Finite Group Databasebeta

← C37×#23488 #22644741

Subgroups — C37×#23488 #22644741

Order 9472.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#537C211yesmaximal, characteristic
2C22×#107×C37C211yesmaximal, characteristic, complemented
3C2×C37×#417C211yesmaximal, characteristic, complemented
4C2×C37×#416C211yesmaximal, characteristic, complemented
5C2×C37×#408C211yesmaximal, characteristic, complemented
6C37×#2311C211yesmaximal, characteristic, complemented
7C37×#2036C214yesmaximal, complemented
8C37×#2033C214yesmaximal, complemented
9C37×#1736C211yesmaximal, characteristic, complemented
10C22×#41×C37C2211yescharacteristic
11C22×#41×C37C2211yescharacteristic
12C22×M16×C37C2211yescharacteristic
13C23×C8×C37C2211yescharacteristic
14C37×#537—24no—
15C37×#537—24no—
16C2×C22⋊Q8×C37—21no—
17C2×C22⋊Q8×C37C2212yes—
18C2×#121×C37—21no—
19C2×#121×C37—22no—
20C2×#121×C37C2212yes—
21C2×#117×C37C2211yescharacteristic
22C22×C4⋊C4×C37C2211yescharacteristic
23C22×C4⋊C4×C37C2211yescharacteristic
24C2×#107×C37—24no—
25C2×#107×C37C2214yes—
26C37×#417—24no—
27C37×#416—24no—
28C2×#103×C37C2212yes—
29C2×#103×C37C2212yes—
30C2×#102×C37C2212yes—
31C2×#102×C37C2212yes—
32C37×#408—24no—
33C2×C22⋊C8×C37C2214yes—
34C37×#391—21no—
35C37×#390—21no—
36C37×#386—22no—
37C37×#380C2214yes—
38C37×#340C2214yes—
39C2×#41×C37—22no—
40C2×#41×C37—22no—
41C2×#41×C37—22no—
42C2×#41×C37—22no—
43C2×#41×C37—22no—
44C2×#41×C37—22no—
45C2×#41×C37—22no—
46C2×#41×C37—22no—
47C2×#41×C37—22no—
48C2×#41×C37—22no—
49C2×#41×C37—22no—
50C2×#41×C37D414yes—
51C22×Q8×C37D411yescharacteristic
52C22×Q8×C37C2311yescharacteristic
53C22×D4×C37—21no—
54C22×D4×C37—22no—
55C22×D4×C37D412yes—
56C22×D4×C37D411yescharacteristic
57C22×D4×C37C2311yescharacteristic
58C23×C4×C37—21no—
59C23×C4×C37—21no—
60C23×C4×C37—21no—
61C23×C4×C37C2312yes—
62C23×C4×C37C2311yescharacteristic
63C2×M16×C37—22no—
64C2×M16×C37—22no—
65C2×M16×C37—22no—
66C22×C8×C37—22no—
67C22×C8×C37—22no—
68C22×C8×C37D414yescomplemented
69C22×C8×C37C2312yes—
70C22×C8×C37C2312yes—
71C22⋊Q8×C37—28no—
72C22⋊Q8×C37—24no—
73C22⋊Q8×C37—24no—
74#121×C37—216no—
75#121×C37—28no—
76#121×C37—24no—
77#121×C37—24no—
78#117×C37—22no—
79#117×C37—22no—
80C2×C4⋊C4×C37—22no—
81C2×C4⋊C4×C37—21no—
82C2×C4⋊C4×C37—22no—
83C2×C4⋊C4×C37—22no—
84C2×C4⋊C4×C37—22no—
85C2×C4⋊C4×C37—21no—
86C2×C4⋊C4×C37D414yes—
87C2×C4⋊C4×C37C2312yes—
88C2×C4⋊C4×C37C2312yes—
89C2×C4⋊C4×C37C2312yes—
90C2×C22⋊C4×C37—22no—
91C2×C22⋊C4×C37—22no—
92C2×C22⋊C4×C37—21no—
93C2×C22⋊C4×C37—24no—
94C2×C42×C37—21no—
95#107×C37—28no—
96#103×C37—24no—
97#103×C37—24no—
98#102×C37—24no—
99#102×C37—24no—
100C22⋊C8×C37—28no—
101#95×C37—24no—
102C24×C37—21no—
103C24×C37C2×D412yes—
104C24×C37C2×D411yescharacteristic
105#41×C37—28no—
106#41×C37—28no—
107#41×C37—44no—
108#41×C37—44no—
109#41×C37—44no—
110#41×C37—44no—
111#41×C37—44no—
112#41×C37—44no—
113C2×Q8×C37—22no—
114C2×Q8×C37—22no—
115C2×Q8×C37—22no—
116C2×Q8×C37—21no—
117C2×Q8×C37—21no—
118C2×Q8×C37—21no—
119C2×Q8×C37—21no—
