Finite Group Databasebeta

← C37×#19031 #22640284

Subgroups — C37×#19031 #22640284

Order 9472.

834 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3563C211yesmaximal, characteristic, complemented
2C37×#3235C211yesmaximal, characteristic, complemented
3C37×#3171C211yesmaximal, characteristic
4C37×#3142C211yesmaximal, characteristic, complemented
5C37×#1664C212yesmaximal, complemented
6C37×#1663C212yesmaximal, complemented
7C37×#1658C212yesmaximal, complemented
8C37×#1657C212yesmaximal, complemented
9C37×#1529C211yesmaximal, characteristic, complemented
10C37×#1528C211yesmaximal, characteristic, complemented
11C37×#1526C211yesmaximal, characteristic, complemented
12C37×#564C2212yes—
13C37×#563C2212yes—
14C37×#557—21no—
15C37×#554—21no—
16C37×#552—21no—
17C37×#552—22no—
18C37×#552—21no—
19C37×#544—21no—
20C37×#544—21no—
21C37×#541—21no—
22C37×#541—22no—
23C37×#539—21no—
24C37×#533C2211yescharacteristic
25C37×#532C2211yescharacteristic
26C37×#529C2211yescharacteristic
27C2×#125×C37C2211yescharacteristic
28C37×#501C2211yescharacteristic
29C37×#501C2212yes—
30C37×#496C2211yescharacteristic
31C37×#491C2212yes—
32C37×#490C2212yes—
33C37×#490C2211yescharacteristic
34C37×#490C2212yes—
35C37×#489C2211yescharacteristic
36C37×#488—22no—
37C37×#482—22no—
38C37×#476—21no—
39C37×#474—21no—
40C37×#469—21no—
41C37×#468—21no—
42C4×SD16×C37—21no—
43C4×D8×C37—21no—
44C37×#432C2211yescharacteristic
45C37×#417—21no—
46C37×#417—21no—
47C4×M16×C37C2211yescharacteristic
48C4×M16×C37C2211yescharacteristic
49C37×#330C2212yes—
50C37×#330C2212yes—
51C37×#330C2212yes—
52C37×#330C2212yes—
53C37×#330C2212yes—
54C37×#329C2212yes—
55C2×#41×C37—21no—
56C2×SD16×C37—21no—
57C2×D8×C37—21no—
58C2×M16×C37—21no—
59C2×M16×C37—21no—
60C4⋊Q8×C37—22no—
61C4⋊Q8×C37—21no—
62C4⋊Q8×C37—22no—
63C4⋊Q8×C37D411yescharacteristic
64C4⋊Q8×C37D411yescharacteristic
65C4⋊Q8×C37C2311yescharacteristic
66#127×C37C2311yescharacteristic
67#126×C37—22no—
68#126×C37—22no—
69#126×C37—22no—
70#126×C37—22no—
71#126×C37—21no—
72#125×C37—22no—
73#125×C37—22no—
74#125×C37—21no—
75#125×C37—22no—
76#125×C37—22no—
77#125×C37—22no—
78#125×C37—22no—
79#125×C37—22no—
80#125×C37—22no—
81#125×C37—21no—
82#125×C37—21no—
83#125×C37D412yes—
84#125×C37C2312yes—
85#125×C37C2311yescharacteristic
86#125×C37C2312yes—
87#124×C37—21no—
88#124×C37C2311yescharacteristic
89#123×C37—22no—
90#123×C37—22no—
91#123×C37—21no—
92C22⋊Q8×C37—22no—
93C22⋊Q8×C37—21no—
94C22⋊Q8×C37—21no—
95C22⋊Q8×C37—22no—
96C22⋊Q8×C37—21no—
97C22⋊Q8×C37—21no—
98C22⋊Q8×C37—22no—
99C22⋊Q8×C37—22no—
100C22⋊Q8×C37D411yescharacteristic
101#121×C37—21no—
102#121×C37—21no—
103#121×C37D411yescharacteristic
104C4×Q8×C37—21no—
105C4×Q8×C37—22no—
106C4×Q8×C37—21no—
107C4×Q8×C37—21no—
108C4×Q8×C37—21no—
109C4×Q8×C37D411yescharacteristic
110C4×D4×C37—22no—
111C4×D4×C37—21no—
112C4×D4×C37—21no—
113C4×D4×C37—21no—
114C4×D4×C37D411yescharacteristic
115#117×C37—22no—
116#117×C37—21no—
117#117×C37—22no—
118#117×C37—21no—
119#117×C37D412yes—
120C2×C4⋊C4×C37—21no—
121C2×C4⋊C4×C37—21no—
122C2×C4⋊C4×C37—21no—
