Finite Group Databasebeta

← C37×#19462 #22640715

Subgroups — C37×#19462 #22640715

Order 9472.

680 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3227C211yesmaximal, characteristic, complemented
2C37×#3109C211yesmaximal, characteristic
3C2×C37×#461C211yesmaximal, characteristic, complemented
4C37×#2974C211yesmaximal, characteristic, complemented
5C37×#1701C212yesmaximal, complemented
6C37×#1691C212yesmaximal, complemented
7C37×#1681C212yesmaximal, complemented
8C37×#1671C212yesmaximal, complemented
9C37×#1573C211yesmaximal, characteristic, complemented
10C37×#1556C211yesmaximal, characteristic, complemented
11C37×#1551C211yesmaximal, characteristic, complemented
12C22×SD16×C37—21no—
13C37×#532C2211yescharacteristic
14C2×C4⋊Q8×C37C2211yescharacteristic
15C2×C4×D4×C37C2211yescharacteristic
16C37×#498C2212yescomplemented
17C37×#486C2212yescomplemented
18C37×#481—21no—
19C37×#481—21no—
20C37×#481—21no—
21C37×#476—22no—
22C37×#476C2211yescharacteristic
23C37×#476C2211yescharacteristic
24C37×#461—21no—
25C37×#461—21no—
26C37×#461C2212yes—
27C37×#461C2212yes—
28C37×#459C2211yescharacteristic
29C37×#459C2211yescharacteristic
30C37×#447—21no—
31C37×#442—21no—
32C37×#442—21no—
33C37×#436C2212yes—
34C37×#435—22no—
35C8×D4×C37C2212yes—
36C37×#432C2211yescharacteristic
37C37×#432C2211yescharacteristic
38C37×#431C2211yescharacteristic
39C2×C4⋊C8×C37C2211yescharacteristic
40C2×#103×C37—21no—
41C37×#407—21no—
42C2×C22⋊C8×C37—21no—
43C37×#335C2212yescomplemented
44C37×#334C2212yescomplemented
45C37×#333C2212yescomplemented
46C37×#332C2212yescomplemented
47C37×#332C2212yescomplemented
48C37×#331C2212yescomplemented
49C2×#41×C37—21no—
50C22×Q8×C37—21no—
51C22×D4×C37D411yescharacteristic
52C23×C4×C37—21no—
53C2×SD16×C37—22no—
54C2×SD16×C37—22no—
55C2×SD16×C37—44no—
56C2×D8×C37—21no—
57C2×D8×C37—21no—
58C2×M16×C37—22no—
59C22×C8×C37—21no—
60C22×C8×C37—22no—
61C4⋊Q8×C37—21no—
62C4⋊Q8×C37—21no—
63C4⋊Q8×C37—21no—
64C4⋊Q8×C37C2312yes—
65C4⋊Q8×C37C2311yescharacteristic
66#127×C37C2311yescharacteristic
67#124×C37—21no—
68#121×C37—21no—
69#121×C37—21no—
70C4×D4×C37—21no—
71C4×D4×C37—21no—
72C4×D4×C37—21no—
73C4×D4×C37—21no—
74C4×D4×C37D411yescharacteristic
75C4×D4×C37D411yescharacteristic
76C4×D4×C37C2311yescharacteristic
77C4×D4×C37C2311yescharacteristic
78C2×C4⋊C4×C37—21no—
79C2×C4⋊C4×C37—21no—
80C2×C4⋊C4×C37D411yescharacteristic
81C2×C22⋊C4×C37—21no—
82C2×C42×C37C2311yescharacteristic
83#106×C37—22no—
84#106×C37—22no—
85#106×C37—21no—
86#106×C37—21no—
87C4⋊C8×C37—21no—
88C4⋊C8×C37D412yes—
89C4⋊C8×C37C2312yes—
90C4⋊C8×C37C2312yes—
91C4⋊C8×C37C2311yescharacteristic
92C4⋊C8×C37C2311yescharacteristic
93#103×C37—22no—
94#103×C37—22no—
95#102×C37—21no—
96#102×C37—21no—
97#102×C37—22no—
98#102×C37—22no—
99#102×C37—21no—
100#102×C37—21no—
101C22⋊C8×C37—21no—
102C22⋊C8×C37—21no—
103C22⋊C8×C37—22no—
104C22⋊C8×C37—22no—
105C22⋊C8×C37D411yescharacteristic
106C22⋊C8×C37D411yescharacteristic
107#97×C37—21no—
108#97×C37—21no—
109C4×C8×C37C2312yes—
