2×Q8×C37—22no—
128C2×Q8×C37—21no—
129C2×Q8×C37—22no—
130C2×Q8×C37C2×D412yes—
131C2×D4×C37—24no—
132C2×D4×C37—21no—
133C2×D4×C37—21no—
134C2×D4×C37—21no—
135C2×D4×C37—21no—
136C2×D4×C37—21no—
137C2×D4×C37—42no—
138C2×D4×C37—42no—
139C2×D4×C37C2×D412yes—
140C22×C4×C37—22no—
141C22×C4×C37—21no—
142C22×C4×C37—21no—
143C22×C4×C37—21no—
144C22×C4×C37—22no—
145C22×C4×C37—22no—
146C22×C4×C37—24no—
147C22×C4×C37—22no—
148C22×C4×C37—22no—
149C22×C4×C37—21no—
150C22×C4×C37—21no—
151C22×C4×C37—21no—
152C22×C4×C37—21no—
153C22×C4×C37—21no—
154C22×C4×C37—21no—
155C22×C4×C37—22no—
156C22×C4×C37—22no—
157C22×C4×C37—41no—
158C22×C4×C37—42no—
159C22×C4×C37—42no—
160C22×C4×C37C2×D411yescharacteristic
161C22×C4×C37C2×D411yescharacteristic
162C22×C4×C37C2×D411yescharacteristic
163C22×C4×C37C2×D411yescharacteristic
164C22×C4×C37C2×D411yescharacteristic
165C22×C4×C37C2411yescharacteristic
166Q16×C37—48no—
167SD16×C37—48no—
168C2×C8×C37—22no—
169C2×C8×C37—22no—
170C2×C8×C37—22no—
171C2×C8×C37—22no—
172C2×C8×C37—22no—
173C2×C8×C37—22no—
174C4⋊C4×C37—24no—
175C4⋊C4×C37—24no—
176C4⋊C4×C37—24no—
177C4⋊C4×C37—24no—
178C4⋊C4×C37—21no—
179C4⋊C4×C37—21no—
180C4⋊C4×C37—22no—
181C4⋊C4×C37—22no—
182C4⋊C4×C37—42no—
183C4⋊C4×C37—44no—
184C4⋊C4×C37—42no—
185C4⋊C4×C37—42no—
186C4⋊C4×C37—42no—
187C4⋊C4×C37C2×D412yes—
188C4⋊C4×C37C2×D412yes—
189C22⋊C4×C37—24no—
190C22⋊C4×C37—24no—
191C22⋊C4×C37—24no—
192C22⋊C4×C37—24no—
193C22⋊C4×C37—24no—
194C22⋊C4×C37—44no—
195C42×C37—22no—
196C42×C37—22no—
197C42×C37—24no—
198C42×C37—21no—
199C42×C37—21no—
200C42×C37C2×D412yes—
201C42×C37C2×D412yes—
202C23×C37—21no—
203C23×C37—21no—
204C23×C37—21no—
205C23×C37—21no—
206C23×C37—21no—
207C23×C37—21no—
208C23×C37—22no—
209C23×C37—41no—
210C23×C37—41no—
211C23×C37—41no—
212C23×C37—41no—
213C23×C37—41no—
214C23×C37—41no—
215C23×C37#13712yes—
216C23×C37C22×D411yescharacteristic
217Q8×C37—22no—
218Q8×C37—22no—
219Q8×C37—41no—
220Q8×C37—41no—
221Q8×C37—42no—
222Q8×C37—42no—
223Q8×C37—42no—
224Q8×C37—42no—
225D4×C37—22no—
226D4×C37—22no—
227D4×C37—48no—
228D4×C37—41no—
229D4×C37—41no—
230C2×C4×C37—21no—
231C2×C4×C37—21no—
232C2×C4×C37—21no—
233C2×C4×C37—21no—
234C2×C4×C37—21no—
235C2×C4×C37—22no—
236C2×C4×C37—22no—
237C2×C4×C37—22no—
238C2×C4×C37—21no—
239C2×C4×C37—21no—
240C2×C4×C37—21no—
241C2×C4×C37—21no—
242C2×C4×C37—21no—
243C2×C4×C37—21no—
244C2×C4×C37—44no—
245C2×C4×C37—41no—
246C2×C4×C37—41no—
247C2×C4×C37—41no—
248C2×C4×C37—41no—
249C2×C4×C37—41no—
250C2×C4×C37—41no—
251C2×C4×C37—41no—
252C2×C4×C37—41no—
253C2×C4×C37—41no—
254C2×C4×C37—41no—
255C2×C4×C37—44no—
256C2×C4×C37—48no—
257C2×C4×C37—42no—
258C2×C4×C37—42no—
259C2×C4×C37—42no—
260C2×C4×C37—42no—
261C2×C4×C37—42no—
262C2×C4×C37—42no—
263C2×C4×C37#12012yes—
264C2×C4×C37#12012yes—
265C2×C4×C37#13512yes—
266C2×C4×C37#13712yes—
267C2×C4×C37C22×D411yescharacteristic
268C2×C4×C37C22×D411yescharacteristic
269C2×C4×C37H(5,𝔽2)11yescharacteristic
270C2×C4×C37H(5,𝔽2)11yescharacteristic
271C2×C4×C37H(5,𝔽2)11yescharacteristic
272C2×C4×C372−1+411yescharacteristic
273C8×C37—42no—
274C8×C37—42no—
275#29665C3711yesmaximal, characteristic, direct product factor, Sylow
276C22×C37—22no—
277C22×C37—22no—
278C22×C37—22no—
279C22×C37—41no—
280C22×C37—41no—
281C22×C37—41no—
282C22×C37—41no—
283C22×C37—42no—
