Finite Group Databasebeta

← C37×#16400 #22637653

Subgroups — C37×#16400 #22637653

Order 9472.

1498 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#400C211yesmaximal, characteristic, complemented
2C2×C37×#385C211yesmaximal, characteristic
3C2×C37×#380C211yesmaximal, characteristic
4C2×C37×#380C211yesmaximal, characteristic
5C37×#1445C211yesmaximal, characteristic, complemented
6C37×#1444C211yesmaximal, characteristic, complemented
7C37×#1441C212yesmaximal, complemented
8C37×#1441C212yesmaximal, complemented
9C37×#1441C211yesmaximal, characteristic, complemented
10C37×#1440C212yesmaximal
11C37×#1439C212yesmaximal
12C24×C4×C37C2211yescharacteristic
13C22×C4⋊C4×C37C2211yescharacteristic
14C22×C4⋊C4×C37C2211yescharacteristic
15C22×C4⋊C4×C37C2211yescharacteristic
16C22×C4⋊C4×C37C2212yes—
17C22×C4⋊C4×C37C2211yescharacteristic
18C22×C4⋊C4×C37C2212yes—
19C22×C22⋊C4×C37C2212yes—
20C22×C22⋊C4×C37C2211yescharacteristic
21C22×C22⋊C4×C37C2211yescharacteristic
22C22×C22⋊C4×C37C2212yes—
23C37×#400—24no—
24C37×#385—24no—
25C37×#380—24no—
26C37×#380—24no—
27C2×#95×C37C2212yes—
28C2×#95×C37C2212yes—
29C2×#95×C37C2212yes—
30C2×#95×C37C2211yescharacteristic
31C2×#95×C37C2211yescharacteristic
32C2×#95×C37C2211yescharacteristic
33C2×#95×C37C2212yes—
34C2×#95×C37C2212yes—
35C2×#95×C37C2212yes—
36C2×#95×C37C2212yes—
37C2×#95×C37C2212yes—
38C2×#95×C37C2211yescharacteristic
39C25×C37C2311yescharacteristic
40C23×C4×C37—22no—
41C23×C4×C37—22no—
42C23×C4×C37—24no—
43C23×C4×C37—42no—
44C23×C4×C37D414yes—
45C23×C4×C37C2312yes—
46C23×C4×C37C2312yes—
47C23×C4×C37C2311yescharacteristic
48C23×C4×C37C2311yescharacteristic
49C23×C4×C37C2312yes—
50C23×C4×C37C2312yes—
51C23×C4×C37C2311yescharacteristic
52C23×C4×C37C2311yescharacteristic
53C23×C4×C37C2311yescharacteristic
54C23×C4×C37C2311yescharacteristic
55C2×C4⋊C4×C37—24no—
56C2×C4⋊C4×C37—22no—
57C2×C4⋊C4×C37—22no—
58C2×C4⋊C4×C37—22no—
59C2×C4⋊C4×C37—22no—
60C2×C4⋊C4×C37—22no—
61C2×C4⋊C4×C37—22no—
62C2×C4⋊C4×C37—24no—
63C2×C4⋊C4×C37—22no—
64C2×C4⋊C4×C37—22no—
65C2×C4⋊C4×C37—24no—
66C2×C4⋊C4×C37—42no—
67C2×C4⋊C4×C37—42no—
68C2×C4⋊C4×C37—42no—
69C2×C4⋊C4×C37—42no—
70C2×C4⋊C4×C37—42no—
71C2×C22⋊C4×C37—22no—
72C2×C22⋊C4×C37—22no—
73C2×C22⋊C4×C37—24no—
74C2×C22⋊C4×C37—22no—
75C2×C22⋊C4×C37—24no—
76C2×C22⋊C4×C37—24no—
77C2×C22⋊C4×C37—42no—
78C2×C22⋊C4×C37—42no—
79C2×C22⋊C4×C37—42no—
80C2×C22⋊C4×C37—42no—
81C2×C22⋊C4×C37Q814yes—
82#95×C37—22no—
83#95×C37—24no—
84#95×C37—22no—
85#95×C37—22no—
86#95×C37—24no—
87#95×C37—24no—
88#95×C37—42no—
89#95×C37—42no—
90#95×C37—42no—
91#95×C37—42no—
92#95×C37—42no—
93#95×C37D414yes—
94C24×C37—21no—
95C24×C37—21no—
96C24×C37—22no—
97C24×C37—22no—
98C24×C37—21no—
99C24×C37—22no—
100C24×C37—22no—
101C24×C37—21no—
102C24×C37C2×D412yes—
103C24×C37C2×Q812yes—
104C24×C37#4112yes—
105C24×C37C2411yescharacteristic
106C22×C4×C37—21no—
107C22×C4×C37—21no—
108C22×C4×C37—22no—
109C22×C4×C37—22no—
110C22×C4×C37—22no—
111C22×C4×C37—21no—
112C22×C4×C37—21no—
113C22×C4×C37—21no—
114C22×C4×C37—21no—
115C22×C4×C37—21no—
116C22×C4×C37—22no—
