Finite Group Databasebeta

← C37×#23901 #22645154

Subgroups — C37×#23901 #22645154

Order 9472.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3484C211yesmaximal, characteristic
2C22×#123×C37C211yesmaximal, characteristic, complemented
3C2×C37×#458C212yesmaximal, complemented
4C37×#2849C212yesmaximal, complemented
5C37×#2022C214yesmaximal, complemented
6C37×#1898C212yesmaximal, complemented
7C37×#1890C212yesmaximal, complemented
8C37×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C37C2211yescharacteristic
10C23×D4×C37C2211yescharacteristic
11C24×C4×C37C2211yescharacteristic, complemented
12C37×#546—24no—
13D42×C37—24no—
14C37×#535—21no—
15C37×#534—21no—
16C37×#534C2212yes—
17C37×#525—21no—
18C2×#123×C37—21no—
19C2×#123×C37—21no—
20C2×#123×C37—28no—
21C2×#123×C37C2212yes—
22C2×#123×C37C2212yescomplemented
23C2×#123×C37C2214yescomplemented
24C2×#120×C37—21no—
25C2×#120×C37C2212yes—
26C37×#518C2211yescharacteristic
27C22×C4⋊C4×C37—21no—
28C22×C22⋊C4×C37—21no—
29C22×C22⋊C4×C37C2211yescharacteristic, complemented
30C37×#458—28no—
31C2×#99×C37C2212yes—
32C2×#99×C37C2214yescomplemented
33C2×#99×C37C2212yescomplemented
34C37×#386—24no—
35C37×#385—24no—
36C37×#381C2214yescomplemented
37C37×#379C2212yescomplemented
38C37×#342C2214yescomplemented
39C25×C37D411yescharacteristic
40C25×C37C2311yescharacteristic
41H(5,𝔽2)×C37—24no—
42H(5,𝔽2)×C37—42no—
43C2×#41×C37—21no—
44C2×#41×C37—22no—
45C22×D4×C37—21no—
46C22×D4×C37—22no—
47C22×D4×C37—24no—
48C22×D4×C37—22no—
49C22×D4×C37—24no—
50C22×D4×C37—42no—
51C22×D4×C37D411yescharacteristic
52C22×D4×C37D411yescharacteristic
53C22×D4×C37D414yes—
54C22×D4×C37C2312yes—
55C22×D4×C37C2311yescharacteristic
56C22×D4×C37C2311yescharacteristic
57C23×C4×C37—21no—
58C23×C4×C37—21no—
59C23×C4×C37—21no—
60C23×C4×C37—24no—
61C23×C4×C37—24no—
62C23×C4×C37—42no—
63C23×C4×C37D411yescharacteristic
64C23×C4×C37C2312yes—
65#127×C37—22no—
66#124×C37—22no—
67#123×C37—28no—
68#123×C37—28no—
69#123×C37—22no—
70#123×C37—24no—
71#123×C37—22no—
72#123×C37—22no—
73#123×C37—24no—
74#123×C37—24no—
75#123×C37—28no—
76#123×C37—44no—
77#123×C37D418yescomplemented
78C22⋊Q8×C37—22no—
79#121×C37—22no—
80#121×C37—24no—
81#121×C37—24no—
82#121×C37—24no—
83#120×C37—24no—
84#120×C37—22no—
85#120×C37—22no—
86#120×C37—22no—
87#120×C37—22no—
88#120×C37—24no—
89#120×C37—22no—
90C4×D4×C37—24no—
91C4×D4×C37—24no—
92#117×C37—21no—
93C2×C4⋊C4×C37—28no—
94C2×C4⋊C4×C37—22no—
95C2×C4⋊C4×C37—22no—
96C2×C4⋊C4×C37—24no—
97C2×C22⋊C4×C37—22no—
98C2×C22⋊C4×C37—28no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37—21no—
101C2×C22⋊C4×C37—22no—
102C2×C22⋊C4×C37—22no—
103C2×C22⋊C4×C37—24no—
104C2×C22⋊C4×C37—24no—
105C2×C22⋊C4×C37—24no—
106C2×C22⋊C4×C37D414yes—
107C2×C22⋊C4×C37C2314yes—
108C2×C22⋊C4×C37C2312yes—
109C2×C22⋊C4×C37C2312yes—
110#99×C37—24no—
111#99×C37—28no—
112#99×C37—42no—
113#95×C37—28no—
114C24×C37—22no—
115C24×C37—22no—
116C24×C37—21no—
117C24×C37—22no—
118C24×C37—22no—
