Finite Group Databasebeta

← C37×#13839 #22635092

Subgroups — C37×#13839 #22635092

Order 9472.

2702 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C37C211yesmaximal, characteristic
2C22×#117×C37C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4×C37C212yesmaximal, complemented
4C2×C4×C22⋊C4×C37C211yesmaximal, characteristic, complemented
5C37×#1441C212yesmaximal, complemented
6C37×#1440C212yesmaximal, complemented
7C37×#1439C212yesmaximal, complemented
8C37×#1438C212yesmaximal
9C4×#95×C37C212yesmaximal, complemented
10C23×D4×C37C2211yescharacteristic
11C24×C4×C37C2211yescharacteristic
12C24×C4×C37C2211yescharacteristic
13C2×C4×D4×C37—24no—
14C2×C4×D4×C37—24no—
15C2×C4×D4×C37C418yescomplemented
16C2×#117×C37—24no—
17C2×#117×C37—28no—
18C22×C4⋊C4×C37C2212yes—
19C22×C4⋊C4×C37C2211yescharacteristic
20C22×C22⋊C4×C37C2212yes—
21C22×C22⋊C4×C37C2214yes—
22C22×C22⋊C4×C37C2212yes—
23C22×C22⋊C4×C37C2211yescharacteristic
24C22×C22⋊C4×C37C2211yescharacteristic
25C22×C42×C37C2212yes—
26C22×C42×C37C2211yescharacteristic
27C22×C42×C37C2212yes—
28C4×C22⋊C4×C37—24no—
29C4×C22⋊C4×C37—28no—
30C2×#95×C37C2214yes—
31C2×#95×C37C2214yes—
32C2×#95×C37C2212yes—
33C2×#95×C37C2214yes—
34C25×C37C2311yescharacteristic
35C25×C37C2311yescharacteristic
36C22×D4×C37—22no—
37C22×D4×C37—22no—
38C22×D4×C37—24no—
39C22×D4×C37—24no—
40C22×D4×C37C2×C414yes—
41C23×C4×C37—24no—
42C23×C4×C37—24no—
43C23×C4×C37—24no—
44C23×C4×C37—22no—
45C23×C4×C37—22no—
46C23×C4×C37—24no—
47C23×C4×C37C2×C414yes—
48C23×C4×C37C2×C414yes—
49C23×C4×C37D414yes—
50C23×C4×C37D414yes—
51C23×C4×C37C2311yescharacteristic
52C23×C4×C37C2311yescharacteristic
53C23×C4×C37C2314yes—
54C23×C4×C37C2314yes—
55C23×C4×C37C2311yescharacteristic
56C23×C4×C37C2311yescharacteristic
57C23×C4×C37C2311yescharacteristic
58C4×D4×C37—232no—
59#117×C37—232no—
60C2×C4⋊C4×C37—28no—
61C2×C4⋊C4×C37—24no—
62C2×C4⋊C4×C37—22no—
63C2×C4⋊C4×C37—22no—
64C2×C4⋊C4×C37C2×C414yes—
65C2×C22⋊C4×C37—24no—
66C2×C22⋊C4×C37—24no—
67C2×C22⋊C4×C37—22no—
68C2×C22⋊C4×C37—22no—
69C2×C22⋊C4×C37—28no—
70C2×C22⋊C4×C37—28no—
71C2×C22⋊C4×C37—28no—
72C2×C22⋊C4×C37—22no—
73C2×C22⋊C4×C37—22no—
74C2×C22⋊C4×C37—28no—
75C2×C22⋊C4×C37—28no—
76C2×C22⋊C4×C37C2×C414yes—
77C2×C22⋊C4×C37C2×C414yes—
78C2×C42×C37—24no—
79C2×C42×C37—24no—
80C2×C42×C37—22no—
81C2×C42×C37—22no—
82C2×C42×C37—24no—
83C2×C42×C37—24no—
84C2×C42×C37—24no—
85C2×C42×C37C2×C414yes—
86C2×C42×C37D418yescomplemented
87#95×C37—28no—
88#95×C37—24no—
89#95×C37—28no—
90#95×C37—28no—
91C24×C37—22no—
92C24×C37—22no—
93C24×C37—21no—
94C24×C37—21no—
95C24×C37—24no—
96C24×C37—24no—
97C24×C37—22no—
98C24×C37—22no—
99C24×C37—22no—
100C24×C37—22no—
101C24×C37—22no—
102C24×C37C22×C412yes—
103C24×C37C22×C412yes—
104C24×C37C2×D412yes—
105C24×C37C2×D412yes—
106C24×C37#4114yes—
107C24×C37C2411yescharacteristic
108C2×D4×C37—216no—
109C2×D4×C37—216no—
110C2×D4×C37—28no—
111C2×D4×C37—28no—
112C2×D4×C37—28no—
113C22×C4×C37—21no—
114C22×C4×C37—21no—
115C22×C4×C37—22no—
116C22×C4×C37—22no—
117C22×C4×C37—22no—
118C22×C4×C37—216no—
119C22×C4×C37—28no—
120C22×C4×C37—216no—
121C22×C4×C37—28no—
122C22×C4×C37—22no—
123C22×C4×C37—24no—
124C22×C4×C37—24no—
125C22×C4×C37—21no—
