Finite Group Databasebeta

← C37×#18136 #22639389

Subgroups — C37×#18136 #22639389

Order 9472.

1196 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×#3459C211yesmaximal, characteristic
2C37×#3004C212yesmaximal, complemented
3C37×#2409C211yesmaximal, characteristic, complemented
4C37×#1603C214yesmaximal, complemented
5C37×#1602C212yesmaximal, complemented
6C37×#1601C212yesmaximal, complemented
7C37×#1513C212yesmaximal, complemented
8C37×#1465C211yesmaximal, characteristic, complemented
9C23×Q8×C37C2211yescharacteristic, complemented
10C2×#137×C37—22no—
11C37×#543C2211yescharacteristic
12C37×#539—22no—
13C37×#539C2212yes—
14C37×#538—22no—
15C37×#538—21no—
16C37×#534—21no—
17C37×#534C2211yescharacteristic
18C37×#529—22no—
19C37×#529—21no—
20C37×#525—22no—
21C37×#525C2212yes—
22C2×C22⋊Q8×C37—21no—
23C2×C4×D4×C37—21no—
24C2×C4×D4×C37C2211yescharacteristic
25C37×#451—22no—
26C37×#451C2214yescomplemented
27C37×#448—22no—
28C37×#448C2214yes—
29C37×#419—22no—
30C37×#407C2212yescomplemented
31C37×#395C2212yescomplemented
32C37×#393C2212yes—
33C37×#391—21no—
34C37×#386—21no—
35C37×#385—21no—
36C37×#380C2211yescharacteristic, complemented
37C37×#328C2214yescomplemented
38C37×#327C2214yes—
39C37×#324C2212yescomplemented
40C37×#323C2212yes—
41C2×#41×C37D412yes—
42C22×Q8×C37—21no—
43C22×Q8×C37—22no—
44C22×Q8×C37—22no—
45C22×Q8×C37—44no—
46C22×Q8×C37C2312yes—
47C22×D4×C37—21no—
48C22×D4×C37D411yescharacteristic
49C23×C4×C37—21no—
50C23×C4×C37—21no—
51C23×C4×C37D411yescharacteristic
52C23×C4×C37C2311yescharacteristic
53#137×C37—48no—
54C2×Q16×C37—24no—
55C2×SD16×C37—24no—
56C2×M16×C37—22no—
57#126×C37—22no—
58#126×C37—22no—
59#126×C37—22no—
60#125×C37—21no—
61#125×C37—22no—
62#124×C37—22no—
63#124×C37—21no—
64#123×C37—22no—
65#123×C37—22no—
66#123×C37—22no—
67#123×C37—22no—
68#123×C37—22no—
69#123×C37—22no—
70C22⋊Q8×C37—22no—
71C22⋊Q8×C37—22no—
72C22⋊Q8×C37—21no—
73C22⋊Q8×C37—22no—
74C22⋊Q8×C37—44no—
75C22⋊Q8×C37D412yes—
76C22⋊Q8×C37C2312yes—
77#121×C37—22no—
78#121×C37—22no—
79#121×C37—22no—
80#121×C37—21no—
81#121×C37D412yes—
82#121×C37C2312yes—
83#120×C37—22no—
84#120×C37—22no—
85#120×C37—22no—
86C4×Q8×C37—21no—
87C4×D4×C37—21no—
88C4×D4×C37—22no—
89C4×D4×C37—22no—
90C4×D4×C37—22no—
91C4×D4×C37—22no—
92C4×D4×C37—22no—
93C4×D4×C37—22no—
94C4×D4×C37—22no—
95#117×C37—22no—
96#117×C37D412yes—
97C2×C4⋊C4×C37—21no—
98C2×C4⋊C4×C37—21no—
99C2×C4⋊C4×C37—22no—
100C2×C4⋊C4×C37C2311yescharacteristic
101C2×C22⋊C4×C37—21no—
102C2×C22⋊C4×C37—21no—
103C2×C22⋊C4×C37—22no—
104C2×C22⋊C4×C37—21no—
105C2×C22⋊C4×C37D411yescharacteristic
106C2×C22⋊C4×C37C2311yescharacteristic
107C2×C42×C37—21no—
108C2×C42×C37D411yescharacteristic
109#103×C37—24no—
110#103×C37—24no—
111C22⋊C8×C37—22no—
112C22⋊C8×C37C2314yes—
113#95×C37—22no—
114#95×C37—22no—
115#95×C37C2312yes—
116C24×C37—21no—
117C24×C37C2×D411yescharacteristic
118C24×C37C2×D411yescharacteristic
119#41×C37—44no—
120C2×Q8×C37—24no—
121C2×Q8×C37—21no—
