Finite Group Databasebeta

← C37×#24112 #22645365

Subgroups — C37×#24112 #22645365

Order 9472.

1518 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3490C211yesmaximal, characteristic
2C37×#2849C212yesmaximal, complemented
3C2×C37×#385C211yesmaximal, characteristic, complemented
4C37×#2062C212yesmaximal, complemented
5C37×#2044C212yesmaximal, complemented
6C37×#1898C212yesmaximal, complemented
7C37×#1867C214yesmaximal, complemented
8C37×#1819C211yesmaximal, characteristic, complemented
9C24×C4×C37C2211yescharacteristic
10C37×#551—22no—
11C37×#551—22no—
12C37×#546—22no—
13C37×#546—22no—
14C37×#543C2211yescharacteristic
15C37×#538—21no—
16C37×#538—21no—
17C37×#534C2211yescharacteristic
18C37×#525—21no—
19C37×#525—21no—
20C2×#123×C37—22no—
21C2×#120×C37C2212yes—
22C37×#518C2212yes—
23C22×C4⋊C4×C37—21no—
24C22×C22⋊C4×C37C2211yescharacteristic
25C22×C22⋊C4×C37C2211yescharacteristic
26C2×#99×C37C2214yes—
27C37×#386—22no—
28C37×#386C2214yes—
29C37×#385—21no—
30C37×#385—21no—
31C37×#385—22no—
32C37×#385C2212yes—
33C37×#385C2214yes—
34C37×#385C2212yes—
35C37×#381—22no—
36C37×#379C2212yes—
37C2×#95×C37—21no—
38C37×#342C2212yes—
39C37×#342C2214yes—
40C37×#342C2212yes—
41C25×C37C2311yescharacteristic
42C2×#41×C37—21no—
43C2×#41×C37—21no—
44C22×D4×C37D412yes—
45C22×D4×C37D412yes—
46C22×D4×C37C2312yes—
47C23×C4×C37—22no—
48C23×C4×C37—21no—
49C23×C4×C37—21no—
50C23×C4×C37—21no—
51C23×C4×C37—22no—
52C23×C4×C37—44no—
53C23×C4×C37D411yescharacteristic
54C23×C4×C37D411yescharacteristic
55C23×C4×C37C2312yes—
56#126×C37—22no—
57#126×C37—22no—
58#124×C37—22no—
59#124×C37—22no—
60#123×C37—22no—
61#123×C37—22no—
62#123×C37—22no—
63#123×C37—22no—
64#123×C37—22no—
65#123×C37—48no—
66C22⋊Q8×C37—21no—
67C22⋊Q8×C37—21no—
68C22⋊Q8×C37—21no—
69C22⋊Q8×C37—21no—
70C22⋊Q8×C37—22no—
71C22⋊Q8×C37—22no—
72#121×C37—21no—
73#121×C37—21no—
74#121×C37—22no—
75#121×C37—22no—
76#121×C37—21no—
77#121×C37—21no—
78#120×C37—22no—
79#120×C37—22no—
80#120×C37—22no—
81#120×C37—22no—
82#120×C37—22no—
83C4×D4×C37—22no—
84C4×D4×C37—22no—
85C4×D4×C37—22no—
86C4×D4×C37—22no—
87C4×D4×C37—22no—
88C4×D4×C37—22no—
89#117×C37—21no—
90#117×C37—21no—
91#117×C37—22no—
92C2×C4⋊C4×C37—24no—
93C2×C4⋊C4×C37—22no—
94C2×C4⋊C4×C37—22no—
95C2×C4⋊C4×C37—44no—
96C2×C22⋊C4×C37—22no—
97C2×C22⋊C4×C37—22no—
98C2×C22⋊C4×C37—22no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37—24no—
101C2×C22⋊C4×C37—22no—
102C2×C22⋊C4×C37—24no—
103C2×C22⋊C4×C37—22no—
104C2×C22⋊C4×C37D412yes—
105C2×C22⋊C4×C37D412yes—
106C2×C22⋊C4×C37D411yescharacteristic
107C2×C22⋊C4×C37D411yescharacteristic
108C2×C22⋊C4×C37C2314yes—
109C2×C22⋊C4×C37C2312yes—
110C2×C22⋊C4×C37C2311yescharacteristic
111C2×C22⋊C4×C37C2311yescharacteristic
112C2×C22⋊C4×C37C2312yes—
113#99×C37—28no—
114#95×C37—22no—
115#95×C37—22no—
116#95×C37—24no—
117#95×C37—24no—
118C24×C37—22no—
119C24×C37—22no—
120C24×C37—42no—
121C24×C37—42no—
122C24×C37C2×D411yescharacteristic
123C24×C37C2×D412yes—
124C24×C37C2×D412yes—
125C24×C37C2×D412yes—
126C24×C37C2×D411yescharacteristic
