Finite Group Databasebeta

← C37×#18114 #22639367

Subgroups — C37×#18114 #22639367

Order 9472.

1246 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3460C211yesmaximal, characteristic, complemented
2C37×#3002C212yesmaximal, complemented
3C37×#2388C211yesmaximal, characteristic, complemented
4C37×#1606C212yesmaximal, complemented
5C37×#1605C211yesmaximal, characteristic, complemented
6C37×#1604C211yesmaximal, characteristic, complemented
7C37×#1600C212yesmaximal, complemented
8C37×#1599C211yesmaximal, characteristic, complemented
9C37×#1598C211yesmaximal, characteristic, complemented
10C37×#1501C212yesmaximal, complemented
11C37×#1459C211yesmaximal, characteristic, complemented
12C22×#41×C37C2211yescharacteristic
13C2×#135×C37—22no—
14C37×#544C2211yescharacteristic
15C37×#542—22no—
16C37×#540—22no—
17C37×#538—21no—
18C37×#538C2212yes—
19C37×#537C2211yescharacteristic
20C37×#535—21no—
21C37×#532—22no—
22C37×#529—22no—
23C37×#525—21no—
24C37×#525C2212yes—
25C4×#41×C37—21no—
26C2×C4×D4×C37—21no—
27C2×C4×D4×C37C2211yescharacteristic
28C37×#452—22no—
29C37×#451C2212yes—
30C37×#450C2212yes—
31C37×#449—22no—
32C37×#448C2212yes—
33C37×#447C2212yes—
34C37×#416—22no—
35C2×C22⋊C8×C37C2212yes—
36C37×#395C2211yescharacteristic
37C37×#394—21no—
38C37×#394C2211yescharacteristic
39C37×#393C2211yescharacteristic
40C37×#392—21no—
41C37×#392C2211yescharacteristic
42C4×C22⋊C4×C37—21no—
43C4×C22⋊C4×C37C2211yescharacteristic
44C37×#328C2212yes—
45C37×#328C2212yes—
46C37×#327C2212yes—
47C37×#327C2212yes—
48C37×#324C2211yescharacteristic
49C37×#324C2211yescharacteristic
50C37×#323C2212yes—
51C2×#41×C37—21no—
52C2×#41×C37—21no—
53C2×#41×C37—22no—
54C2×#41×C37—22no—
55C2×#41×C37—44no—
56C2×#41×C37D412yescomplemented
57C22×Q8×C37C2311yescharacteristic
58C22×D4×C37—21no—
59C22×D4×C37—21no—
60C22×D4×C37D411yescharacteristic
61C22×D4×C37C2311yescharacteristic
62C23×C4×C37—21no—
63C23×C4×C37D411yescharacteristic
64C23×C4×C37C2311yescharacteristic
65#135×C37—48no—
66C2×Q16×C37—22no—
67C2×SD16×C37—22no—
68C2×SD16×C37—22no—
69C2×D8×C37—22no—
70C22×C8×C37—22no—
71C4⋊Q8×C37—22no—
72#127×C37—22no—
73#126×C37—22no—
74#126×C37—22no—
75#125×C37—22no—
76#124×C37—22no—
77#124×C37—22no—
78#124×C37—22no—
79#123×C37—22no—
80#123×C37—22no—
81#123×C37—22no—
82#123×C37—22no—
83C22⋊Q8×C37—21no—
84C22⋊Q8×C37—21no—
85C22⋊Q8×C37—22no—
86C22⋊Q8×C37D412yescomplemented
87C22⋊Q8×C37C2312yes—
88#121×C37—21no—
89#121×C37—22no—
90#121×C37—22no—
91#121×C37—22no—
92#121×C37—22no—
93#121×C37—22no—
94#121×C37—21no—
95#121×C37D412yescomplemented
96#121×C37C2312yes—
97#120×C37—22no—
98#120×C37—22no—
99C4×Q8×C37—21no—
100C4×Q8×C37—21no—
101C4×D4×C37—22no—
102C4×D4×C37—21no—
103C4×D4×C37—21no—
104C4×D4×C37—22no—
105C4×D4×C37—22no—
106C4×D4×C37—22no—
107C4×D4×C37—22no—
108C4×D4×C37—22no—
109C4×D4×C37—44no—
110#117×C37—21no—
111#117×C37—22no—
112#117×C37—21no—
113#117×C37D412yescomplemented
114C2×C4⋊C4×C37—21no—
115C2×C4⋊C4×C37C2311yescharacteristic