119C24×C37—44no—
120C24×C37—41no—
121C24×C37—44no—
122C24×C37C2×D411yescharacteristic
123C24×C37C2×D414yes—
124C24×C37C2×D411yescharacteristic
125C24×C37C2×D412yes—
126C24×C37C2×D411yescharacteristic
127C24×C37C2×D411yescharacteristic
128C24×C37C2411yescharacteristic
129#41×C37—48no—
130#41×C37—42no—
131#41×C37—42no—
132C2×Q8×C37—21no—
133C2×D4×C37—22no—
134C2×D4×C37—22no—
135C2×D4×C37—22no—
136C2×D4×C37—21no—
137C2×D4×C37—24no—
138C2×D4×C37—22no—
139C2×D4×C37—22no—
140C2×D4×C37—22no—
141C2×D4×C37—24no—
142C2×D4×C37—22no—
143C2×D4×C37—22no—
144C2×D4×C37—22no—
145C2×D4×C37—22no—
146C2×D4×C37—22no—
147C2×D4×C37—24no—
148C2×D4×C37—24no—
149C2×D4×C37—24no—
150C2×D4×C37—22no—
151C2×D4×C37—41no—
152C2×D4×C37—41no—
153C2×D4×C37—416no—
154C2×D4×C37—48no—
155C2×D4×C37—42no—
156C2×D4×C37—42no—
157C2×D4×C37—48no—
158C2×D4×C37—84no—
159C2×D4×C37C2×D414yes—
160C22×C4×C37—24no—
161C22×C4×C37—24no—
162C22×C4×C37—21no—
163C22×C4×C37—22no—
164C22×C4×C37—24no—
165C22×C4×C37—22no—
166C22×C4×C37—24no—
167C22×C4×C37—22no—
168C22×C4×C37—22no—
169C22×C4×C37—22no—
170C22×C4×C37—24no—
171C22×C4×C37—22no—
172C22×C4×C37—22no—
173C22×C4×C37—24no—
174C22×C4×C37—41no—
175C22×C4×C37—416no—
176C22×C4×C37—44no—
177C22×C4×C37—42no—
178C22×C4×C37—42no—
179C22×C4×C37—42no—
180C22×C4×C37—42no—
181C22×C4×C37C2×D412yes—
182C22×C4×C37C2×D411yescharacteristic
183C22×C4×C37C2×D411yescharacteristic
184C22×C4×C37C2×D414yes—
185C22×C4×C37C2×D414yescomplemented
186C4⋊C4×C37—28no—
187C4⋊C4×C37—22no—
188C4⋊C4×C37—24no—
189C4⋊C4×C37—28no—
190C4⋊C4×C37—48no—
191C22⋊C4×C37—24no—
192C22⋊C4×C37—28no—
193C22⋊C4×C37—22no—
194C22⋊C4×C37—24no—
195C22⋊C4×C37—24no—
196C22⋊C4×C37—28no—
197C22⋊C4×C37—24no—
198C22⋊C4×C37—24no—
199C22⋊C4×C37—22no—
200C22⋊C4×C37—22no—
201C22⋊C4×C37—22no—
202C22⋊C4×C37—24no—
203C22⋊C4×C37—24no—
204C22⋊C4×C37—48no—
205C22⋊C4×C37—48no—
206C22⋊C4×C37C2×D414yescomplemented
207C42×C37—22no—
208C23×C37—21no—
209C23×C37—21no—
210C23×C37—24no—
211C23×C37—21no—
212C23×C37—21no—
213C23×C37—21no—
214C23×C37—28no—
215C23×C37—28no—
216C23×C37—21no—
217C23×C37—21no—
218C23×C37—21no—
219C23×C37—22no—
220C23×C37—21no—
221C23×C37—24no—
222C23×C37—44no—
223C23×C37—42no—
224C23×C37—42no—
225C23×C37—44no—
226C23×C37—48no—
227C23×C37—48no—
228C23×C37—42no—
229C23×C37—42no—
230C23×C37—48no—
231C23×C37—42no—
232C23×C37—44no—
233C23×C37—44no—
234C23×C37—82no—
235C23×C37—84no—
236C23×C37—82no—
237C23×C37#12012yes—
238C23×C37#12012yes—
239C23×C37#12012yes—
240C23×C37C22×D411yescharacteristic
241C23×C37C22×D412yes—
242C23×C37C22×D412yes—
243C23×C37H(5,𝔽2)12yes—
244Q8×C37—42no—
245D4×C37—28no—
246D4×C37—44no—
247D4×C37—42no—
248D4×C37—44no—
249D4×C37—44no—
250D4×C37—42no—
251D4×C37—42no—
252D4×C37—44no—
253D4×C37—44no—
254D4×C37—48no—
255D4×C37—84no—
256D4×C37—84no—
257C2×C4×C37—21no—
258C2×C4×C37—21no—
259C2×C4×C37—24no—
260C2×C4×C37—24no—
261C2×C4×C37—21no—
262C2×C4×C37—24no—
263C2×C4×C37—22no—
264C2×C4×C37—21no—
265C2×C4×C37—21no—
