Finite Group Databasebeta

← C37×#28776 #22650029

Subgroups — C37×#28776 #22650029

Order 9472.

774 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#2442C211yesmaximal, characteristic
2C37×#2440C211yesmaximal, characteristic
3C37×#2085C211yesmaximal, characteristic, complemented
4C37×#2084C211yesmaximal, characteristic, complemented
5C37×#2059C211yesmaximal, characteristic, complemented
6C37×#2058C211yesmaximal, characteristic, complemented
7C37×#1993C211yesmaximal, characteristic, complemented
8C37×#1992C211yesmaximal, characteristic, complemented
9C37×#1944C211yesmaximal, characteristic, complemented
10C37×#1943C211yesmaximal, characteristic, complemented
11C37×#1853C211yesmaximal, characteristic
12C37×#1852C211yesmaximal, characteristic
13C37×#1807C211yesmaximal, characteristic
14C37×#1752C211yesmaximal, characteristic
15C2×C37×#340C211yesmaximal, characteristic
16C23×C8×C37C2211yescharacteristic
17C2×C4⋊Q8×C37C2211yescharacteristic
18C2×#125×C37C2211yescharacteristic
19C2×C22⋊Q8×C37—21no—
20C22×C4⋊C4×C37C2211yescharacteristic
21C22×C4⋊C4×C37C2211yescharacteristic
22C2×#107×C37C2211yescharacteristic
23C2×#107×C37C2211yescharacteristic
24C2×#106×C37C2211yescharacteristic
25C2×#106×C37C2211yescharacteristic
26C2×#106×C37C2211yescharacteristic
27C2×#106×C37C2211yescharacteristic
28C2×C22⋊C8×C37C2211yescharacteristic
29C2×#97×C37C2211yescharacteristic
30C2×#97×C37C2211yescharacteristic
31C37×#400—21no—
32C37×#398—21no—
33C37×#398—21no—
34C37×#395—21no—
35C37×#389C2211yescharacteristic
36C37×#389C2211yescharacteristic
37C37×#389C2211yescharacteristic
38C37×#389C2211yescharacteristic
39C37×#385—21no—
40C37×#385—21no—
41C37×#384C2211yescharacteristic
42C37×#382C2211yescharacteristic
43C37×#380—21no—
44C37×#380C2211yescharacteristic
45C37×#380C2211yescharacteristic
46C4×C22⋊C4×C37C2211yescharacteristic
47C37×#340C2211yescharacteristic
48C37×#340C2211yescharacteristic
49C37×#340C2211yescharacteristic
50C37×#340C2211yescharacteristic
51C37×#340C2211yescharacteristic
52C37×#340C2211yescharacteristic
53C37×#340C2211yescharacteristic
54C37×#340C2211yescharacteristic
55C37×#340C2211yescharacteristic
56C37×#340C2211yescharacteristic
57C37×#336C2211yescharacteristic
58C37×#336C2211yescharacteristic
59C22×Q8×C37—21no—
60C23×C4×C37—21no—
61C23×C4×C37—21no—
62C23×C4×C37C2311yescharacteristic
63C22×C8×C37—41no—
64C22×C8×C37—41no—
65C22×C8×C37Q814yescomplemented
66C22×C8×C37C2311yescharacteristic
67C22×C8×C37C2311yescharacteristic
68C22×C8×C37C2311yescharacteristic
69C22×C8×C37C2311yescharacteristic
70C4⋊Q8×C37—22no—
71C4⋊Q8×C37—41no—
72#125×C37—22no—
73#125×C37—41no—
74C22⋊Q8×C37—44no—
75C2×C4⋊C4×C37—22no—
76C2×C4⋊C4×C37—22no—
77C2×C4⋊C4×C37—21no—
78C2×C4⋊C4×C37—21no—
79C2×C4⋊C4×C37—21no—
80C2×C4⋊C4×C37—21no—
81C2×C4⋊C4×C37—21no—