116C2×C22⋊C4×C37—21no—
117C2×C22⋊C4×C37—21no—
118C2×C22⋊C4×C37—21no—
119C2×C22⋊C4×C37—21no—
120C2×C22⋊C4×C37—21no—
121C2×C22⋊C4×C37D411yescharacteristic
122C2×C22⋊C4×C37C2311yescharacteristic
123C2×C42×C37—22no—
124C2×C42×C37—21no—
125C2×C42×C37D411yescharacteristic
126#103×C37—22no—
127#103×C37—22no—
128#102×C37—22no—
129#102×C37—22no—
130C22⋊C8×C37—22no—
131C22⋊C8×C37C2312yes—
132C22⋊C8×C37C2312yes—
133#95×C37—21no—
134#95×C37—21no—
135#95×C37C2311yescharacteristic
136#95×C37C2311yescharacteristic
137C24×C37—21no—
138C24×C37C2×D411yescharacteristic
139C24×C37C2×D411yescharacteristic
140#41×C37—28no—
141#41×C37—42no—
142#41×C37—44no—
143#41×C37—44no—
144#41×C37—44no—
145#41×C37—44no—
146#41×C37—42no—
147#41×C37—82no—
148C2×Q8×C37—21no—
149C2×Q8×C37—21no—
150C2×Q8×C37—22no—
151C2×Q8×C37—42no—
152C2×Q8×C37C2×D412yes—
153C2×D4×C37—21no—
154C2×D4×C37—22no—
155C2×D4×C37—22no—
156C2×D4×C37—22no—
157C2×D4×C37—22no—
158C2×D4×C37—21no—
159C2×D4×C37—22no—
160C2×D4×C37—22no—
161C2×D4×C37—22no—
162C2×D4×C37—22no—
163C2×D4×C37—41no—
164C2×D4×C37—42no—
165C2×D4×C37—42no—
166C2×D4×C37—42no—
167C2×D4×C37—42no—
168C2×D4×C37—42no—
169C2×D4×C37—42no—
170C2×D4×C37—42no—
171C2×D4×C37C2×D412yes—
172C22×C4×C37—22no—
173C22×C4×C37—24no—
174C22×C4×C37—21no—
175C22×C4×C37—21no—
176C22×C4×C37—22no—
177C22×C4×C37—22no—
178C22×C4×C37—21no—
179C22×C4×C37—21no—
180C22×C4×C37—21no—
181C22×C4×C37—42no—
182C22×C4×C37—42no—
183C22×C4×C37—42no—
184C22×C4×C37—41no—
185C22×C4×C37—41no—
186C22×C4×C37C2×D411yescharacteristic
187C22×C4×C37C2×D412yes—
188C22×C4×C37C2×D411yescharacteristic
189C22×C4×C37C2×D411yescharacteristic
190C22×C4×C37C2×D411yescharacteristic
191C22×C4×C37C2×D412yes—
192C22×C4×C37C2411yescharacteristic
193Q16×C37—44no—
194SD16×C37—44no—
195SD16×C37—44no—
196D8×C37—44no—
197C2×C8×C37—22no—
198C2×C8×C37—22no—
199C2×C8×C37—44no—
200C4⋊C4×C37—21no—
201C4⋊C4×C37—21no—
202C4⋊C4×C37—22no—
203C4⋊C4×C37—21no—
204C4⋊C4×C37—21no—
205C4⋊C4×C37—22no—
206C4⋊C4×C37—22no—
207C4⋊C4×C37—22no—
208C4⋊C4×C37—22no—
209C4⋊C4×C37—42no—
210C4⋊C4×C37C2×D412yes—
211C4⋊C4×C37C2×D412yes—
212C22⋊C4×C37—22no—
213C22⋊C4×C37—22no—
214C22⋊C4×C37—22no—
215C22⋊C4×C37—22no—
216C22⋊C4×C37—22no—
217C22⋊C4×C37—22no—
218C22⋊C4×C37—22no—
219C22⋊C4×C37—22no—
220C22⋊C4×C37—22no—
221C22⋊C4×C37—22no—
222C22⋊C4×C37—22no—
223C22⋊C4×C37—42no—
224C22⋊C4×C37—42no—
225C22⋊C4×C37—42no—
226C22⋊C4×C37—42no—
227C22⋊C4×C37—42no—
228C42×C37—22no—
229C42×C37—22no—
230C42×C37—21no—
231C42×C37—21no—
232C42×C37—22no—
233C42×C37—42no—
234C23×C37—21no—
235C23×C37—21no—
236C23×C37—22no—
237C23×C37—21no—
238C23×C37—22no—
239C23×C37—21no—
240C23×C37—21no—
241C23×C37—41no—
242C23×C37—41no—
243C23×C37—41no—
244C23×C37—41no—
245C23×C37—41no—
246C23×C37—41no—
247C23×C37—41no—
248C23×C37—41no—
249C23×C37—41no—
250C23×C37—41no—
251C23×C37#12012yes—
252C23×C37C22×D411yescharacteristic
253Q8×C37—24no—
254Q8×C37—42no—
255Q8×C37—42no—
256Q8×C37—42no—
257Q8×C37—81no—
