Finite Group Databasebeta

← C37×#10972 #22632225

Subgroups — C37×#10972 #22632225

Order 9472.

1634 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#534C211yesmaximal, characteristic
2C22×#99×C37C212yesmaximal, complemented
3C22×#99×C37C211yesmaximal, characteristic, complemented
4C37×#2308C211yesmaximal, characteristic, complemented
5C2×C37×#342C212yesmaximal, complemented
6C23×D4×C37C2211yescharacteristic
7C23×D4×C37C2211yescharacteristic
8C37×#534—24no—
9C37×#534—24no—
10C2×#123×C37—21no—
11C2×#123×C37C412yescomplemented
12C2×#121×C37—22no—
13C2×#120×C37—21no—
14C2×#120×C37C412yescomplemented
15C22×C22⋊C4×C37C2212yes—
16C22×C22⋊C4×C37C2212yes—
17C22×C22⋊C4×C37C2211yescharacteristic
18C2×#99×C37—22no—
19C2×#99×C37—24no—
20C2×#99×C37—24no—
21C2×#99×C37—24no—
22C2×#99×C37—24no—
23C37×#386—24no—
24C37×#381—21no—
25C37×#379—21no—
26C37×#342—24no—
27C25×C37C2×C411yescharacteristic
28C25×C37D411yescharacteristic
29C25×C37C2311yescharacteristic
30C22×D4×C37—21no—
31C22×D4×C37—21no—
32C22×D4×C37—22no—
33C22×D4×C37—22no—
34C22×D4×C37—22no—
35C22×D4×C37—22no—
36C22×D4×C37—22no—
37C22×D4×C37—22no—
38C22×D4×C37—22no—
39C22×D4×C37—22no—
40C22×D4×C37—22no—
41C22×D4×C37—22no—
42C22×D4×C37—22no—
43C22×D4×C37—22no—
44C22×D4×C37C2×C412yes—
45C22×D4×C37D412yes—
46C22×D4×C37D414yes—
47C23×C4×C37—21no—
48C23×C4×C37—22no—
49C23×C4×C37—22no—
50C23×C4×C37C2×C411yescharacteristic
51C23×C4×C37D411yescharacteristic
52#123×C37—28no—
53#123×C37—24no—
54#123×C37—24no—
55#121×C37—216no—
56#120×C37—28no—
57#120×C37—24no—
58#120×C37—24no—
59C2×C4⋊C4×C37—22no—
60C2×C22⋊C4×C37—22no—
61C2×C22⋊C4×C37—21no—
62C2×C22⋊C4×C37—22no—
63C2×C22⋊C4×C37—22no—
64C2×C22⋊C4×C37—22no—
65C2×C22⋊C4×C37—22no—
66C2×C22⋊C4×C37—22no—
67C2×C22⋊C4×C37—22no—
68C2×C22⋊C4×C37—22no—
69C2×C22⋊C4×C37—22no—
70C2×C22⋊C4×C37—22no—
71C2×C22⋊C4×C37—22no—
72C2×C22⋊C4×C37—22no—
73C2×C22⋊C4×C37—22no—
74C2×C22⋊C4×C37—24no—
75C2×C22⋊C4×C37C2×C412yes—
76C2×C22⋊C4×C37D414yes—
77C2×C22⋊C4×C37D414yescomplemented
78C2×C22⋊C4×C37D414yescomplemented
79#99×C37—216no—
80#99×C37—28no—
81#95×C37—24no—
82C24×C37—21no—
83C24×C37—21no—
84C24×C37—21no—
85C24×C37—22no—
86C24×C37—22no—
87C24×C37—22no—
88C24×C37—22no—
89C24×C37—21no—
90C24×C37—82no—
91C24×C37—82no—
92C24×C37—82no—
93C24×C37C22⋊C412yes—
94C24×C37C22⋊C412yescomplemented
95C24×C37C22×C411yescharacteristic
96C24×C37C2×D411yescharacteristic
97C24×C37C2×D411yescharacteristic
98C24×C37C2×D412yes—
99C24×C37C2×D412yes—
100C24×C37C2×D412yes—
101C24×C37C2×D412yes—
102C24×C37#4112yes—
103C24×C37#4112yes—
104C2×D4×C37—22no—
105C2×D4×C37—22no—
106C2×D4×C37—24no—
107C2×D4×C37—24no—
108C2×D4×C37—22no—
109C2×D4×C37—24no—
110C2×D4×C37—24no—
111C2×D4×C37—24no—
112C2×D4×C37—24no—
113C2×D4×C37—24no—
114C2×D4×C37—24no—
115C2×D4×C37—24no—
