Finite Group Databasebeta

← C37×#18321 #22639574

Subgroups — C37×#18321 #22639574

Order 9472.

982 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#539C211yesmaximal, characteristic
2C37×#3195C211yesmaximal, characteristic, complemented
3C37×#3193C211yesmaximal, characteristic
4C37×#2583C211yesmaximal, characteristic, complemented
5C37×#1609C214yesmaximal, complemented
6C37×#1603C214yesmaximal, complemented
7C37×#1508C211yesmaximal, characteristic, complemented
8C37×#1507C211yesmaximal, characteristic
9C37×#1463C211yesmaximal, characteristic, complemented
10C37×#539—22no—
11C37×#539—22no—
12C37×#539—22no—
13C37×#539C2214yes—
14C2×#126×C37—21no—
15C2×#125×C37—21no—
16C2×#123×C37—21no—
17C2×#123×C37—21no—
18C2×C22⋊Q8×C37—21no—
19C2×C22⋊Q8×C37C2211yescharacteristic
20C2×C22⋊Q8×C37C2211yescharacteristic
21C2×#121×C37C2211yescharacteristic
22C2×C4×D4×C37—21no—
23C22×C4⋊C4×C37C2211yescharacteristic
24C37×#484—21no—
25C37×#483—21no—
26C37×#483C2212yescomplemented
27C37×#480C2212yes—
28C37×#470—22no—
29C37×#468—22no—
30C37×#451—21no—
31C37×#451C2212yescomplemented
32C37×#450—21no—
33C37×#448C2212yes—
34C37×#407C2211yescharacteristic
35C37×#407C2211yescharacteristic
36C37×#398C2212yescomplemented
37C37×#395C2212yescomplemented
38C37×#389—22no—
39C37×#385—21no—
40C37×#385—21no—
41C4×C22⋊C4×C37C2211yescharacteristic
42C37×#328C2214yescomplemented
43C37×#327C2214yes—
44C37×#324C2214yescomplemented
45C22×Q8×C37D411yescharacteristic
46C22×Q8×C37C2311yescharacteristic
47C22×D4×C37—21no—
48C22×D4×C37D411yescharacteristic
49C23×C4×C37—21no—
50C23×C4×C37D411yescharacteristic
51C23×C4×C37D411yescharacteristic
52C23×C4×C37C2311yescharacteristic
53C2×Q16×C37—22no—
54C2×SD16×C37—22no—
55C2×M16×C37—21no—
56C2×M16×C37—21no—
57#126×C37—24no—
58#126×C37—24no—
59#125×C37—24no—
60#125×C37—24no—
61#123×C37—24no—
62#123×C37—24no—
63#123×C37—24no—
64#123×C37—24no—
65C22⋊Q8×C37—24no—
66C22⋊Q8×C37—24no—
67C22⋊Q8×C37—21no—
68C22⋊Q8×C37—21no—
69C22⋊Q8×C37—21no—
70C22⋊Q8×C37—21no—
71C22⋊Q8×C37—21no—
72C22⋊Q8×C37—21no—
73C22⋊Q8×C37D412yes—
74C22⋊Q8×C37C2312yes—
75#121×C37—21no—
76#121×C37—21no—
77#121×C37D412yes—
78#121×C37C2312yes—
79C4×D4×C37—24no—
80C4×D4×C37—24no—
81C2×C4⋊C4×C37—21no—
82C2×C4⋊C4×C37—21no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37—21no—
85C2×C4⋊C4×C37—21no—
86C2×C4⋊C4×C37—21no—
87C2×C4⋊C4×C37—21no—
88C2×C4⋊C4×C37—21no—
89C2×C4⋊C4×C37—21no—
90C2×C4⋊C4×C37—22no—
91C2×C4⋊C4×C37—22no—
92C2×C4⋊C4×C37D412yes—
93C2×C4⋊C4×C37D411yescharacteristic
94C2×C4⋊C4×C37D411yescharacteristic
95C2×C4⋊C4×C37C2311yescharacteristic
96C2×C4⋊C4×C37C2312yes—
97C2×C22⋊C4×C37—21no—
98C2×C22⋊C4×C37—21no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37—21no—
101C2×C22⋊C4×C37—21no—
102C2×C22⋊C4×C37—21no—
103C2×C22⋊C4×C37—21no—
104C2×C42×C37—21no—
105C2×C42×C37—21no—