82C2×C4⋊C4×C37—21no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37—21no—
85C2×C4⋊C4×C37—21no—
86C2×C4⋊C4×C37—21no—
87C2×C4⋊C4×C37—21no—
88C2×C4⋊C4×C37—41no—
89C2×C4⋊C4×C37—41no—
90C2×C4⋊C4×C37C2311yescharacteristic
91C2×C4⋊C4×C37C2311yescharacteristic
92C2×C4⋊C4×C37C2311yescharacteristic
93C2×C4⋊C4×C37C2311yescharacteristic
94C2×C4⋊C4×C37C2311yescharacteristic
95C2×C4⋊C4×C37C2311yescharacteristic
96C2×C4⋊C4×C37C2311yescharacteristic
97C2×C4⋊C4×C37C2311yescharacteristic
98C2×C22⋊C4×C37—21no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37D411yescharacteristic
101C2×C22⋊C4×C37D411yescharacteristic
102C2×C42×C37C2311yescharacteristic
103C2×C42×C37C2311yescharacteristic
104#107×C37—22no—
105#107×C37—41no—
106#106×C37—22no—
107#106×C37—41no—
108#106×C37—41no—
109#106×C37—41no—
110C22⋊C8×C37Q814yescomplemented
111#97×C37—41no—
112#97×C37—41no—
113#95×C37—21no—
114#95×C37—21no—
115#95×C37—21no—
116#95×C37—21no—
117#95×C37—21no—
118#95×C37—21no—
119#95×C37—21no—
120#95×C37—21no—
121#95×C37D411yescharacteristic
122#95×C37D411yescharacteristic
123C24×C37C2×D411yescharacteristic
124C2×Q8×C37—42no—
125C2×Q8×C37—42no—
126C2×Q8×C37—81no—
127C22×C4×C37—21no—
128C22×C4×C37—21no—
129C22×C4×C37—21no—
130C22×C4×C37—21no—
131C22×C4×C37—21no—
132C22×C4×C37—21no—
133C22×C4×C37—21no—
134C22×C4×C37—21no—
135C22×C4×C37—21no—
136C22×C4×C37—21no—
137C22×C4×C37—21no—
138C22×C4×C37—21no—
139C22×C4×C37—42no—
140C22×C4×C37—42no—
141C22×C4×C37—42no—
142C22×C4×C37—42no—
143C22×C4×C37—81no—
144C22×C4×C37—81no—
145C22×C4×C37C2×D411yescharacteristic
146C22×C4×C37C2×D411yescharacteristic
147C22×C4×C37C2×D411yescharacteristic
148C22×C4×C37C2×D411yescharacteristic
149C22×C4×C37C2×D411yescharacteristic
150C22×C4×C37C2×Q812yes—
151C22×C4×C37C2×Q812yes—
152C22×C4×C37C2411yescharacteristic
153C2×C8×C37—21no—
154C2×C8×C37—21no—
155C2×C8×C37—21no—
156C2×C8×C37—21no—
157C2×C8×C37—21no—
158C2×C8×C37—21no—
159C2×C8×C37—21no—
160C2×C8×C37—21no—
161C2×C8×C37—44no—
162C2×C8×C37C2×Q812yes—
163C2×C8×C37C2×Q812yes—
164C2×C8×C37C2×Q812yes—
165C2×C8×C37C2×Q812yes—
166C4⋊C4×C37—21no—
167C4⋊C4×C37—21no—
168C4⋊C4×C37—21no—
169C4⋊C4×C37—21no—
170C4⋊C4×C37—21no—
171C4⋊C4×C37—21no—
172C4⋊C4×C37—21no—
173C4⋊C4×C37—21no—
174C4⋊C4×C37—21no—
175C4⋊C4×C37—21no—
176C4⋊C4×C37—21no—
177C4⋊C4×C37—21no—
178C4⋊C4×C37—21no—
179C4⋊C4×C37—21no—
180C4⋊C4×C37—42no—
181C4⋊C4×C37—42no—
182C4⋊C4×C37—42no—
183C4⋊C4×C37—42no—
184C4⋊C4×C37—42no—
185C4⋊C4×C37—42no—
186C4⋊C4×C37—42no—
187C4⋊C4×C37—42no—
188C4⋊C4×C37—81no—
189C4⋊C4×C37—81no—
190C4⋊C4×C37—81no—
191C4⋊C4×C37#4112yes—
192C4⋊C4×C37#4112yes—
193C22⋊C4×C37—22no—
194C22⋊C4×C37—22no—
195C22⋊C4×C37—42no—
196C22⋊C4×C37—42no—
197C42×C37—21no—
198C42×C37—21no—
199C42×C37—21no—
200C42×C37—21no—
201C23×C37—21no—
202C23×C37—21no—
203C23×C37—21no—
204C23×C37—21no—
205C23×C37C4⋊Q812yes—