116C2×D4×C37—22no—
117C2×D4×C37—24no—
118C2×D4×C37—28no—
119C2×D4×C37—41no—
120C2×D4×C37—41no—
121C2×D4×C37—42no—
122C2×D4×C37—42no—
123C2×D4×C37—42no—
124C2×D4×C37—42no—
125C2×D4×C37—42no—
126C2×D4×C37—42no—
127C2×D4×C37—42no—
128C2×D4×C37—42no—
129C2×D4×C37—42no—
130C2×D4×C37—41no—
131C2×D4×C37—41no—
132C2×D4×C37—42no—
133C2×D4×C37—42no—
134C2×D4×C37—42no—
135C2×D4×C37—42no—
136C2×D4×C37—42no—
137C2×D4×C37—84no—
138C22×C4×C37—22no—
139C22×C4×C37—21no—
140C22×C4×C37—21no—
141C22×C4×C37—22no—
142C22×C4×C37—22no—
143C22×C4×C37—21no—
144C22×C4×C37—22no—
145C22×C4×C37—22no—
146C22×C4×C37—21no—
147C22×C4×C37—24no—
148C22×C4×C37—24no—
149C22×C4×C37—24no—
150C22×C4×C37—42no—
151C22×C4×C37—42no—
152C22×C4×C37—42no—
153C22×C4×C37—42no—
154C22×C4×C37—82no—
155C22×C4×C37—84no—
156C22×C4×C37—84no—
157C22×C4×C37C22⋊C412yes—
158C22×C4×C37C22⋊C412yescomplemented
159C4⋊C4×C37—28no—
160C22⋊C4×C37—28no—
161C22⋊C4×C37—28no—
162C22⋊C4×C37—22no—
163C22⋊C4×C37—24no—
164C22⋊C4×C37—24no—
165C22⋊C4×C37—24no—
166C22⋊C4×C37—24no—
167C22⋊C4×C37—24no—
168C22⋊C4×C37—42no—
169C22⋊C4×C37—42no—
170C22⋊C4×C37—42no—
171C22⋊C4×C37—42no—
172C22⋊C4×C37—84no—
173C22⋊C4×C37C22⋊C414yescomplemented
174C23×C37—22no—
175C23×C37—22no—
176C23×C37—22no—
177C23×C37—22no—
178C23×C37—24no—
179C23×C37—44no—
180C23×C37—42no—
181C23×C37—44no—
182C23×C37—42no—
183C23×C37—42no—
184C23×C37—44no—
185C23×C37—44no—
186C23×C37—44no—
187C23×C37—41no—
188C23×C37—41no—
189C23×C37—42no—
190C23×C37—42no—
191C23×C37—42no—
192C23×C37—82no—
193C23×C37—84no—
194C23×C37—82no—
195C23×C37—82no—
196C23×C37—82no—
197C23×C37—82no—
198C23×C37—82no—
199C23×C37—82no—
200C23×C37—84no—
201C23×C37—84no—
202C23×C37—84no—
203C23×C37C2×C22⋊C411yescharacteristic
204C23×C37C2×C22⋊C411yescharacteristic
205C23×C37C2×C22⋊C411yescharacteristic
206C23×C37C4×D412yes—
207C23×C37C4×D412yes—
208C23×C37#12011yescharacteristic
209C23×C37#12012yes—
210C23×C37#12012yes—
211C23×C37#12011yescharacteristic
212C23×C37#12012yes—
213C23×C37#12112yes—
214C23×C37#12112yes—
215C23×C37#12112yes—
216C23×C37#12112yes—
217C23×C37#12112yes—
218C23×C37#12112yes—
219C23×C37#12311yescharacteristic
220C23×C37#12311yescharacteristic
221C23×C37#12312yes—
222C23×C37#12412yes—
223C23×C37#12712yes—
224D4×C37—44no—
225D4×C37—44no—
226D4×C37—44no—
227D4×C37—44no—
228D4×C37—44no—
229D4×C37—44no—
230D4×C37—44no—
231D4×C37—44no—
232D4×C37—88no—
233C2×C4×C37—24no—
234C2×C4×C37—22no—
235C2×C4×C37—22no—
236C2×C4×C37—22no—
237C2×C4×C37—24no—
238C2×C4×C37—22no—
239C2×C4×C37—44no—
240C2×C4×C37—41no—
241C2×C4×C37—41no—
242C2×C4×C37—42no—
243C2×C4×C37—42no—
244C2×C4×C37—42no—
245C2×C4×C37—48no—
246C2×C4×C37—44no—
247C2×C4×C37—44no—
248C2×C4×C37—84no—
249C2×C4×C37—84no—
250C2×C4×C37—84no—
251C2×C4×C37—84no—
252C2×C4×C37—84no—