126C22×C4×C37—21no—
127C22×C4×C37—24no—
128C22×C4×C37—24no—
129C22×C4×C37—21no—
130C22×C4×C37—21no—
131C22×C4×C37—24no—
132C22×C4×C37—24no—
133C22×C4×C37—24no—
134C22×C4×C37—24no—
135C22×C4×C37—24no—
136C22×C4×C37—24no—
137C22×C4×C37—24no—
138C22×C4×C37—24no—
139C22×C4×C37—24no—
140C22×C4×C37—22no—
141C22×C4×C37—28no—
142C22×C4×C37—216no—
143C22×C4×C37—44no—
144C22×C4×C37—48no—
145C22×C4×C37—44no—
146C22×C4×C37C22⋊C4116yescomplemented
147C22×C4×C37C22×C412yes—
148C22×C4×C37C22×C412yes—
149C22×C4×C37C22×C412yes—
150C22×C4×C37C22×C412yes—
151C22×C4×C37C22×C412yes—
152C22×C4×C37C2×D418yes—
153C22×C4×C37C2×D412yes—
154C22×C4×C37C2×D412yes—
155C22×C4×C37C2×D412yes—
156C22×C4×C37C2×D412yes—
157C22×C4×C37C2×D414yes—
158C22×C4×C37#4118yes—
159C22×C4×C37#4114yes—
160C22×C4×C37#4114yes—
161C4⋊C4×C37—216no—
162C4⋊C4×C37—28no—
163C22⋊C4×C37—232no—
164C22⋊C4×C37—216no—
165C22⋊C4×C37—216no—
166C22⋊C4×C37—28no—
167C22⋊C4×C37—28no—
168C42×C37—28no—
169C42×C37—216no—
170C42×C37—48no—
171C23×C37—22no—
172C23×C37—22no—
173C23×C37—22no—
174C23×C37—22no—
175C23×C37—24no—
176C23×C37—24no—
177C23×C37—24no—
178C23×C37—24no—
179C23×C37—22no—
180C23×C37—24no—
181C23×C37—22no—
182C23×C37—24no—
183C23×C37—28no—
184C23×C37—24no—
185C23×C37—24no—
186C23×C37—24no—
187C23×C37—24no—
188C23×C37—28no—
189C23×C37—24no—
190C23×C37—41no—
191C23×C37—42no—
192C23×C37—41no—
193C23×C37—44no—
194C23×C37—42no—
195C23×C37—44no—
196C23×C37—44no—
197C23×C37—44no—
198C23×C37—44no—
199C23×C37—44no—
200C23×C37—42no—
201C23×C37C2×C22⋊C414yes—
202C23×C37#11714yescomplemented
203C23×C37C23×C411yescharacteristic
204C23×C37C22×D411yescharacteristic
205C23×C37C22×D411yescharacteristic
206C23×C37C22×D412yes—
207C23×C37C2×#4112yes—
208C23×C37C2×#4112yes—
209C23×C37C2×#4112yes—
210C23×C37C2×#4114yes—
211D4×C37—232no—
212C2×C4×C37—22no—
213C2×C4×C37—22no—
214C2×C4×C37—24no—
215C2×C4×C37—22no—
216C2×C4×C37—24no—
217C2×C4×C37—22no—
218C2×C4×C37—24no—
219C2×C4×C37—24no—
220C2×C4×C37—232no—
221C2×C4×C37—22no—
222C2×C4×C37—22no—
223C2×C4×C37—24no—
224C2×C4×C37—24no—
225C2×C4×C37—28no—
226C2×C4×C37—28no—
227C2×C4×C37—28no—
228C2×C4×C37—28no—
229C2×C4×C37—28no—
230C2×C4×C37—28no—
231C2×C4×C37—41no—
232C2×C4×C37—44no—
233C2×C4×C37—42no—
234C2×C4×C37—41no—
235C2×C4×C37—44no—
236C2×C4×C37—44no—
237C2×C4×C37—44no—
238C2×C4×C37—42no—
239C2×C4×C37—44no—
240C2×C4×C37—44no—
241C2×C4×C37—44no—
242C2×C4×C37—44no—
243C2×C4×C37—42no—
244C2×C4×C37—416no—
245C2×C4×C37C2×C22⋊C414yes—
246C2×C4×C37C2×C22⋊C414yes—
247C2×C4×C37#11714yescomplemented
248C2×C4×C37C4×D418yes—
249C2×C4×C37#12014yes—
250C2×C4×C37#12118yes—
251C2×C4×C37#12314yes—
252#13839C3711yesmaximal, characteristic, direct product factor, Sylow
253C22×C37—24no—
254C22×C37—24no—
255C22×C37—24no—
256C22×C37—28no—
257C22×C37—24no—
258C22×C37—44no—
259C22×C37—44no—
260C22×C37—48no—
261C22×C37—48no—
262C22×C37—44no—
263C22×C37—44no—
264C22×C37—42no—
265C22×C37—44no—
266C22×C37—44no—
267C22×C37—44no—
268C22×C37—42no—
269C22×C37C22×C22⋊C411yescharacteristic
270C22×C37C2×#11711yescharacteristic
271C22×C37C2×#11711yescharacteristic
272C22×C37C2×C4×D412yes—
273C22×C37C4×#4112yes—
274C22×C37C2×#12011yescharacteristic
275C22×C37C2×#12112yes—
