Finite Group Databasebeta

← C37×#10754 #22632007

Subgroups — C37×#10754 #22632007

Order 9472.

1522 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#518C211yesmaximal, characteristic
2C22×#99×C37C211yesmaximal, characteristic
3C37×#2308C211yesmaximal, characteristic
4C2×C37×#342C212yesmaximal, complemented
5C2×C37×#342C212yesmaximal, complemented
6C23×D4×C37C2211yescharacteristic
7C37×#518—22no—
8C37×#518—24no—
9C37×#518C414yescomplemented
10C2×C4×D4×C37—24no—
11C2×#117×C37—21no—
12C22×C22⋊C4×C37—21no—
13C22×C22⋊C4×C37C2211yescharacteristic
14C22×C22⋊C4×C37C2212yes—
15C22×C22⋊C4×C37C2211yescharacteristic
16C22×C22⋊C4×C37C2212yes—
17C2×#99×C37—22no—
18C2×#99×C37—22no—
19C2×#99×C37C414yescomplemented
20C37×#386—22no—
21C37×#386C414yescomplemented
22C37×#381—21no—
23C37×#379—21no—
24C37×#342—24no—
25C37×#342—24no—
26C25×C37D411yescharacteristic
27C25×C37C2311yescharacteristic
28C22×D4×C37—22no—
29C22×D4×C37—22no—
30C22×D4×C37—22no—
31C22×D4×C37—22no—
32C22×D4×C37C2×C412yes—
33C22×D4×C37C2×C412yes—
34C22×D4×C37C2×C412yes—
35C22×D4×C37D412yes—
36C22×D4×C37D412yes—
37C22×D4×C37Q812yes—
38C23×C4×C37—22no—
39C23×C4×C37—22no—
40C23×C4×C37—21no—
41C23×C4×C37—22no—
42C23×C4×C37C2×C412yescomplemented
43C23×C4×C37D411yescharacteristic
44C4×D4×C37—216no—
45C4×D4×C37—216no—
46#117×C37—24no—
47#117×C37—24no—
48C2×C4⋊C4×C37—22no—
49C2×C22⋊C4×C37—22no—
50C2×C22⋊C4×C37—24no—
51C2×C22⋊C4×C37—24no—
52C2×C22⋊C4×C37—22no—
53C2×C22⋊C4×C37—22no—
54C2×C22⋊C4×C37—22no—
55C2×C22⋊C4×C37—22no—
56C2×C22⋊C4×C37—22no—
57C2×C22⋊C4×C37—22no—
58C2×C22⋊C4×C37—22no—
59C2×C22⋊C4×C37—22no—
60C2×C22⋊C4×C37—22no—
61C2×C22⋊C4×C37—22no—
62C2×C22⋊C4×C37—22no—
63C2×C22⋊C4×C37—21no—
64C2×C22⋊C4×C37—21no—
65C2×C22⋊C4×C37—21no—
66C2×C22⋊C4×C37—22no—
67C2×C22⋊C4×C37—24no—
68C2×C22⋊C4×C37C2×C414yes—
69C2×C22⋊C4×C37C2×C412yescomplemented
70C2×C22⋊C4×C37C2×C414yescomplemented
71C2×C22⋊C4×C37D414yes—
72C2×C22⋊C4×C37D414yes—
73C2×C42×C37—22no—
74#99×C37—28no—
75#95×C37—24no—
76C24×C37—21no—
77C24×C37—21no—
78C24×C37—21no—
79C24×C37—21no—
80C24×C37—82no—
81C24×C37—82no—
82C24×C37C22⋊C412yes—
83C24×C37C22⋊C412yes—
84C24×C37C4⋊C412yes—
85C24×C37C22×C411yescharacteristic
86C24×C37C22×C411yescharacteristic
87C24×C37C22×C411yescharacteristic
88C24×C37C2×D411yescharacteristic
89C24×C37C2×D411yescharacteristic
90C24×C37C2×D412yes—
91C24×C37C2×D411yescharacteristic
92C24×C37C2×D412yes—
93C24×C37C2×Q811yescharacteristic
94C24×C37#4112yes—
95C24×C37#4112yes—
96C2×D4×C37—24no—
97C2×D4×C37—24no—
98C2×D4×C37—24no—
99C2×D4×C37—24no—
100C2×D4×C37—24no—
101C2×D4×C37—24no—
102C2×D4×C37—24no—
103C2×D4×C37—24no—
104C2×D4×C37—24no—
105C2×D4×C37—24no—
106C2×D4×C37—24no—
107C2×D4×C37—24no—
108C2×D4×C37C4214yes—
109C2×D4×C37C22⋊C414yes—
110C2×D4×C37C4⋊C414yes—
111C2×D4×C37C4⋊C414yes—
112C22×C4×C37—21no—
113C22×C4×C37—21no—
114C22×C4×C37—21no—
115C22×C4×C37—21no—
116C22×C4×C37—24no—
