Finite Group Databasebeta

← C37×#29682 #22650935

Subgroups — C37×#29682 #22650935

Order 9472.

1152 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3460C211yesmaximal, characteristic, complemented
2C37×#3040C212yesmaximal, complemented
3C37×#2544C211yesmaximal, characteristic, complemented
4C37×#2028C211yesmaximal, characteristic, complemented
5C37×#2027C211yesmaximal, characteristic, complemented
6C37×#2007C211yesmaximal, characteristic, complemented
7C37×#2006C211yesmaximal, characteristic, complemented
8C37×#2004C212yesmaximal, complemented
9C37×#2002C212yesmaximal, complemented
10C37×#1806C212yesmaximal, complemented
11C37×#1800C211yesmaximal, characteristic, complemented
12C2×#137×C37C2212yes—
13C2×AGL(1,ℤ/8)×C37C2212yes—
14C37×#544C2211yescharacteristic
15C37×#542—22no—
16C37×#540—22no—
17C37×#538—22no—
18C37×#537C2211yescharacteristic
19C37×#535C2212yes—
20C37×#532—22no—
21C2×#127×C37C2211yescharacteristic
22C37×#529—22no—
23C2×#124×C37C2211yescharacteristic
24C2×#124×C37C2211yescharacteristic
25C37×#525—22no—
26C4×#41×C37C2212yes—
27C2×C4×D4×C37C2211yescharacteristic
28C37×#464—22no—
29C37×#464—22no—
30C37×#463—22no—
31C37×#463—22no—
32C37×#423C2212yes—
33C2×#104×C37C2212yes—
34C2×#104×C37C2212yes—
35C37×#419C2212yes—
36C37×#418C2212yes—
37C2×#100×C37C2212yes—
38C37×#394—22no—
39C37×#391C2211yescharacteristic
40C37×#390C2211yescharacteristic
41C37×#386—22no—
42C4×C4⋊C4×C37C2211yescharacteristic
43C37×#343C2211yescharacteristic
44C37×#343C2211yescharacteristic
45C37×#339C2212yes—
46C37×#339C2212yes—
47C2×#41×C37—22no—
48C2×#41×C37C2312yes—
49C22×Q8×C37C2311yescharacteristic
50C22×D4×C37—22no—
51C22×D4×C37C2311yescharacteristic
52C22×D4×C37C2311yescharacteristic
53C23×C4×C37D412yes—
54#137×C37—42no—
55#137×C37—42no—
56AGL(1,ℤ/8)×C37—42no—
57AGL(1,ℤ/8)×C37—42no—
58C2×Q16×C37—22no—
59C2×SD16×C37—22no—
60C2×SD16×C37—22no—
61C2×D8×C37—22no—
62C2×M16×C37C2312yes—
63C2×M16×C37C2312yes—
64C4⋊Q8×C37—22no—
65#127×C37—22no—
66#127×C37—42no—
67#126×C37—24no—
68#125×C37—22no—
69#124×C37—24no—
70#124×C37—22no—
71#124×C37—22no—
72#124×C37—22no—
73#124×C37—42no—
74#123×C37—24no—
75#123×C37—24no—
76C22⋊Q8×C37—22no—
77C22⋊Q8×C37—22no—
78C22⋊Q8×C37—22no—
79#121×C37—22no—
80#121×C37—24no—
81#121×C37—22no—
82#121×C37—22no—
83#121×C37—22no—
84#121×C37—22no—
85#120×C37—24no—
86C4×Q8×C37D412yescomplemented
87C4×Q8×C37D412yescomplemented
88C4×D4×C37—24no—
89C4×D4×C37—22no—
90C4×D4×C37—22no—
91C4×D4×C37—22no—
92C4×D4×C37—22no—
93C4×D4×C37D412yescomplemented
94C4×D4×C37D412yescomplemented
95#117×C37—22no—
96#117×C37—22no—
97#117×C37C2312yes—
98C2×C4⋊C4×C37—21no—
99C2×C4⋊C4×C37C2311yescharacteristic
100C2×C22⋊C4×C37—22no—
101C2×C22⋊C4×C37—21no—
102C2×C22⋊C4×C37—21no—
103C2×C22⋊C4×C37—22no—
104C2×C22⋊C4×C37D412yes—
105C2×C42×C37—22no—
106C2×C42×C37C2311yescharacteristic
107C2×C42×C37C2311yescharacteristic
108C2×C42×C37C2311yescharacteristic
109C4⋊C8×C37—22no—
110C4⋊C8×C37—22no—
111#104×C37—42no—
112#104×C37—42no—
113#103×C37—22no—
114#103×C37—22no—
115#102×C37—22no—
116#102×C37—22no—
117#100×C37—42no—
118#95×C37—21no—
