Finite Group Databasebeta

← C37×#29689 #22650942

Subgroups — C37×#29689 #22650942

Order 9472.

1100 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#529C211yesmaximal, characteristic
2C2×C37×#475C211yesmaximal, characteristic, complemented
3C2×C37×#474C211yesmaximal, characteristic, complemented
4C37×#2857C211yesmaximal, characteristic, complemented
5C37×#2059C212yesmaximal, complemented
6C37×#2042C212yesmaximal, complemented
7C37×#2036C212yesmaximal, complemented
8C37×#2034C212yesmaximal
9C37×#1803C211yesmaximal, characteristic, complemented
10C37×#1802C211yesmaximal, characteristic, complemented
11C37×#1797C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C37C2211yescharacteristic
13C37×#529—24no—
14C37×#529—24no—
15C2×#126×C37—21no—
16C2×#125×C37C2211yescharacteristic
17C2×#124×C37C2211yescharacteristic
18C2×#123×C37—21no—
19C2×C22⋊Q8×C37—21no—
20C2×C22⋊Q8×C37C2211yescharacteristic
21C2×#121×C37C2211yescharacteristic
22C2×C4×Q8×C37C2211yescharacteristic
23C2×C4×D4×C37—21no—
24C2×C4×D4×C37C2211yescharacteristic
25C2×#117×C37—21no—
26C22×C42×C37C2211yescharacteristic
27C37×#475—24no—
28C37×#474—24no—
29C2×#107×C37C2212yes—
30C2×#106×C37C2212yes—
31C2×C4⋊C8×C37C2211yescharacteristic
32C2×C4⋊C8×C37C2211yescharacteristic
33C2×#103×C37C2212yes—
34C2×#103×C37C2212yes—
35C2×#102×C37C2212yes—
36C2×#102×C37C2212yes—
37C2×C22⋊C8×C37C2212yes—
38C37×#391—21no—
39C37×#389—21no—
40C37×#389C2212yes—
41C37×#384C2211yescharacteristic
42C37×#380—21no—
43C37×#380C2212yes—
44C37×#340C2212yes—
45C37×#340C2212yes—
46C37×#336C2212yes—
47C22×Q8×C37—21no—
48C22×Q8×C37C2311yescharacteristic
49C22×D4×C37—21no—
50C22×D4×C37C2311yescharacteristic
51C23×C4×C37—21no—
52C23×C4×C37D411yescharacteristic
53C23×C4×C37D411yescharacteristic
54C23×C4×C37C2311yescharacteristic
55C22×C8×C37C2312yes—
56C22×C8×C37C2312yes—
57C4⋊Q8×C37—22no—
58C4⋊Q8×C37—22no—
59#126×C37—28no—
60#125×C37—22no—
61#125×C37—22no—
62#124×C37—22no—
63#124×C37—22no—
64#123×C37—28no—
65C22⋊Q8×C37—22no—
66C22⋊Q8×C37—22no—
67C22⋊Q8×C37—44no—
68#121×C37—22no—
69#121×C37—22no—
70C4×Q8×C37—22no—
71C4×Q8×C37D414yes—
72C4×D4×C37—28no—
73C4×D4×C37—22no—
74C4×D4×C37D414yes—
75#117×C37—28no—
76C2×C4⋊C4×C37—21no—
77C2×C4⋊C4×C37—21no—
78C2×C4⋊C4×C37—21no—
79C2×C4⋊C4×C37—21no—
80C2×C4⋊C4×C37—21no—
81C2×C4⋊C4×C37—22no—
82C2×C4⋊C4×C37—21no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37C2312yes—
85C2×C4⋊C4×C37C2312yes—
86C2×C4⋊C4×C37C2311yescharacteristic
87C2×C4⋊C4×C37C2311yescharacteristic
88C2×C22⋊C4×C37—21no—
89C2×C22⋊C4×C37—21no—
90C2×C22⋊C4×C37—21no—
91C2×C22⋊C4×C37—21no—
92C2×C22⋊C4×C37—21no—
93C2×C22⋊C4×C37—22no—
94C2×C42×C37—21no—
95C2×C42×C37—22no—
96C2×C42×C37—22no—
97C2×C42×C37D411yescharacteristic
98C2×C42×C37D411yescharacteristic
99C2×C42×C37C2311yescharacteristic
100C2×C42×C37C2311yescharacteristic
101#107×C37—24no—
102#106×C37—24no—
103C4⋊C8×C37—22no—
104C4⋊C8×C37—22no—
105#103×C37—24no—
106#103×C37—42no—
107#102×C37—24no—
108#102×C37—42no—
109C22⋊C8×C37—42no—
110#95×C37—22no—
111#95×C37—22no—
112C24×C37—21no—
113C24×C37C2×D411yescharacteristic
114C2×Q8×C37—21no—
115C2×Q8×C37—21no—
116C2×Q8×C37—21no—
117C2×Q8×C37—21no—
118C2×Q8×C37—21no—
119C2×Q8×C37—42no—
