Finite Group Databasebeta

← C37×#16399 #22637652

Subgroups — C37×#16399 #22637652

Order 9472.

1538 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#398C211yesmaximal, characteristic, complemented
2C2×C37×#397C212yesmaximal
3C2×C37×#385C211yesmaximal, characteristic
4C2×C37×#380C211yesmaximal, characteristic
5C2×C37×#380C211yesmaximal, characteristic
6C37×#1445C211yesmaximal, characteristic, complemented
7C37×#1442C211yesmaximal, characteristic, complemented
8C37×#1441C211yesmaximal, characteristic, complemented
9C37×#1441C212yesmaximal, complemented
10C37×#1441C212yesmaximal, complemented
11C37×#1440C212yesmaximal
12C24×C4×C37C2211yescharacteristic
13C22×C4⋊C4×C37C2211yescharacteristic
14C22×C4⋊C4×C37C2211yescharacteristic
15C22×C4⋊C4×C37C2211yescharacteristic
16C22×C4⋊C4×C37C2212yes—
17C22×C4⋊C4×C37C2212yes—
18C22×C4⋊C4×C37C2211yescharacteristic
19C22×C22⋊C4×C37C2212yes—
20C22×C22⋊C4×C37C2212yes—
21C22×C22⋊C4×C37C2211yescharacteristic
22C22×C22⋊C4×C37C2211yescharacteristic
23C37×#398—24no—
24C37×#397—28no—
25C37×#385—24no—
26C37×#380—24no—
27C37×#380—24no—
28C2×#95×C37C2212yes—
29C2×#95×C37C2212yes—
30C2×#95×C37C2212yes—
31C2×#95×C37C2212yes—
32C2×#95×C37C2212yes—
33C2×#95×C37C2211yescharacteristic
34C2×#95×C37C2211yescharacteristic
35C2×#95×C37C2211yescharacteristic
36C2×#95×C37C2212yes—
37C2×#95×C37C2212yes—
38C2×#95×C37C2212yes—
39C2×#95×C37C2211yescharacteristic
40C25×C37C2311yescharacteristic
41C23×C4×C37—22no—
42C23×C4×C37—22no—
43C23×C4×C37—24no—
44C23×C4×C37—42no—
45C23×C4×C37D414yes—
46C23×C4×C37C2311yescharacteristic
47C23×C4×C37C2312yes—
48C23×C4×C37C2312yes—
49C23×C4×C37C2311yescharacteristic
50C23×C4×C37C2312yes—
51C23×C4×C37C2311yescharacteristic
52C23×C4×C37C2312yes—
53C23×C4×C37C2311yescharacteristic
54C23×C4×C37C2311yescharacteristic
55C23×C4×C37C2311yescharacteristic
56C2×C4⋊C4×C37—24no—
57C2×C4⋊C4×C37—22no—
58C2×C4⋊C4×C37—22no—
59C2×C4⋊C4×C37—22no—
60C2×C4⋊C4×C37—22no—
61C2×C4⋊C4×C37—24no—
62C2×C4⋊C4×C37—24no—
63C2×C4⋊C4×C37—22no—
64C2×C4⋊C4×C37—22no—
65C2×C4⋊C4×C37—24no—
66C2×C4⋊C4×C37—22no—
67C2×C4⋊C4×C37—22no—
68C2×C4⋊C4×C37—24no—
69C2×C4⋊C4×C37—42no—
70C2×C4⋊C4×C37—42no—
71C2×C4⋊C4×C37—42no—
72C2×C22⋊C4×C37—22no—
73C2×C22⋊C4×C37—22no—
74C2×C22⋊C4×C37—24no—
75C2×C22⋊C4×C37—22no—
76C2×C22⋊C4×C37—24no—
77C2×C22⋊C4×C37—24no—
78C2×C22⋊C4×C37—24no—
79C2×C22⋊C4×C37—24no—
80C2×C22⋊C4×C37—42no—
81C2×C22⋊C4×C37—42no—
82C2×C22⋊C4×C37D414yes—
83#95×C37—22no—
84#95×C37—24no—
85#95×C37—22no—
86#95×C37—22no—
87#95×C37—24no—
88#95×C37—24no—
89#95×C37—24no—
90#95×C37—24no—
91#95×C37—42no—
92#95×C37—42no—
93#95×C37—42no—
94#95×C37D414yes—
95C24×C37—21no—
96C24×C37—21no—
97C24×C37—22no—
98C24×C37—21no—
99C24×C37—22no—
100C24×C37—22no—
101C24×C37—22no—
102C24×C37—21no—
103C24×C37C2×D412yes—
104C24×C37C2×D412yes—
105C24×C37#4112yes—
106C24×C37C2411yescharacteristic
107C22×C4×C37—21no—
108C22×C4×C37—21no—
109C22×C4×C37—22no—
110C22×C4×C37—22no—
111C22×C4×C37—22no—
112C22×C4×C37—21no—
113C22×C4×C37—21no—
114C22×C4×C37—22no—
115C22×C4×C37—21no—
116C22×C4×C37—21no—
117C22×C4×C37—21no—
118C22×C4×C37—22no—
119C22×C4×C37—21no—
120C22×C4×C37—22no—