119#95×C37—22no—
120C24×C37—21no—
121C24×C37C2×D412yes—
122#41×C37—22no—
123#41×C37—22no—
124#41×C37—22no—
125#41×C37—22no—
126#41×C37—84no—
127C2×Q8×C37—22no—
128C2×Q8×C37—22no—
129C2×Q8×C37—22no—
130C2×Q8×C37—21no—
131C2×Q8×C37—21no—
132C2×Q8×C37C2×D412yes—
133C2×D4×C37—22no—
134C2×D4×C37—24no—
135C2×D4×C37—24no—
136C2×D4×C37—22no—
137C2×D4×C37—22no—
138C2×D4×C37—22no—
139C2×D4×C37—21no—
140C2×D4×C37—22no—
141C2×D4×C37—22no—
142C2×D4×C37—21no—
143C2×D4×C37—21no—
144C2×D4×C37—21no—
145C2×D4×C37—44no—
146C2×D4×C37—44no—
147C2×D4×C37—82no—
148C2×D4×C37C2×D412yes—
149C22×C4×C37—21no—
150C22×C4×C37—22no—
151C22×C4×C37—22no—
152C22×C4×C37—22no—
153C22×C4×C37—22no—
154C22×C4×C37—22no—
155C22×C4×C37—21no—
156C22×C4×C37—21no—
157C22×C4×C37—82no—
158C22×C4×C37C2×D412yes—
159C22×C4×C37C2×D411yescharacteristic
160C22×C4×C37C2×D411yescharacteristic
161C22×C4×C37C2411yescharacteristic
162Q16×C37—44no—
163SD16×C37—44no—
164SD16×C37—44no—
165D8×C37—44no—
166M16×C37—22no—
167M16×C37—22no—
168M16×C37—22no—
169M16×C37—22no—
170C2×C8×C37—22no—
171C2×C8×C37—22no—
172C4⋊C4×C37—22no—
173C4⋊C4×C37—22no—
174C4⋊C4×C37—24no—
175C4⋊C4×C37—22no—
176C4⋊C4×C37—22no—
177C4⋊C4×C37—22no—
178C4⋊C4×C37—42no—
179C4⋊C4×C37C2×D412yes—
180C4⋊C4×C37C2×D412yes—
181C22⋊C4×C37—24no—
182C22⋊C4×C37—24no—
183C22⋊C4×C37—24no—
184C22⋊C4×C37—22no—
185C22⋊C4×C37—24no—
186C22⋊C4×C37—22no—
187C22⋊C4×C37—22no—
188C22⋊C4×C37—44no—
189C22⋊C4×C37—41no—
190C22⋊C4×C37—42no—
191C22⋊C4×C37—41no—
192C22⋊C4×C37—82no—
193C42×C37—22no—
194C42×C37—22no—
195C42×C37—22no—
196C42×C37—22no—
197C42×C37—22no—
198C42×C37C2×D412yes—
199C42×C37C2×D412yes—
200C23×C37—22no—
201C23×C37—22no—
202C23×C37—22no—
203C23×C37—22no—
204C23×C37—42no—
205C23×C37—42no—
206C23×C37—42no—
207C23×C37—42no—
208C23×C37—41no—
209C23×C37—41no—
210C23×C37—41no—
211C23×C37C22×D411yescharacteristic
212Q8×C37—22no—
213Q8×C37—22no—
214Q8×C37—42no—
215Q8×C37—42no—
216Q8×C37—82no—
217D4×C37—22no—
218D4×C37—22no—
219D4×C37—42no—
220D4×C37—42no—
221D4×C37—42no—
222D4×C37—42no—
223D4×C37—42no—
224D4×C37—42no—
225D4×C37—84no—
226D4×C37—82no—
227D4×C37—84no—
228D4×C37—88no—
229C2×C4×C37—22no—
230C2×C4×C37—22no—
231C2×C4×C37—22no—
232C2×C4×C37—22no—
233C2×C4×C37—22no—
234C2×C4×C37—41no—
235C2×C4×C37—41no—
236C2×C4×C37—44no—
237C2×C4×C37—44no—
238C2×C4×C37—41no—
239C2×C4×C37—41no—
240C2×C4×C37—42no—
241C2×C4×C37—42no—
242C2×C4×C37—42no—
243C2×C4×C37—41no—
244C2×C4×C37—41no—
245C2×C4×C37—41no—
246C2×C4×C37—41no—
247C2×C4×C37—41no—
248C2×C4×C37—41no—
249C2×C4×C37—82no—
250C2×C4×C37—82no—
251C2×C4×C37#12011yescharacteristic
252C2×C4×C37#12012yes—
253C2×C4×C37#12011yescharacteristic
254C2×C4×C37C22×D412yes—
255C2×C4×C37H(5,𝔽2)11yescharacteristic
256C2×C4×C37H(5,𝔽2)12yes—
257C2×C4×C372−1+411yescharacteristic
258C8×C37—42no—
259C8×C37—42no—
260#29682C3711yesmaximal, characteristic, direct product factor, Sylow
261C22×C37—21no—
262C22×C37—21no—
263C22×C37—44no—
264C22×C37—42no—
265C22×C37—42no—
266C22×C37—42no—
267C22×C37—42no—
268C22×C37—42no—
269C22×C37—42no—
270C22×C37—82no—
271C22×C37—82no—
272C22×C37—82no—