276C22×C37C2×#12311yescharacteristic
277C22×C37#52514yes—
278C22×C37#52512yes—
279C22×C37#52512yes—
280C22×C37#52512yes—
281C22×C37#52914yes—
282C22×C37#52914yes—
283C22×C37#53214yes—
284C22×C37#53212yes—
285C4×C37—24no—
286C4×C37—24no—
287C4×C37—48no—
288C4×C37—48no—
289C4×C37—42no—
290C4×C37—42no—
291C4×C37#38618yes—
292C22×C4×D4C2×C3711yescharacteristic
293C22×#117C2×C3711yescharacteristic, complemented
294C2×C4×C22⋊C4C2×C3712yescomplemented
295C2×C4×C22⋊C4C2×C3711yescharacteristic, complemented
296#1441C2×C3712yescomplemented
297#1440C2×C3712yescomplemented
298#1439C2×C3712yescomplemented
299#1438C2×C3712yes—
300C4×#95C2×C3712yescomplemented
301C2×C37—44no—
302C2×C37—44no—
303C2×C37#228011yescharacteristic
304C2×C37C2×#38611yescharacteristic
305C2×C37#229012yes—
306C2×C37#229312yes—
307C2×C37#229711yescharacteristic
308C2×C37#238812yes—
309C2×C37#239214yes—
310C2×C37#239512yes—
311C23×D4C22×C3711yescharacteristic
312C24×C4C22×C3711yescharacteristic
313C24×C4C22×C3711yescharacteristic
314C2×C4×D4—24no—
315C2×C4×D4—24no—
316C2×C4×D4C4×C3718yescomplemented
317C2×#117—24no—
318C2×#117—28no—
319C22×C4⋊C4C22×C3712yes—
320C22×C4⋊C4C22×C3711yescharacteristic
321C22×C22⋊C4C22×C3712yes—
322C22×C22⋊C4C22×C3714yes—
323C22×C22⋊C4C22×C3712yes—
324C22×C22⋊C4C22×C3711yescharacteristic
325C22×C22⋊C4C22×C3711yescharacteristic
326C22×C42C22×C3712yes—
327C22×C42C22×C3711yescharacteristic
328C22×C42C22×C3712yes—
329C4×C22⋊C4—24no—
330C4×C22⋊C4—28no—
331C2×#95C22×C3714yes—
332C2×#95C22×C3714yes—
333C2×#95C22×C3712yes—
334C2×#95C22×C3714yes—
335C37#1383911yescharacteristic, direct product factor, Sylow
336C25C23×C3711yescharacteristic
337C25C23×C3711yescharacteristic
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—24no—
341C22×D4—24no—
342C22×D4C2×C4×C3714yes—
343C23×C4—24no—
344C23×C4—24no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—24no—
348C23×C4—24no—
349C23×C4C2×C4×C3714yes—
350C23×C4C2×C4×C3714yes—
351C23×C4D4×C3714yes—
352C23×C4D4×C3714yes—
353C23×C4C23×C3711yescharacteristic
354C23×C4C23×C3711yescharacteristic
355C23×C4C23×C3714yes—
356C23×C4C23×C3714yes—
357C23×C4C23×C3711yescharacteristic
358C23×C4C23×C3711yescharacteristic
359C23×C4C23×C3711yescharacteristic
360C4×D4—232no—
361#117—232no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4C2×C4×C3714yes—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—28no—
378C2×C22⋊C4C2×C4×C3714yes—
379C2×C22⋊C4C2×C4×C3714yes—
380C2×C42—24no—
381C2×C42—24no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—24no—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42C2×C4×C3714yes—
388C2×C42D4×C3718yescomplemented
389#95—28no—
390#95—24no—
391#95—28no—
392#95—28no—
393C24—22no—
394C24—22no—
395C24—21no—
396C24—21no—
397C24—24no—
398C24—24no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24C22×C4×C3712yes—
405C24C22×C4×C3712yes—
406C24C2×D4×C3712yes—
407C24C2×D4×C3712yes—
408C24#41×C3714yes—
409C24C24×C3711yescharacteristic
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—28no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—216no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4C22⋊C4×C37116yescomplemented
449C22×C4C22×C4×C3712yes—