206C23×C37AGL(1,ℤ/8)12yes—
207C23×C37#13712yes—
208C23×C37C22×D411yescharacteristic
209Q8×C37—84no—
210C2×C4×C37—21no—
211C2×C4×C37—21no—
212C2×C4×C37—22no—
213C2×C4×C37—22no—
214C2×C4×C37—22no—
215C2×C4×C37—22no—
216C2×C4×C37—21no—
217C2×C4×C37—21no—
218C2×C4×C37—21no—
219C2×C4×C37—21no—
220C2×C4×C37—22no—
221C2×C4×C37—22no—
222C2×C4×C37—21no—
223C2×C4×C37—21no—
224C2×C4×C37—21no—
225C2×C4×C37—21no—
226C2×C4×C37—21no—
227C2×C4×C37—21no—
228C2×C4×C37—21no—
229C2×C4×C37—21no—
230C2×C4×C37—41no—
231C2×C4×C37—41no—
232C2×C4×C37—41no—
233C2×C4×C37—41no—
234C2×C4×C37—41no—
235C2×C4×C37—41no—
236C2×C4×C37—41no—
237C2×C4×C37—41no—
238C2×C4×C37—41no—
239C2×C4×C37—41no—
240C2×C4×C37—41no—
241C2×C4×C37—41no—
242C2×C4×C37—41no—
243C2×C4×C37—41no—
244C2×C4×C37—41no—
245C2×C4×C37—41no—
246C2×C4×C37—41no—
247C2×C4×C37—41no—
248C2×C4×C37—41no—
249C2×C4×C37—41no—
250C2×C4×C37—41no—
251C2×C4×C37—41no—
252C2×C4×C37—84no—
253C2×C4×C37—84no—
254C2×C4×C37C4⋊Q812yes—
255C2×C4×C37C22×Q811yescharacteristic
256C2×C4×C37C22×Q811yescharacteristic
257C2×C4×C37C2×#4111yescharacteristic
258C2×C4×C37C2×#4111yescharacteristic
259C2×C4×C37H(5,𝔽2)11yescharacteristic
260C2×C4×C37H(5,𝔽2)11yescharacteristic
261C8×C37—22no—
262C8×C37—22no—
263C8×C37—22no—
264C8×C37—22no—
265#28776C3711yesmaximal, characteristic, direct product factor, Sylow
266C22×C37—22no—
267C22×C37—22no—
268C22×C37—22no—
269C22×C37—22no—
270C22×C37—22no—
271C22×C37—22no—
272C22×C37#50114yes—
273C22×C37C2×C4⋊Q811yescharacteristic
274C22×C37#54611yescharacteristic
275C22×C37#54611yescharacteristic
276C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C37C2×#13711yescharacteristic
278C22×C37#57711yescharacteristic
279C22×C37#57711yescharacteristic
280C4×C37—22no—
281C4×C37—41no—
282C4×C37—42no—
283C4×C37—41no—
284C4×C37—42no—
285C4×C37—41no—
286C4×C37—41no—
287C4×C37—81no—
288C4×C37—81no—
289C4×C37—81no—
290C4×C37—81no—
291C4×C37—81no—
292C4×C37—81no—
293C4×C37#53311yescharacteristic
294C4×C37#54311yescharacteristic
295C4×C37#55411yescharacteristic
296C4×C37#55411yescharacteristic
297#2442C2×C3711yescharacteristic
298#2440C2×C3711yescharacteristic
299#2085C2×C3711yescharacteristic, complemented
300#2084C2×C3711yescharacteristic, complemented
301#2059C2×C3711yescharacteristic, complemented
302#2058C2×C3711yescharacteristic, complemented
303#1993C2×C3711yescharacteristic, complemented
304#1992C2×C3711yescharacteristic, complemented
305#1944C2×C3711yescharacteristic, complemented
306#1943C2×C3711yescharacteristic, complemented
307#1853C2×C3711yescharacteristic
308#1852C2×C3711yescharacteristic
309#1807C2×C3711yescharacteristic
310#1752C2×C3711yescharacteristic
311C2×#340C2×C3711yescharacteristic
312C2×C37—24no—
313C2×C37#254911yescharacteristic
314C2×C37C2×#50111yescharacteristic
315C2×C37#316311yescharacteristic
