Finite Group Databasebeta

← C37×#24054 #22645307

Subgroups — C37×#24054 #22645307

Order 9472.

1358 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#547C211yesmaximal, characteristic
2C2×C37×#484C211yesmaximal, characteristic, complemented
3C2×C37×#417C211yesmaximal, characteristic, complemented
4C37×#2486C211yesmaximal, characteristic, complemented
5C37×#2043C212yesmaximal, complemented
6C37×#2036C212yesmaximal, complemented
7C37×#1894C211yesmaximal, characteristic, complemented
8C37×#1870C211yesmaximal, characteristic, complemented
9C37×#1862C212yesmaximal, complemented
10C37×#1788C212yesmaximal, complemented
11C37×#1788C211yesmaximal, characteristic, complemented
12C22×#41×C37C2211yescharacteristic
13C22×#41×C37C2211yescharacteristic
14C22×M16×C37C2211yescharacteristic
15C37×#547—24no—
16C37×#547—24no—
17C2×C4⋊Q8×C37C2211yescharacteristic
18C2×#123×C37—21no—
19C2×#123×C37—21no—
20C2×#123×C37—21no—
21C2×C22⋊Q8×C37C2211yescharacteristic
22C2×C22⋊Q8×C37C2211yescharacteristic
23C2×#121×C37—21no—
24C2×C4×D4×C37C2211yescharacteristic
25C2×C4×D4×C37C2211yescharacteristic
26C22×C4⋊C4×C37—21no—
27C22×C4⋊C4×C37C2211yescharacteristic
28C37×#484—24no—
29C2×#107×C37C2212yes—
30C37×#417—24no—
31C2×#103×C37C2212yes—
32C2×#103×C37C2212yes—
33C2×#103×C37C2212yes—
34C2×#103×C37C2212yes—
35C2×#102×C37C2212yes—
36C2×#102×C37C2212yes—
37C2×C22⋊C8×C37C2211yescharacteristic
38C2×C22⋊C8×C37C2212yes—
39C37×#398—21no—
40C37×#397—21no—
41C37×#389C2212yes—
42C37×#381—21no—
43C37×#380C2211yescharacteristic
44C37×#380C2212yes—
45C37×#340C2212yes—
46C37×#336C2212yes—
47C2×#41×C37—22no—
48C2×#41×C37—22no—
49C2×#41×C37—22no—
50C2×#41×C37—24no—
51C2×#41×C37—24no—
52C2×#41×C37—42no—
53C2×#41×C37—42no—
54C2×#41×C37D414yes—
55C22×Q8×C37C2311yescharacteristic
56C22×Q8×C37C2311yescharacteristic
57C22×D4×C37—21no—
58C22×D4×C37—21no—
59C22×D4×C37D411yescharacteristic
60C22×D4×C37C2311yescharacteristic
61C23×C4×C37—21no—
62C23×C4×C37—21no—
63C23×C4×C37—21no—
64C23×C4×C37—21no—
65C23×C4×C37C2311yescharacteristic
66C23×C4×C37C2311yescharacteristic
67C2×M16×C37—22no—
68C2×M16×C37—42no—
69C22×C8×C37C2312yes—
70C22×C8×C37C2312yes—
71C4⋊Q8×C37—22no—
72C4⋊Q8×C37—22no—
73#123×C37—28no—
74#123×C37—28no—
75#123×C37—44no—
76C22⋊Q8×C37—22no—
77C22⋊Q8×C37—22no—
78C22⋊Q8×C37—22no—
79C22⋊Q8×C37D414yes—
80#121×C37—28no—
81C4×D4×C37—22no—
82C4×D4×C37—22no—
83C4×D4×C37—22no—
84C4×D4×C37—22no—
85C2×C4⋊C4×C37—28no—
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×C37—22no—
93C2×C4⋊C4×C37—21no—
94C2×C4⋊C4×C37D411yescharacteristic
95C2×C4⋊C4×C37C2312yes—
96C2×C4⋊C4×C37C2312yes—
97C2×C4⋊C4×C37C2311yescharacteristic
98C2×C22⋊C4×C37—21no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37—21no—
101C2×C22⋊C4×C37—21no—
102C2×C22⋊C4×C37—22no—
103C2×C22⋊C4×C37D411yescharacteristic
104C2×C22⋊C4×C37D411yescharacteristic
105C2×C42×C37C2311yescharacteristic
106#107×C37—24no—
107#103×C37—24no—
108#103×C37—24no—
109#103×C37—42no—
110#103×C37—42no—
111#102×C37—24no—
112#102×C37—24no—
113C22⋊C8×C37—22no—
114C22⋊C8×C37—42no—
115#95×C37—22no—
116#95×C37—21no—
117#95×C37—22no—
118C24×C37—21no—
119C24×C37C2×D411yescharacteristic
120C24×C37C2×D411yescharacteristic
