Finite Group Databasebeta

← C37×#24147 #22645400

Subgroups — C37×#24147 #22645400

Order 9472.

1006 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3509C211yesmaximal, characteristic
2C37×#3078C211yesmaximal, characteristic, complemented
3C4×#137×C37C211yesmaximal, characteristic
4C37×#2546C211yesmaximal, characteristic, complemented
5C37×#2038C212yesmaximal
6C37×#2005C212yesmaximal
7C37×#1986C211yesmaximal, characteristic, complemented
8C37×#1966C211yesmaximal, characteristic, complemented
9C37×#1886C212yesmaximal, complemented
10C37×#1857C212yesmaximal, complemented
11C37×#1806C211yesmaximal, characteristic
12C2×#137×C37C2211yescharacteristic
13C37×#557—21no—
14C37×#557—21no—
15C37×#556—21no—
16C37×#556—21no—
17C37×#552—22no—
18C37×#552—22no—
19C37×#541—22no—
20C37×#536C2212yes—
21C37×#533—21no—
22C37×#533—21no—
23C37×#533C2211yescharacteristic
24C2×C4⋊Q8×C37C2212yes—
25C2×C4⋊Q8×C37C2211yescharacteristic
26C37×#519C2212yes—
27C4×#41×C37C2211yescharacteristic
28C2×C4×Q8×C37C2211yescharacteristic
29C37×#479—22no—
30C37×#477—21no—
31C37×#475—21no—
32C37×#441—21no—
33C37×#440—21no—
34C4×Q16×C37—21no—
35C4×SD16×C37—21no—
36C37×#428C2211yescharacteristic
37C37×#428C2212yes—
38C37×#423C2211yescharacteristic
39C2×#104×C37C2212yes—
40C37×#419C2211yescharacteristic
41C37×#419C2212yes—
42C37×#417C2211yescharacteristic
43C2×#101×C37C2212yes—
44C2×#101×C37C2212yes—
45C4×M16×C37C2211yescharacteristic
46C37×#398—22no—
47C37×#391C2212yes—
48C37×#389—22no—
49C4×C4⋊C4×C37C2211yescharacteristic
50C37×#339C2212yes—
51C37×#339C2212yes—
52C37×#339C2212yes—
53C2×#41×C37C2311yescharacteristic
54C22×Q8×C37C2311yescharacteristic
55C22×Q8×C37C2312yes—
56#137×C37—22no—
57#137×C37—22no—
58C2×Q16×C37—21no—
59C2×SD16×C37—21no—
60C2×M16×C37C2311yescharacteristic
61C2×M16×C37C2312yes—
62C2×M16×C37C2311yescharacteristic
63C4⋊Q8×C37—22no—
64C4⋊Q8×C37—24no—
65C4⋊Q8×C37—21no—
66C4⋊Q8×C37—21no—
67C4⋊Q8×C37—21no—
68C4⋊Q8×C37—21no—
69C4⋊Q8×C37—42no—
70C4⋊Q8×C37D411yescharacteristic
71C4⋊Q8×C37D411yescharacteristic
72#126×C37—24no—
73#126×C37—24no—
74#125×C37—22no—
75#125×C37—22no—
76#125×C37—24no—
77#125×C37—24no—
78#125×C37D412yes—
79#124×C37—22no—
80#123×C37—24no—
81C22⋊Q8×C37—22no—
82C22⋊Q8×C37—22no—
83C22⋊Q8×C37—21no—
84C22⋊Q8×C37—21no—
85C22⋊Q8×C37—22no—
86C22⋊Q8×C37—22no—
87C4×Q8×C37—22no—
88C4×Q8×C37—22no—
89C4×Q8×C37—22no—
90C4×Q8×C37—22no—
91C4×Q8×C37—22no—
92C4×Q8×C37—22no—
93C4×Q8×C37—21no—
94C4×Q8×C37—21no—
95C4×Q8×C37—21no—
96C4×Q8×C37—21no—
97C4×Q8×C37—21no—
98C4×Q8×C37—21no—
99C4×Q8×C37D411yescharacteristic
100C4×Q8×C37D411yescharacteristic
101C4×D4×C37—21no—
102C4×D4×C37—21no—
103C4×D4×C37D411yescharacteristic
104C4×D4×C37D411yescharacteristic
105#117×C37—22no—
106#117×C37—22no—
107#117×C37—24no—
108#117×C37—21no—
109#117×C37C2311yescharacteristic
110#117×C37C2312yes—
111C2×C4⋊C4×C37—21no—
112C2×C4⋊C4×C37—24no—
113C2×C4⋊C4×C37—21no—
114C2×C4⋊C4×C37D412yes—
115C2×C4⋊C4×C37D412yes—
116C2×C4⋊C4×C37C2311yescharacteristic
117C2×C42×C37—21no—
118C2×C42×C37—21no—
119C2×C42×C37C2312yes—
120C2×C42×C37C2311yescharacteristic
121#107×C37—21no—
122#107×C37—22no—
123#106×C37—21no—
124#106×C37—22no—
125C4⋊C8×C37—21no—
126C4⋊C8×C37—21no—
127#104×C37—42no—
128#103×C37—21no—
