Finite Group Databasebeta

← C37×#18865 #22640118

Subgroups — C37×#18865 #22640118

Order 9472.

922 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3506C211yesmaximal, characteristic
2C37×#3254C211yesmaximal, characteristic, complemented
3C37×#3231C211yesmaximal, characteristic, complemented
4C37×#3110C211yesmaximal, characteristic, complemented
5C37×#1654C212yesmaximal, complemented
6C37×#1647C212yesmaximal, complemented
7C37×#1646C212yesmaximal, complemented
8C37×#1646C212yesmaximal, complemented
9C37×#1559C211yesmaximal, characteristic, complemented
10C37×#1496C211yesmaximal, characteristic, complemented
11C37×#1486C211yesmaximal, characteristic, complemented
12C37×#559—22no—
13C37×#556—21no—
14C37×#555C2212yes—
15C37×#553—22no—
16C37×#550—21no—
17C37×#547C2212yes—
18C37×#538—22no—
19C37×#535—21no—
20C37×#533C2211yescharacteristic
21C37×#533C2211yescharacteristic
22C37×#532C2211yescharacteristic
23C37×#525—22no—
24C37×#520—21no—
25C2×C4×D4×C37C2211yescharacteristic
26C37×#500C2211yescharacteristic, complemented
27C37×#498C2211yescharacteristic, complemented
28C37×#487C2211yescharacteristic, complemented
29C37×#487C2211yescharacteristic, complemented
30C37×#485—22no—
31C37×#484—21no—
32C37×#479—22no—
33C37×#477—21no—
34C37×#474C2212yes—
35C37×#474C2212yescomplemented
36C37×#468—22no—
37C37×#461C2212yes—
38C37×#461C2212yescomplemented
39C37×#460—21no—
40C37×#450—21no—
41C37×#440—22no—
42C37×#432C2211yescharacteristic
43C37×#431C2211yescharacteristic
44C37×#423C2211yescharacteristic
45C37×#333C2212yescomplemented
46C37×#332C2212yescomplemented
47C37×D4⋊C8C2212yes—
48C37×D4⋊C8C2212yes—
49C37×D4⋊C8C2212yescomplemented
50C37×D4⋊C8C2212yescomplemented
51C2×#41×C37—22no—
52C2×#41×C37—21no—
53C22×D4×C37D411yescharacteristic
54C23×C4×C37—21no—
55C2×SD16×C37—22no—
56C2×SD16×C37—22no—
57C2×M16×C37—21no—
58C4⋊Q8×C37D411yescharacteristic
59C4⋊Q8×C37C2311yescharacteristic
60C4⋊Q8×C37C2311yescharacteristic
61C4⋊Q8×C37C2311yescharacteristic
62C4⋊Q8×C37C2311yescharacteristic
63#127×C37—22no—
64#127×C37D411yescharacteristic
65#126×C37—24no—
66#125×C37—21no—
67#125×C37—21no—
68#124×C37—22no—
69#124×C37—24no—
70#124×C37—21no—
71#123×C37—24no—
72#123×C37—24no—
73C22⋊Q8×C37—21no—
74C22⋊Q8×C37—22no—
75C22⋊Q8×C37—21no—
76C22⋊Q8×C37—21no—
77C22⋊Q8×C37D412yes—
78#121×C37—24no—
79#121×C37—22no—
80#121×C37—22no—
81#121×C37—22no—
82#121×C37—21no—
83#121×C37D412yes—
84#120×C37—24no—
85C4×Q8×C37—22no—
86C4×Q8×C37—21no—
87C4×Q8×C37—21no—
88C4×Q8×C37—21no—
89C4×Q8×C37D412yes—
90C4×D4×C37—24no—
91C4×D4×C37—22no—
92C4×D4×C37—24no—
93C4×D4×C37—21no—
94C4×D4×C37—21no—
95C4×D4×C37—21no—
96C4×D4×C37D412yes—
97C4×D4×C37C2312yes—
98C4×D4×C37C2312yes—
99#117×C37—21no—
100#117×C37—22no—
101#117×C37—22no—
102#117×C37—21no—
103C2×C4⋊C4×C37—22no—
104C2×C4⋊C4×C37D411yescharacteristic
105C2×C22⋊C4×C37—21no—
106C2×C42×C37C2311yescharacteristic
107#107×C37—22no—
108#107×C37—22no—
109#107×C37—22no—
110#106×C37—22no—
111C4⋊C8×C37—21no—
112C4⋊C8×C37—21no—
113C4⋊C8×C37C2312yes—
114C4⋊C8×C37C2312yes—
115#103×C37—22no—
116#103×C37—22no—
117#103×C37—22no—
118#103×C37—22no—
119#102×C37—22no—