120C2×Q8×C37—42no—
121C2×Q8×C37—42no—
122C2×Q8×C37C2×D412yes—
123C2×D4×C37—24no—
124C2×D4×C37—21no—
125C2×D4×C37—21no—
126C2×D4×C37—21no—
127C2×D4×C37—21no—
128C2×D4×C37—21no—
129C2×D4×C37—42no—
130C2×D4×C37—42no—
131C2×D4×C37C2×D412yes—
132C22×C4×C37—22no—
133C22×C4×C37—21no—
134C22×C4×C37—21no—
135C22×C4×C37—21no—
136C22×C4×C37—22no—
137C22×C4×C37—22no—
138C22×C4×C37—24no—
139C22×C4×C37—22no—
140C22×C4×C37—22no—
141C22×C4×C37—21no—
142C22×C4×C37—21no—
143C22×C4×C37—21no—
144C22×C4×C37—21no—
145C22×C4×C37—21no—
146C22×C4×C37—21no—
147C22×C4×C37—22no—
148C22×C4×C37—22no—
149C22×C4×C37—41no—
150C22×C4×C37—42no—
151C22×C4×C37—42no—
152C22×C4×C37C2×D411yescharacteristic
153C22×C4×C37C2×D411yescharacteristic
154C22×C4×C37C2×D411yescharacteristic
155C22×C4×C37C2×D411yescharacteristic
156C22×C4×C37C2×D411yescharacteristic
157C22×C4×C37C2411yescharacteristic
158C2×C8×C37—22no—
159C2×C8×C37—22no—
160C2×C8×C37—22no—
161C2×C8×C37—22no—
162C2×C8×C37—22no—
163C2×C8×C37—22no—
164C4⋊C4×C37—24no—
165C4⋊C4×C37—24no—
166C4⋊C4×C37—24no—
167C4⋊C4×C37—24no—
168C4⋊C4×C37—21no—
169C4⋊C4×C37—21no—
170C4⋊C4×C37—22no—
171C4⋊C4×C37—22no—
172C4⋊C4×C37—22no—
173C4⋊C4×C37—22no—
174C4⋊C4×C37—42no—
175C4⋊C4×C37—44no—
176C4⋊C4×C37—42no—
177C4⋊C4×C37—42no—
178C4⋊C4×C37C2×D412yes—
179C4⋊C4×C37C2×D412yes—
180C22⋊C4×C37—24no—
181C22⋊C4×C37—24no—
182C22⋊C4×C37—24no—
183C22⋊C4×C37—24no—
184C22⋊C4×C37—24no—
185C22⋊C4×C37—44no—
186C42×C37—22no—
187C42×C37—22no—
188C42×C37—24no—
189C42×C37—21no—
190C42×C37—21no—
191C42×C37C2×D412yes—
192C42×C37C2×D412yes—
193C23×C37—21no—
194C23×C37—21no—
195C23×C37—21no—
196C23×C37—21no—
197C23×C37—21no—
198C23×C37—21no—
199C23×C37—22no—
200C23×C37—41no—
201C23×C37—41no—
202C23×C37—41no—
203C23×C37—41no—
204C23×C37—41no—
205C23×C37—41no—
206C23×C37#13712yes—
207C23×C37C22×D411yescharacteristic
208Q8×C37—22no—
209Q8×C37—22no—
210Q8×C37—41no—
211Q8×C37—41no—
212Q8×C37—84no—
213D4×C37—22no—
214D4×C37—22no—
215D4×C37—48no—
216D4×C37—41no—
217D4×C37—41no—
218C2×C4×C37—21no—
219C2×C4×C37—21no—
220C2×C4×C37—21no—
221C2×C4×C37—21no—
222C2×C4×C37—21no—
223C2×C4×C37—22no—
224C2×C4×C37—22no—
225C2×C4×C37—22no—
226C2×C4×C37—21no—
227C2×C4×C37—21no—
228C2×C4×C37—21no—
229C2×C4×C37—21no—
230C2×C4×C37—21no—
231C2×C4×C37—21no—
232C2×C4×C37—44no—
233C2×C4×C37—41no—
234C2×C4×C37—41no—
235C2×C4×C37—41no—
236C2×C4×C37—41no—
237C2×C4×C37—41no—
238C2×C4×C37—41no—
239C2×C4×C37—41no—
240C2×C4×C37—41no—
241C2×C4×C37—41no—
242C2×C4×C37—41no—
243C2×C4×C37—44no—
244C2×C4×C37—48no—
245C2×C4×C37—42no—
246C2×C4×C37—42no—
247C2×C4×C37—42no—
248C2×C4×C37—42no—
249C2×C4×C37—42no—
250C2×C4×C37—42no—
251C2×C4×C37#12012yes—
252C2×C4×C37#12012yes—
253C2×C4×C37#13512yes—
254C2×C4×C37AGL(1,ℤ/8)12yes—
255C2×C4×C37C22×D411yescharacteristic
256C2×C4×C37C22×D411yescharacteristic
257C2×C4×C37H(5,𝔽2)11yescharacteristic
258C2×C4×C372−1+411yescharacteristic
259C2×C4×C372−1+411yescharacteristic
260C2×C4×C372−1+411yescharacteristic
261C8×C37—42no—
262C8×C37—42no—
263#29689C3711yesmaximal, characteristic, direct product factor, Sylow
264C22×C37—22no—
265C22×C37—22no—
266C22×C37—22no—
267C22×C37—41no—
268C22×C37—41no—
269C22×C37—41no—
270C22×C37—41no—
