Finite Group Databasebeta

← C37×#25519 #22646772

Subgroups — C37×#25519 #22646772

Order 9472.

1092 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3492C211yesmaximal, characteristic
2C37×#3056C212yesmaximal, complemented
3C37×#2502C211yesmaximal, characteristic, complemented
4C37×#2060C211yesmaximal, characteristic, complemented
5C37×#2039C211yesmaximal, characteristic, complemented
6C37×#2028C211yesmaximal, characteristic, complemented
7C37×#2007C211yesmaximal, characteristic, complemented
8C37×#1882C212yesmaximal, complemented
9C37×#1871C212yesmaximal, complemented
10C37×#1859C212yesmaximal, complemented
11C2×C37×#343C211yesmaximal, characteristic, complemented
12C2×#137×C37C2212yes—
13C22×M16×C37C2212yes—
14C37×#554—22no—
15C37×#553—22no—
16C37×#552—22no—
17C37×#547—22no—
18C37×#544C2211yescharacteristic
19C37×#539—22no—
20C37×#537C2211yescharacteristic
21C37×#525—22no—
22C37×#525C2212yes—
23C2×C22⋊Q8×C37C2211yescharacteristic
24C37×#520C2212yes—
25C22×C4⋊C4×C37C2211yescharacteristic
26C37×#470—22no—
27C37×#468—22no—
28C37×#467—22no—
29C37×#465—22no—
30C37×#428C2212yes—
31C2×#104×C37C2212yes—
32C2×#104×C37C2212yes—
33C37×#419C2212yes—
34C37×#417C2212yes—
35C37×#398—22no—
36C37×#391C2211yescharacteristic
37C37×#391C2211yescharacteristic
38C37×#385—22no—
39C37×#384C2211yescharacteristic
40C37×#384C2211yescharacteristic
41C4×C22⋊C4×C37C2211yescharacteristic
42C37×#343C2212yes—
43C37×#343C2211yescharacteristic
44C37×#343C2211yescharacteristic
45C37×#339C2212yes—
46C37×#339C2212yes—
47C2×#41×C37—22no—
48C2×#41×C37C2312yes—
49C22×Q8×C37C2311yescharacteristic
50C23×C4×C37D412yes—
51C23×C4×C37C2311yescharacteristic
52#137×C37—24no—
53#137×C37—42no—
54C2×Q16×C37—22no—
55C2×SD16×C37—22no—
56C2×M16×C37—22no—
57C2×M16×C37—24no—
58C2×M16×C37—42no—
59C2×M16×C37C2312yes—
60C2×M16×C37C2312yes—
61C22×C8×C37—22no—
62C4⋊Q8×C37—22no—
63#126×C37—24no—
64#125×C37—24no—
65#125×C37—22no—
66#123×C37—24no—
67#123×C37—24no—
68#123×C37—22no—
69C22⋊Q8×C37—22no—
70C22⋊Q8×C37—24no—
71C22⋊Q8×C37—22no—
72C22⋊Q8×C37—22no—
73C22⋊Q8×C37—22no—
74C22⋊Q8×C37—22no—
75C22⋊Q8×C37D412yes—
76C22⋊Q8×C37D412yes—
77#121×C37—22no—
78#121×C37—22no—
79#121×C37—22no—
80#121×C37D412yes—
81#121×C37D412yes—
82#120×C37—22no—
83C4×Q8×C37—22no—
84C4×D4×C37—24no—
85C4×D4×C37—22no—
86C4×D4×C37—24no—
87C4×D4×C37—22no—
88C4×D4×C37—22no—
89#117×C37—22no—
90#117×C37—22no—
91#117×C37C2312yes—
92C2×C4⋊C4×C37—22no—
93C2×C4⋊C4×C37—22no—
94C2×C4⋊C4×C37—22no—
95C2×C4⋊C4×C37—21no—
96C2×C4⋊C4×C37—22no—
97C2×C4⋊C4×C37—22no—
98C2×C4⋊C4×C37D412yes—
99C2×C4⋊C4×C37C2311yescharacteristic
100C2×C4⋊C4×C37C2311yescharacteristic
101C2×C4⋊C4×C37C2311yescharacteristic
102C2×C22⋊C4×C37—21no—
103C2×C22⋊C4×C37—21no—
104C2×C42×C37C2311yescharacteristic
105C2×C42×C37C2311yescharacteristic
106#107×C37—22no—
107#106×C37—22no—
108#104×C37—24no—
109#104×C37—42no—
110#103×C37—22no—