117C22×C4×C37—28no—
118C22×C4×C37—24no—
119C22×C4×C37—24no—
120C22×C4×C37—22no—
121C22×C4×C37—22no—
122C22×C4×C37—24no—
123C22×C4×C37—24no—
124C22×C4×C37—42no—
125C22×C4×C37—42no—
126C22×C4×C37—42no—
127C22×C4×C37—84no—
128C22×C4×C37—84no—
129C22×C4×C37—82no—
130C22×C4×C37—82no—
131C22×C4×C37—84no—
132C22×C4×C37C22⋊C412yes—
133C22×C4×C37C22⋊C412yes—
134C22×C4×C37C4⋊C412yes—
135C4⋊C4×C37—28no—
136C22⋊C4×C37—28no—
137C22⋊C4×C37—24no—
138C22⋊C4×C37—24no—
139C22⋊C4×C37—24no—
140C22⋊C4×C37—24no—
141C22⋊C4×C37—24no—
142C22⋊C4×C37—24no—
143C22⋊C4×C37—41no—
144C22⋊C4×C37—42no—
145C22⋊C4×C37—41no—
146C22⋊C4×C37—42no—
147C22⋊C4×C37—42no—
148C22⋊C4×C37—41no—
149C22⋊C4×C37—41no—
150C22⋊C4×C37—84no—
151C22⋊C4×C37C22⋊C418yescomplemented
152C42×C37—28no—
153C23×C37—22no—
154C23×C37—22no—
155C23×C37—22no—
156C23×C37—22no—
157C23×C37—22no—
158C23×C37—22no—
159C23×C37—42no—
160C23×C37—42no—
161C23×C37—44no—
162C23×C37—44no—
163C23×C37—42no—
164C23×C37—44no—
165C23×C37—44no—
166C23×C37—82no—
167C23×C37—81no—
168C23×C37—81no—
169C23×C37—82no—
170C23×C37—81no—
171C23×C37—81no—
172C23×C37—82no—
173C23×C37—82no—
174C23×C37—82no—
175C23×C37—82no—
176C23×C37—82no—
177C23×C37#9514yes—
178C23×C37C2×C4211yescharacteristic
179C23×C37C2×C22⋊C411yescharacteristic
180C23×C37C2×C22⋊C411yescharacteristic
181C23×C37C2×C22⋊C411yescharacteristic
182C23×C37C2×C22⋊C412yes—
183C23×C37C2×C4⋊C411yescharacteristic
184C23×C37C2×C4⋊C411yescharacteristic
185C23×C37C2×C4⋊C411yescharacteristic
186C23×C37#11712yes—
187C23×C37C4×D412yes—
188C23×C37C4×D412yes—
189C23×C37C4×D412yes—
190C23×C37C4×D412yes—
191C23×C37#12011yescharacteristic
192C23×C37#12011yescharacteristic
193C23×C37#12012yes—
194C23×C37#12112yes—
195C23×C37#12112yes—
196C23×C37C22⋊Q812yes—
197C23×C37C22⋊Q812yes—
198C23×C37#12311yescharacteristic
199C23×C37#12311yescharacteristic
200C23×C37#12312yes—
201D4×C37—28no—
202D4×C37—28no—
203D4×C37—28no—
204D4×C37—28no—
205C2×C4×C37—22no—
206C2×C4×C37—22no—
207C2×C4×C37—22no—
208C2×C4×C37—22no—
209C2×C4×C37—22no—
210C2×C4×C37—22no—
211C2×C4×C37—28no—
212C2×C4×C37—48no—
213C2×C4×C37—44no—
214C2×C4×C37—44no—
215C2×C4×C37—42no—
216C2×C4×C37—44no—
217C2×C4×C37—42no—
218C2×C4×C37—84no—
219C2×C4×C37—84no—
220C2×C4×C37—88no—
221C2×C4×C37—84no—
222C2×C4×C37—84no—
223C2×C4×C37—84no—
224C2×C4×C37—84no—
225C2×C4×C37—84no—
226C2×C4×C37#9514yes—
227#10754C3711yesmaximal, characteristic, direct product factor, Sylow
228C22×C37—22no—
229C22×C37—22no—
230C22×C37—24no—
231C22×C37—24no—
232C22×C37—22no—
233C22×C37—24no—
234C22×C37—42no—
235C22×C37—42no—
236C22×C37—42no—
237C22×C37—42no—
238C22×C37—42no—
239C22×C37—41no—
240C22×C37—41no—
241C22×C37—41no—
242C22×C37—41no—
243C22×C37—41no—
244C22×C37—41no—
245C22×C37—41no—
246C22×C37—41no—
247C22×C37—42no—
248C22×C37—42no—
249C22×C37—42no—
250C22×C37—42no—
251C22×C37—42no—
252C22×C37—44no—
253C22×C37—44no—
254C22×C37—44no—