120C2×Q8×C37—21no—
121C2×Q8×C37C2×D412yes—
122C2×D4×C37—24no—
123C2×D4×C37—24no—
124C2×D4×C37—24no—
125C2×D4×C37—24no—
126C2×D4×C37—28no—
127C2×D4×C37—22no—
128C2×D4×C37—22no—
129C2×D4×C37—22no—
130C2×D4×C37—21no—
131C2×D4×C37—21no—
132C2×D4×C37—21no—
133C2×D4×C37—21no—
134C2×D4×C37—21no—
135C2×D4×C37—42no—
136C2×D4×C37—42no—
137C2×D4×C37—84no—
138C2×D4×C37C2×D412yes—
139C22×C4×C37—22no—
140C22×C4×C37—21no—
141C22×C4×C37—24no—
142C22×C4×C37—24no—
143C22×C4×C37—22no—
144C22×C4×C37—22no—
145C22×C4×C37—22no—
146C22×C4×C37—22no—
147C22×C4×C37—22no—
148C22×C4×C37—22no—
149C22×C4×C37—22no—
150C22×C4×C37—22no—
151C22×C4×C37—42no—
152C22×C4×C37—42no—
153C22×C4×C37—42no—
154C22×C4×C37—82no—
155C22×C4×C37—82no—
156C22×C4×C37C2×D412yes—
157C22×C4×C37C2×D411yescharacteristic
158C22×C4×C37C2×D412yes—
159C22×C4×C37C2×D412yes—
160C22×C4×C37C2×D412yes—
161C22×C4×C37#4112yes—
162C22×C4×C37#4112yes—
163C22×C4×C37#4112yes—
164C22×C4×C37C2411yescharacteristic
165M16×C37—44no—
166C2×C8×C37—28no—
167C2×C8×C37—22no—
168C2×C8×C37—22no—
169C2×C8×C37—22no—
170C2×C8×C37—22no—
171C2×C8×C37—22no—
172C2×C8×C37C2×D414yes—
173C4⋊C4×C37—28no—
174C4⋊C4×C37—24no—
175C4⋊C4×C37—22no—
176C4⋊C4×C37—22no—
177C4⋊C4×C37—22no—
178C4⋊C4×C37—22no—
179C4⋊C4×C37—22no—
180C4⋊C4×C37—42no—
181C4⋊C4×C37C2×D414yes—
182C22⋊C4×C37—28no—
183C22⋊C4×C37—28no—
184C22⋊C4×C37—42no—
185C22⋊C4×C37—84no—
186C42×C37—42no—
187C23×C37—22no—
188C23×C37—22no—
189C23×C37—22no—
190C23×C37—22no—
191C23×C37—22no—
192C23×C37—22no—
193C23×C37—24no—
194C23×C37—24no—
195C23×C37—41no—
196C23×C37—41no—
197C23×C37—42no—
198C23×C37—42no—
199C23×C37#12012yes—
200C23×C37#12112yes—
201C23×C37#12312yes—
202C23×C37C22×D411yescharacteristic
203Q8×C37—22no—
204Q8×C37—22no—
205Q8×C37—41no—
206Q8×C37—41no—
207Q8×C37—44no—
208D4×C37—22no—
209D4×C37—22no—
210D4×C37—48no—
211D4×C37—48no—
212D4×C37—41no—
213D4×C37—41no—
214D4×C37—44no—
215D4×C37—88no—
216C2×C4×C37—24no—
217C2×C4×C37—22no—
218C2×C4×C37—22no—
219C2×C4×C37—22no—
220C2×C4×C37—22no—
221C2×C4×C37—41no—
222C2×C4×C37—41no—
223C2×C4×C37—44no—
224C2×C4×C37—42no—
225C2×C4×C37—42no—
226C2×C4×C37—42no—
227C2×C4×C37—42no—
228C2×C4×C37—48no—
229C2×C4×C37—42no—
230C2×C4×C37—42no—
231C2×C4×C37—42no—
232C2×C4×C37—44no—
233C2×C4×C37—42no—
234C2×C4×C37—84no—
235C2×C4×C37—84no—
236C2×C4×C37#12012yes—
237C2×C4×C37#12114yes—
238C2×C4×C37#12114yes—
239C2×C4×C37#12112yes—
240C2×C4×C37#12314yes—
241C2×C4×C37#12312yes—
242C2×C4×C37#12414yes—
243C2×C4×C37C22×D411yescharacteristic
244C2×C4×C37C22×D411yescharacteristic
245C2×C4×C37C22×D411yescharacteristic
246C2×C4×C37C2×#4111yescharacteristic
247C2×C4×C37C2×#4111yescharacteristic
248C2×C4×C37C2×#4111yescharacteristic
249C8×C37—24no—
250C8×C37—42no—
251#23488C3711yesmaximal, characteristic, direct product factor, Sylow
252C22×C37—24no—
253C22×C37—22no—
254C22×C37—22no—
255C22×C37—24no—
256C22×C37—44no—
257C22×C37—42no—
258C22×C37—42no—
259C22×C37—42no—
260C22×C37—42no—
261C22×C37—42no—
262C22×C37—42no—
263C22×C37—44no—