111#103×C37—22no—
112#103×C37—22no—
113#102×C37—22no—
114#95×C37—21no—
115#95×C37—22no—
116#95×C37—22no—
117C24×C37C2×D411yescharacteristic
118#41×C37—22no—
119#41×C37—22no—
120#41×C37—22no—
121#41×C37—22no—
122#41×C37—84no—
123C2×Q8×C37—22no—
124C2×Q8×C37—22no—
125C2×Q8×C37—22no—
126C2×Q8×C37—21no—
127C2×Q8×C37—21no—
128C2×Q8×C37C2×D412yes—
129C2×D4×C37—22no—
130C2×D4×C37—22no—
131C2×D4×C37—22no—
132C2×D4×C37—22no—
133C2×D4×C37C2×D412yes—
134C22×C4×C37—22no—
135C22×C4×C37—21no—
136C22×C4×C37—21no—
137C22×C4×C37—22no—
138C22×C4×C37—21no—
139C22×C4×C37—21no—
140C22×C4×C37—24no—
141C22×C4×C37—22no—
142C22×C4×C37—22no—
143C22×C4×C37—21no—
144C22×C4×C37—21no—
145C22×C4×C37—82no—
146C22×C4×C37C2×D411yescharacteristic
147C22×C4×C37C2×D412yes—
148C22×C4×C37C2×D412yes—
149C22×C4×C37C2×D412yes—
150C22×C4×C37C2×D412yes—
151C22×C4×C37C2411yescharacteristic
152Q16×C37—44no—
153SD16×C37—44no—
154M16×C37—22no—
155M16×C37—22no—
156M16×C37—22no—
157M16×C37—44no—
158M16×C37#4114yescomplemented
159C2×C8×C37—22no—
160C2×C8×C37—22no—
161C2×C8×C37—44no—
162C4⋊C4×C37—24no—
163C4⋊C4×C37—22no—
164C4⋊C4×C37—22no—
165C4⋊C4×C37—24no—
166C4⋊C4×C37—24no—
167C4⋊C4×C37—22no—
168C4⋊C4×C37—22no—
169C4⋊C4×C37—22no—
170C4⋊C4×C37—22no—
171C4⋊C4×C37—22no—
172C4⋊C4×C37—42no—
173C4⋊C4×C37—82no—
174C4⋊C4×C37—82no—
175C4⋊C4×C37C2×D412yes—
176C4⋊C4×C37C2×D412yes—
177C22⋊C4×C37—24no—
178C22⋊C4×C37—24no—
179C22⋊C4×C37—24no—
180C22⋊C4×C37—22no—
181C22⋊C4×C37—22no—
182C22⋊C4×C37—22no—
183C22⋊C4×C37—22no—
184C22⋊C4×C37—22no—
185C22⋊C4×C37—41no—
186C22⋊C4×C37—41no—
187C42×C37—24no—
188C42×C37—22no—
189C42×C37—22no—
190C42×C37—21no—
191C42×C37—21no—
192C23×C37—21no—
193C23×C37—21no—
194C23×C37—22no—
195C23×C37—22no—
196C23×C37—41no—
197C23×C37#12011yescharacteristic
198C23×C37#12011yescharacteristic
199C23×C37C22×D411yescharacteristic
200Q8×C37—22no—
201Q8×C37—22no—
202Q8×C37—42no—
203Q8×C37—42no—
204Q8×C37—82no—
205D4×C37—22no—
206D4×C37—22no—
207D4×C37—44no—
208D4×C37—42no—
209D4×C37—42no—
210D4×C37—44no—
211D4×C37—82no—
212C2×C4×C37—21no—
213C2×C4×C37—21no—
214C2×C4×C37—22no—
215C2×C4×C37—22no—
216C2×C4×C37—22no—
217C2×C4×C37—22no—
218C2×C4×C37—22no—
219C2×C4×C37—22no—
220C2×C4×C37—22no—
221C2×C4×C37—22no—
222C2×C4×C37—22no—
223C2×C4×C37—22no—
224C2×C4×C37—22no—
225C2×C4×C37—41no—
226C2×C4×C37—41no—
227C2×C4×C37—44no—
228C2×C4×C37—41no—
229C2×C4×C37—41no—
230C2×C4×C37—42no—
231C2×C4×C37—42no—
232C2×C4×C37—42no—
233C2×C4×C37—42no—
234C2×C4×C37—42no—
235C2×C4×C37—41no—
236C2×C4×C37—41no—
237C2×C4×C37—41no—
238C2×C4×C37—41no—
239C2×C4×C37—82no—
240C2×C4×C37—84no—
241C2×C4×C37#12012yes—
242C2×C4×C37C22×D412yes—
243C2×C4×C37C2×#4112yes—
244C2×C4×C37H(5,𝔽2)11yescharacteristic
245C2×C4×C372−1+411yescharacteristic
246C8×C37—24no—
247C8×C37—42no—
248#25519C3711yesmaximal, characteristic, direct product factor, Sylow
249C22×C37—21no—
250C22×C37—21no—
251C22×C37—21no—