129#103×C37—21no—
130#103×C37—21no—
131#103×C37—22no—
132#103×C37—22no—
133#102×C37—21no—
134#101×C37—24no—
135#101×C37—42no—
136#97×C37—21no—
137C4×C8×C37—21no—
138#95×C37—21no—
139#95×C37—24no—
140#41×C37—21no—
141#41×C37—21no—
142#41×C37—21no—
143#41×C37—21no—
144C2×Q8×C37—21no—
145C2×Q8×C37—21no—
146C2×Q8×C37—22no—
147C2×Q8×C37—22no—
148C2×Q8×C37—22no—
149C2×Q8×C37—22no—
150C2×Q8×C37—22no—
151C2×Q8×C37—21no—
152C2×Q8×C37—22no—
153C2×Q8×C37—21no—
154C2×Q8×C37—21no—
155C2×Q8×C37—21no—
156C2×Q8×C37—21no—
157C2×Q8×C37C2×D411yescharacteristic
158C2×Q8×C37#4112yes—
159C2×D4×C37—21no—
160C2×D4×C37C2×D411yescharacteristic
161C22×C4×C37—22no—
162C22×C4×C37—21no—
163C22×C4×C37—21no—
164C22×C4×C37—21no—
165C22×C4×C37—21no—
166C22×C4×C37—21no—
167C22×C4×C37C2×D414yes—
168C22×C4×C37C2×D411yescharacteristic
169C22×C4×C37C2×D411yescharacteristic
170C22×C4×C37C2411yescharacteristic
171Q16×C37—24no—
172SD16×C37—24no—
173M16×C37—21no—
174M16×C37—22no—
175M16×C37—21no—
176M16×C37—22no—
177M16×C37—21no—
178M16×C37#4112yes—
179C2×C8×C37—21no—
180C2×C8×C37—21no—
181C2×C8×C37—22no—
182C4⋊C4×C37—24no—
183C4⋊C4×C37—24no—
184C4⋊C4×C37—24no—
185C4⋊C4×C37—24no—
186C4⋊C4×C37—24no—
187C4⋊C4×C37—22no—
188C4⋊C4×C37—22no—
189C4⋊C4×C37—22no—
190C4⋊C4×C37—22no—
191C4⋊C4×C37—21no—
192C4⋊C4×C37—21no—
193C4⋊C4×C37—21no—
194C4⋊C4×C37—21no—
195C4⋊C4×C37—22no—
196C4⋊C4×C37—22no—
197C4⋊C4×C37—21no—
198C4⋊C4×C37—21no—
199C4⋊C4×C37—48no—
200C4⋊C4×C37—41no—
201C4⋊C4×C37—41no—
202C4⋊C4×C37C2×D411yescharacteristic
203C4⋊C4×C37C2×D412yes—
204C4⋊C4×C37C2×D411yescharacteristic
205C4⋊C4×C37C2×D412yes—
206C22⋊C4×C37—24no—
207C22⋊C4×C37—24no—
208C22⋊C4×C37—21no—
209C22⋊C4×C37—22no—
210C22⋊C4×C37—21no—
211C22⋊C4×C37—21no—
212C42×C37—24no—
213C42×C37—24no—
214C42×C37—21no—
215C42×C37—21no—
216C42×C37—21no—
217C42×C37—22no—
218C42×C37—22no—
219C42×C37—22no—
220C42×C37—41no—
221C42×C37—41no—
222C42×C37C2×D411yescharacteristic
223C42×C37C2×D411yescharacteristic
224C42×C37C2×D411yescharacteristic
225C42×C37C2×D411yescharacteristic
226C23×C37—21no—
227C23×C37C22×D411yescharacteristic
228Q8×C37—22no—
229Q8×C37—21no—
230Q8×C37—21no—
231Q8×C37—22no—
232Q8×C37—42no—
233Q8×C37—42no—
234Q8×C37—42no—
235Q8×C37—41no—
236Q8×C37—42no—
237Q8×C37—42no—
238Q8×C37—41no—
239Q8×C37—42no—
240D4×C37—21no—
241D4×C37—21no—
242D4×C37—41no—
243D4×C37—41no—
244C2×C4×C37—22no—
245C2×C4×C37—24no—
246C2×C4×C37—21no—
247C2×C4×C37—21no—
248C2×C4×C37—22no—
249C2×C4×C37—22no—
250C2×C4×C37—24no—
251C2×C4×C37—21no—
252C2×C4×C37—22no—
253C2×C4×C37—24no—
254C2×C4×C37—24no—
255C2×C4×C37—41no—
256C2×C4×C37—42no—
257C2×C4×C37—42no—
258C2×C4×C37—41no—
259C2×C4×C37—42no—
260C2×C4×C37—41no—
261C2×C4×C37—41no—
262C2×C4×C37—41no—
263C2×C4×C37—42no—
264C2×C4×C37—42no—
265C2×C4×C37—41no—
266C2×C4×C37#12011yescharacteristic
267C2×C4×C37#12011yescharacteristic
268C2×C4×C37#12011yescharacteristic
269C2×C4×C37#12011yescharacteristic
270C2×C4×C37C22×D411yescharacteristic
271C2×C4×C37C22×D411yescharacteristic
272C2×C4×C37C2×#4111yescharacteristic
273C2×C4×C37C2×#4111yescharacteristic
274C2×C4×C37H(5,𝔽2)11yescharacteristic
275C2×C4×C372−1+411yescharacteristic
276C8×C37—22no—
277C8×C37—41no—
278C8×C37—42no—