121C22×C4×C37—22no—
122C22×C4×C37—22no—
123C22×C4×C37—21no—
124C22×C4×C37—22no—
125C22×C4×C37—22no—
126C22×C4×C37—22no—
127C22×C4×C37—21no—
128C22×C4×C37—28no—
129C22×C4×C37—28no—
130C22×C4×C37—22no—
131C22×C4×C37—21no—
132C22×C4×C37—22no—
133C22×C4×C37—21no—
134C22×C4×C37—22no—
135C22×C4×C37—22no—
136C22×C4×C37—22no—
137C22×C4×C37—22no—
138C22×C4×C37—22no—
139C22×C4×C37—21no—
140C22×C4×C37—22no—
141C22×C4×C37—22no—
142C22×C4×C37—22no—
143C22×C4×C37—21no—
144C22×C4×C37—22no—
145C22×C4×C37—22no—
146C22×C4×C37—21no—
147C22×C4×C37—22no—
148C22×C4×C37—22no—
149C22×C4×C37—22no—
150C22×C4×C37—22no—
151C22×C4×C37—22no—
152C22×C4×C37—22no—
153C22×C4×C37—22no—
154C22×C4×C37—22no—
155C22×C4×C37—22no—
156C22×C4×C37—22no—
157C22×C4×C37—22no—
158C22×C4×C37—22no—
159C22×C4×C37—22no—
160C22×C4×C37—21no—
161C22×C4×C37—44no—
162C22×C4×C37—48no—
163C22×C4×C37—48no—
164C22×C4×C37C2×D412yes—
165C22×C4×C37C2×D412yes—
166C22×C4×C37C2×D414yes—
167C22×C4×C37C2×D412yes—
168C22×C4×C37C2×D412yes—
169C22×C4×C37C2×D412yes—
170C22×C4×C37#4112yes—
171C22×C4×C37#4114yes—
172C4⋊C4×C37—48no—
173C4⋊C4×C37—48no—
174C4⋊C4×C37—44no—
175C4⋊C4×C37—44no—
176C4⋊C4×C37—44no—
177C4⋊C4×C37—44no—
178C22⋊C4×C37—28no—
179C22⋊C4×C37—48no—
180C22⋊C4×C37—44no—
181C22⋊C4×C37—48no—
182C23×C37—22no—
183C23×C37—22no—
184C23×C37—22no—
185C23×C37—22no—
186C23×C37—22no—
187C23×C37—24no—
188C23×C37—22no—
189C23×C37—24no—
190C23×C37—22no—
191C23×C37—22no—
192C23×C37—24no—
193C23×C37—22no—
194C23×C37—24no—
195C23×C37—22no—
196C23×C37—42no—
197C23×C37—42no—
198C23×C37—42no—
199C23×C37—42no—
200C23×C37—42no—
201C23×C37—42no—
202C23×C37—42no—
203C23×C37—42no—
204C23×C37#12114yes—
205C23×C37C22×D411yescharacteristic
206C23×C37C22×D411yescharacteristic
207C23×C37C22×D411yescharacteristic
208C23×C37C2×#4111yescharacteristic
209C23×C37C2×#4111yescharacteristic
210C23×C37C2×#4112yes—
211C23×C37H(5,𝔽2)12yes—
212C23×C37H(5,𝔽2)11yescharacteristic
213C23×C372−1+412yes—
214C23×C372−1+412yes—
215C23×C372−1+411yescharacteristic
216C2×C4×C37—22no—
217C2×C4×C37—22no—
218C2×C4×C37—22no—
219C2×C4×C37—22no—
220C2×C4×C37—24no—
221C2×C4×C37—24no—
222C2×C4×C37—24no—
223C2×C4×C37—24no—
224C2×C4×C37—22no—
225C2×C4×C37—22no—
226C2×C4×C37—24no—
227C2×C4×C37—24no—
228C2×C4×C37—22no—
229C2×C4×C37—24no—
230C2×C4×C37—22no—
231C2×C4×C37—24no—
232C2×C4×C37—416no—
233C2×C4×C37—41no—
234C2×C4×C37—41no—
235C2×C4×C37—41no—
236C2×C4×C37—41no—
237C2×C4×C37—42no—
238C2×C4×C37—42no—
239C2×C4×C37—42no—
240C2×C4×C37—41no—
241C2×C4×C37—42no—
242C2×C4×C37—41no—
243C2×C4×C37—42no—
244C2×C4×C37—42no—
245C2×C4×C37—42no—
246C2×C4×C37—41no—
247C2×C4×C37—42no—
248C2×C4×C37—41no—
249C2×C4×C37—41no—
250C2×C4×C37—41no—
251C2×C4×C37—42no—
252C2×C4×C37—42no—
253C2×C4×C37—42no—
254C2×C4×C37—41no—
255C2×C4×C37—42no—
256C2×C4×C37—42no—
257C2×C4×C37—42no—
258C2×C4×C37—42no—
259C2×C4×C37—41no—
260C2×C4×C37—42no—
261C2×C4×C37—42no—
262C2×C4×C37—42no—
263C2×C4×C37—42no—
264C2×C4×C37—42no—
265C2×C4×C37—42no—
266C2×C4×C37—42no—
267C2×C4×C37—42no—
268C2×C4×C37—42no—
269C2×C4×C37—42no—
270C2×C4×C37—42no—
271C2×C4×C37—42no—
272C2×C4×C37—42no—
273C2×C4×C37—42no—
274C2×C4×C37—42no—