122C2×Q8×C37—28no—
123C2×Q8×C37—24no—
124C2×Q8×C37—44no—
125C2×Q8×C37—42no—
126C2×Q8×C37—44no—
127C2×Q8×C37—48no—
128C2×Q8×C37—41no—
129C2×Q8×C37—41no—
130C2×Q8×C37C2×D412yes—
131C2×D4×C37—22no—
132C2×D4×C37—22no—
133C2×D4×C37—22no—
134C2×D4×C37—22no—
135C2×D4×C37—21no—
136C2×D4×C37—22no—
137C2×D4×C37—22no—
138C2×D4×C37—41no—
139C2×D4×C37—42no—
140C2×D4×C37—42no—
141C2×D4×C37C2×D412yes—
142C22×C4×C37—21no—
143C22×C4×C37—22no—
144C22×C4×C37—21no—
145C22×C4×C37—21no—
146C22×C4×C37—22no—
147C22×C4×C37—21no—
148C22×C4×C37—22no—
149C22×C4×C37—21no—
150C22×C4×C37—22no—
151C22×C4×C37—42no—
152C22×C4×C37—44no—
153C22×C4×C37—42no—
154C22×C4×C37—42no—
155C22×C4×C37—82no—
156C22×C4×C37C2×D411yescharacteristic
157C22×C4×C37C2×D411yescharacteristic
158C22×C4×C37C2×D412yes—
159C22×C4×C37C2×D411yescharacteristic
160C22×C4×C37C2×D411yescharacteristic
161C22×C4×C37C2×D412yes—
162C22×C4×C37C2411yescharacteristic
163Q16×C37—48no—
164SD16×C37—48no—
165M16×C37—44no—
166C2×C8×C37—24no—
167C4⋊C4×C37—22no—
168C4⋊C4×C37—22no—
169C4⋊C4×C37—22no—
170C4⋊C4×C37—21no—
171C4⋊C4×C37—21no—
172C4⋊C4×C37—22no—
173C4⋊C4×C37—22no—
174C4⋊C4×C37—22no—
175C4⋊C4×C37—22no—
176C4⋊C4×C37—42no—
177C4⋊C4×C37—44no—
178C4⋊C4×C37C2×D412yes—
179C4⋊C4×C37C2×D412yes—
180C22⋊C4×C37—22no—
181C22⋊C4×C37—22no—
182C22⋊C4×C37—22no—
183C22⋊C4×C37—22no—
184C22⋊C4×C37—22no—
185C22⋊C4×C37—22no—
186C22⋊C4×C37—22no—
187C22⋊C4×C37—22no—
188C22⋊C4×C37—22no—
189C22⋊C4×C37—22no—
190C22⋊C4×C37—22no—
191C22⋊C4×C37—22no—
192C22⋊C4×C37—44no—
193C22⋊C4×C37—42no—
194C42×C37—22no—
195C42×C37—22no—
196C42×C37—21no—
197C42×C37—21no—
198C23×C37—21no—
199C23×C37—21no—
200C23×C37—21no—
201C23×C37—21no—
202C23×C37—22no—
203C23×C37—41no—
204C23×C37—41no—
205C23×C37—41no—
206C23×C37—41no—
207C23×C37—42no—
208C23×C37—81no—
209C23×C37—81no—
210C23×C37#12012yes—
211C23×C37#13712yescomplemented
212C23×C37C22×D411yescharacteristic
213Q8×C37—28no—
214Q8×C37—42no—
215Q8×C37—44no—
216Q8×C37—44no—
217Q8×C37—44no—
218Q8×C37—81no—
219D4×C37—42no—
220D4×C37—44no—
221D4×C37—44no—
222D4×C37—42no—
223D4×C37—42no—
224D4×C37—82no—
225D4×C37—82no—
226D4×C37—81no—
227C2×C4×C37—22no—
228C2×C4×C37—24no—
229C2×C4×C37—24no—
230C2×C4×C37—21no—
231C2×C4×C37—22no—
232C2×C4×C37—21no—
233C2×C4×C37—22no—
234C2×C4×C37—21no—
235C2×C4×C37—22no—
236C2×C4×C37—21no—
237C2×C4×C37—21no—
238C2×C4×C37—42no—
239C2×C4×C37—41no—
240C2×C4×C37—42no—
241C2×C4×C37—42no—
242C2×C4×C37—48no—
243C2×C4×C37—42no—
244C2×C4×C37—42no—
245C2×C4×C37—44no—
246C2×C4×C37—41no—
247C2×C4×C37—41no—
248C2×C4×C37—42no—
249C2×C4×C37—42no—
250C2×C4×C37—42no—
251C2×C4×C37—81no—
252C2×C4×C37—81no—
253C2×C4×C37—82no—
254C2×C4×C37—81no—
255C2×C4×C37#12011yescharacteristic
256C2×C4×C37#12011yescharacteristic
257C2×C4×C37C22×D412yes—
258C8×C37—44no—
259#18136C3711yesmaximal, characteristic, direct product factor, Sylow
260C22×C37—21no—
261C22×C37—21no—
262C22×C37—22no—
263C22×C37—22no—
264C22×C37—22no—