110C24×C37—21no—
111#41×C37—82no—
112C2×Q8×C37—22no—
113C2×Q8×C37—21no—
114C2×Q8×C37—42no—
115C2×Q8×C37—42no—
116C2×Q8×C37—41no—
117C2×D4×C37—22no—
118C2×D4×C37—21no—
119C2×D4×C37—21no—
120C2×D4×C37—21no—
121C2×D4×C37—21no—
122C2×D4×C37—41no—
123C2×D4×C37—41no—
124C2×D4×C37C2×D411yescharacteristic
125C2×D4×C37C2×D411yescharacteristic
126C22×C4×C37—21no—
127C22×C4×C37—21no—
128C22×C4×C37—21no—
129C22×C4×C37—21no—
130C22×C4×C37—21no—
131C22×C4×C37—41no—
132C22×C4×C37—41no—
133C22×C4×C37—82no—
134C22×C4×C37C2×D411yescharacteristic
135C22×C4×C37C2×D411yescharacteristic
136C22×C4×C37#4111yescharacteristic
137SD16×C37—44no—
138SD16×C37—44no—
139D8×C37—42no—
140D8×C37—42no—
141M16×C37—82no—
142C2×C8×C37—22no—
143C2×C8×C37—21no—
144C2×C8×C37—21no—
145C2×C8×C37—22no—
146C2×C8×C37—42no—
147C2×C8×C37—82no—
148C2×C8×C37C2×D412yes—
149C2×C8×C37C2×D412yes—
150C4⋊C4×C37—21no—
151C4⋊C4×C37—22no—
152C4⋊C4×C37—21no—
153C4⋊C4×C37—21no—
154C4⋊C4×C37—22no—
155C4⋊C4×C37—41no—
156C4⋊C4×C37—41no—
157C4⋊C4×C37C2×D411yescharacteristic
158C4⋊C4×C37C2×D411yescharacteristic
159C22⋊C4×C37—21no—
160C22⋊C4×C37—21no—
161C22⋊C4×C37—21no—
162C22⋊C4×C37—21no—
163C22⋊C4×C37—41no—
164C22⋊C4×C37—41no—
165C42×C37C2×D411yescharacteristic
166C42×C37C2×D411yescharacteristic
167C42×C37#4111yescharacteristic
168C42×C37C2411yescharacteristic
169C23×C37—21no—
170C23×C37—21no—
171C23×C37—42no—
172C23×C37—42no—
173C23×C37—41no—
174C23×C37—81no—
175C23×C37#12111yescharacteristic
176Q8×C37—42no—
177Q8×C37—42no—
178Q8×C37—81no—
179Q8×C37—82no—
180D4×C37—21no—
181D4×C37—21no—
182D4×C37—21no—
183D4×C37—42no—
184D4×C37—41no—
185D4×C37—41no—
186D4×C37—42no—
187D4×C37—81no—
188D4×C37—82no—
189D4×C37AGL(1,ℤ/8)12yes—
190C2×C4×C37—22no—
191C2×C4×C37—21no—
192C2×C4×C37—21no—
193C2×C4×C37—22no—
194C2×C4×C37—21no—
195C2×C4×C37—21no—
196C2×C4×C37—42no—
197C2×C4×C37—42no—
198C2×C4×C37—41no—
199C2×C4×C37—42no—
200C2×C4×C37—81no—
201C2×C4×C37—81no—
202C2×C4×C37—82no—
203C2×C4×C37—81no—
204C2×C4×C37—82no—
205C2×C4×C37#12111yescharacteristic
206C2×C4×C37#12111yescharacteristic
207C2×C4×C37#12111yescharacteristic
208C2×C4×C37C22×D411yescharacteristic
209C2×C4×C37C22×D411yescharacteristic
210C2×C4×C37C2×#4111yescharacteristic
211C8×C37—22no—
212C8×C37—22no—
213C8×C37—42no—
214C8×C37—41no—
215C8×C37—41no—
216C8×C37—42no—
217#19462C3711yesmaximal, characteristic, direct product factor, Sylow
218C22×C37—21no—
219C22×C37—21no—
220C22×C37—22no—
221C22×C37—41no—
222C22×C37—41no—
223C22×C37—41no—
224C22×C37—41no—
225C22×C37—41no—
226C22×C37—84no—
227C22×C37—81no—
228C22×C37—81no—
229C22×C37—81no—
230C22×C37#47112yes—
231C22×C37C2×#12111yescharacteristic
232C4×C37—21no—
233C4×C37—21no—
234C4×C37—21no—
235C4×C37—21no—
236C4×C37—41no—
237C4×C37—41no—
238C4×C37—82no—
239C4×C37—81no—
240C4×C37C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C37#57511yescharacteristic
242C4×C37#57712yes—
243#3227C2×C3711yescharacteristic, complemented
244#3109C2×C3711yescharacteristic
245C2×#461C2×C3711yescharacteristic, complemented