121#41×C37—28no—
122#41×C37—44no—
123#41×C37—48no—
124#41×C37—48no—
125#41×C37—44no—
126#41×C37—44no—
127C2×Q8×C37—21no—
128C2×Q8×C37—21no—
129C2×Q8×C37—22no—
130C2×Q8×C37—22no—
131C2×Q8×C37—22no—
132C2×Q8×C37—22no—
133C2×Q8×C37C2×D412yes—
134C2×Q8×C37C2×D412yes—
135C2×D4×C37—24no—
136C2×D4×C37—22no—
137C2×D4×C37—24no—
138C2×D4×C37—22no—
139C2×D4×C37—21no—
140C2×D4×C37—22no—
141C2×D4×C37—41no—
142C2×D4×C37—44no—
143C2×D4×C37—44no—
144C2×D4×C37C2×D412yes—
145C2×D4×C37#4114yes—
146C22×C4×C37—22no—
147C22×C4×C37—22no—
148C22×C4×C37—24no—
149C22×C4×C37—24no—
150C22×C4×C37—21no—
151C22×C4×C37—21no—
152C22×C4×C37—21no—
153C22×C4×C37—24no—
154C22×C4×C37—21no—
155C22×C4×C37—21no—
156C22×C4×C37—22no—
157C22×C4×C37—22no—
158C22×C4×C37—22no—
159C22×C4×C37—22no—
160C22×C4×C37—44no—
161C22×C4×C37—44no—
162C22×C4×C37—42no—
163C22×C4×C37—42no—
164C22×C4×C37—42no—
165C22×C4×C37—42no—
166C22×C4×C37—42no—
167C22×C4×C37C2×D411yescharacteristic
168C22×C4×C37C2×D411yescharacteristic
169C22×C4×C37C2×D411yescharacteristic
170C22×C4×C37C2×D412yes—
171C22×C4×C37C2×D412yes—
172C22×C4×C37C2×D411yescharacteristic
173C22×C4×C37C2411yescharacteristic
174M16×C37—44no—
175C2×C8×C37—22no—
176C2×C8×C37—22no—
177C2×C8×C37—22no—
178C2×C8×C37—22no—
179C2×C8×C37—22no—
180C2×C8×C37—22no—
181C4⋊C4×C37—24no—
182C4⋊C4×C37—24no—
183C4⋊C4×C37—24no—
184C4⋊C4×C37—22no—
185C4⋊C4×C37—24no—
186C4⋊C4×C37—21no—
187C4⋊C4×C37—22no—
188C4⋊C4×C37—22no—
189C4⋊C4×C37—22no—
190C4⋊C4×C37—22no—
191C4⋊C4×C37—42no—
192C4⋊C4×C37—44no—
193C4⋊C4×C37C2×D412yes—
194C22⋊C4×C37—24no—
195C22⋊C4×C37—24no—
196C22⋊C4×C37—24no—
197C22⋊C4×C37—22no—
198C22⋊C4×C37—22no—
199C22⋊C4×C37—42no—
200C22⋊C4×C37—44no—
201C42×C37—21no—
202C42×C37—21no—
203C23×C37—22no—
204C23×C37—22no—
205C23×C37—24no—
206C23×C37—21no—
207C23×C37—21no—
208C23×C37—22no—
209C23×C37—21no—
210C23×C37—22no—
211C23×C37—21no—
212C23×C37—21no—
213C23×C37—22no—
214C23×C37—41no—
215C23×C37—41no—
216C23×C37—42no—
217C23×C37#12012yes—
218C23×C37C22×D411yescharacteristic
219Q8×C37—22no—
220Q8×C37—22no—
221Q8×C37—24no—
222Q8×C37—41no—
223Q8×C37—41no—
224Q8×C37—42no—
225D4×C37—22no—
226D4×C37—24no—
227D4×C37—44no—
228D4×C37—48no—
229D4×C37—48no—
230D4×C37#12314yescomplemented
231C2×C4×C37—21no—
232C2×C4×C37—22no—
233C2×C4×C37—21no—
234C2×C4×C37—22no—
235C2×C4×C37—22no—
236C2×C4×C37—24no—
237C2×C4×C37—21no—
238C2×C4×C37—22no—
239C2×C4×C37—21no—
240C2×C4×C37—22no—
241C2×C4×C37—22no—
242C2×C4×C37—22no—
243C2×C4×C37—22no—
244C2×C4×C37—48no—
245C2×C4×C37—48no—
246C2×C4×C37—48no—
247C2×C4×C37—42no—
248C2×C4×C37—42no—
249C2×C4×C37—41no—
250C2×C4×C37—41no—
251C2×C4×C37—41no—
252C2×C4×C37—41no—
253C2×C4×C37—42no—
254C2×C4×C37—42no—
255C2×C4×C37—42no—
256C2×C4×C37—42no—
257C2×C4×C37—42no—
258C2×C4×C37—42no—
259C2×C4×C37—42no—
260C2×C4×C37—84no—
261C2×C4×C37#12012yes—
262C2×C4×C37AGL(1,ℤ/8)12yes—
263C2×C4×C37C22×D411yescharacteristic
264C2×C4×C37C22×D411yescharacteristic
265C2×C4×C37C2×#4112yes—
266C2×C4×C37H(5,𝔽2)12yes—
267C8×C37—42no—
268C8×C37—42no—
269#24054C3711yesmaximal, characteristic, direct product factor, Sylow
270C22×C37—22no—
271C22×C37—24no—
272C22×C37—22no—
273C22×C37—22no—
274C22×C37—42no—
275C22×C37—44no—