123C2×C4⋊C4×C37D411yescharacteristic
124C2×C4⋊C4×C37D411yescharacteristic
125C2×C42×C37C2311yescharacteristic
126#107×C37—21no—
127#107×C37—22no—
128#106×C37—21no—
129#106×C37—22no—
130C4⋊C8×C37—21no—
131#103×C37—21no—
132#103×C37—22no—
133#103×C37—22no—
134#102×C37—21no—
135#102×C37—22no—
136#102×C37—22no—
137#102×C37—21no—
138#102×C37—21no—
139#102×C37—22no—
140#102×C37—22no—
141C22⋊C8×C37—21no—
142#97×C37C2312yes—
143#97×C37C2312yes—
144#97×C37C2311yescharacteristic
145#97×C37C2311yescharacteristic
146C4×C8×C37—21no—
147C4×C8×C37—21no—
148#41×C37—44no—
149C2×Q8×C37—22no—
150C2×Q8×C37—21no—
151C2×Q8×C37—21no—
152C2×Q8×C37—21no—
153C2×Q8×C37C2×D411yescharacteristic
154C2×D4×C37—21no—
155C2×D4×C37—21no—
156C2×D4×C37—21no—
157C2×D4×C37—21no—
158C2×D4×C37—41no—
159C2×D4×C37C2×D411yescharacteristic
160C22×C4×C37—21no—
161C22×C4×C37—22no—
162C22×C4×C37—21no—
163C22×C4×C37—21no—
164C22×C4×C37—41no—
165C22×C4×C37C2×D411yescharacteristic
166C22×C4×C37C2×D411yescharacteristic
167C22×C4×C37C2×D411yescharacteristic
168SD16×C37—42no—
169D8×C37—42no—
170M16×C37—42no—
171M16×C37—42no—
172C2×C8×C37—21no—
173C2×C8×C37—21no—
174C2×C8×C37—22no—
175C2×C8×C37—22no—
176C4⋊C4×C37—22no—
177C4⋊C4×C37—22no—
178C4⋊C4×C37—22no—
179C4⋊C4×C37—22no—
180C4⋊C4×C37—22no—
181C4⋊C4×C37—22no—
182C4⋊C4×C37—22no—
183C4⋊C4×C37—22no—
184C4⋊C4×C37—21no—
185C4⋊C4×C37—21no—
186C4⋊C4×C37—22no—
187C4⋊C4×C37—22no—
188C4⋊C4×C37—21no—
189C4⋊C4×C37—21no—
190C4⋊C4×C37—21no—
191C4⋊C4×C37—21no—
192C4⋊C4×C37—22no—
193C4⋊C4×C37—22no—
194C4⋊C4×C37—21no—
195C4⋊C4×C37—21no—
196C4⋊C4×C37—22no—
197C4⋊C4×C37—22no—
198C4⋊C4×C37—21no—
199C4⋊C4×C37—21no—
200C4⋊C4×C37C2×D412yes—
201C4⋊C4×C37C2×D412yes—
202C4⋊C4×C37C2×D412yes—
203C4⋊C4×C37C2×D411yescharacteristic
204C4⋊C4×C37C2×D412yes—
205C4⋊C4×C37C2×D411yescharacteristic
206C22⋊C4×C37—22no—
207C22⋊C4×C37—22no—
208C22⋊C4×C37—22no—
209C22⋊C4×C37—21no—
210C22⋊C4×C37—22no—
211C22⋊C4×C37—21no—
212C22⋊C4×C37—21no—
213C22⋊C4×C37—21no—
214C22⋊C4×C37—21no—
215C22⋊C4×C37—41no—
216C22⋊C4×C37—41no—
217C42×C37—22no—
218C42×C37—22no—
219C42×C37—22no—
220C42×C37—22no—
221C42×C37—21no—
222C42×C37C2×D411yescharacteristic
223C42×C37C2×D411yescharacteristic
224C42×C37C2×D411yescharacteristic
225C42×C37C2411yescharacteristic
226C23×C37—21no—
227C23×C37—41no—
228C23×C37#12011yescharacteristic
229Q8×C37—41no—
230Q8×C37—42no—
231Q8×C37—42no—
232Q8×C37—81no—
233Q8×C37—82no—
234D4×C37—41no—
235D4×C37—41no—
236D4×C37—41no—
237D4×C37—42no—
238D4×C37—42no—
239D4×C37—42no—
240D4×C37—82no—
241C2×C4×C37—22no—
242C2×C4×C37—22no—
243C2×C4×C37—21no—
244C2×C4×C37—21no—
245C2×C4×C37—21no—
246C2×C4×C37—22no—
247C2×C4×C37—22no—
248C2×C4×C37—22no—
249C2×C4×C37—41no—
250C2×C4×C37—41no—
251C2×C4×C37—42no—
252C2×C4×C37—42no—
253C2×C4×C37—41no—
254C2×C4×C37—81no—
255C2×C4×C37—82no—
256C2×C4×C37—82no—
257C2×C4×C37#12011yescharacteristic
258C2×C4×C37#12011yescharacteristic
259C2×C4×C37#12011yescharacteristic
260C2×C4×C37C22×D411yescharacteristic
261C2×C4×C37C22×D411yescharacteristic
262C2×C4×C37C22×D411yescharacteristic
263C8×C37—22no—