273C22×C37—81no—
274C22×C37—81no—
275C22×C37—81no—
276C22×C37C2×#12011yescharacteristic
277C22×C37#53511yescharacteristic
278C22×C37#53711yescharacteristic
279C4×C37—22no—
280C4×C37—21no—
281C4×C37—21no—
282C4×C37—42no—
283C4×C37—42no—
284C4×C37—42no—
285C4×C37—42no—
286C4×C37—42no—
287C4×C37—81no—
288C4×C37—81no—
289C4×C37#53512yes—
290C4×C37#53712yes—
291#3460C2×C3711yescharacteristic, complemented
292#3040C2×C3712yescomplemented
293#2544C2×C3711yescharacteristic, complemented
294#2028C2×C3711yescharacteristic, complemented
295#2027C2×C3711yescharacteristic, complemented
296#2007C2×C3711yescharacteristic, complemented
297#2006C2×C3711yescharacteristic, complemented
298#2004C2×C3712yescomplemented
299#2002C2×C3712yescomplemented
300#1806C2×C3712yescomplemented
301#1800C2×C3711yescharacteristic, complemented
302C2×C37—21no—
303C2×C37—21no—
304C2×C37—82no—
305C2×C37—82no—
306C2×C37—161no—
307C2×C37#261511yescharacteristic
308C2×C37#301811yescharacteristic
309C2×C37#301911yescharacteristic
310C2×#137C22×C3712yes—
311C2×AGL(1,ℤ/8)C22×C3712yes—
312#544C22×C3711yescharacteristic
313#542—22no—
314#540—22no—
315#538—22no—
316#537C22×C3711yescharacteristic
317#535C22×C3712yes—
318#532—22no—
319C2×#127C22×C3711yescharacteristic
320#529—22no—
321C2×#124C22×C3711yescharacteristic
322C2×#124C22×C3711yescharacteristic
323#525—22no—
324C4×#41C22×C3712yes—
325C2×C4×D4C22×C3711yescharacteristic
326#464—22no—
327#464—22no—
328#463—22no—
329#463—22no—
330#423C22×C3712yes—
331C2×#104C22×C3712yes—
332C2×#104C22×C3712yes—
333#419C22×C3712yes—
334#418C22×C3712yes—
335C2×#100C22×C3712yes—
336#394—22no—
337#391C22×C3711yescharacteristic
338#390C22×C3711yescharacteristic
339#386—22no—
340C4×C4⋊C4C22×C3711yescharacteristic
341#343C22×C3711yescharacteristic
342#343C22×C3711yescharacteristic
343#339C22×C3712yes—
344#339C22×C3712yes—
345C37#2968211yescharacteristic, direct product factor, Sylow
346C2×#41—22no—
347C2×#41C23×C3712yes—
348C22×Q8C23×C3711yescharacteristic
349C22×D4—22no—
350C22×D4C23×C3711yescharacteristic
351C22×D4C23×C3711yescharacteristic
352C23×C4D4×C3712yes—
353#137—42no—
354#137—42no—
355AGL(1,ℤ/8)—42no—
356AGL(1,ℤ/8)—42no—
357C2×Q16—22no—
358C2×SD16—22no—
359C2×SD16—22no—
360C2×D8—22no—
361C2×M16C23×C3712yes—
362C2×M16C23×C3712yes—
363C4⋊Q8—22no—
364#127—22no—
365#127—42no—
366#126—24no—
367#125—22no—
368#124—24no—
369#124—22no—
370#124—22no—
371#124—22no—
372#124—42no—
373#123—24no—
374#123—24no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378#121—22no—
379#121—24no—
380#121—22no—
381#121—22no—
382#121—22no—
383#121—22no—
384#120—24no—
385C4×Q8D4×C3712yescomplemented
386C4×Q8D4×C3712yescomplemented
387C4×D4—24no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4D4×C3712yescomplemented
393C4×D4D4×C3712yescomplemented
394#117—22no—
395#117—22no—
396#117C23×C3712yes—
397C2×C4⋊C4—21no—
398C2×C4⋊C4C23×C3711yescharacteristic
399C2×C22⋊C4—22no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4D4×C3712yes—
404C2×C42—22no—
405C2×C42C23×C3711yescharacteristic
406C2×C42C23×C3711yescharacteristic