106#107×C37—22no—
107#106×C37—22no—
108#103×C37—22no—
109#103×C37—22no—
110#103×C37—22no—
111#102×C37—22no—
112C22⋊C8×C37—21no—
113C22⋊C8×C37—21no—
114C22⋊C8×C37C2312yes—
115C22⋊C8×C37C2312yes—
116#95×C37—22no—
117#95×C37—22no—
118#95×C37C2312yes—
119C24×C37—21no—
120C24×C37C2×D411yescharacteristic
121C2×Q8×C37—21no—
122C2×Q8×C37—21no—
123C2×Q8×C37—21no—
124C2×Q8×C37—81no—
125C2×Q8×C37—81no—
126C2×Q8×C37C2×D412yes—
127C2×D4×C37—22no—
128C2×D4×C37—22no—
129C2×D4×C37—21no—
130C2×D4×C37—44no—
131C2×D4×C37—81no—
132C2×D4×C37C2×D412yes—
133C22×C4×C37—21no—
134C22×C4×C37—21no—
135C22×C4×C37—21no—
136C22×C4×C37—21no—
137C22×C4×C37—21no—
138C22×C4×C37—21no—
139C22×C4×C37—22no—
140C22×C4×C37—24no—
141C22×C4×C37—21no—
142C22×C4×C37—21no—
143C22×C4×C37—21no—
144C22×C4×C37—42no—
145C22×C4×C37—81no—
146C22×C4×C37—81no—
147C22×C4×C37—82no—
148C22×C4×C37C2×D412yes—
149C22×C4×C37C2×D412yes—
150C22×C4×C37C2×D411yescharacteristic
151C22×C4×C37C2×D411yescharacteristic
152C22×C4×C37C2×D411yescharacteristic
153C22×C4×C37C2×D411yescharacteristic
154C22×C4×C37C2×D411yescharacteristic
155C22×C4×C37C2411yescharacteristic
156Q16×C37—82no—
157SD16×C37—82no—
158M16×C37—81no—
159M16×C37—81no—
160C2×C8×C37—22no—
161C2×C8×C37—22no—
162C4⋊C4×C37—24no—
163C4⋊C4×C37—24no—
164C4⋊C4×C37—24no—
165C4⋊C4×C37—22no—
166C4⋊C4×C37—22no—
167C4⋊C4×C37—22no—
168C4⋊C4×C37—21no—
169C4⋊C4×C37—21no—
170C4⋊C4×C37—22no—
171C4⋊C4×C37—22no—
172C4⋊C4×C37—22no—
173C4⋊C4×C37—21no—
174C4⋊C4×C37—21no—
175C4⋊C4×C37—44no—
176C4⋊C4×C37—41no—
177C4⋊C4×C37—41no—
178C4⋊C4×C37—82no—
179C4⋊C4×C37C2×D412yes—
180C4⋊C4×C37C2×D412yes—
181C22⋊C4×C37—24no—
182C22⋊C4×C37—24no—
183C22⋊C4×C37—24no—
184C22⋊C4×C37—24no—
185C22⋊C4×C37—22no—
186C22⋊C4×C37—22no—
187C22⋊C4×C37—22no—
188C22⋊C4×C37—22no—
189C22⋊C4×C37—41no—
190C22⋊C4×C37—41no—
191C42×C37—24no—
192C42×C37—81no—
193C23×C37—22no—
194C23×C37—22no—
195C23×C37—22no—
196C23×C37—41no—
197C23×C37—42no—
198C23×C37—42no—
199C23×C37—42no—
200C23×C37#12011yescharacteristic
201C23×C37#12011yescharacteristic
202C23×C37C22×D411yescharacteristic
203Q8×C37—42no—
204Q8×C37—42no—
205Q8×C37—42no—
206Q8×C37—81no—
207D4×C37—42no—
208D4×C37—44no—
209D4×C37—44no—
210D4×C37—81no—
211C2×C4×C37—24no—
212C2×C4×C37—22no—
213C2×C4×C37—22no—
214C2×C4×C37—22no—
215C2×C4×C37—22no—
216C2×C4×C37—22no—
217C2×C4×C37—22no—
218C2×C4×C37—22no—
219C2×C4×C37—41no—
220C2×C4×C37—41no—
221C2×C4×C37—41no—
222C2×C4×C37—41no—
223C2×C4×C37—41no—
224C2×C4×C37—41no—
225C2×C4×C37—42no—
226C2×C4×C37—42no—
227C2×C4×C37—42no—
228C2×C4×C37—42no—
229C2×C4×C37—42no—
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—42no—
238C2×C4×C37—81no—
239C2×C4×C37—84no—
240C2×C4×C37#12011yescharacteristic
241C2×C4×C37#12011yescharacteristic
242C2×C4×C37C22×D411yescharacteristic
243C2×C4×C37C22×D411yescharacteristic