271C22×C37—42no—
272C22×C37—42no—
273C22×C37—42no—
274C22×C37—81no—
275C22×C37—81no—
276C22×C37—81no—
277C22×C37—81no—
278C22×C37C2×#12011yescharacteristic
279C22×C37#53611yescharacteristic
280C22×C37#53711yescharacteristic
281C22×C37C2×#13511yescharacteristic
282C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C37C2×#13711yescharacteristic
284C22×C37#57511yescharacteristic
285C4×C37—22no—
286C4×C37—41no—
287C4×C37—42no—
288C4×C37—41no—
289C4×C37—42no—
290C4×C37—41no—
291C4×C37—42no—
292C4×C37—82no—
293C4×C37—82no—
294C4×C37#44914yes—
295C4×C37#53611yescharacteristic
296C4×C37#53611yescharacteristic
297C4×C37#53711yescharacteristic
298C4×C37#53711yescharacteristic
299C2×#529C2×C3711yescharacteristic
300C2×#475C2×C3711yescharacteristic, complemented
301C2×#474C2×C3711yescharacteristic, complemented
302#2857C2×C3711yescharacteristic, complemented
303#2059C2×C3712yescomplemented
304#2042C2×C3712yescomplemented
305#2036C2×C3712yescomplemented
306#2034C2×C3712yes—
307#1803C2×C3711yescharacteristic, complemented
308#1802C2×C3711yescharacteristic, complemented
309#1797C2×C3711yescharacteristic, complemented
310C2×C37—42no—
311C2×C37—82no—
312C2×C37#261611yescharacteristic
313C2×C37C2×#44911yescharacteristic
314C2×C37#300711yescharacteristic
315C2×C37#323311yescharacteristic
316C2×C37#323911yescharacteristic
317C2×C37#324811yescharacteristic
318C2×C37#325111yescharacteristic
319C2×C4⋊Q8C22×C3711yescharacteristic
320#529—24no—
321#529—24no—
322C2×#126—21no—
323C2×#125C22×C3711yescharacteristic
324C2×#124C22×C3711yescharacteristic
325C2×#123—21no—
326C2×C22⋊Q8—21no—
327C2×C22⋊Q8C22×C3711yescharacteristic
328C2×#121C22×C3711yescharacteristic
329C2×C4×Q8C22×C3711yescharacteristic
330C2×C4×D4—21no—
331C2×C4×D4C22×C3711yescharacteristic
332C2×#117—21no—
333C22×C42C22×C3711yescharacteristic
334#475—24no—
335#474—24no—
336C2×#107C22×C3712yes—
337C2×#106C22×C3712yes—
338C2×C4⋊C8C22×C3711yescharacteristic
339C2×C4⋊C8C22×C3711yescharacteristic
340C2×#103C22×C3712yes—
341C2×#103C22×C3712yes—
342C2×#102C22×C3712yes—
343C2×#102C22×C3712yes—
344C2×C22⋊C8C22×C3712yes—
345#391—21no—
346#389—21no—
347#389C22×C3712yes—
348#384C22×C3711yescharacteristic
349#380—21no—
350#380C22×C3712yes—
351#340C22×C3712yes—
352#340C22×C3712yes—
353#336C22×C3712yes—
354C37#2968911yescharacteristic, direct product factor, Sylow
355C22×Q8—21no—
356C22×Q8C23×C3711yescharacteristic
357C22×D4—21no—
358C22×D4C23×C3711yescharacteristic
359C23×C4—21no—
360C23×C4D4×C3711yescharacteristic
361C23×C4D4×C3711yescharacteristic
362C23×C4C23×C3711yescharacteristic
363C22×C8C23×C3712yes—
364C22×C8C23×C3712yes—
365C4⋊Q8—22no—
366C4⋊Q8—22no—
367#126—28no—
368#125—22no—
369#125—22no—
370#124—22no—
371#124—22no—
372#123—28no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—44no—
376#121—22no—
377#121—22no—
378C4×Q8—22no—
379C4×Q8D4×C3714yes—
380C4×D4—28no—
381C4×D4—22no—
382C4×D4D4×C3714yes—
383#117—28no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4C23×C3712yes—
393C2×C4⋊C4C23×C3712yes—
394C2×C4⋊C4C23×C3711yescharacteristic
395C2×C4⋊C4C23×C3711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—22no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42D4×C3711yescharacteristic