264C22×C37—82no—
265C22×C37—82no—
266C22×C37#39414yes—
267C22×C37C2×#12011yescharacteristic
268C22×C37C2×#12111yescharacteristic
269C22×C37C2×#12311yescharacteristic
270C22×C37#57612yes—
271C22×C37#57812yes—
272C4×C37—42no—
273C4×C37—42no—
274C4×C37—44no—
275C4×C37—82no—
276C4×C37—82no—
277C4×C37#39414yes—
278C4×C37C2×#12111yescharacteristic
279C4×C37C2×#12111yescharacteristic
280C4×C37C2×#12311yescharacteristic
281C4×C37C2×#12411yescharacteristic
282C2×#537C2×C3711yescharacteristic
283C22×#107C2×C3711yescharacteristic, complemented
284C2×#417C2×C3711yescharacteristic, complemented
285C2×#416C2×C3711yescharacteristic, complemented
286C2×#408C2×C3711yescharacteristic, complemented
287#2311C2×C3711yescharacteristic, complemented
288#2036C2×C3714yescomplemented
289#2033C2×C3714yescomplemented
290#1736C2×C3711yescharacteristic, complemented
291C2×C37—24no—
292C2×C37—82no—
293C2×C37—82no—
294C2×C37C2×#39411yescharacteristic
295C2×C37#300811yescharacteristic
296C2×C37#301111yescharacteristic
297C2×C37#306112yes—
298C2×C37#309812yes—
299C22×#41C22×C3711yescharacteristic
300C22×#41C22×C3711yescharacteristic
301C22×M16C22×C3711yescharacteristic
302C23×C8C22×C3711yescharacteristic
303#537—24no—
304#537—24no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8C22×C3712yes—
307C2×#121—21no—
308C2×#121—22no—
309C2×#121C22×C3712yes—
310C2×#117C22×C3711yescharacteristic
311C22×C4⋊C4C22×C3711yescharacteristic
312C22×C4⋊C4C22×C3711yescharacteristic
313C2×#107—24no—
314C2×#107C22×C3714yes—
315#417—24no—
316#416—24no—
317C2×#103C22×C3712yes—
318C2×#103C22×C3712yes—
319C2×#102C22×C3712yes—
320C2×#102C22×C3712yes—
321#408—24no—
322C2×C22⋊C8C22×C3714yes—
323#391—21no—
324#390—21no—
325#386—22no—
326#380C22×C3714yes—
327#340C22×C3714yes—
328C37#2348811yescharacteristic, direct product factor, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41D4×C3714yes—
341C22×Q8D4×C3711yescharacteristic
342C22×Q8C23×C3711yescharacteristic
343C22×D4—21no—
344C22×D4—22no—
345C22×D4D4×C3712yes—
346C22×D4D4×C3711yescharacteristic
347C22×D4C23×C3711yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—21no—
351C23×C4C23×C3712yes—
352C23×C4C23×C3711yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—22no—
356C22×C8—22no—
357C22×C8—22no—
358C22×C8D4×C3714yescomplemented
359C22×C8C23×C3712yes—
360C22×C8C23×C3712yes—
361C22⋊Q8—28no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364#121—216no—
365#121—28no—
366#121—24no—
367#121—24no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4D4×C3714yes—
377C2×C4⋊C4C23×C3712yes—
378C2×C4⋊C4C23×C3712yes—
379C2×C4⋊C4C23×C3712yes—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—24no—
384C2×C42—21no—
385#107—28no—
386#103—24no—
387#103—24no—
388#102—24no—
389#102—24no—
390C22⋊C8—28no—
391#95—24no—
392C24—21no—
393C24C2×D4×C3712yes—
394C24C2×D4×C3711yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401#41—44no—
402#41—44no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×D4×C3712yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—84no—