127C24×C37C2411yescharacteristic
128#41×C37—82no—
129#41×C37—82no—
130C2×Q8×C37—21no—
131C2×Q8×C37—21no—
132C2×D4×C37—22no—
133C2×D4×C37—22no—
134C2×D4×C37—22no—
135C2×D4×C37—22no—
136C2×D4×C37—24no—
137C2×D4×C37—24no—
138C2×D4×C37—21no—
139C2×D4×C37—21no—
140C2×D4×C37—22no—
141C2×D4×C37—22no—
142C2×D4×C37—44no—
143C2×D4×C37—44no—
144C22×C4×C37—24no—
145C22×C4×C37—24no—
146C22×C4×C37—28no—
147C22×C4×C37—21no—
148C22×C4×C37—21no—
149C22×C4×C37—22no—
150C22×C4×C37—24no—
151C22×C4×C37—22no—
152C22×C4×C37—22no—
153C22×C4×C37—22no—
154C22×C4×C37—21no—
155C22×C4×C37—21no—
156C22×C4×C37—24no—
157C22×C4×C37—44no—
158C22×C4×C37—48no—
159C22×C4×C37—44no—
160C22×C4×C37—42no—
161C22×C4×C37—42no—
162C22×C4×C37—42no—
163C22×C4×C37—44no—
164C22×C4×C37—42no—
165C22×C4×C37—41no—
166C22×C4×C37—41no—
167C22×C4×C37—42no—
168C22×C4×C37—42no—
169C22×C4×C37C2×D412yes—
170C22×C4×C37C2×D411yescharacteristic
171C22×C4×C37C2×D412yes—
172C22×C4×C37C2×D411yescharacteristic
173C22×C4×C37C2×D411yescharacteristic
174C22×C4×C37C2×D412yes—
175C22×C4×C37C2×D411yescharacteristic
176C4⋊C4×C37—22no—
177C4⋊C4×C37—22no—
178C4⋊C4×C37—21no—
179C4⋊C4×C37—21no—
180C4⋊C4×C37—21no—
181C4⋊C4×C37—21no—
182C4⋊C4×C37—22no—
183C4⋊C4×C37—22no—
184C4⋊C4×C37—44no—
185C4⋊C4×C37—44no—
186C4⋊C4×C37—48no—
187C22⋊C4×C37—24no—
188C22⋊C4×C37—22no—
189C22⋊C4×C37—22no—
190C22⋊C4×C37—22no—
191C22⋊C4×C37—22no—
192C22⋊C4×C37—22no—
193C22⋊C4×C37—22no—
194C22⋊C4×C37—22no—
195C22⋊C4×C37—22no—
196C22⋊C4×C37—22no—
197C22⋊C4×C37—22no—
198C22⋊C4×C37—22no—
199C22⋊C4×C37—22no—
200C22⋊C4×C37—22no—
201C22⋊C4×C37—22no—
202C22⋊C4×C37—22no—
203C22⋊C4×C37—24no—
204C22⋊C4×C37—24no—
205C22⋊C4×C37—48no—
206C22⋊C4×C37—44no—
207C22⋊C4×C37—44no—
208C22⋊C4×C37#4114yescomplemented
209C42×C37—22no—
210C42×C37—22no—
211C23×C37—21no—
212C23×C37—21no—
213C23×C37—22no—
214C23×C37—22no—
215C23×C37—21no—
216C23×C37—21no—
217C23×C37—24no—
218C23×C37—24no—
219C23×C37—24no—
220C23×C37—24no—
221C23×C37—22no—
222C23×C37—24no—
223C23×C37—41no—
224C23×C37—42no—
225C23×C37—41no—
226C23×C37—41no—
227C23×C37—42no—
228C23×C37—41no—
229C23×C37—48no—
230C23×C37—48no—
231C23×C37—42no—
232C23×C37—42no—
233C23×C37#12012yes—
234C23×C37#12011yescharacteristic
235C23×C37#12011yescharacteristic
236C23×C37C22×D411yescharacteristic
237C23×C37C22×D412yes—
238C23×C37C2×#4112yes—
239C23×C372−1+412yes—
240Q8×C37—81no—
241Q8×C37—81no—
242D4×C37—42no—
243D4×C37—42no—
244D4×C37—44no—
245D4×C37—44no—
246D4×C37—44no—
247D4×C37—82no—
248D4×C37—82no—
249D4×C37—81no—
250D4×C37—81no—
251C2×C4×C37—28no—
252C2×C4×C37—24no—
253C2×C4×C37—22no—
254C2×C4×C37—21no—
255C2×C4×C37—21no—
256C2×C4×C37—22no—
257C2×C4×C37—22no—
258C2×C4×C37—21no—
259C2×C4×C37—21no—
260C2×C4×C37—24no—
261C2×C4×C37—24no—
262C2×C4×C37—42no—
263C2×C4×C37—48no—
264C2×C4×C37—42no—
265C2×C4×C37—42no—
266C2×C4×C37—44no—
267C2×C4×C37—44no—
268C2×C4×C37—48no—
269C2×C4×C37—42no—
270C2×C4×C37—44no—
271C2×C4×C37—42no—
272C2×C4×C37—42no—
273C2×C4×C37—41no—
274C2×C4×C37—41no—
275C2×C4×C37—42no—
276C2×C4×C37—41no—
277C2×C4×C37—41no—