265C22×C37—22no—
266C22×C37—42no—
267C22×C37—42no—
268C22×C37—42no—
269C22×C37—41no—
270C22×C37—41no—
271C22×C37—42no—
272C22×C37—42no—
273C22×C37—42no—
274C22×C37—81no—
275C22×C37—84no—
276C22×C37C2×#12011yescharacteristic
277C22×C37C2×#13711yescharacteristic
278C22×C37#57511yescharacteristic
279C4×C37—22no—
280C4×C37—21no—
281C4×C37—41no—
282C4×C37—42no—
283C4×C37—41no—
284C4×C37—42no—
285C4×C37—44no—
286C4×C37—41no—
287C4×C37—42no—
288C4×C37—82no—
289C4×C37B(4,𝔽2)12yes—
290#3459C2×C3711yescharacteristic
291#3004C2×C3712yescomplemented
292#2409C2×C3711yescharacteristic, complemented
293#1603C2×C3714yescomplemented
294#1602C2×C3712yescomplemented
295#1601C2×C3712yescomplemented
296#1513C2×C3712yescomplemented
297#1465C2×C3711yescharacteristic, complemented
298C2×C37—21no—
299C2×C37—21no—
300C2×C37—42no—
301C2×C37—42no—
302C2×C37—82no—
303C2×C37#300711yescharacteristic
304C2×C37#302211yescharacteristic
305C2×C37C2×B(4,𝔽2)11yescharacteristic
306C23×Q8C22×C3711yescharacteristic, complemented
307C2×#137—22no—
308#543C22×C3711yescharacteristic
309#539—22no—
310#539C22×C3712yes—
311#538—22no—
312#538—21no—
313#534—21no—
314#534C22×C3711yescharacteristic
315#529—22no—
316#529—21no—
317#525—22no—
318#525C22×C3712yes—
319C2×C22⋊Q8—21no—
320C2×C4×D4—21no—
321C2×C4×D4C22×C3711yescharacteristic
322#451—22no—
323#451C22×C3714yescomplemented
324#448—22no—
325#448C22×C3714yes—
326#419—22no—
327#407C22×C3712yescomplemented
328#395C22×C3712yescomplemented
329#393C22×C3712yes—
330#391—21no—
331#386—21no—
332#385—21no—
333#380C22×C3711yescharacteristic, complemented
334#328C22×C3714yescomplemented
335#327C22×C3714yes—
336#324C22×C3712yescomplemented
337#323C22×C3712yes—
338C37#1813611yescharacteristic, direct product factor, Sylow
339C2×#41D4×C3712yes—
340C22×Q8—21no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—44no—
344C22×Q8C23×C3712yes—
345C22×D4—21no—
346C22×D4D4×C3711yescharacteristic
347C23×C4—21no—
348C23×C4—21no—
349C23×C4D4×C3711yescharacteristic
350C23×C4C23×C3711yescharacteristic
351#137—48no—
352C2×Q16—24no—
353C2×SD16—24no—
354C2×M16—22no—
355#126—22no—
356#126—22no—
357#126—22no—
358#125—21no—
359#125—22no—
360#124—22no—
361#124—21no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—44no—
373C22⋊Q8D4×C3712yes—
374C22⋊Q8C23×C3712yes—
375#121—22no—
376#121—22no—
377#121—22no—
378#121—21no—
379#121D4×C3712yes—
380#121C23×C3712yes—
381#120—22no—
382#120—22no—
383#120—22no—
384C4×Q8—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—21no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393#117—22no—
394#117D4×C3712yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4C23×C3711yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4D4×C3711yescharacteristic
404C2×C22⋊C4C23×C3711yescharacteristic
405C2×C42—21no—
406C2×C42D4×C3711yescharacteristic
407#103—24no—
408#103—24no—
409C22⋊C8—22no—
410C22⋊C8C23×C3714yes—
411#95—22no—
412#95—22no—
413#95C23×C3712yes—