276C22×C37—42no—
277C22×C37—42no—
278C22×C37—42no—
279C22×C37—44no—
280C22×C37—42no—
281C22×C37—41no—
282C22×C37—41no—
283C22×C37—41no—
284C22×C37—42no—
285C22×C37C2×#12011yescharacteristic
286C22×C37#54612yes—
287C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C37#57511yescharacteristic
289C22×C37#57812yes—
290C4×C37—22no—
291C4×C37—22no—
292C4×C37—41no—
293C4×C37—42no—
294C4×C37—42no—
295C4×C37—42no—
296C4×C37—41no—
297C4×C37—82no—
298C4×C37—82no—
299C4×C37C2×#12311yescharacteristic
300C4×C37#53411yescharacteristic
301C4×C37#54612yes—
302C2×#547C2×C3711yescharacteristic
303C2×#484C2×C3711yescharacteristic, complemented
304C2×#417C2×C3711yescharacteristic, complemented
305#2486C2×C3711yescharacteristic, complemented
306#2043C2×C3712yescomplemented
307#2036C2×C3712yescomplemented
308#1894C2×C3711yescharacteristic, complemented
309#1870C2×C3711yescharacteristic, complemented
310#1862C2×C3712yescomplemented
311#1788C2×C3712yescomplemented
312#1788C2×C3711yescharacteristic, complemented
313C2×C37—42no—
314C2×C37—44no—
315C2×C37—81no—
316C2×C37#240511yescharacteristic
317C2×C37#300611yescharacteristic
318C2×C37#301111yescharacteristic
319C2×C37#331512yes—
320C2×C37#333012yes—
321C22×#41C22×C3711yescharacteristic
322C22×#41C22×C3711yescharacteristic
323C22×M16C22×C3711yescharacteristic
324#547—24no—
325#547—24no—
326C2×C4⋊Q8C22×C3711yescharacteristic
327C2×#123—21no—
328C2×#123—21no—
329C2×#123—21no—
330C2×C22⋊Q8C22×C3711yescharacteristic
331C2×C22⋊Q8C22×C3711yescharacteristic
332C2×#121—21no—
333C2×C4×D4C22×C3711yescharacteristic
334C2×C4×D4C22×C3711yescharacteristic
335C22×C4⋊C4—21no—
336C22×C4⋊C4C22×C3711yescharacteristic
337#484—24no—
338C2×#107C22×C3712yes—
339#417—24no—
340C2×#103C22×C3712yes—
341C2×#103C22×C3712yes—
342C2×#103C22×C3712yes—
343C2×#103C22×C3712yes—
344C2×#102C22×C3712yes—
345C2×#102C22×C3712yes—
346C2×C22⋊C8C22×C3711yescharacteristic
347C2×C22⋊C8C22×C3712yes—
348#398—21no—
349#397—21no—
350#389C22×C3712yes—
351#381—21no—
352#380C22×C3711yescharacteristic
353#380C22×C3712yes—
354#340C22×C3712yes—
355#336C22×C3712yes—
356C37#2405411yescharacteristic, direct product factor, Sylow
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—24no—
361C2×#41—24no—
362C2×#41—42no—
363C2×#41—42no—
364C2×#41D4×C3714yes—
365C22×Q8C23×C3711yescharacteristic
366C22×Q8C23×C3711yescharacteristic
367C22×D4—21no—
368C22×D4—21no—
369C22×D4D4×C3711yescharacteristic
370C22×D4C23×C3711yescharacteristic
371C23×C4—21no—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—21no—
375C23×C4C23×C3711yescharacteristic
376C23×C4C23×C3711yescharacteristic
377C2×M16—22no—
378C2×M16—42no—
379C22×C8C23×C3712yes—
380C22×C8C23×C3712yes—
381C4⋊Q8—22no—
382C4⋊Q8—22no—
383#123—28no—
384#123—28no—
385#123—44no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8D4×C3714yes—
390#121—28no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C2×C4⋊C4—28no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4D4×C3711yescharacteristic
405C2×C4⋊C4C23×C3712yes—
406C2×C4⋊C4C23×C3712yes—
407C2×C4⋊C4C23×C3711yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4D4×C3711yescharacteristic
414C2×C22⋊C4D4×C3711yescharacteristic