258D4×C37—24no—
259D4×C37—44no—
260D4×C37—42no—
261D4×C37—42no—
262D4×C37—42no—
263D4×C37—48no—
264D4×C37—48no—
265D4×C37—42no—
266D4×C37—42no—
267D4×C37—82no—
268D4×C37—82no—
269D4×C37—82no—
270D4×C37—81no—
271D4×C37—82no—
272C2×C4×C37—21no—
273C2×C4×C37—21no—
274C2×C4×C37—22no—
275C2×C4×C37—24no—
276C2×C4×C37—22no—
277C2×C4×C37—22no—
278C2×C4×C37—21no—
279C2×C4×C37—22no—
280C2×C4×C37—22no—
281C2×C4×C37—22no—
282C2×C4×C37—21no—
283C2×C4×C37—42no—
284C2×C4×C37—41no—
285C2×C4×C37—44no—
286C2×C4×C37—44no—
287C2×C4×C37—41no—
288C2×C4×C37—41no—
289C2×C4×C37—41no—
290C2×C4×C37—41no—
291C2×C4×C37—44no—
292C2×C4×C37—42no—
293C2×C4×C37—44no—
294C2×C4×C37—41no—
295C2×C4×C37—41no—
296C2×C4×C37—41no—
297C2×C4×C37—41no—
298C2×C4×C37—41no—
299C2×C4×C37—81no—
300C2×C4×C37—82no—
301C2×C4×C37#12011yescharacteristic
302C2×C4×C37#12011yescharacteristic
303C2×C4×C37#13512yescomplemented
304C2×C4×C37C22×D412yes—
305C8×C37—42no—
306C8×C37—42no—
307#18114C3711yesmaximal, characteristic, direct product factor, Sylow
308C22×C37—21no—
309C22×C37—21no—
310C22×C37—22no—
311C22×C37—22no—
312C22×C37—22no—
313C22×C37—42no—
314C22×C37—42no—
315C22×C37—42no—
316C22×C37—42no—
317C22×C37—41no—
318C22×C37—42no—
319C22×C37—41no—
320C22×C37—42no—
321C22×C37—41no—
322C22×C37—42no—
323C22×C37—81no—
324C22×C37—81no—
325C22×C37—81no—
326C22×C37—81no—
327C22×C37—81no—
328C22×C37—81no—
329C22×C37—81no—
330C22×C37C2×#12011yescharacteristic
331C22×C37C2×#13511yescharacteristic
332C22×C37#57511yescharacteristic
333C4×C37—22no—
334C4×C37—22no—
335C4×C37—21no—
336C4×C37—42no—
337C4×C37—42no—
338C4×C37—41no—
339C4×C37—42no—
340C4×C37—42no—
341C4×C37—41no—
342C4×C37—42no—
343C4×C37—81no—
344C4×C37B(4,𝔽2)12yes—
345#3460C2×C3711yescharacteristic, complemented
346#3002C2×C3712yescomplemented
347#2388C2×C3711yescharacteristic, complemented
348#1606C2×C3712yescomplemented
349#1605C2×C3711yescharacteristic, complemented
350#1604C2×C3711yescharacteristic, complemented
351#1600C2×C3712yescomplemented
352#1599C2×C3711yescharacteristic, complemented
353#1598C2×C3711yescharacteristic, complemented
354#1501C2×C3712yescomplemented
355#1459C2×C3711yescharacteristic, complemented
356C2×C37—21no—
357C2×C37—21no—
358C2×C37—42no—
359C2×C37—82no—
360C2×C37—81no—
361C2×C37—82no—
362C2×C37#300211yescharacteristic
363C2×C37#301811yescharacteristic
364C2×C37C2×B(4,𝔽2)11yescharacteristic
365C22×#41C22×C3711yescharacteristic
366C2×#135—22no—
367#544C22×C3711yescharacteristic
368#542—22no—
369#540—22no—
370#538—21no—
371#538C22×C3712yes—
372#537C22×C3711yescharacteristic
373#535—21no—
374#532—22no—
375#529—22no—
376#525—21no—
377#525C22×C3712yes—
378C4×#41—21no—
379C2×C4×D4—21no—
380C2×C4×D4C22×C3711yescharacteristic
381#452—22no—
382#451C22×C3712yes—
383#450C22×C3712yes—
384#449—22no—
385#448C22×C3712yes—
386#447C22×C3712yes—
387#416—22no—
388C2×C22⋊C8C22×C3712yes—
389#395C22×C3711yescharacteristic
390#394—21no—
391#394C22×C3711yescharacteristic