246#2974C2×C3711yescharacteristic, complemented
247#1701C2×C3712yescomplemented
248#1691C2×C3712yescomplemented
249#1681C2×C3712yescomplemented
250#1671C2×C3712yescomplemented
251#1573C2×C3711yescharacteristic, complemented
252#1556C2×C3711yescharacteristic, complemented
253#1551C2×C3711yescharacteristic, complemented
254C2×C37—22no—
255C2×C37—42no—
256C2×C37—81no—
257C2×C37—161no—
258C2×C37#304211yescharacteristic
259C2×C37#305511yescharacteristic
260C2×C37C2×#47111yescharacteristic
261C22×SD16—21no—
262#532C22×C3711yescharacteristic
263C2×C4⋊Q8C22×C3711yescharacteristic
264C2×C4×D4C22×C3711yescharacteristic
265#498C22×C3712yescomplemented
266#486C22×C3712yescomplemented
267#481—21no—
268#481—21no—
269#481—21no—
270#476—22no—
271#476C22×C3711yescharacteristic
272#476C22×C3711yescharacteristic
273#461—21no—
274#461—21no—
275#461C22×C3712yes—
276#461C22×C3712yes—
277#459C22×C3711yescharacteristic
278#459C22×C3711yescharacteristic
279#447—21no—
280#442—21no—
281#442—21no—
282#436C22×C3712yes—
283#435—22no—
284C8×D4C22×C3712yes—
285#432C22×C3711yescharacteristic
286#432C22×C3711yescharacteristic
287#431C22×C3711yescharacteristic
288C2×C4⋊C8C22×C3711yescharacteristic
289C2×#103—21no—
290#407—21no—
291C2×C22⋊C8—21no—
292#335C22×C3712yescomplemented
293#334C22×C3712yescomplemented
294#333C22×C3712yescomplemented
295#332C22×C3712yescomplemented
296#332C22×C3712yescomplemented
297#331C22×C3712yescomplemented
298C37#1946211yescharacteristic, direct product factor, Sylow
299C2×#41—21no—
300C22×Q8—21no—
301C22×D4D4×C3711yescharacteristic
302C23×C4—21no—
303C2×SD16—22no—
304C2×SD16—22no—
305C2×SD16—44no—
306C2×D8—21no—
307C2×D8—21no—
308C2×M16—22no—
309C22×C8—21no—
310C22×C8—22no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8C23×C3712yes—
315C4⋊Q8C23×C3711yescharacteristic
316#127C23×C3711yescharacteristic
317#124—21no—
318#121—21no—
319#121—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4D4×C3711yescharacteristic
325C4×D4D4×C3711yescharacteristic
326C4×D4C23×C3711yescharacteristic
327C4×D4C23×C3711yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4D4×C3711yescharacteristic
331C2×C22⋊C4—21no—
332C2×C42C23×C3711yescharacteristic
333#106—22no—
334#106—22no—
335#106—21no—
336#106—21no—
337C4⋊C8—21no—
338C4⋊C8D4×C3712yes—
339C4⋊C8C23×C3712yes—
340C4⋊C8C23×C3712yes—
341C4⋊C8C23×C3711yescharacteristic
342C4⋊C8C23×C3711yescharacteristic
343#103—22no—
344#103—22no—
345#102—21no—
346#102—21no—
347#102—22no—
348#102—22no—
349#102—21no—
350#102—21no—
351C22⋊C8—21no—
352C22⋊C8—21no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C22⋊C8D4×C3711yescharacteristic
356C22⋊C8D4×C3711yescharacteristic
357#97—21no—
358#97—21no—
359C4×C8C23×C3712yes—
360C24—21no—
361#41—82no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—41no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4C2×D4×C3711yescharacteristic
375C2×D4C2×D4×C3711yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—82no—
384C22×C4C2×D4×C3711yescharacteristic