255C22×C37—44no—
256C22×C37—82no—
257C22×C37—82no—
258C22×C37—82no—
259C22×C37C2×#9511yescharacteristic
260C22×C37C4×C22⋊C412yes—
261C22×C37#37911yescharacteristic
262C22×C37#38012yes—
263C22×C37#38111yescharacteristic
264C22×C37#38511yescharacteristic
265C22×C37#38511yescharacteristic
266C22×C37#38512yes—
267C22×C37#38611yescharacteristic
268C22×C37#38612yes—
269C22×C37#38611yescharacteristic
270C4×C37—24no—
271C4×C37—48no—
272C4×C37—82no—
273C4×C37—84no—
274C4×C37—82no—
275C2×#518C2×C3711yescharacteristic
276C22×#99C2×C3711yescharacteristic
277#2308C2×C3711yescharacteristic
278C2×#342C2×C3712yescomplemented
279C2×#342C2×C3712yescomplemented
280C2×C37—21no—
281C2×C37—21no—
282C2×C37—22no—
283C2×C37—21no—
284C2×C37—21no—
285C2×C37—21no—
286C2×C37—21no—
287C2×C37—22no—
288C2×C37—22no—
289C2×C37—22no—
290C2×C37—44no—
291C2×C37C22⋊#9511yescharacteristic
292C2×C37#191312yes—
293C23×D4C22×C3711yescharacteristic
294#518—22no—
295#518—24no—
296#518C4×C3714yescomplemented
297C2×C4×D4—24no—
298C2×#117—21no—
299C22×C22⋊C4—21no—
300C22×C22⋊C4C22×C3711yescharacteristic
301C22×C22⋊C4C22×C3711yescharacteristic
302C22×C22⋊C4C22×C3712yes—
303C22×C22⋊C4C22×C3712yes—
304C2×#99—22no—
305C2×#99—22no—
306C2×#99C4×C3714yescomplemented
307#386—22no—
308#386C4×C3714yescomplemented
309#381—21no—
310#379—21no—
311#342—24no—
312#342—24no—
313C37#1075411yescharacteristic, direct product factor, Sylow
314C25D4×C3711yescharacteristic
315C25C23×C3711yescharacteristic
316C22×D4—22no—
317C22×D4—22no—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4C2×C4×C3712yes—
321C22×D4C2×C4×C3712yes—
322C22×D4C2×C4×C3712yes—
323C22×D4D4×C3712yes—
324C22×D4D4×C3712yes—
325C22×D4Q8×C3712yes—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—21no—
329C23×C4—22no—
330C23×C4C2×C4×C3712yescomplemented
331C23×C4D4×C3711yescharacteristic
332C4×D4—216no—
333C4×D4—216no—
334#117—24no—
335#117—24no—
336C2×C4⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4C2×C4×C3714yes—
357C2×C22⋊C4C2×C4×C3712yescomplemented
358C2×C22⋊C4C2×C4×C3714yescomplemented
359C2×C22⋊C4D4×C3714yes—
360C2×C22⋊C4D4×C3714yes—
361C2×C42—22no—
362#99—28no—
363#95—24no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—82no—
369C24—82no—
370C24C22⋊C4×C3712yes—
371C24C22⋊C4×C3712yes—
372C24C4⋊C4×C3712yes—
373C24C22×C4×C3711yescharacteristic
374C24C22×C4×C3711yescharacteristic
375C24C22×C4×C3711yescharacteristic
376C24C2×D4×C3711yescharacteristic
377C24C2×D4×C3711yescharacteristic
378C24C2×D4×C3712yes—
379C24C2×D4×C3711yescharacteristic
380C24C2×D4×C3712yes—
381C24C2×Q8×C3711yescharacteristic
382C24#41×C3712yes—
383C24#41×C3712yes—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4C42×C3714yes—
397C2×D4C22⋊C4×C3714yes—
398C2×D4C4⋊C4×C3714yes—
399C2×D4C4⋊C4×C3714yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—84no—
420C22×C4C22⋊C4×C3712yes—
421C22×C4C22⋊C4×C3712yes—
422C22×C4C4⋊C4×C3712yes—