244C8×C37—42no—
245C8×C37—42no—
246#18321C3711yesmaximal, characteristic, direct product factor, Sylow
247C22×C37—21no—
248C22×C37—21no—
249C22×C37—21no—
250C22×C37—21no—
251C22×C37—21no—
252C22×C37—21no—
253C22×C37—21no—
254C22×C37—44no—
255C22×C37—42no—
256C22×C37—42no—
257C22×C37—42no—
258C22×C37—84no—
259C22×C37#45612yes—
260C22×C37C2×#12011yescharacteristic
261C22×C37#57512yes—
262C4×C37—21no—
263C4×C37—21no—
264C4×C37—21no—
265C4×C37—21no—
266C4×C37—44no—
267C4×C37—42no—
268C4×C37—42no—
269C4×C37—42no—
270C4×C37—42no—
271C4×C37—42no—
272C4×C37—81no—
273C4×C37—81no—
274C2×#539C2×C3711yescharacteristic
275#3195C2×C3711yescharacteristic, complemented
276#3193C2×C3711yescharacteristic
277#2583C2×C3711yescharacteristic, complemented
278#1609C2×C3714yescomplemented
279#1603C2×C3714yescomplemented
280#1508C2×C3711yescharacteristic, complemented
281#1507C2×C3711yescharacteristic
282#1463C2×C3711yescharacteristic, complemented
283C2×C37—22no—
284C2×C37—22no—
285C2×C37—41no—
286C2×C37—82no—
287C2×C37#300511yescharacteristic
288C2×C37C2×#45611yescharacteristic
289C2×C37#302611yescharacteristic
290#539—22no—
291#539—22no—
292#539—22no—
293#539C22×C3714yes—
294C2×#126—21no—
295C2×#125—21no—
296C2×#123—21no—
297C2×#123—21no—
298C2×C22⋊Q8—21no—
299C2×C22⋊Q8C22×C3711yescharacteristic
300C2×C22⋊Q8C22×C3711yescharacteristic
301C2×#121C22×C3711yescharacteristic
302C2×C4×D4—21no—
303C22×C4⋊C4C22×C3711yescharacteristic
304#484—21no—
305#483—21no—
306#483C22×C3712yescomplemented
307#480C22×C3712yes—
308#470—22no—
309#468—22no—
310#451—21no—
311#451C22×C3712yescomplemented
312#450—21no—
313#448C22×C3712yes—
314#407C22×C3711yescharacteristic
315#407C22×C3711yescharacteristic
316#398C22×C3712yescomplemented
317#395C22×C3712yescomplemented
318#389—22no—
319#385—21no—
320#385—21no—
321C4×C22⋊C4C22×C3711yescharacteristic
322#328C22×C3714yescomplemented
323#327C22×C3714yes—
324#324C22×C3714yescomplemented
325C37#1832111yescharacteristic, direct product factor, Sylow
326C22×Q8D4×C3711yescharacteristic
327C22×Q8C23×C3711yescharacteristic
328C22×D4—21no—
329C22×D4D4×C3711yescharacteristic
330C23×C4—21no—
331C23×C4D4×C3711yescharacteristic
332C23×C4D4×C3711yescharacteristic
333C23×C4C23×C3711yescharacteristic
334C2×Q16—22no—
335C2×SD16—22no—
336C2×M16—21no—
337C2×M16—21no—
338#126—24no—
339#126—24no—
340#125—24no—
341#125—24no—
342#123—24no—
343#123—24no—
344#123—24no—
345#123—24no—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8—21no—
354C22⋊Q8D4×C3712yes—
355C22⋊Q8C23×C3712yes—
356#121—21no—
357#121—21no—
358#121D4×C3712yes—
359#121C23×C3712yes—
360C4×D4—24no—
361C4×D4—24no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4D4×C3712yes—
374C2×C4⋊C4D4×C3711yescharacteristic
375C2×C4⋊C4D4×C3711yescharacteristic
376C2×C4⋊C4C23×C3711yescharacteristic