316C2×C37#330611yescharacteristic
317C2×C37#331011yescharacteristic
318C2×C37#331611yescharacteristic
319C2×C37#332011yescharacteristic
320C23×C8C22×C3711yescharacteristic
321C2×C4⋊Q8C22×C3711yescharacteristic
322C2×#125C22×C3711yescharacteristic
323C2×C22⋊Q8—21no—
324C22×C4⋊C4C22×C3711yescharacteristic
325C22×C4⋊C4C22×C3711yescharacteristic
326C2×#107C22×C3711yescharacteristic
327C2×#107C22×C3711yescharacteristic
328C2×#106C22×C3711yescharacteristic
329C2×#106C22×C3711yescharacteristic
330C2×#106C22×C3711yescharacteristic
331C2×#106C22×C3711yescharacteristic
332C2×C22⋊C8C22×C3711yescharacteristic
333C2×#97C22×C3711yescharacteristic
334C2×#97C22×C3711yescharacteristic
335#400—21no—
336#398—21no—
337#398—21no—
338#395—21no—
339#389C22×C3711yescharacteristic
340#389C22×C3711yescharacteristic
341#389C22×C3711yescharacteristic
342#389C22×C3711yescharacteristic
343#385—21no—
344#385—21no—
345#384C22×C3711yescharacteristic
346#382C22×C3711yescharacteristic
347#380—21no—
348#380C22×C3711yescharacteristic
349#380C22×C3711yescharacteristic
350C4×C22⋊C4C22×C3711yescharacteristic
351#340C22×C3711yescharacteristic
352#340C22×C3711yescharacteristic
353#340C22×C3711yescharacteristic
354#340C22×C3711yescharacteristic
355#340C22×C3711yescharacteristic
356#340C22×C3711yescharacteristic
357#340C22×C3711yescharacteristic
358#340C22×C3711yescharacteristic
359#340C22×C3711yescharacteristic
360#340C22×C3711yescharacteristic
361#336C22×C3711yescharacteristic
362#336C22×C3711yescharacteristic
363C37#2877611yescharacteristic, direct product factor, Sylow
364C22×Q8—21no—
365C23×C4—21no—
366C23×C4—21no—
367C23×C4C23×C3711yescharacteristic
368C22×C8—41no—
369C22×C8—41no—
370C22×C8Q8×C3714yescomplemented
371C22×C8C23×C3711yescharacteristic
372C22×C8C23×C3711yescharacteristic
373C22×C8C23×C3711yescharacteristic
374C22×C8C23×C3711yescharacteristic
375C4⋊Q8—22no—
376C4⋊Q8—41no—
377#125—22no—
378#125—41no—
379C22⋊Q8—44no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—41no—
394C2×C4⋊C4—41no—
395C2×C4⋊C4C23×C3711yescharacteristic
396C2×C4⋊C4C23×C3711yescharacteristic
397C2×C4⋊C4C23×C3711yescharacteristic
398C2×C4⋊C4C23×C3711yescharacteristic
399C2×C4⋊C4C23×C3711yescharacteristic
400C2×C4⋊C4C23×C3711yescharacteristic
401C2×C4⋊C4C23×C3711yescharacteristic
402C2×C4⋊C4C23×C3711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4D4×C3711yescharacteristic
406C2×C22⋊C4D4×C3711yescharacteristic
407C2×C42C23×C3711yescharacteristic
408C2×C42C23×C3711yescharacteristic
409#107—22no—
410#107—41no—
411#106—22no—
412#106—41no—
413#106—41no—
414#106—41no—
415C22⋊C8Q8×C3714yescomplemented
416#97—41no—
417#97—41no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95D4×C3711yescharacteristic
427#95D4×C3711yescharacteristic
428C24C2×D4×C3711yescharacteristic