279#24147C3711yesmaximal, characteristic, direct product factor, Sylow
280C22×C37—21no—
281C22×C37—21no—
282C22×C37—41no—
283C22×C37—41no—
284C22×C37—41no—
285C22×C37C2×#12011yescharacteristic
286C22×C37#54611yescharacteristic
287C22×C37#54711yescharacteristic
288C4×C37—21no—
289C4×C37—21no—
290C4×C37—21no—
291C4×C37—44no—
292C4×C37—44no—
293C4×C37—41no—
294C4×C37—42no—
295C4×C37—41no—
296C4×C37—42no—
297C4×C37—41no—
298C4×C37—42no—
299C4×C37—81no—
300C4×C37#45612yes—
301C4×C37#52511yescharacteristic
302C4×C37#53711yescharacteristic
303C4×C37#54811yescharacteristic
304C4×C37#55011yescharacteristic
305#3509C2×C3711yescharacteristic
306#3078C2×C3711yescharacteristic, complemented
307C4×#137C2×C3711yescharacteristic
308#2546C2×C3711yescharacteristic, complemented
309#2038C2×C3712yes—
310#2005C2×C3712yes—
311#1986C2×C3711yescharacteristic, complemented
312#1966C2×C3711yescharacteristic, complemented
313#1886C2×C3712yescomplemented
314#1857C2×C3712yescomplemented
315#1806C2×C3711yescharacteristic
316C2×C37—21no—
317C2×C37—21no—
318C2×C37—81no—
319C2×C37#240711yescharacteristic
320C2×C37C2×#45611yescharacteristic
321C2×C37#302011yescharacteristic
322C2×#137C22×C3711yescharacteristic
323#557—21no—
324#557—21no—
325#556—21no—
326#556—21no—
327#552—22no—
328#552—22no—
329#541—22no—
330#536C22×C3712yes—
331#533—21no—
332#533—21no—
333#533C22×C3711yescharacteristic
334C2×C4⋊Q8C22×C3712yes—
335C2×C4⋊Q8C22×C3711yescharacteristic
336#519C22×C3712yes—
337C4×#41C22×C3711yescharacteristic
338C2×C4×Q8C22×C3711yescharacteristic
339#479—22no—
340#477—21no—
341#475—21no—
342#441—21no—
343#440—21no—
344C4×Q16—21no—
345C4×SD16—21no—
346#428C22×C3712yes—
347#428C22×C3711yescharacteristic
348#423C22×C3711yescharacteristic
349C2×#104C22×C3712yes—
350#419C22×C3712yes—
351#419C22×C3711yescharacteristic
352#417C22×C3711yescharacteristic
353C2×#101C22×C3712yes—
354C2×#101C22×C3712yes—
355C4×M16C22×C3711yescharacteristic
356#398—22no—
357#391C22×C3712yes—
358#389—22no—
359C4×C4⋊C4C22×C3711yescharacteristic
360#339C22×C3712yes—
361#339C22×C3712yes—
362#339C22×C3712yes—
363C37#2414711yescharacteristic, direct product factor, Sylow
364C2×#41C23×C3711yescharacteristic
365C22×Q8C23×C3711yescharacteristic
366C22×Q8C23×C3712yes—
367#137—22no—
368#137—22no—
369C2×Q16—21no—
370C2×SD16—21no—
371C2×M16C23×C3711yescharacteristic
372C2×M16C23×C3712yes—
373C2×M16C23×C3711yescharacteristic
374C4⋊Q8—22no—
375C4⋊Q8—24no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—21no—
380C4⋊Q8—42no—
381C4⋊Q8D4×C3711yescharacteristic
382C4⋊Q8D4×C3711yescharacteristic
383#126—24no—
384#126—24no—
385#125—22no—
386#125—22no—
387#125—24no—
388#125—24no—
389#125D4×C3712yes—
390#124—22no—
391#123—24no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8D4×C3711yescharacteristic
411C4×Q8D4×C3711yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4D4×C3711yescharacteristic
415C4×D4D4×C3711yescharacteristic
416#117—22no—
417#117—22no—
418#117—24no—
419#117—21no—
420#117C23×C3711yescharacteristic
421#117C23×C3712yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—24no—
425C2×C4⋊C4D4×C3712yes—
426C2×C4⋊C4D4×C3712yes—
427C2×C4⋊C4C23×C3711yescharacteristic
428C2×C42—21no—
429C2×C42—21no—
430C2×C42C23×C3712yes—