392#393C22×C3711yescharacteristic
393#392—21no—
394#392C22×C3711yescharacteristic
395C4×C22⋊C4—21no—
396C4×C22⋊C4C22×C3711yescharacteristic
397#328C22×C3712yes—
398#328C22×C3712yes—
399#327C22×C3712yes—
400#327C22×C3712yes—
401#324C22×C3711yescharacteristic
402#324C22×C3711yescharacteristic
403#323C22×C3712yes—
404C37#1811411yescharacteristic, direct product factor, Sylow
405C2×#41—21no—
406C2×#41—21no—
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—44no—
410C2×#41D4×C3712yescomplemented
411C22×Q8C23×C3711yescharacteristic
412C22×D4—21no—
413C22×D4—21no—
414C22×D4D4×C3711yescharacteristic
415C22×D4C23×C3711yescharacteristic
416C23×C4—21no—
417C23×C4D4×C3711yescharacteristic
418C23×C4C23×C3711yescharacteristic
419#135—48no—
420C2×Q16—22no—
421C2×SD16—22no—
422C2×SD16—22no—
423C2×D8—22no—
424C22×C8—22no—
425C4⋊Q8—22no—
426#127—22no—
427#126—22no—
428#126—22no—
429#125—22no—
430#124—22no—
431#124—22no—
432#124—22no—
433#123—22no—
434#123—22no—
435#123—22no—
436#123—22no—
437C22⋊Q8—21no—
438C22⋊Q8—21no—
439C22⋊Q8—22no—
440C22⋊Q8D4×C3712yescomplemented
441C22⋊Q8C23×C3712yes—
442#121—21no—
443#121—22no—
444#121—22no—
445#121—22no—
446#121—22no—
447#121—22no—
448#121—21no—
449#121D4×C3712yescomplemented
450#121C23×C3712yes—
451#120—22no—
452#120—22no—
453C4×Q8—21no—
454C4×Q8—21no—
455C4×D4—21no—
456C4×D4—21no—
457C4×D4—22no—
458C4×D4—22no—
459C4×D4—22no—
460C4×D4—22no—
461C4×D4—22no—
462C4×D4—22no—
463C4×D4—44no—
464#117—22no—
465#117—21no—
466#117—21no—
467#117D4×C3712yescomplemented
468C2×C4⋊C4—21no—
469C2×C4⋊C4C23×C3711yescharacteristic
470C2×C22⋊C4—21no—
471C2×C22⋊C4—21no—
472C2×C22⋊C4—21no—
473C2×C22⋊C4—21no—
474C2×C22⋊C4—21no—
475C2×C22⋊C4D4×C3711yescharacteristic
476C2×C22⋊C4C23×C3711yescharacteristic
477C2×C42—22no—
478C2×C42—21no—
479C2×C42D4×C3711yescharacteristic
480#103—22no—
481#103—22no—
482#102—22no—
483#102—22no—
484C22⋊C8—22no—
485C22⋊C8C23×C3712yes—
486C22⋊C8C23×C3712yes—
487#95—21no—
488#95—21no—
489#95C23×C3711yescharacteristic
490#95C23×C3711yescharacteristic
491C24—21no—
492C24C2×D4×C3711yescharacteristic
493C24C2×D4×C3711yescharacteristic
494#41—28no—
495#41—42no—
496#41—44no—
497#41—44no—
498#41—44no—
499#41—44no—
500#41—42no—
501#41—82no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—22no—
505C2×Q8—42no—
506C2×Q8C2×D4×C3712yes—
507C2×D4—21no—
508C2×D4—22no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—21no—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—41no—
518C2×D4—42no—
519C2×D4—42no—
520C2×D4—42no—
521C2×D4—42no—
522C2×D4—42no—
523C2×D4—42no—
524C2×D4—42no—
525C2×D4C2×D4×C3712yes—
526C22×C4—22no—
527C22×C4—24no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—21no—
535C22×C4—42no—
536C22×C4—42no—
537C22×C4—42no—
538C22×C4—41no—
539C22×C4—41no—
540C22×C4C2×D4×C3711yescharacteristic
541C22×C4C2×D4×C3712yes—
542C22×C4C2×D4×C3711yescharacteristic
543C22×C4C2×D4×C3711yescharacteristic
544C22×C4C2×D4×C3711yescharacteristic
545C22×C4C2×D4×C3712yes—