423C4⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—41no—
432C22⋊C4—42no—
433C22⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—84no—
439C22⋊C4C22⋊C4×C3718yescomplemented
440C42—28no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—82no—
455C23—81no—
456C23—81no—
457C23—82no—
458C23—81no—
459C23—81no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23#95×C3714yes—
466C23C2×C42×C3711yescharacteristic
467C23C2×C22⋊C4×C3711yescharacteristic
468C23C2×C22⋊C4×C3711yescharacteristic
469C23C2×C22⋊C4×C3711yescharacteristic
470C23C2×C22⋊C4×C3712yes—
471C23C2×C4⋊C4×C3711yescharacteristic
472C23C2×C4⋊C4×C3711yescharacteristic
473C23C2×C4⋊C4×C3711yescharacteristic
474C23#117×C3712yes—
475C23C4×D4×C3712yes—
476C23C4×D4×C3712yes—
477C23C4×D4×C3712yes—
478C23C4×D4×C3712yes—
479C23#120×C3711yescharacteristic
480C23#120×C3711yescharacteristic
481C23#120×C3712yes—
482C23#121×C3712yes—
483C23#121×C3712yes—
484C23C22⋊Q8×C3712yes—
485C23C22⋊Q8×C3712yes—
486C23#123×C3711yescharacteristic
487C23#123×C3711yescharacteristic
488C23#123×C3712yes—
489D4—28no—
490D4—28no—
491D4—28no—
492D4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—88no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4#95×C3714yes—
515C22—22no—
516C22—22no—
517C22—24no—
518C22—24no—
519C22—22no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C2×#95×C3711yescharacteristic
547C22C4×C22⋊C4×C3712yes—
548C22C37×#37911yescharacteristic
549C22C37×#38012yes—
550C22C37×#38111yescharacteristic
551C22C37×#38511yescharacteristic
552C22C37×#38511yescharacteristic
553C22C37×#38512yes—
554C22C37×#38611yescharacteristic
555C22C37×#38612yes—
556C22C37×#38611yescharacteristic
557C4—24no—
558C4—48no—
559C4—82no—
560C4—84no—
561C4—82no—
562C2—21no—
563C2—21no—
564C2—22no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—22no—
571C2—22no—
572C2—44no—
573C2C37×C22⋊#9511yescharacteristic
574C2C37×#191312yes—
subgroupG/H
center, socleC22×C37#386
commutatorC23C2×C42×C37
Frattini, cosocleC25C23×C37
prime radicalC23×D4×C37C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C25×C37 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C37×#342C2semidirect
5. C2×C37×#342C2semidirect
9. C37×#518C4semidirect
19. C2×#99×C37C4semidirect
21. C37×#386C4semidirect
42. C23×C4×C37C2×C4semidirect
69. C2×C22⋊C4×C37C2×C4semidirect
70. C2×C22⋊C4×C37C2×C4semidirect
151. C22⋊C4×C37C22⋊C4semidirect
227. #10754C37direct (N × H)
278. C2×#342C2×C37semidirect
279. C2×#342C2×C37semidirect
296. #518C4×C37semidirect
306. C2×#99C4×C37semidirect
308. #386C4×C37semidirect
313. C37#10754direct (N × H)
330. C23×C4C2×C4×C37semidirect
357. C2×C22⋊C4C2×C4×C37semidirect
358. C2×C22⋊C4C2×C4×C37semidirect
439. C22⋊C4C22⋊C4×C37semidirect

Maximal subgroups

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

subgroupindex
C2×C37×#5182
C22×#99×C372
C37×#23082
C2×C37×#3422
#1075437