431C2×C42C23×C3711yescharacteristic
432#107—21no—
433#107—22no—
434#106—21no—
435#106—22no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438#104—42no—
439#103—21no—
440#103—21no—
441#103—21no—
442#103—22no—
443#103—22no—
444#102—21no—
445#101—24no—
446#101—42no—
447#97—21no—
448C4×C8—21no—
449#95—21no—
450#95—24no—
451#41—21no—
452#41—21no—
453#41—21no—
454#41—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×D4×C3711yescharacteristic
469C2×Q8#41×C3712yes—
470C2×D4—21no—
471C2×D4C2×D4×C3711yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4C2×D4×C3714yes—
479C22×C4C2×D4×C3711yescharacteristic
480C22×C4C2×D4×C3711yescharacteristic
481C22×C4C24×C3711yescharacteristic
482Q16—24no—
483SD16—24no—
484M16—21no—
485M16—22no—
486M16—21no—
487M16—22no—
488M16—21no—
489M16#41×C3712yes—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—48no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C3711yescharacteristic
514C4⋊C4C2×D4×C3712yes—
515C4⋊C4C2×D4×C3711yescharacteristic
516C4⋊C4C2×D4×C3712yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—21no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C42—24no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—41no—
532C42—41no—
533C42C2×D4×C3711yescharacteristic
534C42C2×D4×C3711yescharacteristic
535C42C2×D4×C3711yescharacteristic
536C42C2×D4×C3711yescharacteristic
537C23—21no—
538C23C22×D4×C3711yescharacteristic
539Q8—22no—
540Q8—21no—
541Q8—21no—
542Q8—22no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—41no—
547Q8—42no—
548Q8—42no—
549Q8—41no—
550Q8—42no—
551D4—21no—
552D4—21no—
553D4—41no—
554D4—41no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4#120×C3711yescharacteristic
578C2×C4#120×C3711yescharacteristic
579C2×C4#120×C3711yescharacteristic
580C2×C4#120×C3711yescharacteristic
581C2×C4C22×D4×C3711yescharacteristic
582C2×C4C22×D4×C3711yescharacteristic
583C2×C4C2×#41×C3711yescharacteristic
584C2×C4C2×#41×C3711yescharacteristic
585C2×C4H(5,𝔽2)×C3711yescharacteristic
586C2×C42−1+4×C3711yescharacteristic
587C8—22no—
588C8—41no—
589C8—42no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×#120×C3711yescharacteristic
596C22C37×#54611yescharacteristic
597C22C37×#54711yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—44no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4C37×#45612yes—
611C4C37×#52511yescharacteristic
612C4C37×#53711yescharacteristic
613C4C37×#54811yescharacteristic
614C4C37×#55011yescharacteristic
615C2—21no—
616C2—21no—
617C2—81no—
618C2C37×#240711yescharacteristic
619C2C2×C37×#45611yescharacteristic
620C2C37×#302011yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×#41×C37C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3078C2semidirect
4. C37×#2546C2semidirect
7. C37×#1986C2semidirect
8. C37×#1966C2semidirect
9. C37×#1886C2semidirect
10. C37×#1857C2semidirect
279. #24147C37direct (N × H)
306. #3078C2×C37semidirect
308. #2546C2×C37semidirect
311. #1986C2×C37semidirect
312. #1966C2×C37semidirect
313. #1886C2×C37semidirect
314. #1857C2×C37semidirect
363. C37#24147direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35092
C37×#30782
C4×#137×C372
C37×#25462
C37×#20382
C37×#20052
C37×#19862
C37×#19662
C37×#18862
C37×#18572
C37×#18062
#2414737