415C2×C42C23×C3711yescharacteristic
416#107—24no—
417#103—24no—
418#103—24no—
419#103—42no—
420#103—42no—
421#102—24no—
422#102—24no—
423C22⋊C8—22no—
424C22⋊C8—42no—
425#95—22no—
426#95—21no—
427#95—22no—
428C24—21no—
429C24C2×D4×C3711yescharacteristic
430C24C2×D4×C3711yescharacteristic
431#41—28no—
432#41—44no—
433#41—48no—
434#41—48no—
435#41—44no—
436#41—44no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8C2×D4×C3712yes—
444C2×Q8C2×D4×C3712yes—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4C2×D4×C3712yes—
455C2×D4#41×C3714yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4C2×D4×C3711yescharacteristic
478C22×C4C2×D4×C3711yescharacteristic
479C22×C4C2×D4×C3711yescharacteristic
480C22×C4C2×D4×C3712yes—
481C22×C4C2×D4×C3712yes—
482C22×C4C2×D4×C3711yescharacteristic
483C22×C4C24×C3711yescharacteristic
484M16—44no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4C2×D4×C3712yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C42—21no—
512C42—21no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23#120×C3712yes—
528C23C22×D4×C3711yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—24no—
532Q8—41no—
533Q8—41no—
534Q8—42no—
535D4—22no—
536D4—24no—
537D4—44no—
538D4—48no—
539D4—48no—
540D4#123×C3714yescomplemented
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4#120×C3712yes—
572C2×C4AGL(1,ℤ/8)×C3712yes—
573C2×C4C22×D4×C3711yescharacteristic
574C2×C4C22×D4×C3711yescharacteristic
575C2×C4C2×#41×C3712yes—
576C2×C4H(5,𝔽2)×C3712yes—
577C8—42no—
578C8—42no—
579C22—22no—
580C22—24no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22C2×#120×C3711yescharacteristic
595C22C37×#54612yes—
596C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
597C22C37×#57511yescharacteristic
598C22C37×#57812yes—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—82no—
607C4—82no—
608C4C2×#123×C3711yescharacteristic
609C4C37×#53411yescharacteristic
610C4C37×#54612yes—
611C2—42no—
612C2—44no—
613C2—81no—
614C2C37×#240511yescharacteristic
615C2C37×#300611yescharacteristic
616C2C37×#301111yescharacteristic
617C2C37×#331512yes—
618C2C37×#333012yes—
subgroupG/H
center, socleC23×C37C22×D4
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×C37×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C37 ◁ C2×C4×D4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C37×#484C2semidirect
3. C2×C37×#417C2semidirect
4. C37×#2486C2semidirect
5. C37×#2043C2semidirect
6. C37×#2036C2semidirect
7. C37×#1894C2semidirect
8. C37×#1870C2semidirect
9. C37×#1862C2semidirect
10. C37×#1788C2semidirect
11. C37×#1788C2semidirect
230. D4×C37#123semidirect
269. #24054C37direct (N × H)
303. C2×#484C2×C37semidirect
304. C2×#417C2×C37semidirect
305. #2486C2×C37semidirect
306. #2043C2×C37semidirect
307. #2036C2×C37semidirect
308. #1894C2×C37semidirect
309. #1870C2×C37semidirect
310. #1862C2×C37semidirect
311. #1788C2×C37semidirect
312. #1788C2×C37semidirect
356. C37#24054direct (N × H)
540. D4#123×C37semidirect

Maximal subgroups

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

subgroupindex
C2×C37×#5472
C2×C37×#4842
C2×C37×#4172
C37×#24862
C37×#20432
C37×#20362
C37×#18942
C37×#18702
C37×#18622
C37×#17882
#2405437