Finite Group Databasebeta

← C37×#18238 #22639491

Subgroups — C37×#18238 #22639491

Order 9472.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3538C211yesmaximal, characteristic
2C37×#3095C211yesmaximal, characteristic, complemented
3C37×#3091C211yesmaximal, characteristic
4C37×#2625C211yesmaximal, characteristic, complemented
5C37×#1617C212yesmaximal, complemented
6C37×#1615C212yesmaximal, complemented
7C37×#1611C212yesmaximal, complemented
8C37×#1609C212yesmaximal, complemented
9C37×#1520C211yesmaximal, characteristic, complemented
10C37×#1513C211yesmaximal, characteristic
11C37×#1472C211yesmaximal, characteristic, complemented
12C37×#563—22no—
13C37×#552—21no—
14C37×#552—21no—
15C37×#551—22no—
16D4×Q8×C37—21no—
17D4×Q8×C37—21no—
18C37×#544—21no—
19C37×#544C2212yes—
20C37×#541—21no—
21C37×#539—21no—
22C37×#539C2212yes—
23C37×#536—21no—
24C37×#525C2211yescharacteristic
25C2×C22⋊Q8×C37C2211yescharacteristic
26C2×C22⋊Q8×C37C2211yescharacteristic
27C2×#117×C37C2211yescharacteristic
28C37×#485—21no—
29C37×#485—22no—
30C37×#484C2212yes—
31C37×#484C2211yescharacteristic
32C37×#483C2212yes—
33C37×#483C2211yescharacteristic
34C37×#482—21no—
35C37×#481C2211yescharacteristic
36C37×#480C2211yescharacteristic
37C37×#428—21no—
38C37×#427—21no—
39C37×#419—21no—
40C37×#416—21no—
41C37×#407C2211yescharacteristic
42C2×C22⋊C8×C37C2211yescharacteristic
43C37×#400C2212yes—
44C37×#398C2212yes—
45C37×#397—22no—
46C37×#391—21no—
47C37×#389—21no—
48C37×#380C2211yescharacteristic
49C37×#328C2212yes—
50C37×#328C2212yes—
51C37×#328C2212yes—
52C37×#327C2212yes—
53C37×#324C2212yes—
54C37×#324C2212yes—
55C2×#41×C37D411yescharacteristic
56C22×Q8×C37—21no—
57C22×Q8×C37D411yescharacteristic
58C22×Q8×C37D411yescharacteristic
59C23×C4×C37C2311yescharacteristic
60C2×M16×C37—21no—
61C22×C8×C37—21no—
62C4⋊Q8×C37—21no—
63C4⋊Q8×C37—22no—
64C4⋊Q8×C37—22no—
65C4⋊Q8×C37—22no—
66#126×C37—22no—
67#126×C37—22no—
68#126×C37—22no—
69#126×C37—22no—
70#125×C37—21no—
71#125×C37—22no—
72#125×C37—22no—
73#125×C37—22no—
74#125×C37—22no—
75#124×C37—22no—
76#123×C37—22no—
77#123×C37—22no—
78#123×C37—22no—
79#123×C37—21no—
80#123×C37—22no—
81C22⋊Q8×C37—21no—
82C22⋊Q8×C37—21no—
83C22⋊Q8×C37—22no—
84C22⋊Q8×C37—22no—
85C22⋊Q8×C37—22no—
86C22⋊Q8×C37—21no—
87C22⋊Q8×C37—21no—
88C22⋊Q8×C37—21no—
89C22⋊Q8×C37—21no—
90C22⋊Q8×C37—21no—
91C22⋊Q8×C37—21no—
92C22⋊Q8×C37—22no—
93C22⋊Q8×C37D412yes—
94C22⋊Q8×C37D411yescharacteristic
95C22⋊Q8×C37C2312yes—
96C22⋊Q8×C37C2311yescharacteristic
97#121×C37D411yescharacteristic
98#121×C37C2311yescharacteristic
99#120×C37—21no—
100C4×Q8×C37—21no—
101C4×Q8×C37—21no—
102C4×D4×C37—22no—
103C4×D4×C37—21no—
104C4×D4×C37—21no—
105C4×D4×C37—22no—
106#117×C37—21no—
107#117×C37—22no—
108#117×C37—22no—
109#117×C37—21no—
110#117×C37—21no—
111#117×C37—21no—
112#117×C37—21no—
113#117×C37D411yescharacteristic
114C2×C4⋊C4×C37—21no—
115C2×C4⋊C4×C37—21no—
116C2×C4⋊C4×C37—22no—
117C2×C4⋊C4×C37D411yescharacteristic
118C2×C4⋊C4×C37D411yescharacteristic
119C2×C4⋊C4×C37C2311yescharacteristic
120C2×C4⋊C4×C37C2311yescharacteristic
121C2×C4⋊C4×C37C2312yes—
122C2×C22⋊C4×C37—21no—
123C2×C22⋊C4×C37—21no—
124C2×C22⋊C4×C37—21no—
125C2×C22⋊C4×C37D412yes—
126C2×C42×C37—21no—
127#107×C37—21no—