278C2×C4×C37—42no—
279C2×C4×C37—44no—
280C2×C4×C37—44no—
281C2×C4×C37—44no—
282C2×C4×C37—81no—
283C2×C4×C37—81no—
284C2×C4×C37—82no—
285C2×C4×C37—82no—
286#24112C3711yesmaximal, characteristic, direct product factor, Sylow
287C22×C37—24no—
288C22×C37—24no—
289C22×C37—21no—
290C22×C37—21no—
291C22×C37—21no—
292C22×C37—22no—
293C22×C37—21no—
294C22×C37—21no—
295C22×C37—22no—
296C22×C37—21no—
297C22×C37—22no—
298C22×C37—22no—
299C22×C37—22no—
300C22×C37—42no—
301C22×C37—42no—
302C22×C37—48no—
303C22×C37—48no—
304C22×C37—42no—
305C22×C37—42no—
306C22×C37—42no—
307C22×C37—42no—
308C22×C37—42no—
309C22×C37—84no—
310C22×C37B(4,𝔽2)12yes—
311C22×C37#45812yes—
312C22×C37C2×#12011yescharacteristic
313C22×C37#52511yescharacteristic
314C22×C37#53611yescharacteristic
315C22×C37#54714yes—
316C4×C37—44no—
317C4×C37—42no—
318C4×C37—44no—
319C4×C37—41no—
320C4×C37—41no—
321C4×C37—42no—
322C4×C37—41no—
323C4×C37—41no—
324C4×C37—42no—
325C4×C37—84no—
326#3490C2×C3711yescharacteristic
327#2849C2×C3712yescomplemented
328C2×#385C2×C3711yescharacteristic, complemented
329#2062C2×C3712yescomplemented
330#2044C2×C3712yescomplemented
331#1898C2×C3712yescomplemented
332#1867C2×C3714yescomplemented
333#1819C2×C3711yescharacteristic, complemented
334C2×C37—21no—
335C2×C37—24no—
336C2×C37—21no—
337C2×C37—22no—
338C2×C37—22no—
339C2×C37—42no—
340C2×C37—82no—
341C2×C37#240611yescharacteristic
342C2×C37C2×B(4,𝔽2)11yescharacteristic
343C2×C37C2×#45811yescharacteristic
344C24×C4C22×C3711yescharacteristic
345#551—22no—
346#551—22no—
347#546—22no—
348#546—22no—
349#543C22×C3711yescharacteristic
350#538—21no—
351#538—21no—
352#534C22×C3711yescharacteristic
353#525—21no—
354#525—21no—
355C2×#123—22no—
356C2×#120C22×C3712yes—
357#518C22×C3712yes—
358C22×C4⋊C4—21no—
359C22×C22⋊C4C22×C3711yescharacteristic
360C22×C22⋊C4C22×C3711yescharacteristic
361C2×#99C22×C3714yes—
362#386—22no—
363#386C22×C3714yes—
364#385—22no—
365#385—21no—
366#385—21no—
367#385C22×C3714yes—
368#385C22×C3712yes—
369#385C22×C3712yes—
370#381—22no—
371#379C22×C3712yes—
372C2×#95—21no—
373#342C22×C3712yes—
374#342C22×C3714yes—
375#342C22×C3712yes—
376C37#2411211yescharacteristic, direct product factor, Sylow
377C25C23×C3711yescharacteristic
378C2×#41—21no—
379C2×#41—21no—
380C22×D4D4×C3712yes—
381C22×D4D4×C3712yes—
382C22×D4C23×C3712yes—
383C23×C4—21no—
384C23×C4—21no—
385C23×C4—22no—
386C23×C4—21no—
387C23×C4—22no—
388C23×C4—44no—
389C23×C4D4×C3711yescharacteristic
390C23×C4D4×C3711yescharacteristic
391C23×C4C23×C3712yes—
392#126—22no—
393#126—22no—
394#124—22no—
395#124—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399#123—22no—
400#123—22no—
401#123—48no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408#121—21no—
409#121—21no—
410#121—22no—
411#121—22no—
412#121—21no—
413#121—21no—
414#120—22no—
415#120—22no—
416#120—22no—
417#120—22no—
418#120—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—22no—
425#117—21no—
426#117—21no—
427#117—22no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—24no—