450C22×C4C22×C4×C3712yes—
451C22×C4C22×C4×C3712yes—
452C22×C4C22×C4×C3712yes—
453C22×C4C22×C4×C3712yes—
454C22×C4C2×D4×C3718yes—
455C22×C4C2×D4×C3712yes—
456C22×C4C2×D4×C3712yes—
457C22×C4C2×D4×C3712yes—
458C22×C4C2×D4×C3712yes—
459C22×C4C2×D4×C3714yes—
460C22×C4#41×C3718yes—
461C22×C4#41×C3714yes—
462C22×C4#41×C3714yes—
463C4⋊C4—216no—
464C4⋊C4—28no—
465C22⋊C4—232no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C42—28no—
471C42—216no—
472C42—48no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—44no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23C2×C22⋊C4×C3714yes—
504C23#117×C3714yescomplemented
505C23C23×C4×C3711yescharacteristic
506C23C22×D4×C3712yes—
507C23C22×D4×C3711yescharacteristic
508C23C22×D4×C3711yescharacteristic
509C23C2×#41×C3712yes—
510C23C2×#41×C3712yes—
511C23C2×#41×C3712yes—
512C23C2×#41×C3714yes—
513D4—232no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—232no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—416no—
547C2×C4C2×C22⋊C4×C3714yes—
548C2×C4C2×C22⋊C4×C3714yes—
549C2×C4#117×C3714yescomplemented
550C2×C4C4×D4×C3718yes—
551C2×C4#120×C3714yes—
552C2×C4#121×C3718yes—
553C2×C4#123×C3714yes—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—28no—
558C22—24no—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—48no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22C22×C22⋊C4×C3711yescharacteristic
571C22C2×#117×C3711yescharacteristic
572C22C2×#117×C3711yescharacteristic
573C22C2×C4×D4×C3712yes—
574C22C4×#41×C3712yes—
575C22C2×#120×C3711yescharacteristic
576C22C2×#121×C3712yes—
577C22C2×#123×C3711yescharacteristic
578C22C37×#52514yes—
579C22C37×#52512yes—
580C22C37×#52512yes—
581C22C37×#52512yes—
582C22C37×#52914yes—
583C22C37×#52914yes—
584C22C37×#53214yes—
585C22C37×#53212yes—
586C4—24no—
587C4—24no—
588C4—48no—
589C4—48no—
590C4—42no—
591C4—42no—
592C4C37×#38618yes—
593C2—44no—
594C2—44no—
595C2C37×#228011yescharacteristic
596C2C2×C37×#38611yescharacteristic
597C2C37×#229012yes—
598C2C37×#229312yes—
599C2C37×#229711yescharacteristic
600C2C37×#238812yes—
601C2C37×#239214yes—
602C2C37×#239512yes—
subgroupG/H
centerC23×C4×C37C23
commutatorC23C23×C4×C37
Frattini, cosocleC24C24×C37
socleC24×C37C24
prime radicalC23×D4×C37C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#117×C37C2semidirect
3. C2×C4×C22⋊C4×C37C2semidirect
4. C2×C4×C22⋊C4×C37C2semidirect
5. C37×#1441C2semidirect
6. C37×#1440C2semidirect
7. C37×#1439C2semidirect
9. C4×#95×C37C2semidirect
15. C2×C4×D4×C37C4semidirect
86. C2×C42×C37D4semidirect
146. C22×C4×C37C22⋊C4semidirect
202. C23×C37#117semidirect
247. C2×C4×C37#117semidirect
252. #13839C37direct (N × H)
293. C22×#117C2×C37semidirect
294. C2×C4×C22⋊C4C2×C37semidirect
295. C2×C4×C22⋊C4C2×C37semidirect
296. #1441C2×C37semidirect
297. #1440C2×C37semidirect
298. #1439C2×C37semidirect
300. C4×#95C2×C37semidirect
316. C2×C4×D4C4×C37semidirect
335. C37#13839direct (N × H)
388. C2×C42D4×C37semidirect
448. C22×C4C22⋊C4×C37semidirect
504. C23#117×C37semidirect
549. C2×C4#117×C37semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D4×C372
C22×#117×C372
C2×C4×C22⋊C4×C372
C37×#14412
C37×#14402
C37×#14392
C37×#14382
C4×#95×C372
#1383937