407C2×C42C23×C3711yescharacteristic
408C4⋊C8—22no—
409C4⋊C8—22no—
410#104—42no—
411#104—42no—
412#103—22no—
413#103—22no—
414#102—22no—
415#102—22no—
416#100—42no—
417#95—21no—
418#95—22no—
419C24—21no—
420C24C2×D4×C3712yes—
421#41—22no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—84no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×D4×C3712yes—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—82no—
447C2×D4C2×D4×C3712yes—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—82no—
457C22×C4C2×D4×C3712yes—
458C22×C4C2×D4×C3711yescharacteristic
459C22×C4C2×D4×C3711yescharacteristic
460C22×C4C24×C3711yescharacteristic
461Q16—44no—
462SD16—44no—
463SD16—44no—
464D8—44no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—22no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4C2×D4×C3712yes—
479C4⋊C4C2×D4×C3712yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—44no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—41no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42C2×D4×C3712yes—
498C42C2×D4×C3712yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×D4×C3711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—42no—
514Q8—42no—
515Q8—82no—
516D4—22no—
517D4—22no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—82no—
525D4—84no—
526D4—84no—
527D4—88no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#120×C3711yescharacteristic
551C2×C4#120×C3712yes—
552C2×C4#120×C3711yescharacteristic
553C2×C4C22×D4×C3712yes—
554C2×C4H(5,𝔽2)×C3711yescharacteristic
555C2×C4H(5,𝔽2)×C3712yes—
556C2×C42−1+4×C3711yescharacteristic
557C8—42no—
558C8—42no—
559C22—21no—
560C22—21no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×#120×C3711yescharacteristic
575C22C37×#53511yescharacteristic
576C22C37×#53711yescharacteristic
577C4—22no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4C37×#53512yes—
588C4C37×#53712yes—
589C2—21no—
590C2—21no—
591C2—82no—
592C2—82no—
593C2—161no—
594C2C37×#261511yescharacteristic
595C2C37×#301811yescharacteristic
596C2C37×#301911yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3460C2semidirect
2. C37×#3040C2semidirect
3. C37×#2544C2semidirect
4. C37×#2028C2semidirect
5. C37×#2027C2semidirect
6. C37×#2007C2semidirect
7. C37×#2006C2semidirect
8. C37×#2004C2semidirect
9. C37×#2002C2semidirect
10. C37×#1806C2semidirect
11. C37×#1800C2semidirect
86. C4×Q8×C37D4semidirect
87. C4×Q8×C37D4semidirect
93. C4×D4×C37D4semidirect
94. C4×D4×C37D4semidirect
260. #29682C37direct (N × H)
291. #3460C2×C37semidirect
292. #3040C2×C37semidirect
293. #2544C2×C37semidirect
294. #2028C2×C37semidirect
295. #2027C2×C37semidirect
296. #2007C2×C37semidirect
297. #2006C2×C37semidirect
298. #2004C2×C37semidirect
299. #2002C2×C37semidirect
300. #1806C2×C37semidirect
301. #1800C2×C37semidirect
345. C37#29682direct (N × H)
385. C4×Q8D4×C37semidirect
386. C4×Q8D4×C37semidirect
392. C4×D4D4×C37semidirect
393. C4×D4D4×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34602
C37×#30402
C37×#25442
C37×#20282
C37×#20272
C37×#20072
C37×#20062
C37×#20042
C37×#20022
C37×#18062
C37×#18002
#2968237