120#102×C37—22no—
121C22⋊C8×C37—21no—
122C22⋊C8×C37—21no—
123#97×C37—21no—
124C4×C8×C37C2311yescharacteristic
125C4×C8×C37C2311yescharacteristic
126C24×C37—21no—
127#41×C37—44no—
128#41×C37—82no—
129#41×C37—82no—
130C2×Q8×C37—21no—
131C2×Q8×C37—21no—
132C2×Q8×C37—21no—
133C2×Q8×C37—21no—
134C2×Q8×C37C2×D412yes—
135C2×D4×C37—22no—
136C2×D4×C37—24no—
137C2×D4×C37—24no—
138C2×D4×C37—21no—
139C2×D4×C37—21no—
140C2×D4×C37—21no—
141C2×D4×C37—42no—
142C2×D4×C37C2×D412yes—
143C2×D4×C37C2×D412yes—
144C22×C4×C37—24no—
145C22×C4×C37—21no—
146C22×C4×C37—22no—
147C22×C4×C37—22no—
148C22×C4×C37—22no—
149C22×C4×C37—81no—
150C22×C4×C37—81no—
151C22×C4×C37C2×D411yescharacteristic
152C22×C4×C37C2×D411yescharacteristic
153C22×C4×C37C2×D411yescharacteristic
154SD16×C37—44no—
155SD16×C37—82no—
156M16×C37—81no—
157C2×C8×C37—21no—
158C2×C8×C37—21no—
159C2×C8×C37—22no—
160C2×C8×C37—22no—
161C4⋊C4×C37—24no—
162C4⋊C4×C37—21no—
163C4⋊C4×C37—22no—
164C4⋊C4×C37—21no—
165C4⋊C4×C37—22no—
166C4⋊C4×C37—21no—
167C4⋊C4×C37—21no—
168C4⋊C4×C37—21no—
169C4⋊C4×C37—21no—
170C4⋊C4×C37—21no—
171C4⋊C4×C37—21no—
172C4⋊C4×C37—21no—
173C4⋊C4×C37—41no—
174C4⋊C4×C37—41no—
175C4⋊C4×C37C2×D412yes—
176C4⋊C4×C37C2×D412yes—
177C4⋊C4×C37C2×D412yes—
178C22⋊C4×C37—24no—
179C22⋊C4×C37—24no—
180C22⋊C4×C37—24no—
181C22⋊C4×C37—22no—
182C22⋊C4×C37—22no—
183C22⋊C4×C37—22no—
184C22⋊C4×C37—22no—
185C22⋊C4×C37—22no—
186C22⋊C4×C37—41no—
187C42×C37—24no—
188C42×C37—22no—
189C42×C37—41no—
190C42×C37C2×D411yescharacteristic
191C42×C37C2×D411yescharacteristic
192C42×C37C2×D411yescharacteristic
193C42×C37C2411yescharacteristic
194C23×C37—22no—
195C23×C37—22no—
196C23×C37—42no—
197C23×C37—81no—
198C23×C37—81no—
199C23×C37#12011yescharacteristic
200Q8×C37—42no—
201Q8×C37—42no—
202Q8×C37—41no—
203Q8×C37—41no—
204Q8×C37—42no—
205Q8×C37—41no—
206Q8×C37—41no—
207D4×C37—22no—
208D4×C37—22no—
209D4×C37—22no—
210D4×C37—22no—
211D4×C37—42no—
212D4×C37—42no—
213D4×C37—44no—
214D4×C37—81no—
215D4×C37—84no—
216D4×C37—84no—
217C2×C4×C37—22no—
218C2×C4×C37—21no—
219C2×C4×C37—21no—
220C2×C4×C37—22no—
221C2×C4×C37—22no—
222C2×C4×C37—22no—
223C2×C4×C37—22no—
224C2×C4×C37—41no—
225C2×C4×C37—44no—
226C2×C4×C37—42no—
227C2×C4×C37—42no—
228C2×C4×C37—41no—
229C2×C4×C37—41no—
230C2×C4×C37—82no—
231C2×C4×C37—81no—
232C2×C4×C37—84no—
233C2×C4×C37#12011yescharacteristic
234C2×C4×C37#12011yescharacteristic
235C2×C4×C37#12011yescharacteristic
236C2×C4×C37C22×D411yescharacteristic
237C2×C4×C37C22×D411yescharacteristic
238C2×C4×C37C22×D411yescharacteristic
239C8×C37—41no—
240C8×C37—41no—
241C8×C37—42no—
242C8×C37—42no—
243#18865C3711yesmaximal, characteristic, direct product factor, Sylow
244C22×C37—21no—
245C22×C37—21no—
246C22×C37—21no—
247C22×C37—42no—
248C22×C37—42no—
249C22×C37—42no—
250C22×C37—42no—
251C22×C37—42no—
252C22×C37—42no—
253C22×C37—84no—
254C22×C37C2×#12011yescharacteristic
255C4×C37—21no—
256C4×C37—21no—
257C4×C37—21no—
258C4×C37—21no—
259C4×C37—42no—
260C4×C37—42no—
261C4×C37—42no—
262C4×C37—42no—
263C4×C37—81no—
264C4×C37—81no—
265C4×C37#57512yes—
266C4×C37#57611yescharacteristic
267C4×C37#57711yescharacteristic
268#3506C2×C3711yescharacteristic