377C2×C4⋊C4C23×C3712yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42—21no—
387#107—22no—
388#106—22no—
389#103—22no—
390#103—22no—
391#103—22no—
392#102—22no—
393C22⋊C8—21no—
394C22⋊C8—21no—
395C22⋊C8C23×C3712yes—
396C22⋊C8C23×C3712yes—
397#95—22no—
398#95—22no—
399#95C23×C3712yes—
400C24—21no—
401C24C2×D4×C3711yescharacteristic
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×D4×C3712yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—81no—
413C2×D4C2×D4×C3712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—82no—
429C22×C4C2×D4×C3712yes—
430C22×C4C2×D4×C3712yes—
431C22×C4C2×D4×C3711yescharacteristic
432C22×C4C2×D4×C3711yescharacteristic
433C22×C4C2×D4×C3711yescharacteristic
434C22×C4C2×D4×C3711yescharacteristic
435C22×C4C2×D4×C3711yescharacteristic
436C22×C4C24×C3711yescharacteristic
437Q16—82no—
438SD16—82no—
439M16—81no—
440M16—81no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—44no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—82no—
460C4⋊C4C2×D4×C3712yes—
461C4⋊C4C2×D4×C3712yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—24no—
473C42—81no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23#120×C3711yescharacteristic
482C23#120×C3711yescharacteristic
483C23C22×D4×C3711yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—81no—
488D4—42no—
489D4—44no—
490D4—44no—
491D4—81no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—84no—
521C2×C4#120×C3711yescharacteristic
522C2×C4#120×C3711yescharacteristic
523C2×C4C22×D4×C3711yescharacteristic
524C2×C4C22×D4×C3711yescharacteristic
525C8—42no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22C37×#45612yes—
540C22C2×#120×C3711yescharacteristic
541C22C37×#57512yes—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C2—22no—
555C2—22no—
556C2—41no—
557C2—82no—
558C2C37×#300511yescharacteristic
559C2C2×C37×#45611yescharacteristic
560C2C37×#302611yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×#121×C37C22
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×#3195C2semidirect
4. C37×#2583C2semidirect
5. C37×#1609C2semidirect
6. C37×#1603C2semidirect
7. C37×#1508C2semidirect
9. C37×#1463C2semidirect
26. C37×#483C22semidirect
31. C37×#451C22semidirect
36. C37×#398C22semidirect
37. C37×#395C22semidirect
42. C37×#328C22semidirect
44. C37×#324C22semidirect
246. #18321C37direct (N × H)
275. #3195C2×C37semidirect
277. #2583C2×C37semidirect
278. #1609C2×C37semidirect
279. #1603C2×C37semidirect
280. #1508C2×C37semidirect
282. #1463C2×C37semidirect
306. #483C22×C37semidirect
311. #451C22×C37semidirect
316. #398C22×C37semidirect
317. #395C22×C37semidirect
322. #328C22×C37semidirect
324. #324C22×C37semidirect
325. C37#18321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#5392
C37×#31952
C37×#31932
C37×#25832
C37×#16092
C37×#16032
C37×#15082
C37×#15072
C37×#14632
#1832137