Finite Group Databasebeta

← C37×#18140 #22639393

Subgroups — C37×#18140 #22639393

Order 9472.

1024 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3459C211yesmaximal, characteristic
2C37×#3092C212yesmaximal, complemented
3C37×#2486C211yesmaximal, characteristic, complemented
4C37×#1609C214yesmaximal, complemented
5C37×#1608C212yesmaximal, complemented
6C37×#1607C212yesmaximal, complemented
7C37×#1513C212yesmaximal, complemented
8C37×#1465C211yesmaximal, characteristic, complemented
9C37×#543C2211yescharacteristic
10C37×#539—22no—
11C37×#539C2212yes—
12C37×#538—22no—
13C37×#538—21no—
14C37×#534—21no—
15C37×#534C2211yescharacteristic
16C37×#529—22no—
17C37×#529—21no—
18C37×#525—22no—
19C37×#525C2212yes—
20C2×#123×C37—21no—
21C2×C4×D4×C37—21no—
22C2×C4×D4×C37C2211yescharacteristic
23C22×C4⋊C4×C37C2211yescharacteristic, complemented
24C37×#484—22no—
25C37×#483C2214yescomplemented
26C37×#481—22no—
27C37×#480C2214yes—
28C37×#428—22no—
29C37×#417—22no—
30C37×#407C2212yescomplemented
31C37×#398C2212yescomplemented
32C37×#397C2212yes—
33C37×#389—21no—
34C37×#381—21no—
35C37×#380—21no—
36C37×#380C2211yescharacteristic, complemented
37C37×#328C2214yescomplemented
38C37×#327C2214yes—
39C37×#324C2212yescomplemented
40C37×#323C2212yes—
41C2×#41×C37D412yes—
42C22×D4×C37—21no—
43C22×D4×C37D411yescharacteristic
44C23×C4×C37—21no—
45C23×C4×C37—21no—
46C23×C4×C37D411yescharacteristic
47C23×C4×C37C2311yescharacteristic
48C2×M16×C37—22no—
49#126×C37—22no—
50#126×C37—22no—
51#126×C37—22no—
52#125×C37—21no—
53#125×C37—22no—
54#124×C37—22no—
55#124×C37—21no—
56#123×C37—22no—
57#123×C37—22no—
58#123×C37—22no—
59#123×C37—22no—
60#123×C37—22no—
61#123×C37—22no—
62#123×C37—44no—
63C22⋊Q8×C37—22no—
64C22⋊Q8×C37—22no—
65C22⋊Q8×C37—21no—
66C22⋊Q8×C37—22no—
67C22⋊Q8×C37D412yes—
68C22⋊Q8×C37C2312yes—
69#121×C37—22no—
70#121×C37—22no—
71#121×C37—22no—
72#121×C37—21no—
73#121×C37D412yes—
74#121×C37C2312yes—
75#120×C37—22no—
76#120×C37—22no—
77#120×C37—22no—
78C4×Q8×C37—21no—
79C4×D4×C37—21no—
80C4×D4×C37—22no—
81C4×D4×C37—22no—
82C4×D4×C37—22no—
83C4×D4×C37—22no—
84C4×D4×C37—22no—
85C4×D4×C37—22no—
86C4×D4×C37—22no—
87#117×C37—22no—
88#117×C37D412yes—
89C2×C4⋊C4×C37—21no—
90C2×C4⋊C4×C37—22no—
91C2×C4⋊C4×C37—22no—
92C2×C4⋊C4×C37—22no—
93C2×C4⋊C4×C37—21no—
94C2×C4⋊C4×C37—21no—
95C2×C4⋊C4×C37C2312yes—
96C2×C4⋊C4×C37C2311yescharacteristic
97C2×C22⋊C4×C37—21no—
98C2×C22⋊C4×C37—21no—
99C2×C22⋊C4×C37—22no—
100C2×C22⋊C4×C37—21no—
101C2×C22⋊C4×C37D411yescharacteristic
102C2×C22⋊C4×C37C2311yescharacteristic
103C2×C42×C37—21no—
104C2×C42×C37D411yescharacteristic
105#107×C37—24no—
106#106×C37—24no—
107#103×C37—24no—
108#102×C37—24no—
109C22⋊C8×C37—22no—
110C22⋊C8×C37C2314yes—
111#95×C37—22no—