253C2×C4×C37—84no—
254#10972C3711yesmaximal, characteristic, direct product factor, Sylow
255C22×C37—22no—
256C22×C37—22no—
257C22×C37—24no—
258C22×C37—24no—
259C22×C37—22no—
260C22×C37—42no—
261C22×C37—42no—
262C22×C37—42no—
263C22×C37—42no—
264C22×C37—42no—
265C22×C37—41no—
266C22×C37—41no—
267C22×C37—42no—
268C22×C37—42no—
269C22×C37—42no—
270C22×C37—42no—
271C22×C37—41no—
272C22×C37—42no—
273C22×C37—41no—
274C22×C37—42no—
275C22×C37—42no—
276C22×C37—44no—
277C22×C37—44no—
278C22×C37—44no—
279C22×C37—42no—
280C22×C37—42no—
281C22×C37—82no—
282C22×C37—84no—
283C22×C37—84no—
284C22×C37—82no—
285C22×C37—84no—
286C22×C37—84no—
287C22×C37—82no—
288C22×C37#37911yescharacteristic
289C22×C37#38611yescharacteristic
290C22×C37#38612yes—
291C22×C37#38611yescharacteristic
292C22×C37#39012yes—
293C22×C37#39212yes—
294C22×C37#39212yes—
295C22×C37#39412yes—
296C22×C37#39412yes—
297C4×C37—44no—
298C4×C37—42no—
299C4×C37—42no—
300C4×C37—84no—
301C4×C37—84no—
302C2×#534C2×C3711yescharacteristic
303C22×#99C2×C3712yescomplemented
304C22×#99C2×C3711yescharacteristic, complemented
305#2308C2×C3711yescharacteristic, complemented
306C2×#342C2×C3712yescomplemented
307C2×C37—21no—
308C2×C37—22no—
309C2×C37—22no—
310C2×C37—21no—
311C2×C37—22no—
312C2×C37—22no—
313C2×C37—22no—
314C2×C37—22no—
315C2×C37—82no—
316C2×C37—82no—
317C2×C37#143811yescharacteristic
318C2×C37#191312yes—
319C23×D4C22×C3711yescharacteristic
320C23×D4C22×C3711yescharacteristic
321#534—24no—
322#534—24no—
323C2×#123—21no—
324C2×#123C4×C3712yescomplemented
325C2×#121—22no—
326C2×#120—21no—
327C2×#120C4×C3712yescomplemented
328C22×C22⋊C4C22×C3712yes—
329C22×C22⋊C4C22×C3712yes—
330C22×C22⋊C4C22×C3711yescharacteristic
331C2×#99—22no—
332C2×#99—24no—
333C2×#99—24no—
334C2×#99—24no—
335C2×#99—24no—
336#386—24no—
337#381—21no—
338#379—21no—
339#342—24no—
340C37#1097211yescharacteristic, direct product factor, Sylow
341C25C2×C4×C3711yescharacteristic
342C25D4×C3711yescharacteristic
343C25C23×C3711yescharacteristic
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4C2×C4×C3712yes—
359C22×D4D4×C3712yes—
360C22×D4D4×C3714yes—
361C23×C4—21no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4C2×C4×C3711yescharacteristic
365C23×C4D4×C3711yescharacteristic
366#123—28no—
367#123—24no—
368#123—24no—
369#121—216no—
370#120—28no—
371#120—24no—
372#120—24no—
373C2×C4⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4C2×C4×C3712yes—
390C2×C22⋊C4D4×C3714yes—
391C2×C22⋊C4D4×C3714yescomplemented
392C2×C22⋊C4D4×C3714yescomplemented
393#99—216no—
394#99—28no—
395#95—24no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—82no—
405C24—82no—
406C24—82no—
407C24C22⋊C4×C3712yes—
408C24C22⋊C4×C3712yescomplemented
409C24C22×C4×C3711yescharacteristic
410C24C2×D4×C3711yescharacteristic