264C8×C37—42no—
265C8×C37—42no—
266C8×C37—41no—
267#19031C3711yesmaximal, characteristic, direct product factor, Sylow
268C22×C37—21no—
269C22×C37—21no—
270C22×C37—21no—
271C22×C37—41no—
272C22×C37—41no—
273C22×C37—41no—
274C22×C37—81no—
275C22×C37—81no—
276C22×C37—81no—
277C22×C37C2×#12011yescharacteristic
278C4×C37—21no—
279C4×C37—21no—
280C4×C37—21no—
281C4×C37—21no—
282C4×C37—21no—
283C4×C37—41no—
284C4×C37—42no—
285C4×C37—42no—
286C4×C37—42no—
287C4×C37—41no—
288C4×C37—42no—
289C4×C37—41no—
290C4×C37—81no—
291C4×C37#45412yes—
292#3563C2×C3711yescharacteristic, complemented
293#3235C2×C3711yescharacteristic, complemented
294#3171C2×C3711yescharacteristic
295#3142C2×C3711yescharacteristic, complemented
296#1664C2×C3712yescomplemented
297#1663C2×C3712yescomplemented
298#1658C2×C3712yescomplemented
299#1657C2×C3712yescomplemented
300#1529C2×C3711yescharacteristic, complemented
301#1528C2×C3711yescharacteristic, complemented
302#1526C2×C3711yescharacteristic, complemented
303C2×C37—41no—
304C2×C37—81no—
305C2×C37—161no—
306C2×C37C2×#45411yescharacteristic
307C2×C37#301811yescharacteristic
308C2×C37#302011yescharacteristic
309#564C22×C3712yes—
310#563C22×C3712yes—
311#557—21no—
312#554—21no—
313#552—21no—
314#552—22no—
315#552—21no—
316#544—21no—
317#544—21no—
318#541—21no—
319#541—22no—
320#539—21no—
321#533C22×C3711yescharacteristic
322#532C22×C3711yescharacteristic
323#529C22×C3711yescharacteristic
324C2×#125C22×C3711yescharacteristic
325#501C22×C3711yescharacteristic
326#501C22×C3712yes—
327#496C22×C3711yescharacteristic
328#491C22×C3712yes—
329#490C22×C3712yes—
330#490C22×C3711yescharacteristic
331#490C22×C3712yes—
332#489C22×C3711yescharacteristic
333#488—22no—
334#482—22no—
335#476—21no—
336#474—21no—
337#469—21no—
338#468—21no—
339C4×SD16—21no—
340C4×D8—21no—
341#432C22×C3711yescharacteristic
342#417—21no—
343#417—21no—
344C4×M16C22×C3711yescharacteristic
345C4×M16C22×C3711yescharacteristic
346#330C22×C3712yes—
347#330C22×C3712yes—
348#330C22×C3712yes—
349#330C22×C3712yes—
350#330C22×C3712yes—
351#329C22×C3712yes—
352C37#1903111yescharacteristic, direct product factor, Sylow
353C2×#41—21no—
354C2×SD16—21no—
355C2×D8—21no—
356C2×M16—21no—
357C2×M16—21no—
358C4⋊Q8—22no—
359C4⋊Q8—21no—
360C4⋊Q8—22no—
361C4⋊Q8D4×C3711yescharacteristic
362C4⋊Q8D4×C3711yescharacteristic
363C4⋊Q8C23×C3711yescharacteristic
364#127C23×C3711yescharacteristic
365#126—22no—
366#126—22no—
367#126—22no—
368#126—22no—
369#126—21no—
370#125—22no—
371#125—22no—
372#125—21no—
373#125—22no—
374#125—22no—
375#125—22no—
376#125—22no—
377#125—22no—
378#125—22no—
379#125—21no—
380#125—21no—
381#125D4×C3712yes—
382#125C23×C3712yes—
383#125C23×C3711yescharacteristic
384#125C23×C3712yes—
385#124—21no—
386#124C23×C3711yescharacteristic
387#123—22no—
388#123—22no—
389#123—21no—
390C22⋊Q8—22no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8D4×C3711yescharacteristic
399#121—21no—
400#121—21no—
401#121D4×C3711yescharacteristic
402C4×Q8—21no—
403C4×Q8—22no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8D4×C3711yescharacteristic