112#95×C37—22no—
113#95×C37C2312yes—
114C24×C37—21no—
115C24×C37C2×D411yescharacteristic
116C24×C37C2×D411yescharacteristic
117#41×C37—44no—
118C2×Q8×C37—21no—
119C2×Q8×C37C2×D412yes—
120C2×D4×C37—22no—
121C2×D4×C37—22no—
122C2×D4×C37—22no—
123C2×D4×C37—22no—
124C2×D4×C37—21no—
125C2×D4×C37—22no—
126C2×D4×C37—22no—
127C2×D4×C37—41no—
128C2×D4×C37—42no—
129C2×D4×C37—42no—
130C2×D4×C37C2×D412yes—
131C22×C4×C37—21no—
132C22×C4×C37—21no—
133C22×C4×C37—21no—
134C22×C4×C37—22no—
135C22×C4×C37—21no—
136C22×C4×C37—22no—
137C22×C4×C37—21no—
138C22×C4×C37—22no—
139C22×C4×C37—22no—
140C22×C4×C37—42no—
141C22×C4×C37—42no—
142C22×C4×C37—42no—
143C22×C4×C37—42no—
144C22×C4×C37—42no—
145C22×C4×C37—82no—
146C22×C4×C37C2×D411yescharacteristic
147C22×C4×C37C2×D411yescharacteristic
148C22×C4×C37C2×D412yes—
149C22×C4×C37C2×D411yescharacteristic
150C22×C4×C37C2×D411yescharacteristic
151C22×C4×C37C2×D412yes—
152C22×C4×C37C2411yescharacteristic
153M16×C37—44no—
154C2×C8×C37—24no—
155C4⋊C4×C37—22no—
156C4⋊C4×C37—22no—
157C4⋊C4×C37—22no—
158C4⋊C4×C37—21no—
159C4⋊C4×C37—21no—
160C4⋊C4×C37—22no—
161C4⋊C4×C37—22no—
162C4⋊C4×C37—22no—
163C4⋊C4×C37—22no—
164C4⋊C4×C37—24no—
165C4⋊C4×C37—42no—
166C4⋊C4×C37—44no—
167C4⋊C4×C37—41no—
168C4⋊C4×C37—41no—
169C4⋊C4×C37C2×D412yes—
170C4⋊C4×C37C2×D412yes—
171C22⋊C4×C37—22no—
172C22⋊C4×C37—22no—
173C22⋊C4×C37—22no—
174C22⋊C4×C37—22no—
175C22⋊C4×C37—22no—
176C22⋊C4×C37—22no—
177C22⋊C4×C37—22no—
178C22⋊C4×C37—22no—
179C22⋊C4×C37—22no—
180C22⋊C4×C37—22no—
181C22⋊C4×C37—22no—
182C22⋊C4×C37—22no—
183C22⋊C4×C37—42no—
184C22⋊C4×C37—44no—
185C42×C37—22no—
186C42×C37—22no—
187C42×C37—21no—
188C42×C37—21no—
189C23×C37—21no—
190C23×C37—21no—
191C23×C37—21no—
192C23×C37—21no—
193C23×C37—22no—
194C23×C37—41no—
195C23×C37—41no—
196C23×C37—41no—
197C23×C37—41no—
198C23×C37—42no—
199C23×C37—81no—
200C23×C37—81no—
201C23×C37#12012yes—
202C23×C37AGL(1,ℤ/8)12yes—
203C23×C37C22×D411yescharacteristic
204Q8×C37—42no—
205Q8×C37—81no—
206D4×C37—42no—
207D4×C37—44no—
208D4×C37—44no—
209D4×C37—42no—
210D4×C37—42no—
211D4×C37—82no—
212D4×C37—82no—
213D4×C37—81no—
214C2×C4×C37—22no—
215C2×C4×C37—24no—
216C2×C4×C37—21no—
217C2×C4×C37—22no—
218C2×C4×C37—21no—
219C2×C4×C37—22no—
220C2×C4×C37—21no—
221C2×C4×C37—22no—
222C2×C4×C37—21no—
223C2×C4×C37—21no—
224C2×C4×C37—42no—
225C2×C4×C37—41no—
226C2×C4×C37—42no—
227C2×C4×C37—42no—
228C2×C4×C37—44no—
229C2×C4×C37—41no—
230C2×C4×C37—41no—
231C2×C4×C37—42no—
232C2×C4×C37—42no—
233C2×C4×C37—42no—
234C2×C4×C37—42no—
235C2×C4×C37—42no—
236C2×C4×C37—42no—
237C2×C4×C37—81no—
238C2×C4×C37—81no—
239C2×C4×C37—82no—
240C2×C4×C37—81no—
241C2×C4×C37—84no—
242C2×C4×C37#12011yescharacteristic
243C2×C4×C37#12011yescharacteristic
244C2×C4×C37C22×D412yes—