275C2×C4×C37#12114yes—
276C2×C4×C37#12314yes—
277#16399C3711yesmaximal, characteristic, direct product factor, Sylow
278C22×C37—24no—
279C22×C37—24no—
280C22×C37—24no—
281C22×C37—42no—
282C22×C37—42no—
283C22×C37—42no—
284C22×C37—44no—
285C22×C37—44no—
286C22×C37—44no—
287C22×C37—44no—
288C22×C37—42no—
289C22×C37C2×#12111yescharacteristic
290C22×C37C2×#12111yescharacteristic
291C22×C37C2×#12311yescharacteristic
292C22×C37#53411yescharacteristic
293C22×C37#53611yescharacteristic
294C22×C37#53612yes—
295C22×C37#53611yescharacteristic
296C22×C37#53712yes—
297C22×C37#53711yescharacteristic
298C22×C37#53912yes—
299C22×C37#54112yes—
300C22×C37#54112yes—
301C22×C37#54212yes—
302C22×C37D4211yescharacteristic
303C22×C37#54612yes—
304C22×C37#54711yescharacteristic
305C22×C37#54712yes—
306C22×C37#54811yescharacteristic
307C22×C37#55112yes—
308C22×C37#55212yes—
309C22×C37#55311yescharacteristic
310C22×C37#56212yes—
311C22×C37#56312yes—
312C4×C37—24no—
313C4×C37—24no—
314C4×C37—42no—
315C4×C37—44no—
316C4×C37—42no—
317C4×C37—82no—
318C4×C37—81no—
319C4×C37—82no—
320C4×C37—82no—
321C4×C37—81no—
322C2×#398C2×C3711yescharacteristic, complemented
323C2×#397C2×C3712yes—
324C2×#385C2×C3711yescharacteristic
325C2×#380C2×C3711yescharacteristic
326C2×#380C2×C3711yescharacteristic
327#1445C2×C3711yescharacteristic, complemented
328#1442C2×C3711yescharacteristic, complemented
329#1441C2×C3711yescharacteristic, complemented
330#1441C2×C3712yescomplemented
331#1441C2×C3712yescomplemented
332#1440C2×C3712yes—
333C2×C37—44no—
334C2×C37#241211yescharacteristic
335C2×C37#241611yescharacteristic
336C2×C37#248611yescharacteristic
337C2×C37#261811yescharacteristic
338C2×C37#262111yescharacteristic
339C2×C37#262511yescharacteristic
340C2×C37#267212yes—
341C2×C37#269012yes—
342C2×C37#272212yes—
343C2×C37#274712yes—
344C2×C37#281411yescharacteristic
345C24×C4C22×C3711yescharacteristic
346C22×C4⋊C4C22×C3711yescharacteristic
347C22×C4⋊C4C22×C3711yescharacteristic
348C22×C4⋊C4C22×C3711yescharacteristic
349C22×C4⋊C4C22×C3712yes—
350C22×C4⋊C4C22×C3712yes—
351C22×C4⋊C4C22×C3711yescharacteristic
352C22×C22⋊C4C22×C3712yes—
353C22×C22⋊C4C22×C3712yes—
354C22×C22⋊C4C22×C3711yescharacteristic
355C22×C22⋊C4C22×C3711yescharacteristic
356#398—24no—
357#397—28no—
358#385—24no—
359#380—24no—
360#380—24no—
361C2×#95C22×C3712yes—
362C2×#95C22×C3712yes—
363C2×#95C22×C3712yes—
364C2×#95C22×C3712yes—
365C2×#95C22×C3712yes—
366C2×#95C22×C3711yescharacteristic
367C2×#95C22×C3711yescharacteristic
368C2×#95C22×C3711yescharacteristic
369C2×#95C22×C3712yes—
370C2×#95C22×C3712yes—
371C2×#95C22×C3712yes—
372C2×#95C22×C3711yescharacteristic
373C37#1639911yescharacteristic, direct product factor, Sylow
374C25C23×C3711yescharacteristic
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—24no—
378C23×C4—42no—
379C23×C4D4×C3714yes—
380C23×C4C23×C3711yescharacteristic
381C23×C4C23×C3712yes—
382C23×C4C23×C3712yes—
383C23×C4C23×C3712yes—
384C23×C4C23×C3711yescharacteristic
385C23×C4C23×C3711yescharacteristic
386C23×C4C23×C3712yes—
387C23×C4C23×C3711yescharacteristic
388C23×C4C23×C3711yescharacteristic
389C23×C4C23×C3711yescharacteristic