546C22×C4C24×C3711yescharacteristic
547Q16—44no—
548SD16—44no—
549SD16—44no—
550D8—44no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—44no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—42no—
564C4⋊C4C2×D4×C3712yes—
565C4⋊C4C2×D4×C3712yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—42no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C42—22no—
583C42—22no—
584C42—21no—
585C42—22no—
586C42—21no—
587C42—42no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—21no—
592C23—22no—
593C23—21no—
594C23—21no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23#120×C3712yes—
606C23C22×D4×C3711yescharacteristic
607Q8—24no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—81no—
612D4—24no—
613D4—44no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—48no—
618D4—48no—
619D4—42no—
620D4—42no—
621D4—82no—
622D4—82no—
623D4—81no—
624D4—82no—
625D4—82no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—81no—
654C2×C4—82no—
655C2×C4#120×C3711yescharacteristic
656C2×C4#120×C3711yescharacteristic
657C2×C4#135×C3712yescomplemented
658C2×C4C22×D4×C3712yes—
659C8—42no—
660C8—42no—
661C22—21no—
662C22—21no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—41no—
671C22—42no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—42no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22C2×#120×C3711yescharacteristic
684C22C2×#135×C3711yescharacteristic
685C22C37×#57511yescharacteristic
686C4—22no—
687C4—22no—
688C4—21no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—81no—
697C4C37×B(4,𝔽2)12yes—
698C2—21no—
699C2—21no—
700C2—42no—
701C2—81no—
702C2—82no—
703C2—82no—
704C2C37×#300211yescharacteristic
705C2C37×#301811yescharacteristic
706C2C2×C37×B(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C4×C37#120
commutator, Frattini, cosocleC22×C4C24×C37
socleC22×C37C2×#120
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3460C2semidirect
2. C37×#3002C2semidirect
3. C37×#2388C2semidirect
4. C37×#1606C2semidirect
5. C37×#1605C2semidirect
6. C37×#1604C2semidirect
7. C37×#1600C2semidirect
8. C37×#1599C2semidirect
9. C37×#1598C2semidirect
10. C37×#1501C2semidirect
11. C37×#1459C2semidirect
56. C2×#41×C37D4semidirect
86. C22⋊Q8×C37D4semidirect
95. #121×C37D4semidirect
113. #117×C37D4semidirect
303. C2×C4×C37#135semidirect
307. #18114C37direct (N × H)
345. #3460C2×C37semidirect
346. #3002C2×C37semidirect
347. #2388C2×C37semidirect
348. #1606C2×C37semidirect
349. #1605C2×C37semidirect
350. #1604C2×C37semidirect
351. #1600C2×C37semidirect
352. #1599C2×C37semidirect
353. #1598C2×C37semidirect
354. #1501C2×C37semidirect
355. #1459C2×C37semidirect
404. C37#18114direct (N × H)
410. C2×#41D4×C37semidirect
440. C22⋊Q8D4×C37semidirect
449. #121D4×C37semidirect
467. #117D4×C37semidirect
657. C2×C4#135×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34602
C37×#30022
C37×#23882
C37×#16062
C37×#16052
C37×#16042
C37×#16002
C37×#15992
C37×#15982
C37×#15012
C37×#14592
#1811437