245C8×C37—44no—
246#18140C3711yesmaximal, characteristic, direct product factor, Sylow
247C22×C37—21no—
248C22×C37—21no—
249C22×C37—22no—
250C22×C37—22no—
251C22×C37—22no—
252C22×C37—22no—
253C22×C37—42no—
254C22×C37—42no—
255C22×C37—42no—
256C22×C37—41no—
257C22×C37—41no—
258C22×C37—42no—
259C22×C37—42no—
260C22×C37—42no—
261C22×C37—81no—
262C22×C37—84no—
263C22×C37C2×#12011yescharacteristic
264C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C37#57511yescharacteristic
266C4×C37—22no—
267C4×C37—21no—
268C4×C37—41no—
269C4×C37—42no—
270C4×C37—41no—
271C4×C37—42no—
272C4×C37—41no—
273C4×C37—42no—
274C4×C37—82no—
275C4×C37—82no—
276C4×C37#45812yes—
277#3459C2×C3711yescharacteristic
278#3092C2×C3712yescomplemented
279#2486C2×C3711yescharacteristic, complemented
280#1609C2×C3714yescomplemented
281#1608C2×C3712yescomplemented
282#1607C2×C3712yescomplemented
283#1513C2×C3712yescomplemented
284#1465C2×C3711yescharacteristic, complemented
285C2×C37—21no—
286C2×C37—21no—
287C2×C37—42no—
288C2×C37—42no—
289C2×C37—82no—
290C2×C37#300611yescharacteristic
291C2×C37#302211yescharacteristic
292C2×C37C2×#45811yescharacteristic
293#543C22×C3711yescharacteristic
294#539—22no—
295#539C22×C3712yes—
296#538—22no—
297#538—21no—
298#534—21no—
299#534C22×C3711yescharacteristic
300#529—22no—
301#529—21no—
302#525—22no—
303#525C22×C3712yes—
304C2×#123—21no—
305C2×C4×D4—21no—
306C2×C4×D4C22×C3711yescharacteristic
307C22×C4⋊C4C22×C3711yescharacteristic, complemented
308#484—22no—
309#483C22×C3714yescomplemented
310#481—22no—
311#480C22×C3714yes—
312#428—22no—
313#417—22no—
314#407C22×C3712yescomplemented
315#398C22×C3712yescomplemented
316#397C22×C3712yes—
317#389—21no—
318#381—21no—
319#380—21no—
320#380C22×C3711yescharacteristic, complemented
321#328C22×C3714yescomplemented
322#327C22×C3714yes—
323#324C22×C3712yescomplemented
324#323C22×C3712yes—
325C37#1814011yescharacteristic, direct product factor, Sylow
326C2×#41D4×C3712yes—
327C22×D4—21no—
328C22×D4D4×C3711yescharacteristic
329C23×C4—21no—
330C23×C4—21no—
331C23×C4D4×C3711yescharacteristic
332C23×C4C23×C3711yescharacteristic
333C2×M16—22no—
334#126—22no—
335#126—22no—
336#126—22no—
337#125—21no—
338#125—22no—
339#124—22no—
340#124—21no—
341#123—22no—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—44no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8D4×C3712yes—
353C22⋊Q8C23×C3712yes—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121D4×C3712yes—
359#121C23×C3712yes—
360#120—22no—
361#120—22no—
362#120—22no—
363C4×Q8—21no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372#117—22no—
373#117D4×C3712yes—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4C23×C3712yes—