117C22×C4×C37—22no—
118C22×C4×C37—21no—
119C22×C4×C37—22no—
120C22×C4×C37—22no—
121C22×C4×C37—21no—
122C22×C4×C37—22no—
123C22×C4×C37—22no—
124C22×C4×C37—22no—
125C22×C4×C37—22no—
126C22×C4×C37—21no—
127C22×C4×C37—28no—
128C22×C4×C37—28no—
129C22×C4×C37—21no—
130C22×C4×C37—22no—
131C22×C4×C37—21no—
132C22×C4×C37—22no—
133C22×C4×C37—22no—
134C22×C4×C37—22no—
135C22×C4×C37—21no—
136C22×C4×C37—22no—
137C22×C4×C37—22no—
138C22×C4×C37—21no—
139C22×C4×C37—22no—
140C22×C4×C37—22no—
141C22×C4×C37—22no—
142C22×C4×C37—22no—
143C22×C4×C37—22no—
144C22×C4×C37—22no—
145C22×C4×C37—22no—
146C22×C4×C37—22no—
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—21no—
155C22×C4×C37—22no—
156C22×C4×C37—22no—
157C22×C4×C37—22no—
158C22×C4×C37—21no—
159C22×C4×C37—22no—
160C22×C4×C37—44no—
161C22×C4×C37—48no—
162C22×C4×C37—48no—
163C22×C4×C37C2×D414yes—
164C22×C4×C37C2×D412yes—
165C22×C4×C37C2×D412yes—
166C22×C4×C37C2×D412yes—
167C22×C4×C37C2×Q812yes—
168C22×C4×C37C2×Q812yes—
169C22×C4×C37#4112yes—
170C22×C4×C37#4114yes—
171C4⋊C4×C37—48no—
172C4⋊C4×C37—44no—
173C4⋊C4×C37—48no—
174C4⋊C4×C37—44no—
175C4⋊C4×C37—44no—
176C4⋊C4×C37—44no—
177C22⋊C4×C37—28no—
178C22⋊C4×C37—48no—
179C22⋊C4×C37—44no—
180C22⋊C4×C37—48no—
181C23×C37—22no—
182C23×C37—22no—
183C23×C37—22no—
184C23×C37—22no—
185C23×C37—22no—
186C23×C37—22no—
187C23×C37—24no—
188C23×C37—24no—
189C23×C37—22no—
190C23×C37—22no—
191C23×C37—22no—
192C23×C37—24no—
193C23×C37—22no—
194C23×C37—24no—
195C23×C37—42no—
196C23×C37—42no—
197C23×C37—42no—
198C23×C37—42no—
199C23×C37—42no—
200C23×C37—42no—
201C23×C37—42no—
202C23×C37—42no—
203C23×C37C22⋊Q814yes—
204C23×C37C22×D411yescharacteristic
205C23×C37C22×D411yescharacteristic
206C23×C37C22×Q811yescharacteristic
207C23×C37C2×#4111yescharacteristic
208C23×C37C2×#4111yescharacteristic
209C23×C37C2×#4112yes—
210C23×C37H(5,𝔽2)12yes—
211C23×C37H(5,𝔽2)12yes—
212C23×C37H(5,𝔽2)11yescharacteristic
213C23×C372−1+412yes—
214C23×C372−1+411yescharacteristic
215C2×C4×C37—22no—
216C2×C4×C37—22no—
217C2×C4×C37—22no—
218C2×C4×C37—22no—
219C2×C4×C37—24no—
220C2×C4×C37—24no—
221C2×C4×C37—24no—
222C2×C4×C37—24no—
223C2×C4×C37—22no—
224C2×C4×C37—22no—
225C2×C4×C37—24no—
226C2×C4×C37—24no—
227C2×C4×C37—24no—
228C2×C4×C37—22no—
229C2×C4×C37—22no—
230C2×C4×C37—24no—
231C2×C4×C37—416no—
232C2×C4×C37—41no—
233C2×C4×C37—41no—
234C2×C4×C37—41no—
235C2×C4×C37—41no—
236C2×C4×C37—42no—
237C2×C4×C37—42no—
238C2×C4×C37—41no—
239C2×C4×C37—41no—
240C2×C4×C37—42no—
241C2×C4×C37—42no—
242C2×C4×C37—42no—
243C2×C4×C37—42no—
244C2×C4×C37—41no—
245C2×C4×C37—41no—
246C2×C4×C37—41no—
247C2×C4×C37—41no—
248C2×C4×C37—42no—
249C2×C4×C37—42no—
250C2×C4×C37—42no—
251C2×C4×C37—41no—
252C2×C4×C37—42no—
253C2×C4×C37—42no—
254C2×C4×C37—42no—
255C2×C4×C37—42no—
256C2×C4×C37—41no—
257C2×C4×C37—42no—
258C2×C4×C37—42no—
259C2×C4×C37—42no—
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#12114yes—
275C2×C4×C37#12314yes—
276#16400C3711yesmaximal, characteristic, direct product factor, Sylow
277C22×C37—24no—
278C22×C37—24no—
279C22×C37—24no—