385C22×C4C2×D4×C3711yescharacteristic
386C22×C4#41×C3711yescharacteristic
387SD16—44no—
388SD16—44no—
389D8—42no—
390D8—42no—
391M16—82no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—42no—
397C2×C8—82no—
398C2×C8C2×D4×C3712yes—
399C2×C8C2×D4×C3712yes—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×D4×C3711yescharacteristic
408C4⋊C4C2×D4×C3711yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C42C2×D4×C3711yescharacteristic
416C42C2×D4×C3711yescharacteristic
417C42#41×C3711yescharacteristic
418C42C24×C3711yescharacteristic
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—81no—
425C23#121×C3711yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—81no—
429Q8—82no—
430D4—21no—
431D4—21no—
432D4—21no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—42no—
437D4—81no—
438D4—82no—
439D4AGL(1,ℤ/8)×C3712yes—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4#121×C3711yescharacteristic
456C2×C4#121×C3711yescharacteristic
457C2×C4#121×C3711yescharacteristic
458C2×C4C22×D4×C3711yescharacteristic
459C2×C4C22×D4×C3711yescharacteristic
460C2×C4C2×#41×C3711yescharacteristic
461C8—22no—
462C8—22no—
463C8—42no—
464C8—41no—
465C8—41no—
466C8—42no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—84no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22C37×#47112yes—
480C22C2×#121×C3711yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—81no—
489C4C2×AGL(1,ℤ/8)×C3711yescharacteristic
490C4C37×#57511yescharacteristic
491C4C37×#57712yes—
492C2—22no—
493C2—42no—
494C2—81no—
495C2—161no—
496C2C37×#304211yescharacteristic
497C2C37×#305511yescharacteristic
498C2C2×C37×#47111yescharacteristic
subgroupG/H
center, socleC22×C37C2×#121
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#3109C2
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×#3227C2semidirect
3. C2×C37×#461C2semidirect
4. C37×#2974C2semidirect
5. C37×#1701C2semidirect
6. C37×#1691C2semidirect
7. C37×#1681C2semidirect
8. C37×#1671C2semidirect
9. C37×#1573C2semidirect
10. C37×#1556C2semidirect
11. C37×#1551C2semidirect
16. C37×#498C22semidirect
17. C37×#486C22semidirect
43. C37×#335C22semidirect
44. C37×#334C22semidirect
45. C37×#333C22semidirect
46. C37×#332C22semidirect
47. C37×#332C22semidirect
48. C37×#331C22semidirect
217. #19462C37direct (N × H)
243. #3227C2×C37semidirect
245. C2×#461C2×C37semidirect
246. #2974C2×C37semidirect
247. #1701C2×C37semidirect
248. #1691C2×C37semidirect
249. #1681C2×C37semidirect
250. #1671C2×C37semidirect
251. #1573C2×C37semidirect
252. #1556C2×C37semidirect
253. #1551C2×C37semidirect
265. #498C22×C37semidirect
266. #486C22×C37semidirect
292. #335C22×C37semidirect
293. #334C22×C37semidirect
294. #333C22×C37semidirect
295. #332C22×C37semidirect
296. #332C22×C37semidirect
297. #331C22×C37semidirect
298. C37#19462direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32272
C37×#31092
C2×C37×#4612
C37×#29742
C37×#17012
C37×#16912
C37×#16812
C37×#16712
C37×#15732
C37×#15562
C37×#15512
#1946237