252C22×C37—21no—
253C22×C37—21no—
254C22×C37—21no—
255C22×C37—21no—
256C22×C37—42no—
257C22×C37—42no—
258C22×C37—42no—
259C22×C37—42no—
260C22×C37—42no—
261C22×C37#45612yes—
262C22×C37C2×#12011yescharacteristic
263C22×C37#53211yescharacteristic
264C22×C37#53711yescharacteristic
265C4×C37—22no—
266C4×C37—42no—
267C4×C37—42no—
268C4×C37—42no—
269C4×C37—42no—
270C4×C37—42no—
271C4×C37—42no—
272C4×C37—42no—
273C4×C37—81no—
274C4×C37—81no—
275C4×C37#54612yes—
276C4×C37#54712yes—
277#3492C2×C3711yescharacteristic
278#3056C2×C3712yescomplemented
279#2502C2×C3711yescharacteristic, complemented
280#2060C2×C3711yescharacteristic, complemented
281#2039C2×C3711yescharacteristic, complemented
282#2028C2×C3711yescharacteristic, complemented
283#2007C2×C3711yescharacteristic, complemented
284#1882C2×C3712yescomplemented
285#1871C2×C3712yescomplemented
286#1859C2×C3712yescomplemented
287C2×#343C2×C3711yescharacteristic, complemented
288C2×C37—21no—
289C2×C37—21no—
290C2×C37—22no—
291C2×C37—41no—
292C2×C37—82no—
293C2×C37#243011yescharacteristic
294C2×C37C2×#45611yescharacteristic
295C2×C37#301911yescharacteristic
296C2×#137C22×C3712yes—
297C22×M16C22×C3712yes—
298#554—22no—
299#553—22no—
300#552—22no—
301#547—22no—
302#544C22×C3711yescharacteristic
303#539—22no—
304#537C22×C3711yescharacteristic
305#525—22no—
306#525C22×C3712yes—
307C2×C22⋊Q8C22×C3711yescharacteristic
308#520C22×C3712yes—
309C22×C4⋊C4C22×C3711yescharacteristic
310#470—22no—
311#468—22no—
312#467—22no—
313#465—22no—
314#428C22×C3712yes—
315C2×#104C22×C3712yes—
316C2×#104C22×C3712yes—
317#419C22×C3712yes—
318#417C22×C3712yes—
319#398—22no—
320#391C22×C3711yescharacteristic
321#391C22×C3711yescharacteristic
322#385—22no—
323#384C22×C3711yescharacteristic
324#384C22×C3711yescharacteristic
325C4×C22⋊C4C22×C3711yescharacteristic
326#343C22×C3711yescharacteristic
327#343C22×C3712yes—
328#343C22×C3711yescharacteristic
329#339C22×C3712yes—
330#339C22×C3712yes—
331C37#2551911yescharacteristic, direct product factor, Sylow
332C2×#41—22no—
333C2×#41C23×C3712yes—
334C22×Q8C23×C3711yescharacteristic
335C23×C4D4×C3712yes—
336C23×C4C23×C3711yescharacteristic
337#137—24no—
338#137—42no—
339C2×Q16—22no—
340C2×SD16—22no—
341C2×M16—22no—
342C2×M16—24no—
343C2×M16—42no—
344C2×M16C23×C3712yes—
345C2×M16C23×C3712yes—
346C22×C8—22no—
347C4⋊Q8—22no—
348#126—24no—
349#125—24no—
350#125—22no—
351#123—24no—
352#123—24no—
353#123—22no—
354C22⋊Q8—22no—
355C22⋊Q8—24no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8D4×C3712yes—
361C22⋊Q8D4×C3712yes—
362#121—22no—
363#121—22no—
364#121—22no—
365#121D4×C3712yes—
366#121D4×C3712yes—
367#120—22no—
368C4×Q8—22no—
369C4×D4—24no—
370C4×D4—22no—
371C4×D4—24no—
372C4×D4—22no—
373C4×D4—22no—
374#117—22no—
375#117—22no—
376#117C23×C3712yes—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4D4×C3712yes—
384C2×C4⋊C4C23×C3711yescharacteristic