280C22×C37—42no—
281C22×C37—42no—
282C22×C37—42no—
283C22×C37—44no—
284C22×C37—44no—
285C22×C37—44no—
286C22×C37—44no—
287C22×C37—42no—
288C22×C37C2×#12111yescharacteristic
289C22×C37C2×C22⋊Q811yescharacteristic
290C22×C37C2×#12311yescharacteristic
291C22×C37#53411yescharacteristic
292C22×C37#53511yescharacteristic
293C22×C37#53611yescharacteristic
294C22×C37#53712yes—
295C22×C37#53711yescharacteristic
296C22×C37#53912yes—
297C22×C37#54112yes—
298C22×C37#54212yes—
299C22×C37#54212yes—
300C22×C37#54412yes—
301C22×C37#54612yes—
302C22×C37#54611yescharacteristic
303C22×C37D4×Q811yescharacteristic
304C22×C37#55112yes—
305C22×C37#55211yescharacteristic
306C22×C37#55312yes—
307C22×C37#55411yescharacteristic
308C22×C37#55412yes—
309C22×C37#56012yes—
310C22×C37#56212yes—
311C4×C37—24no—
312C4×C37—24no—
313C4×C37—42no—
314C4×C37—44no—
315C4×C37—42no—
316C4×C37—82no—
317C4×C37—81no—
318C4×C37—82no—
319C4×C37—81no—
320C4×C37—82no—
321C2×#400C2×C3711yescharacteristic, complemented
322C2×#385C2×C3711yescharacteristic
323C2×#380C2×C3711yescharacteristic
324C2×#380C2×C3711yescharacteristic
325#1445C2×C3711yescharacteristic, complemented
326#1444C2×C3711yescharacteristic, complemented
327#1441C2×C3711yescharacteristic, complemented
328#1441C2×C3712yescomplemented
329#1441C2×C3712yescomplemented
330#1440C2×C3712yes—
331#1439C2×C3712yes—
332C2×C37—44no—
333C2×C37#241211yescharacteristic
334C2×C37#241711yescharacteristic
335C2×C37#248711yescharacteristic
336C2×C37#262011yescharacteristic
337C2×C37#262511yescharacteristic
338C2×C37#262711yescharacteristic
339C2×C37#269112yes—
340C2×C37#270412yes—
341C2×C37#274212yes—
342C2×C37#274712yes—
343C2×C37#281311yescharacteristic
344C24×C4C22×C3711yescharacteristic
345C22×C4⋊C4C22×C3711yescharacteristic
346C22×C4⋊C4C22×C3711yescharacteristic
347C22×C4⋊C4C22×C3711yescharacteristic
348C22×C4⋊C4C22×C3712yes—
349C22×C4⋊C4C22×C3711yescharacteristic
350C22×C4⋊C4C22×C3712yes—
351C22×C22⋊C4C22×C3712yes—
352C22×C22⋊C4C22×C3711yescharacteristic
353C22×C22⋊C4C22×C3711yescharacteristic
354C22×C22⋊C4C22×C3712yes—
355#400—24no—
356#385—24no—
357#380—24no—
358#380—24no—
359C2×#95C22×C3712yes—
360C2×#95C22×C3712yes—
361C2×#95C22×C3712yes—
362C2×#95C22×C3711yescharacteristic
363C2×#95C22×C3711yescharacteristic
364C2×#95C22×C3711yescharacteristic
365C2×#95C22×C3712yes—
366C2×#95C22×C3712yes—
367C2×#95C22×C3712yes—
368C2×#95C22×C3712yes—
369C2×#95C22×C3712yes—
370C2×#95C22×C3711yescharacteristic
371C37#1640011yescharacteristic, direct product factor, Sylow
372C25C23×C3711yescharacteristic
373C23×C4—22no—
374C23×C4—22no—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4D4×C3714yes—
378C23×C4C23×C3711yescharacteristic
379C23×C4C23×C3711yescharacteristic
380C23×C4C23×C3712yes—
381C23×C4C23×C3712yes—
382C23×C4C23×C3712yes—
383C23×C4C23×C3712yes—
384C23×C4C23×C3711yescharacteristic
385C23×C4C23×C3711yescharacteristic
386C23×C4C23×C3711yescharacteristic
387C23×C4C23×C3711yescharacteristic
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—42no—
401C2×C4⋊C4—42no—