406C2×C42D4×C3711yescharacteristic
407C2×C42C23×C3711yescharacteristic
408C2×C42C23×C3711yescharacteristic
409#107—24no—
410#106—24no—
411C4⋊C8—22no—
412C4⋊C8—22no—
413#103—24no—
414#103—42no—
415#102—24no—
416#102—42no—
417C22⋊C8—42no—
418#95—22no—
419#95—22no—
420C24—21no—
421C24C2×D4×C3711yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×D4×C3712yes—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×D4×C3712yes—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4C2×D4×C3711yescharacteristic
461C22×C4C2×D4×C3711yescharacteristic
462C22×C4C2×D4×C3711yescharacteristic
463C22×C4C2×D4×C3711yescharacteristic
464C22×C4C2×D4×C3711yescharacteristic
465C22×C4C24×C3711yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4C2×D4×C3712yes—
487C4⋊C4C2×D4×C3712yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—21no—
498C42—21no—
499C42C2×D4×C3712yes—
500C42C2×D4×C3712yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23#137×C3712yes—
515C23C22×D4×C3711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—84no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4#120×C3712yes—
560C2×C4#120×C3712yes—
561C2×C4#135×C3712yes—
562C2×C4AGL(1,ℤ/8)×C3712yes—
563C2×C4C22×D4×C3711yescharacteristic
564C2×C4C22×D4×C3711yescharacteristic
565C2×C4H(5,𝔽2)×C3711yescharacteristic
566C2×C42−1+4×C3711yescharacteristic
567C2×C42−1+4×C3711yescharacteristic
568C2×C42−1+4×C3711yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#120×C3711yescharacteristic
586C22C37×#53611yescharacteristic
587C22C37×#53711yescharacteristic
588C22C2×#135×C3711yescharacteristic
589C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
590C22C2×#137×C3711yescharacteristic
591C22C37×#57511yescharacteristic
592C4—22no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—42no—
599C4—82no—
600C4—82no—
601C4C37×#44914yes—
602C4C37×#53611yescharacteristic
603C4C37×#53611yescharacteristic
604C4C37×#53711yescharacteristic
605C4C37×#53711yescharacteristic
606C2—42no—
607C2—82no—
608C2C37×#261611yescharacteristic
609C2C2×C37×#44911yescharacteristic
610C2C37×#300711yescharacteristic
611C2C37×#323311yescharacteristic
612C2C37×#323911yescharacteristic
613C2C37×#324811yescharacteristic
614C2C37×#325111yescharacteristic
subgroupG/H
center, socleC23×C37C22×D4
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×#121×C37C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C37 ◁ C22×C42×C37 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C37×#475C2semidirect
3. C2×C37×#474C2semidirect
4. C37×#2857C2semidirect
5. C37×#2059C2semidirect
6. C37×#2042C2semidirect
7. C37×#2036C2semidirect
9. C37×#1803C2semidirect
10. C37×#1802C2semidirect
11. C37×#1797C2semidirect
263. #29689C37direct (N × H)
300. C2×#475C2×C37semidirect
301. C2×#474C2×C37semidirect
302. #2857C2×C37semidirect
303. #2059C2×C37semidirect
304. #2042C2×C37semidirect
305. #2036C2×C37semidirect
307. #1803C2×C37semidirect
308. #1802C2×C37semidirect
309. #1797C2×C37semidirect
354. C37#29689direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#5292
C2×C37×#4752
C2×C37×#4742
C37×#28572
C37×#20592
C37×#20422
C37×#20362
C37×#20342
C37×#18032
C37×#18022
C37×#17972
#2968937