428C2×D4C2×D4×C3712yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4C2×D4×C3712yes—
447C22×C4C2×D4×C3711yescharacteristic
448C22×C4C2×D4×C3712yes—
449C22×C4C2×D4×C3712yes—
450C22×C4C2×D4×C3712yes—
451C22×C4#41×C3712yes—
452C22×C4#41×C3712yes—
453C22×C4#41×C3712yes—
454C22×C4C24×C3711yescharacteristic
455M16—44no—
456C2×C8—28no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8C2×D4×C3714yes—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4C2×D4×C3714yes—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—42no—
475C22⋊C4—84no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23#120×C3712yes—
490C23#121×C3712yes—
491C23#123×C3712yes—
492C23C22×D4×C3711yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—44no—
498D4—22no—
499D4—22no—
500D4—48no—
501D4—48no—
502D4—41no—
503D4—41no—
504D4—44no—
505D4—88no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4#120×C3712yes—
527C2×C4#121×C3714yes—
528C2×C4#121×C3714yes—
529C2×C4#121×C3712yes—
530C2×C4#123×C3714yes—
531C2×C4#123×C3712yes—
532C2×C4#124×C3714yes—
533C2×C4C22×D4×C3711yescharacteristic
534C2×C4C22×D4×C3711yescharacteristic
535C2×C4C22×D4×C3711yescharacteristic
536C2×C4C2×#41×C3711yescharacteristic
537C2×C4C2×#41×C3711yescharacteristic
538C2×C4C2×#41×C3711yescharacteristic
539C8—24no—
540C8—42no—
541C22—22no—
542C22—22no—
543C22—24no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22C37×#39414yes—
556C22C2×#120×C3711yescharacteristic
557C22C2×#121×C3711yescharacteristic
558C22C2×#123×C3711yescharacteristic
559C22C37×#57612yes—
560C22C37×#57812yes—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—82no—
565C4—82no—
566C4C37×#39414yes—
567C4C2×#121×C3711yescharacteristic
568C4C2×#121×C3711yescharacteristic
569C4C2×#123×C3711yescharacteristic
570C4C2×#124×C3711yescharacteristic
571C2—24no—
572C2—82no—
573C2—82no—
574C2C2×C37×#39411yescharacteristic
575C2C37×#300811yescharacteristic
576C2C37×#301111yescharacteristic
577C2C37×#306112yes—
578C2C37×#309812yes—
subgroupG/H
center, socleC23×C37C22×D4
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×C37×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C37 ◁ C22×#41×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#107×C37C2semidirect
3. C2×C37×#417C2semidirect
4. C2×C37×#416C2semidirect
5. C2×C37×#408C2semidirect
6. C37×#2311C2semidirect
7. C37×#2036C2semidirect
8. C37×#2033C2semidirect
9. C37×#1736C2semidirect
68. C22×C8×C37D4semidirect
251. #23488C37direct (N × H)
283. C22×#107C2×C37semidirect
284. C2×#417C2×C37semidirect
285. C2×#416C2×C37semidirect
286. C2×#408C2×C37semidirect
287. #2311C2×C37semidirect
288. #2036C2×C37semidirect
289. #2033C2×C37semidirect
290. #1736C2×C37semidirect
328. C37#23488direct (N × H)
358. C22×C8D4×C37semidirect

Maximal subgroups

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

subgroupindex
C2×C37×#5372
C22×#107×C372
C2×C37×#4172
C2×C37×#4162
C2×C37×#4082
C37×#23112
C37×#20362
C37×#20332
C37×#17362
#2348837