402C2×C4⋊C4—42no—
403C2×C4⋊C4—42no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—42no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4Q8×C3714yes—
415#95—22no—
416#95—24no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—42no—
422#95—42no—
423#95—42no—
424#95—42no—
425#95—42no—
426#95D4×C3714yes—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—21no—
435C24C2×D4×C3712yes—
436C24C2×Q8×C3712yes—
437C24#41×C3712yes—
438C24C24×C3711yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4C2×D4×C3714yes—
497C22×C4C2×D4×C3712yes—
498C22×C4C2×D4×C3712yes—
499C22×C4C2×D4×C3712yes—
500C22×C4C2×Q8×C3712yes—
501C22×C4C2×Q8×C3712yes—
502C22×C4#41×C3712yes—
503C22×C4#41×C3714yes—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C22⋊C4—28no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C22⋊Q8×C3714yes—
537C23C22×D4×C3711yescharacteristic
538C23C22×D4×C3711yescharacteristic
539C23C22×Q8×C3711yescharacteristic
540C23C2×#41×C3711yescharacteristic
541C23C2×#41×C3711yescharacteristic
542C23C2×#41×C3712yes—
543C23H(5,𝔽2)×C3712yes—
544C23H(5,𝔽2)×C3712yes—
545C23H(5,𝔽2)×C3711yescharacteristic
546C232−1+4×C3712yes—
547C232−1+4×C3711yescharacteristic
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—416no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
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#121×C3714yes—
608C2×C4#123×C3714yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22C2×#121×C3711yescharacteristic
621C22C2×C22⋊Q8×C3711yescharacteristic
622C22C2×#123×C3711yescharacteristic
623C22C37×#53411yescharacteristic
624C22C37×#53511yescharacteristic
625C22C37×#53611yescharacteristic
626C22C37×#53712yes—
627C22C37×#53711yescharacteristic
628C22C37×#53912yes—
629C22C37×#54112yes—
630C22C37×#54212yes—
631C22C37×#54212yes—
632C22C37×#54412yes—
633C22C37×#54612yes—
634C22C37×#54611yescharacteristic
635C22D4×Q8×C3711yescharacteristic
636C22C37×#55112yes—
637C22C37×#55211yescharacteristic
638C22C37×#55312yes—
639C22C37×#55411yescharacteristic
640C22C37×#55412yes—
641C22C37×#56012yes—
642C22C37×#56212yes—
643C4—24no—
644C4—24no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C2—44no—
654C2C37×#241211yescharacteristic
655C2C37×#241711yescharacteristic
656C2C37×#248711yescharacteristic
657C2C37×#262011yescharacteristic
658C2C37×#262511yescharacteristic
659C2C37×#262711yescharacteristic
660C2C37×#269112yes—
661C2C37×#270412yes—
662C2C37×#274212yes—
663C2C37×#274712yes—
664C2C37×#281311yescharacteristic
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×#400C2semidirect
5. C37×#1445C2semidirect
6. C37×#1444C2semidirect
7. C37×#1441C2semidirect
8. C37×#1441C2semidirect
9. C37×#1441C2semidirect
276. #16400C37direct (N × H)
321. C2×#400C2×C37semidirect
325. #1445C2×C37semidirect
326. #1444C2×C37semidirect
327. #1441C2×C37semidirect
328. #1441C2×C37semidirect
329. #1441C2×C37semidirect
371. C37#16400direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#4002
C2×C37×#3852
C2×C37×#3802
C37×#14452
C37×#14442
C37×#14412
C37×#14402
C37×#14392
#1640037