284C22×C37—42no—
285C22×C37—42no—
286C22×C37—81no—
287C22×C37—81no—
288C22×C37—81no—
289C22×C37—81no—
290C22×C37C2×#12011yescharacteristic
291C22×C37#53511yescharacteristic
292C22×C37#53711yescharacteristic
293C22×C37C2×#13511yescharacteristic
294C22×C37C2×#13711yescharacteristic
295C22×C37C2×#13711yescharacteristic
296C22×C37#57511yescharacteristic
297C4×C37—22no—
298C4×C37—41no—
299C4×C37—42no—
300C4×C37—41no—
301C4×C37—42no—
302C4×C37—41no—
303C4×C37—42no—
304C4×C37—82no—
305C4×C37—82no—
306C4×C37#45214yes—
307C4×C37#53511yescharacteristic
308C4×C37#53511yescharacteristic
309C4×C37#53711yescharacteristic
310C4×C37#53711yescharacteristic
311C2×#529C2×C3711yescharacteristic
312C2×#462C2×C3711yescharacteristic, complemented
313C2×#461C2×C3711yescharacteristic, complemented
314#2857C2×C3711yescharacteristic, complemented
315#2026C2×C3712yescomplemented
316#2015C2×C3712yescomplemented
317#2001C2×C3712yescomplemented
318#2000C2×C3712yes—
319#1803C2×C3711yescharacteristic, complemented
320#1802C2×C3711yescharacteristic, complemented
321#1797C2×C3711yescharacteristic, complemented
322C2×C37—42no—
323C2×C37—82no—
324C2×C37#261511yescharacteristic
325C2×C37C2×#45211yescharacteristic
326C2×C37#300711yescharacteristic
327C2×C37#321211yescharacteristic
328C2×C37#321811yescharacteristic
329C2×C37#324811yescharacteristic
330C2×C37#325211yescharacteristic
331C22×Q16C22×C3712yes—
332C22×SD16C22×C3712yes—
333C2×C4⋊Q8C22×C3711yescharacteristic
334#529—24no—
335#529—24no—
336C2×#126—21no—
337C2×#125C22×C3711yescharacteristic
338C2×#124C22×C3711yescharacteristic
339C2×#123—21no—
340C2×C22⋊Q8C22×C3711yescharacteristic
341C2×C22⋊Q8C22×C3712yes—
342C2×#121C22×C3711yescharacteristic
343C2×C4×Q8C22×C3711yescharacteristic
344C2×C4×D4—21no—
345C2×C4×D4C22×C3711yescharacteristic
346C2×#117—21no—
347C22×C42C22×C3711yescharacteristic
348#462—24no—
349#461—24no—
350C2×C4⋊C8C22×C3711yescharacteristic
351C2×C4⋊C8C22×C3711yescharacteristic
352C2×#103C22×C3712yes—
353C2×#103C22×C3712yes—
354C2×#103C22×C3712yes—
355C2×#102C22×C3712yes—
356C2×C22⋊C8C22×C3712yes—
357#391C22×C3712yes—
358#389—21no—
359#389—21no—
360#384C22×C3711yescharacteristic
361#380—21no—
362#380—21no—
363#340C22×C3712yes—
364#340C22×C3712yes—
365#336C22×C3712yes—
366C37#2966511yescharacteristic, direct product factor, Sylow
367C22×Q8C23×C3712yes—
368C22×Q8C23×C3711yescharacteristic
369C22×D4—21no—
370C22×D4C23×C3711yescharacteristic
371C23×C4—21no—
372C23×C4D4×C3711yescharacteristic
373C23×C4D4×C3711yescharacteristic
374C23×C4C23×C3711yescharacteristic
375C2×Q16—24no—
376C2×Q16—42no—
377C2×Q16—42no—
378C2×SD16—24no—
379C2×SD16—42no—
380C2×SD16—42no—
381C22×C8C23×C3712yes—
382C22×C8C23×C3712yes—
383C4⋊Q8—22no—
384C4⋊Q8—22no—
385#126—28no—
386#125—22no—
387#125—22no—
388#124—22no—
389#124—22no—
390#123—28no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—42no—
394C22⋊Q8—42no—
395#121—22no—
396#121—22no—
397C4×Q8—22no—
398C4×Q8D4×C3714yes—
399C4×D4—28no—
400C4×D4—22no—
401C4×D4D4×C3714yes—
402#117—28no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4C23×C3712yes—
413C2×C4⋊C4C23×C3711yescharacteristic
414C2×C4⋊C4C23×C3711yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—22no—
421C2×C42—21no—
422C2×C42—22no—
423C2×C42—22no—
424C2×C42D4×C3711yescharacteristic