128#107×C37—21no—
129#107×C37—22no—
130#106×C37—21no—
131#106×C37—21no—
132#106×C37—22no—
133#103×C37—21no—
134#103×C37—21no—
135#103×C37—22no—
136#103×C37—22no—
137#102×C37—21no—
138#102×C37—21no—
139C22⋊C8×C37—21no—
140C22⋊C8×C37—21no—
141C22⋊C8×C37C2312yes—
142C22⋊C8×C37C2311yescharacteristic
143C22⋊C8×C37C2311yescharacteristic
144#95×C37—22no—
145#95×C37—22no—
146#95×C37—22no—
147#95×C37C2312yes—
148C24×C37C2×D411yescharacteristic
149#41×C37—42no—
150C2×Q8×C37—22no—
151C2×Q8×C37—21no—
152C2×Q8×C37—21no—
153C2×Q8×C37—21no—
154C2×Q8×C37—42no—
155C2×Q8×C37—42no—
156C2×Q8×C37—82no—
157C2×Q8×C37C2×D412yes—
158C2×Q8×C37C2×D411yescharacteristic
159C2×D4×C37—21no—
160C2×D4×C37—21no—
161C2×D4×C37—21no—
162C2×D4×C37C2×D411yescharacteristic
163C22×C4×C37—21no—
164C22×C4×C37—22no—
165C22×C4×C37—21no—
166C22×C4×C37—21no—
167C22×C4×C37—21no—
168C22×C4×C37—22no—
169C22×C4×C37—21no—
170C22×C4×C37—42no—
171C22×C4×C37C2×D412yes—
172C22×C4×C37C2×D411yescharacteristic
173C22×C4×C37C2×D411yescharacteristic
174C22×C4×C37C2×D411yescharacteristic
175C22×C4×C37C2×D412yes—
176C22×C4×C37C2×D411yescharacteristic
177C22×C4×C37C2×D411yescharacteristic
178C22×C4×C37C2411yescharacteristic
179M16×C37—42no—
180C2×C8×C37—21no—
181C2×C8×C37—21no—
182C2×C8×C37—22no—
183C2×C8×C37—42no—
184C4⋊C4×C37—22no—
185C4⋊C4×C37—22no—
186C4⋊C4×C37—22no—
187C4⋊C4×C37—22no—
188C4⋊C4×C37—22no—
189C4⋊C4×C37—22no—
190C4⋊C4×C37—22no—
191C4⋊C4×C37—22no—
192C4⋊C4×C37—22no—
193C4⋊C4×C37—21no—
194C4⋊C4×C37—21no—
195C4⋊C4×C37—21no—
196C4⋊C4×C37—21no—
197C4⋊C4×C37—22no—
198C4⋊C4×C37—22no—
199C4⋊C4×C37—22no—
200C4⋊C4×C37—21no—
201C4⋊C4×C37—21no—
202C4⋊C4×C37—21no—
203C4⋊C4×C37—21no—
204C4⋊C4×C37—21no—
205C4⋊C4×C37—42no—
206C4⋊C4×C37—82no—
207C4⋊C4×C37C2×D412yes—
208C4⋊C4×C37C2×D411yescharacteristic
209C4⋊C4×C37C2×D411yescharacteristic
210C22⋊C4×C37—22no—
211C22⋊C4×C37—22no—
212C22⋊C4×C37—22no—
213C22⋊C4×C37—22no—
214C22⋊C4×C37—22no—
215C22⋊C4×C37—22no—
216C22⋊C4×C37—21no—
217C22⋊C4×C37—21no—
218C22⋊C4×C37—22no—
219C22⋊C4×C37—21no—
220C22⋊C4×C37—21no—
221C22⋊C4×C37—21no—
222C22⋊C4×C37—41no—
223C22⋊C4×C37—41no—
224C22⋊C4×C37—81no—
225C42×C37—22no—
226C42×C37—22no—
227C42×C37—22no—
228C42×C37—21no—
229C42×C37—41no—
230C42×C37—41no—
231C23×C37—21no—
232C23×C37—21no—
233C23×C37—21no—
234C23×C37—41no—
235C23×C37—41no—
236C23×C37#12011yescharacteristic
237C23×C37#12011yescharacteristic
238C23×C37C22×D411yescharacteristic
239Q8×C37—42no—
240Q8×C37—41no—
241Q8×C37—42no—
242Q8×C37—42no—
243Q8×C37—81no—
244Q8×C37—82no—
245D4×C37—41no—
246D4×C37—42no—
247D4×C37—41no—
248D4×C37—41no—
249D4×C37—81no—
250C2×C4×C37—22no—
251C2×C4×C37—22no—
252C2×C4×C37—22no—
253C2×C4×C37—21no—
254C2×C4×C37—21no—
255C2×C4×C37—22no—
256C2×C4×C37—22no—
257C2×C4×C37—21no—
258C2×C4×C37—22no—
259C2×C4×C37—22no—
260C2×C4×C37—22no—
261C2×C4×C37—22no—
262C2×C4×C37—41no—
263C2×C4×C37—41no—
264C2×C4×C37—41no—
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—41no—
272C2×C4×C37—41no—
273C2×C4×C37—42no—
274C2×C4×C37—41no—
275C2×C4×C37—41no—
276C2×C4×C37—81no—
277C2×C4×C37—82no—
278C2×C4×C37#12011yescharacteristic
279C2×C4×C37#12011yescharacteristic
280C2×C4×C37C22×D411yescharacteristic
281C2×C4×C37C22×D411yescharacteristic