381C2×C4⋊C4C23×C3711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4D4×C3711yescharacteristic
387C2×C22⋊C4C23×C3711yescharacteristic
388C2×C42—21no—
389C2×C42D4×C3711yescharacteristic
390#107—24no—
391#106—24no—
392#103—24no—
393#102—24no—
394C22⋊C8—22no—
395C22⋊C8C23×C3714yes—
396#95—22no—
397#95—22no—
398#95C23×C3712yes—
399C24—21no—
400C24C2×D4×C3711yescharacteristic
401C24C2×D4×C3711yescharacteristic
402#41—44no—
403C2×Q8—21no—
404C2×Q8C2×D4×C3712yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×D4×C3712yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4C2×D4×C3711yescharacteristic
432C22×C4C2×D4×C3711yescharacteristic
433C22×C4C2×D4×C3712yes—
434C22×C4C2×D4×C3711yescharacteristic
435C22×C4C2×D4×C3711yescharacteristic
436C22×C4C2×D4×C3712yes—
437C22×C4C24×C3711yescharacteristic
438M16—44no—
439C2×C8—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4C2×D4×C3712yes—
455C4⋊C4C2×D4×C3712yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23#120×C3712yes—
487C23AGL(1,ℤ/8)×C3712yes—
488C23C22×D4×C3711yescharacteristic
489Q8—42no—
490Q8—81no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—82no—
497D4—82no—
498D4—81no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—84no—
527C2×C4#120×C3711yescharacteristic
528C2×C4#120×C3711yescharacteristic
529C2×C4C22×D4×C3712yes—
530C8—44no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—84no—
547C22C2×#120×C3711yescharacteristic
548C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
549C22C37×#57511yescharacteristic
550C4—22no—
551C4—21no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4C37×#45812yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—42no—
565C2—82no—
566C2C37×#300611yescharacteristic
567C2C37×#302211yescharacteristic
568C2C2×C37×#45811yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3092C2semidirect
3. C37×#2486C2semidirect
4. C37×#1609C2semidirect
5. C37×#1608C2semidirect
6. C37×#1607C2semidirect
7. C37×#1513C2semidirect
8. C37×#1465C2semidirect
23. C22×C4⋊C4×C37C22semidirect
25. C37×#483C22semidirect
30. C37×#407C22semidirect
31. C37×#398C22semidirect
36. C37×#380C22semidirect
37. C37×#328C22semidirect
39. C37×#324C22semidirect
246. #18140C37direct (N × H)
278. #3092C2×C37semidirect
279. #2486C2×C37semidirect
280. #1609C2×C37semidirect
281. #1608C2×C37semidirect
282. #1607C2×C37semidirect
283. #1513C2×C37semidirect
284. #1465C2×C37semidirect
307. C22×C4⋊C4C22×C37semidirect
309. #483C22×C37semidirect
314. #407C22×C37semidirect
315. #398C22×C37semidirect
320. #380C22×C37semidirect
321. #328C22×C37semidirect
323. #324C22×C37semidirect
325. C37#18140direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#34592
C37×#30922
C37×#24862
C37×#16092
C37×#16082
C37×#16072
C37×#15132
C37×#14652
#1814037