385C2×C4⋊C4C23×C3711yescharacteristic
386C2×C4⋊C4C23×C3711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C42C23×C3711yescharacteristic
390C2×C42C23×C3711yescharacteristic
391#107—22no—
392#106—22no—
393#104—24no—
394#104—42no—
395#103—22no—
396#103—22no—
397#103—22no—
398#102—22no—
399#95—21no—
400#95—22no—
401#95—22no—
402C24C2×D4×C3711yescharacteristic
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—84no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8C2×D4×C3712yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4C2×D4×C3712yes—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—82no—
431C22×C4C2×D4×C3711yescharacteristic
432C22×C4C2×D4×C3712yes—
433C22×C4C2×D4×C3712yes—
434C22×C4C2×D4×C3712yes—
435C22×C4C2×D4×C3712yes—
436C22×C4C24×C3711yescharacteristic
437Q16—44no—
438SD16—44no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—44no—
443M16#41×C3714yescomplemented
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—44no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C2×D4×C3712yes—
461C4⋊C4C2×D4×C3712yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23#120×C3711yescharacteristic
483C23#120×C3711yescharacteristic
484C23C22×D4×C3711yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—42no—
488Q8—42no—
489Q8—82no—
490D4—22no—
491D4—22no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—44no—
496D4—82no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4#120×C3712yes—
527C2×C4C22×D4×C3712yes—
528C2×C4C2×#41×C3712yes—
529C2×C4H(5,𝔽2)×C3711yescharacteristic
530C2×C42−1+4×C3711yescharacteristic
531C8—24no—
532C8—42no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22C37×#45612yes—
546C22C2×#120×C3711yescharacteristic
547C22C37×#53211yescharacteristic
548C22C37×#53711yescharacteristic
549C4—22no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4C37×#54612yes—
560C4C37×#54712yes—
561C2—21no—
562C2—21no—
563C2—22no—
564C2—41no—
565C2—82no—
566C2C37×#243011yescharacteristic
567C2C2×C37×#45611yescharacteristic
568C2C37×#301911yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3056C2semidirect
3. C37×#2502C2semidirect
4. C37×#2060C2semidirect
5. C37×#2039C2semidirect
6. C37×#2028C2semidirect
7. C37×#2007C2semidirect
8. C37×#1882C2semidirect
9. C37×#1871C2semidirect
10. C37×#1859C2semidirect
11. C2×C37×#343C2semidirect
158. M16×C37#41semidirect
248. #25519C37direct (N × H)
278. #3056C2×C37semidirect
279. #2502C2×C37semidirect
280. #2060C2×C37semidirect
281. #2039C2×C37semidirect
282. #2028C2×C37semidirect
283. #2007C2×C37semidirect
284. #1882C2×C37semidirect
285. #1871C2×C37semidirect
286. #1859C2×C37semidirect
287. C2×#343C2×C37semidirect
331. C37#25519direct (N × H)
443. M16#41×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34922
C37×#30562
C37×#25022
C37×#20602
C37×#20392
C37×#20282
C37×#20072
C37×#18822
C37×#18712
C37×#18592
C2×C37×#3432
#2551937