266C2×C4×C37—28no—
267C2×C4×C37—24no—
268C2×C4×C37—44no—
269C2×C4×C37—44no—
270C2×C4×C37—48no—
271C2×C4×C37—44no—
272C2×C4×C37—48no—
273C2×C4×C37—48no—
274C2×C4×C37—42no—
275C2×C4×C37—42no—
276C2×C4×C37—42no—
277C2×C4×C37—44no—
278C2×C4×C37—44no—
279C2×C4×C37—48no—
280C2×C4×C37—44no—
281C2×C4×C37—82no—
282C2×C4×C37#12012yes—
283#23901C3711yesmaximal, characteristic, direct product factor, Sylow
284C22×C37—24no—
285C22×C37—24no—
286C22×C37—22no—
287C22×C37—24no—
288C22×C37—22no—
289C22×C37—22no—
290C22×C37—22no—
291C22×C37—22no—
292C22×C37—24no—
293C22×C37—42no—
294C22×C37—42no—
295C22×C37—416no—
296C22×C37—42no—
297C22×C37—42no—
298C22×C37—42no—
299C22×C37—42no—
300C22×C37—42no—
301C22×C37—42no—
302C22×C37—41no—
303C22×C37—44no—
304C22×C37—42no—
305C22×C37—81no—
306C22×C37—88no—
307C22×C37—84no—
308C22×C37—81no—
309C22×C37—162no—
310C22×C37#45814yescomplemented
311C22×C37C2×#12011yescharacteristic
312C22×C37C2×#12011yescharacteristic
313C22×C37#53411yescharacteristic
314C22×C37D4214yes—
315C4×C37—22no—
316C4×C37—44no—
317C4×C37—44no—
318C4×C37—41no—
319C4×C37—42no—
320C4×C37—44no—
321C4×C37—84no—
322#3484C2×C3711yescharacteristic
323C22×#123C2×C3711yescharacteristic, complemented
324C2×#458C2×C3712yescomplemented
325#2849C2×C3712yescomplemented
326#2022C2×C3714yescomplemented
327#1898C2×C3712yescomplemented
328#1890C2×C3712yescomplemented
329#1781C2×C3711yescharacteristic, complemented
330C2×C37—21no—
331C2×C37—24no—
332C2×C37—21no—
333C2×C37—24no—
334C2×C37—42no—
335C2×C37—42no—
336C2×C37—81no—
337C2×C37—82no—
338C2×C37#240311yescharacteristic
339C2×C37C2×#45812yes—
340C2×H(5,𝔽2)C22×C3711yescharacteristic
341C23×D4C22×C3711yescharacteristic
342C24×C4C22×C3711yescharacteristic, complemented
343#546—24no—
344D42—24no—
345#535—21no—
346#534—21no—
347#534C22×C3712yes—
348#525—21no—
349C2×#123—21no—
350C2×#123—21no—
351C2×#123—28no—
352C2×#123C22×C3712yes—
353C2×#123C22×C3712yescomplemented
354C2×#123C22×C3714yescomplemented
355C2×#120—21no—
356C2×#120C22×C3712yes—
357#518C22×C3711yescharacteristic
358C22×C4⋊C4—21no—
359C22×C22⋊C4—21no—
360C22×C22⋊C4C22×C3711yescharacteristic, complemented
361#458—28no—
362C2×#99C22×C3712yes—
363C2×#99C22×C3714yescomplemented
364C2×#99C22×C3712yescomplemented
365#386—24no—
366#385—24no—
367#381C22×C3714yescomplemented
368#379C22×C3712yescomplemented
369#342C22×C3714yescomplemented
370C37#2390111yescharacteristic, direct product factor, Sylow
371C25D4×C3711yescharacteristic
372C25C23×C3711yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C3711yescharacteristic
384C22×D4D4×C3711yescharacteristic
385C22×D4D4×C3714yes—
386C22×D4C23×C3712yes—
387C22×D4C23×C3711yescharacteristic
388C22×D4C23×C3711yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C3711yescharacteristic
396C23×C4C23×C3712yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C3718yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C3714yes—
439C2×C22⋊C4C23×C3714yes—