390C2×C4⋊C4—24no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—42no—
405C2×C4⋊C4—42no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4D4×C3714yes—
417#95—22no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—42no—
426#95—42no—
427#95—42no—
428#95D4×C3714yes—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24C2×D4×C3712yes—
438C24C2×D4×C3712yes—
439C24#41×C3712yes—
440C24C24×C3711yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
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—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—44no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4C2×D4×C3712yes—
499C22×C4C2×D4×C3712yes—
500C22×C4C2×D4×C3714yes—
501C22×C4C2×D4×C3712yes—
502C22×C4C2×D4×C3712yes—
503C22×C4C2×D4×C3712yes—
504C22×C4#41×C3712yes—
505C22×C4#41×C3714yes—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#121×C3714yes—
539C23C22×D4×C3711yescharacteristic
540C23C22×D4×C3711yescharacteristic
541C23C22×D4×C3711yescharacteristic
542C23C2×#41×C3711yescharacteristic
543C23C2×#41×C3711yescharacteristic
544C23C2×#41×C3712yes—
545C23H(5,𝔽2)×C3712yes—
546C23H(5,𝔽2)×C3711yescharacteristic
547C232−1+4×C3712yes—
548C232−1+4×C3712yes—
549C232−1+4×C3711yescharacteristic
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—416no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4#121×C3714yes—
610C2×C4#123×C3714yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22C2×#121×C3711yescharacteristic
623C22C2×#121×C3711yescharacteristic
624C22C2×#123×C3711yescharacteristic
625C22C37×#53411yescharacteristic
626C22C37×#53611yescharacteristic
627C22C37×#53612yes—
628C22C37×#53611yescharacteristic
629C22C37×#53712yes—
630C22C37×#53711yescharacteristic
631C22C37×#53912yes—
632C22C37×#54112yes—
633C22C37×#54112yes—
634C22C37×#54212yes—
635C22D42×C3711yescharacteristic
636C22C37×#54612yes—
637C22C37×#54711yescharacteristic
638C22C37×#54712yes—
639C22C37×#54811yescharacteristic
640C22C37×#55112yes—
641C22C37×#55212yes—
642C22C37×#55311yescharacteristic
643C22C37×#56212yes—
644C22C37×#56312yes—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—81no—
655C2—44no—
656C2C37×#241211yescharacteristic
657C2C37×#241611yescharacteristic
658C2C37×#248611yescharacteristic
659C2C37×#261811yescharacteristic
660C2C37×#262111yescharacteristic
661C2C37×#262511yescharacteristic
662C2C37×#267212yes—
663C2C37×#269012yes—
664C2C37×#272212yes—
665C2C37×#274712yes—
666C2C37×#281411yescharacteristic
subgroupG/H
center, socleC24×C37C24
commutator, Frattini, cosocleC24C24×C37
prime radicalC25×C37C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C37×#398C2semidirect
6. C37×#1445C2semidirect
7. C37×#1442C2semidirect
8. C37×#1441C2semidirect
9. C37×#1441C2semidirect
10. C37×#1441C2semidirect
277. #16399C37direct (N × H)
322. C2×#398C2×C37semidirect
327. #1445C2×C37semidirect
328. #1442C2×C37semidirect
329. #1441C2×C37semidirect
330. #1441C2×C37semidirect
331. #1441C2×C37semidirect
373. C37#16399direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#3982
C2×C37×#3972
C2×C37×#3852
C2×C37×#3802
C37×#14452
C37×#14422
C37×#14412
C37×#14402
#1639937