282C8×C37—41no—
283C8×C37—41no—
284C8×C37—42no—
285#18238C3711yesmaximal, characteristic, direct product factor, Sylow
286C22×C37—21no—
287C22×C37—21no—
288C22×C37—21no—
289C22×C37—21no—
290C22×C37—21no—
291C22×C37—21no—
292C22×C37—21no—
293C22×C37—21no—
294C22×C37—41no—
295C22×C37—41no—
296C22×C37—41no—
297C22×C37—81no—
298C22×C37—81no—
299C22×C37C2×#12011yescharacteristic
300C22×C37#57711yescharacteristic
301C22×C37#57811yescharacteristic
302C4×C37—21no—
303C4×C37—21no—
304C4×C37—21no—
305C4×C37—42no—
306C4×C37—41no—
307C4×C37—42no—
308C4×C37—41no—
309C4×C37—41no—
310C4×C37—42no—
311C4×C37—42no—
312C4×C37—82no—
313C4×C37—81no—
314C4×C37—81no—
315C4×C37#45812yes—
316#3538C2×C3711yescharacteristic
317#3095C2×C3711yescharacteristic, complemented
318#3091C2×C3711yescharacteristic
319#2625C2×C3711yescharacteristic, complemented
320#1617C2×C3712yescomplemented
321#1615C2×C3712yescomplemented
322#1611C2×C3712yescomplemented
323#1609C2×C3712yescomplemented
324#1520C2×C3711yescharacteristic, complemented
325#1513C2×C3711yescharacteristic
326#1472C2×C3711yescharacteristic, complemented
327C2×C37—21no—
328C2×C37—21no—
329C2×C37—41no—
330C2×C37—41no—
331C2×C37—81no—
332C2×C37#301311yescharacteristic
333C2×C37#302011yescharacteristic
334C2×C37C2×#45811yescharacteristic
335#563—22no—
336#552—21no—
337#552—21no—
338#551—22no—
339D4×Q8—21no—
340D4×Q8—21no—
341#544—21no—
342#544C22×C3712yes—
343#541—21no—
344#539—21no—
345#539C22×C3712yes—
346#536—21no—
347#525C22×C3711yescharacteristic
348C2×C22⋊Q8C22×C3711yescharacteristic
349C2×C22⋊Q8C22×C3711yescharacteristic
350C2×#117C22×C3711yescharacteristic
351#485—21no—
352#485—22no—
353#484C22×C3711yescharacteristic
354#484C22×C3712yes—
355#483C22×C3711yescharacteristic
356#483C22×C3712yes—
357#482—21no—
358#481C22×C3711yescharacteristic
359#480C22×C3711yescharacteristic
360#428—21no—
361#427—21no—
362#419—21no—
363#416—21no—
364#407C22×C3711yescharacteristic
365C2×C22⋊C8C22×C3711yescharacteristic
366#400C22×C3712yes—
367#398C22×C3712yes—
368#397—22no—
369#391—21no—
370#389—21no—
371#380C22×C3711yescharacteristic
372#328C22×C3712yes—
373#328C22×C3712yes—
374#328C22×C3712yes—
375#327C22×C3712yes—
376#324C22×C3712yes—
377#324C22×C3712yes—
378C37#1823811yescharacteristic, direct product factor, Sylow
379C2×#41D4×C3711yescharacteristic
380C22×Q8—21no—
381C22×Q8D4×C3711yescharacteristic
382C22×Q8D4×C3711yescharacteristic
383C23×C4C23×C3711yescharacteristic
384C2×M16—21no—
385C22×C8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—22no—
388C4⋊Q8—22no—
389C4⋊Q8—22no—
390#126—22no—
391#126—22no—
392#126—22no—
393#126—22no—
394#125—21no—
395#125—22no—
396#125—22no—
397#125—22no—
398#125—22no—
399#124—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403#123—21no—
404#123—22no—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8D4×C3712yes—
418C22⋊Q8D4×C3711yescharacteristic
419C22⋊Q8C23×C3712yes—
420C22⋊Q8C23×C3711yescharacteristic
421#121D4×C3711yescharacteristic
422#121C23×C3711yescharacteristic
423#120—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×D4—22no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—22no—
430#117—21no—
431#117—22no—
432#117—22no—
433#117—21no—
434#117—21no—
435#117—21no—