411C24C2×D4×C3711yescharacteristic
412C24C2×D4×C3712yes—
413C24C2×D4×C3712yes—
414C24C2×D4×C3712yes—
415C24C2×D4×C3712yes—
416C24#41×C3712yes—
417C24#41×C3712yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—84no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4C22⋊C4×C3712yes—
472C22×C4C22⋊C4×C3712yescomplemented
473C4⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—22no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—84no—
487C22⋊C4C22⋊C4×C3714yescomplemented
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—82no—
507C23—84no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23C2×C22⋊C4×C3711yescharacteristic
518C23C2×C22⋊C4×C3711yescharacteristic
519C23C2×C22⋊C4×C3711yescharacteristic
520C23C4×D4×C3712yes—
521C23C4×D4×C3712yes—
522C23#120×C3711yescharacteristic
523C23#120×C3712yes—
524C23#120×C3712yes—
525C23#120×C3711yescharacteristic
526C23#120×C3712yes—
527C23#121×C3712yes—
528C23#121×C3712yes—
529C23#121×C3712yes—
530C23#121×C3712yes—
531C23#121×C3712yes—
532C23#121×C3712yes—
533C23#123×C3711yescharacteristic
534C23#123×C3711yescharacteristic
535C23#123×C3712yes—
536C23#124×C3712yes—
537C23#127×C3712yes—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—44no—
546D4—88no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—82no—
595C22—84no—
596C22—84no—
597C22—82no—
598C22—84no—
599C22—84no—
600C22—82no—
601C22C37×#37911yescharacteristic
602C22C37×#38611yescharacteristic
603C22C37×#38612yes—
604C22C37×#38611yescharacteristic
605C22C37×#39012yes—
606C22C37×#39212yes—
607C22C37×#39212yes—
608C22C37×#39412yes—
609C22C37×#39412yes—
610C4—44no—
611C4—42no—
612C4—42no—
613C4—84no—
614C4—84no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—22no—
622C2—22no—
623C2—82no—
624C2—82no—
625C2C37×#143811yescharacteristic
626C2C37×#191312yes—
subgroupG/H
center, socleC22×C37#386
commutatorC24C22×C4×C37
Frattini, cosocleC25C23×C37
prime radicalC2×C37×#534C2
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. C22×#99×C37C2semidirect
3. C22×#99×C37C2semidirect
4. C37×#2308C2semidirect
5. C2×C37×#342C2semidirect
11. C2×#123×C37C4semidirect
14. C2×#120×C37C4semidirect
77. C2×C22⋊C4×C37D4semidirect
78. C2×C22⋊C4×C37D4semidirect
94. C24×C37C22⋊C4semidirect
158. C22×C4×C37C22⋊C4semidirect
173. C22⋊C4×C37C22⋊C4semidirect
254. #10972C37direct (N × H)
303. C22×#99C2×C37semidirect
304. C22×#99C2×C37semidirect
305. #2308C2×C37semidirect
306. C2×#342C2×C37semidirect
324. C2×#123C4×C37semidirect
327. C2×#120C4×C37semidirect
340. C37#10972direct (N × H)
391. C2×C22⋊C4D4×C37semidirect
392. C2×C22⋊C4D4×C37semidirect
408. C24C22⋊C4×C37semidirect
472. C22×C4C22⋊C4×C37semidirect
487. 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×#5342
C22×#99×C372
C37×#23082
C2×C37×#3422
#1097237