425C2×C42D4×C3711yescharacteristic
426C2×C42C23×C3711yescharacteristic
427C2×C42C23×C3711yescharacteristic
428C4⋊C8—22no—
429C4⋊C8—22no—
430#103—24no—
431#103—24no—
432#103—42no—
433#102—42no—
434C22⋊C8—42no—
435#95—22no—
436#95—22no—
437C24—21no—
438C24C2×D4×C3711yescharacteristic
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8C2×D4×C3712yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4C2×D4×C3712yes—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4C2×D4×C3711yescharacteristic
481C22×C4C2×D4×C3711yescharacteristic
482C22×C4C2×D4×C3711yescharacteristic
483C22×C4C2×D4×C3711yescharacteristic
484C22×C4C2×D4×C3711yescharacteristic
485C22×C4C24×C3711yescharacteristic
486Q16—48no—
487SD16—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4C2×D4×C3712yes—
508C4⋊C4C2×D4×C3712yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—44no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42C2×D4×C3712yes—
521C42C2×D4×C3712yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23#137×C3712yes—
536C23C22×D4×C3711yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545D4—22no—
546D4—22no—
547D4—48no—
548D4—41no—
549D4—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4#120×C3712yes—
584C2×C4#120×C3712yes—
585C2×C4#135×C3712yes—
586C2×C4#137×C3712yes—
587C2×C4C22×D4×C3711yescharacteristic
588C2×C4C22×D4×C3711yescharacteristic
589C2×C4H(5,𝔽2)×C3711yescharacteristic
590C2×C4H(5,𝔽2)×C3711yescharacteristic
591C2×C4H(5,𝔽2)×C3711yescharacteristic
592C2×C42−1+4×C3711yescharacteristic
593C8—42no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C2×#120×C3711yescharacteristic
610C22C37×#53511yescharacteristic
611C22C37×#53711yescharacteristic
612C22C2×#135×C3711yescharacteristic
613C22C2×#137×C3711yescharacteristic
614C22C2×#137×C3711yescharacteristic
615C22C37×#57511yescharacteristic
616C4—22no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—82no—
624C4—82no—
625C4C37×#45214yes—
626C4C37×#53511yescharacteristic
627C4C37×#53511yescharacteristic
628C4C37×#53711yescharacteristic
629C4C37×#53711yescharacteristic
630C2—42no—
631C2—82no—
632C2C37×#261511yescharacteristic
633C2C2×C37×#45211yescharacteristic
634C2C37×#300711yescharacteristic
635C2C37×#321211yescharacteristic
636C2C37×#321811yescharacteristic
637C2C37×#324811yescharacteristic
638C2C37×#325211yescharacteristic
subgroupG/H
center, socleC23×C37C22×D4
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×#121×C37C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C37 ◁ C22×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C37×#462C2semidirect
3. C2×C37×#461C2semidirect
4. C37×#2857C2semidirect
5. C37×#2026C2semidirect
6. C37×#2015C2semidirect
7. C37×#2001C2semidirect
9. C37×#1803C2semidirect
10. C37×#1802C2semidirect
11. C37×#1797C2semidirect
275. #29665C37direct (N × H)
312. C2×#462C2×C37semidirect
313. C2×#461C2×C37semidirect
314. #2857C2×C37semidirect
315. #2026C2×C37semidirect
316. #2015C2×C37semidirect
317. #2001C2×C37semidirect
319. #1803C2×C37semidirect
320. #1802C2×C37semidirect
321. #1797C2×C37semidirect
366. C37#29665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#5292
C2×C37×#4622
C2×C37×#4612
C37×#28572
C37×#20262
C37×#20152
C37×#20012
C37×#20002
C37×#18032
C37×#18022
C37×#17972
#2966537