436#117—21no—
437#117D4×C3711yescharacteristic
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4D4×C3711yescharacteristic
442C2×C4⋊C4D4×C3711yescharacteristic
443C2×C4⋊C4C23×C3711yescharacteristic
444C2×C4⋊C4C23×C3711yescharacteristic
445C2×C4⋊C4C23×C3712yes—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4D4×C3712yes—
450C2×C42—21no—
451#107—21no—
452#107—21no—
453#107—22no—
454#106—21no—
455#106—21no—
456#106—22no—
457#103—21no—
458#103—21no—
459#103—22no—
460#103—22no—
461#102—21no—
462#102—21no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8C23×C3712yes—
466C22⋊C8C23×C3711yescharacteristic
467C22⋊C8C23×C3711yescharacteristic
468#95—22no—
469#95—22no—
470#95—22no—
471#95C23×C3712yes—
472C24C2×D4×C3711yescharacteristic
473#41—42no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—82no—
481C2×Q8C2×D4×C3712yes—
482C2×Q8C2×D4×C3711yescharacteristic
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×D4×C3711yescharacteristic
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4C2×D4×C3712yes—
496C22×C4C2×D4×C3711yescharacteristic
497C22×C4C2×D4×C3711yescharacteristic
498C22×C4C2×D4×C3711yescharacteristic
499C22×C4C2×D4×C3712yes—
500C22×C4C2×D4×C3711yescharacteristic
501C22×C4C2×D4×C3711yescharacteristic
502C22×C4C24×C3711yescharacteristic
503M16—42no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—22no—
507C2×C8—42no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4C2×D4×C3712yes—
532C4⋊C4C2×D4×C3711yescharacteristic
533C4⋊C4C2×D4×C3711yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—81no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—41no—
554C42—41no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23#120×C3711yescharacteristic
561C23#120×C3711yescharacteristic
562C23C22×D4×C3711yescharacteristic
563Q8—42no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—81no—
568Q8—82no—
569D4—41no—
570D4—42no—
571D4—41no—
572D4—41no—
573D4—81no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4#120×C3711yescharacteristic
603C2×C4#120×C3711yescharacteristic
604C2×C4C22×D4×C3711yescharacteristic
605C2×C4C22×D4×C3711yescharacteristic
606C8—41no—
607C8—41no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—81no—
621C22—81no—
622C22C2×#120×C3711yescharacteristic
623C22C37×#57711yescharacteristic
624C22C37×#57811yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4C37×#45812yes—
639C2—21no—
640C2—21no—
641C2—41no—
642C2—41no—
643C2—81no—
644C2C37×#301311yescharacteristic
645C2C37×#302011yescharacteristic
646C2C2×C37×#45811yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3095C2semidirect
4. C37×#2625C2semidirect
5. C37×#1617C2semidirect
6. C37×#1615C2semidirect
7. C37×#1611C2semidirect
8. C37×#1609C2semidirect
9. C37×#1520C2semidirect
11. C37×#1472C2semidirect
285. #18238C37direct (N × H)
317. #3095C2×C37semidirect
319. #2625C2×C37semidirect
320. #1617C2×C37semidirect
321. #1615C2×C37semidirect
322. #1611C2×C37semidirect
323. #1609C2×C37semidirect
324. #1520C2×C37semidirect
326. #1472C2×C37semidirect
378. C37#18238direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35382
C37×#30952
C37×#30912
C37×#26252
C37×#16172
C37×#16152
C37×#16112
C37×#16092
C37×#15202
C37×#15132
C37×#14722
#1823837