269#3254C2×C3711yescharacteristic, complemented
270#3231C2×C3711yescharacteristic, complemented
271#3110C2×C3711yescharacteristic, complemented
272#1654C2×C3712yescomplemented
273#1647C2×C3712yescomplemented
274#1646C2×C3712yescomplemented
275#1646C2×C3712yescomplemented
276#1559C2×C3711yescharacteristic, complemented
277#1496C2×C3711yescharacteristic, complemented
278#1486C2×C3711yescharacteristic, complemented
279C2×C37—41no—
280C2×C37—82no—
281C2×C37—82no—
282C2×C37#300511yescharacteristic
283C2×C37#301211yescharacteristic
284C2×C37#302211yescharacteristic
285#559—22no—
286#556—21no—
287#555C22×C3712yes—
288#553—22no—
289#550—21no—
290#547C22×C3712yes—
291#538—22no—
292#535—21no—
293#533C22×C3711yescharacteristic
294#533C22×C3711yescharacteristic
295#532C22×C3711yescharacteristic
296#525—22no—
297#520—21no—
298C2×C4×D4C22×C3711yescharacteristic
299#500C22×C3711yescharacteristic, complemented
300#498C22×C3711yescharacteristic, complemented
301#487C22×C3711yescharacteristic, complemented
302#487C22×C3711yescharacteristic, complemented
303#485—22no—
304#484—21no—
305#479—22no—
306#477—21no—
307#474C22×C3712yes—
308#474C22×C3712yescomplemented
309#468—22no—
310#461C22×C3712yes—
311#461C22×C3712yescomplemented
312#460—21no—
313#450—21no—
314#440—22no—
315#432C22×C3711yescharacteristic
316#431C22×C3711yescharacteristic
317#423C22×C3711yescharacteristic
318#333C22×C3712yescomplemented
319#332C22×C3712yescomplemented
320D4⋊C8C22×C3712yes—
321D4⋊C8C22×C3712yes—
322D4⋊C8C22×C3712yescomplemented
323D4⋊C8C22×C3712yescomplemented
324C37#1886511yescharacteristic, direct product factor, Sylow
325C2×#41—22no—
326C2×#41—21no—
327C22×D4D4×C3711yescharacteristic
328C23×C4—21no—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×M16—21no—
332C4⋊Q8D4×C3711yescharacteristic
333C4⋊Q8C23×C3711yescharacteristic
334C4⋊Q8C23×C3711yescharacteristic
335C4⋊Q8C23×C3711yescharacteristic
336C4⋊Q8C23×C3711yescharacteristic
337#127—22no—
338#127D4×C3711yescharacteristic
339#126—24no—
340#125—21no—
341#125—21no—
342#124—22no—
343#124—24no—
344#124—21no—
345#123—24no—
346#123—24no—
347C22⋊Q8—21no—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8D4×C3712yes—
352#121—24no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—21no—
357#121D4×C3712yes—
358#120—24no—
359C4×Q8—22no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8D4×C3712yes—
364C4×D4—24no—
365C4×D4—22no—
366C4×D4—24no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4D4×C3712yes—
371C4×D4C23×C3712yes—
372C4×D4C23×C3712yes—
373#117—21no—
374#117—22no—
375#117—22no—
376#117—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4D4×C3711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C42C23×C3711yescharacteristic
381#107—22no—
382#107—22no—
383#107—22no—
384#106—22no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8C23×C3712yes—
388C4⋊C8C23×C3712yes—
389#103—22no—
390#103—22no—
391#103—22no—
392#103—22no—
393#102—22no—
394#102—22no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397#97—21no—
398C4×C8C23×C3711yescharacteristic
399C4×C8C23×C3711yescharacteristic
400C24—21no—
401#41—44no—