414C24—21no—
415C24C2×D4×C3711yescharacteristic
416C24C2×D4×C3711yescharacteristic
417#41—44no—
418C2×Q8—24no—
419C2×Q8—21no—
420C2×Q8—28no—
421C2×Q8—24no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—48no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8C2×D4×C3712yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—41no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×D4×C3712yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4C2×D4×C3711yescharacteristic
455C22×C4C2×D4×C3711yescharacteristic
456C22×C4C2×D4×C3712yes—
457C22×C4C2×D4×C3711yescharacteristic
458C22×C4C2×D4×C3711yescharacteristic
459C22×C4C2×D4×C3712yes—
460C22×C4C24×C3711yescharacteristic
461Q16—48no—
462SD16—48no—
463M16—44no—
464C2×C8—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4C2×D4×C3712yes—
477C4⋊C4C2×D4×C3712yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—44no—
491C22⋊C4—42no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—81no—
507C23—81no—
508C23#120×C3712yes—
509C23#137×C3712yescomplemented
510C23C22×D4×C3711yescharacteristic
511Q8—28no—
512Q8—42no—
513Q8—44no—
514Q8—44no—
515Q8—44no—
516Q8—81no—
517D4—42no—
518D4—44no—
519D4—44no—
520D4—42no—
521D4—42no—
522D4—82no—
523D4—82no—
524D4—81no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4#120×C3711yescharacteristic
554C2×C4#120×C3711yescharacteristic
555C2×C4C22×D4×C3712yes—
556C8—44no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—84no—
573C22C2×#120×C3711yescharacteristic
574C22C2×#137×C3711yescharacteristic
575C22C37×#57511yescharacteristic
576C4—22no—
577C4—21no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—44no—
583C4—41no—
584C4—42no—
585C4—82no—
586C4C37×B(4,𝔽2)12yes—
587C2—21no—
588C2—21no—
589C2—42no—
590C2—42no—
591C2—82no—
592C2C37×#300711yescharacteristic
593C2C37×#302211yescharacteristic
594C2C2×C37×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#3459C2
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
2. C37×#3004C2semidirect
3. C37×#2409C2semidirect
4. C37×#1603C2semidirect
5. C37×#1602C2semidirect
6. C37×#1601C2semidirect
7. C37×#1513C2semidirect
8. C37×#1465C2semidirect
9. C23×Q8×C37C22semidirect
26. C37×#451C22semidirect
30. C37×#407C22semidirect
31. C37×#395C22semidirect
36. C37×#380C22semidirect
37. C37×#328C22semidirect
39. C37×#324C22semidirect
211. C23×C37#137semidirect
259. #18136C37direct (N × H)
291. #3004C2×C37semidirect
292. #2409C2×C37semidirect
293. #1603C2×C37semidirect
294. #1602C2×C37semidirect
295. #1601C2×C37semidirect
296. #1513C2×C37semidirect
297. #1465C2×C37semidirect
306. C23×Q8C22×C37semidirect
323. #451C22×C37semidirect
327. #407C22×C37semidirect
328. #395C22×C37semidirect
333. #380C22×C37semidirect
334. #328C22×C37semidirect
336. #324C22×C37semidirect
338. C37#18136direct (N × H)
509. C23#137×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34592
C37×#30042
C37×#24092
C37×#16032
C37×#16022
C37×#16012
C37×#15132
C37×#14652
#1813637