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—81no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4C2×D4×C3711yescharacteristic
451C22×C4C2×D4×C3711yescharacteristic
452C22×C4C2×D4×C3711yescharacteristic
453C22×C4C2×D4×C3711yescharacteristic
454C22×C4C2×D4×C3711yescharacteristic
455C22×C4C2×Q8×C3712yes—
456C22×C4C2×Q8×C3712yes—
457C22×C4C24×C3711yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8C2×Q8×C3712yes—
468C2×C8C2×Q8×C3712yes—
469C2×C8C2×Q8×C3712yes—
470C2×C8C2×Q8×C3712yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4#41×C3712yes—
497C4⋊C4#41×C3712yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23C4⋊Q8×C3712yes—
511C23AGL(1,ℤ/8)×C3712yes—
512C23#137×C3712yes—
513C23C22×D4×C3711yescharacteristic
514Q8—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C4⋊Q8×C3712yes—
560C2×C4C22×Q8×C3711yescharacteristic
561C2×C4C22×Q8×C3711yescharacteristic
562C2×C4C2×#41×C3711yescharacteristic
563C2×C4C2×#41×C3711yescharacteristic
564C2×C4H(5,𝔽2)×C3711yescharacteristic
565C2×C4H(5,𝔽2)×C3711yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22C37×#50114yes—
577C22C2×C4⋊Q8×C3711yescharacteristic
578C22C37×#54611yescharacteristic
579C22C37×#54611yescharacteristic
580C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
581C22C2×#137×C3711yescharacteristic
582C22C37×#57711yescharacteristic
583C22C37×#57711yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4C37×#53311yescharacteristic
598C4C37×#54311yescharacteristic
599C4C37×#55411yescharacteristic
600C4C37×#55411yescharacteristic
601C2—24no—
602C2C37×#254911yescharacteristic
603C2C2×C37×#50111yescharacteristic
604C2C37×#316311yescharacteristic
605C2C37×#330611yescharacteristic
606C2C37×#331011yescharacteristic
607C2C37×#331611yescharacteristic
608C2C37×#332011yescharacteristic
subgroupG/H
center, socleC23×C37C22×D4
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC24×C37C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C37 ◁ C4×C22⋊C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
3. C37×#2085C2semidirect
4. C37×#2084C2semidirect
5. C37×#2059C2semidirect
6. C37×#2058C2semidirect
7. C37×#1993C2semidirect
8. C37×#1992C2semidirect
9. C37×#1944C2semidirect
10. C37×#1943C2semidirect
65. C22×C8×C37Q8semidirect
110. C22⋊C8×C37Q8semidirect
265. #28776C37direct (N × H)
299. #2085C2×C37semidirect
300. #2084C2×C37semidirect
301. #2059C2×C37semidirect
302. #2058C2×C37semidirect
303. #1993C2×C37semidirect
304. #1992C2×C37semidirect
305. #1944C2×C37semidirect
306. #1943C2×C37semidirect
363. C37#28776direct (N × H)
370. C22×C8Q8×C37semidirect
415. C22⋊C8Q8×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#24422
C37×#24402
C37×#20852
C37×#20842
C37×#20592
C37×#20582
C37×#19932
C37×#19922
C37×#19442
C37×#19432
C37×#18532
C37×#18522
C37×#18072
C37×#17522
C2×C37×#3402
#2877637