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4D4×C3711yescharacteristic
413#117—22no—
414#117—21no—
415#117—22no—
416#117—21no—
417#117D4×C3712yes—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4D4×C3711yescharacteristic
422C2×C4⋊C4D4×C3711yescharacteristic
423C2×C42C23×C3711yescharacteristic
424#107—21no—
425#107—22no—
426#106—21no—
427#106—22no—
428C4⋊C8—21no—
429#103—21no—
430#103—22no—
431#103—22no—
432#102—21no—
433#102—22no—
434#102—22no—
435#102—21no—
436#102—21no—
437#102—22no—
438#102—22no—
439C22⋊C8—21no—
440#97C23×C3712yes—
441#97C23×C3712yes—
442#97C23×C3711yescharacteristic
443#97C23×C3711yescharacteristic
444C4×C8—21no—
445C4×C8—21no—
446#41—44no—
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×D4×C3711yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—41no—
457C2×D4C2×D4×C3711yescharacteristic
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4C2×D4×C3711yescharacteristic
464C22×C4C2×D4×C3711yescharacteristic
465C22×C4C2×D4×C3711yescharacteristic
466SD16—42no—
467D8—42no—
468M16—42no—
469M16—42no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4C2×D4×C3712yes—
499C4⋊C4C2×D4×C3712yes—
500C4⋊C4C2×D4×C3712yes—
501C4⋊C4C2×D4×C3711yescharacteristic
502C4⋊C4C2×D4×C3712yes—
503C4⋊C4C2×D4×C3711yescharacteristic
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42C2×D4×C3711yescharacteristic
521C42C2×D4×C3711yescharacteristic
522C42C2×D4×C3711yescharacteristic
523C42C24×C3711yescharacteristic
524C23—21no—
525C23—41no—
526C23#120×C3711yescharacteristic
527Q8—41no—
528Q8—42no—
529Q8—42no—
530Q8—81no—
531Q8—82no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#120×C3711yescharacteristic
556C2×C4#120×C3711yescharacteristic
557C2×C4#120×C3711yescharacteristic
558C2×C4C22×D4×C3711yescharacteristic
559C2×C4C22×D4×C3711yescharacteristic
560C2×C4C22×D4×C3711yescharacteristic
561C8—22no—
562C8—42no—
563C8—41no—
564C8—42no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×#120×C3711yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—81no—
588C4C37×#45412yes—
589C2—41no—
590C2—81no—
591C2—161no—
592C2C2×C37×#45411yescharacteristic
593C2C37×#301811yescharacteristic
594C2C37×#302011yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#3171C2
LCSG ▷ C42 ▷ 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×#3563C2semidirect
2. C37×#3235C2semidirect
4. C37×#3142C2semidirect
5. C37×#1664C2semidirect
6. C37×#1663C2semidirect
7. C37×#1658C2semidirect
8. C37×#1657C2semidirect
9. C37×#1529C2semidirect
10. C37×#1528C2semidirect
11. C37×#1526C2semidirect
267. #19031C37direct (N × H)
292. #3563C2×C37semidirect
293. #3235C2×C37semidirect
295. #3142C2×C37semidirect
296. #1664C2×C37semidirect
297. #1663C2×C37semidirect
298. #1658C2×C37semidirect
299. #1657C2×C37semidirect
300. #1529C2×C37semidirect
301. #1528C2×C37semidirect
302. #1526C2×C37semidirect
352. C37#19031direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35632
C37×#32352
C37×#31712
C37×#31422
C37×#16642
C37×#16632
C37×#16582
C37×#16572
C37×#15292
C37×#15282
C37×#15262
#1903137