440C2×C22⋊C4C23×C3712yes—
441C2×C22⋊C4C23×C3712yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C3711yescharacteristic
455C24C2×D4×C3714yes—
456C24C2×D4×C3711yescharacteristic
457C24C2×D4×C3712yes—
458C24C2×D4×C3711yescharacteristic
459C24C2×D4×C3711yescharacteristic
460C24C24×C3711yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C3714yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C3712yes—
514C22×C4C2×D4×C3711yescharacteristic
515C22×C4C2×D4×C3711yescharacteristic
516C22×C4C2×D4×C3714yes—
517C22×C4C2×D4×C3714yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C3714yescomplemented
539C42—22no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—28no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C3712yes—
570C23#120×C3712yes—
571C23#120×C3712yes—
572C23C22×D4×C3711yescharacteristic
573C23C22×D4×C3712yes—
574C23C22×D4×C3712yes—
575C23H(5,𝔽2)×C3712yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—44no—
586D4—48no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C3712yes—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C37×#45814yescomplemented
642C22C2×#120×C3711yescharacteristic
643C22C2×#120×C3711yescharacteristic
644C22C37×#53411yescharacteristic
645C22D42×C3714yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—21no—
656C2—24no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C37×#240311yescharacteristic
662C2C2×C37×#45812yes—
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC24C24×C37
prime radicalC37×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C25×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C37C2semidirect
3. C2×C37×#458C2semidirect
4. C37×#2849C2semidirect
5. C37×#2022C2semidirect
6. C37×#1898C2semidirect
7. C37×#1890C2semidirect
8. C37×#1781C2semidirect
11. C24×C4×C37C22semidirect
22. C2×#123×C37C22semidirect
23. C2×#123×C37C22semidirect
29. C22×C22⋊C4×C37C22semidirect
32. C2×#99×C37C22semidirect
33. C2×#99×C37C22semidirect
36. C37×#381C22semidirect
37. C37×#379C22semidirect
38. C37×#342C22semidirect
77. #123×C37D4semidirect
185. C22×C4×C37C2×D4semidirect
206. C22⋊C4×C37C2×D4semidirect
283. #23901C37direct (N × H)
310. C22×C37#458semidirect
323. C22×#123C2×C37semidirect
324. C2×#458C2×C37semidirect
325. #2849C2×C37semidirect
326. #2022C2×C37semidirect
327. #1898C2×C37semidirect
328. #1890C2×C37semidirect
329. #1781C2×C37semidirect
342. C24×C4C22×C37semidirect
353. C2×#123C22×C37semidirect
354. C2×#123C22×C37semidirect
360. C22×C22⋊C4C22×C37semidirect
363. C2×#99C22×C37semidirect
364. C2×#99C22×C37semidirect
367. #381C22×C37semidirect
368. #379C22×C37semidirect
369. #342C22×C37semidirect
370. C37#23901direct (N × H)
409. #123D4×C37semidirect
517. C22×C4C2×D4×C37semidirect
538. C22⋊C4C2×D4×C37semidirect
641. C22C37×#458semidirect

Maximal subgroups

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

subgroupindex
C37×#34842
C22×#123×C372
C2×C37×#4582
C37×#28492
C37×#20222
C37×#18982
C37×#18902
C37×#17812
#2390137