431C2×C4⋊C4—44no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—24no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4D4×C3711yescharacteristic
441C2×C22⋊C4D4×C3711yescharacteristic
442C2×C22⋊C4D4×C3712yes—
443C2×C22⋊C4D4×C3712yes—
444C2×C22⋊C4C23×C3714yes—
445C2×C22⋊C4C23×C3711yescharacteristic
446C2×C22⋊C4C23×C3711yescharacteristic
447C2×C22⋊C4C23×C3712yes—
448C2×C22⋊C4C23×C3712yes—
449#99—28no—
450#95—22no—
451#95—22no—
452#95—24no—
453#95—24no—
454C24—22no—
455C24—22no—
456C24—42no—
457C24—42no—
458C24C2×D4×C3711yescharacteristic
459C24C2×D4×C3712yes—
460C24C2×D4×C3712yes—
461C24C2×D4×C3712yes—
462C24C2×D4×C3711yescharacteristic
463C24C24×C3711yescharacteristic
464#41—82no—
465#41—82no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—44no—
479C2×D4—44no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4C2×D4×C3712yes—
506C22×C4C2×D4×C3711yescharacteristic
507C22×C4C2×D4×C3712yes—
508C22×C4C2×D4×C3711yescharacteristic
509C22×C4C2×D4×C3711yescharacteristic
510C22×C4C2×D4×C3712yes—
511C22×C4C2×D4×C3711yescharacteristic
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4#41×C3714yescomplemented
545C42—22no—
546C42—22no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—24no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—41no—
565C23—48no—
566C23—48no—
567C23—42no—
568C23—42no—
569C23#120×C3712yes—
570C23#120×C3711yescharacteristic
571C23#120×C3711yescharacteristic
572C23C22×D4×C3711yescharacteristic
573C23C22×D4×C3712yes—
574C23C2×#41×C3712yes—
575C232−1+4×C3712yes—
576Q8—81no—
577Q8—81no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—44no—
583D4—82no—
584D4—82no—
585D4—81no—
586D4—81no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4—82no—
622C22—24no—
623C22—24no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—48no—
638C22—48no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—84no—
645C22C37×B(4,𝔽2)12yes—
646C22C37×#45812yes—
647C22C2×#120×C3711yescharacteristic
648C22C37×#52511yescharacteristic
649C22C37×#53611yescharacteristic
650C22C37×#54714yes—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—84no—
661C2—21no—
662C2—24no—
663C2—21no—
664C2—22no—
665C2—22no—
666C2—42no—
667C2—82no—
668C2C37×#240611yescharacteristic
669C2C2×C37×B(4,𝔽2)11yescharacteristic
670C2C2×C37×#45811yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC24C24×C37
prime radicalC37×#3490C2
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. C37×#2849C2semidirect
3. C2×C37×#385C2semidirect
4. C37×#2062C2semidirect
5. C37×#2044C2semidirect
6. C37×#1898C2semidirect
7. C37×#1867C2semidirect
8. C37×#1819C2semidirect
208. C22⋊C4×C37#41semidirect
286. #24112C37direct (N × H)
327. #2849C2×C37semidirect
328. C2×#385C2×C37semidirect
329. #2062C2×C37semidirect
330. #2044C2×C37semidirect
331. #1898C2×C37semidirect
332. #1867C2×C37semidirect
333. #1819C2×C37semidirect
376. C37#24112direct (N × H)
544. C22⋊C4#41×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34902
C37×#28492
C2×C37×#3852
C37×#20622
C37×#20442
C37×#18982
C37×#18672
C37×#18192
#2411237