402#41—82no—
403#41—82no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×D4×C3712yes—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4C2×D4×C3712yes—
417C2×D4C2×D4×C3712yes—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4C2×D4×C3711yescharacteristic
426C22×C4C2×D4×C3711yescharacteristic
427C22×C4C2×D4×C3711yescharacteristic
428SD16—44no—
429SD16—82no—
430M16—81no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4C2×D4×C3712yes—
450C4⋊C4C2×D4×C3712yes—
451C4⋊C4C2×D4×C3712yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C42—24no—
462C42—22no—
463C42—41no—
464C42C2×D4×C3711yescharacteristic
465C42C2×D4×C3711yescharacteristic
466C42C2×D4×C3711yescharacteristic
467C42C24×C3711yescharacteristic
468C23—22no—
469C23—22no—
470C23—42no—
471C23—81no—
472C23—81no—
473C23#120×C3711yescharacteristic
474Q8—42no—
475Q8—42no—
476Q8—41no—
477Q8—41no—
478Q8—42no—
479Q8—41no—
480Q8—41no—
481D4—22no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—42no—
486D4—42no—
487D4—44no—
488D4—81no—
489D4—84no—
490D4—84no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—84no—
507C2×C4#120×C3711yescharacteristic
508C2×C4#120×C3711yescharacteristic
509C2×C4#120×C3711yescharacteristic
510C2×C4C22×D4×C3711yescharacteristic
511C2×C4C22×D4×C3711yescharacteristic
512C2×C4C22×D4×C3711yescharacteristic
513C8—41no—
514C8—41no—
515C8—42no—
516C8—42no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—84no—
527C22C2×#120×C3711yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—81no—
537C4—81no—
538C4C37×#57512yes—
539C4C37×#57611yescharacteristic
540C4C37×#57711yescharacteristic
541C2—41no—
542C2—82no—
543C2—82no—
544C2C37×#300511yescharacteristic
545C2C37×#301211yescharacteristic
546C2C37×#302211yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#3506C2
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
2. C37×#3254C2semidirect
3. C37×#3231C2semidirect
4. C37×#3110C2semidirect
5. C37×#1654C2semidirect
6. C37×#1647C2semidirect
7. C37×#1646C2semidirect
8. C37×#1646C2semidirect
9. C37×#1559C2semidirect
10. C37×#1496C2semidirect
11. C37×#1486C2semidirect
26. C37×#500C22semidirect
27. C37×#498C22semidirect
28. C37×#487C22semidirect
29. C37×#487C22semidirect
35. C37×#474C22semidirect
38. C37×#461C22semidirect
45. C37×#333C22semidirect
46. C37×#332C22semidirect
49. C37×D4⋊C8C22semidirect
50. C37×D4⋊C8C22semidirect
243. #18865C37direct (N × H)
269. #3254C2×C37semidirect
270. #3231C2×C37semidirect
271. #3110C2×C37semidirect
272. #1654C2×C37semidirect
273. #1647C2×C37semidirect
274. #1646C2×C37semidirect
275. #1646C2×C37semidirect
276. #1559C2×C37semidirect
277. #1496C2×C37semidirect
278. #1486C2×C37semidirect
299. #500C22×C37semidirect
300. #498C22×C37semidirect
301. #487C22×C37semidirect
302. #487C22×C37semidirect
308. #474C22×C37semidirect
311. #461C22×C37semidirect
318. #333C22×C37semidirect
319. #332C22×C37semidirect
322. D4⋊C8C22×C37semidirect
323. D4⋊C8C22×C37semidirect
324. C37#18865direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35062
C37×#32542
C37×#32312
C37×#31102
C37×#16542
C37×#16472
C37×#16462
C37×#15592
C37×#14962
C37×#14862
#1886537