Finite Group Databasebeta

← C37×#18356 #22639609

Subgroups — C37×#18356 #22639609

Order 9472.

1054 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3491C211yesmaximal, characteristic, complemented
2C37×#3217C212yesmaximal, complemented
3C37×#2619C211yesmaximal, characteristic, complemented
4C37×#1597C214yesmaximal, complemented
5C37×#1596C212yesmaximal, complemented
6C37×#1595C212yesmaximal, complemented
7C37×#1512C212yesmaximal, complemented
8C37×#1464C211yesmaximal, characteristic, complemented
9C37×#552—22no—
10C37×#552—22no—
11C37×#551—22no—
12C37×#551—22no—
13C37×#543C2211yescharacteristic
14C37×#539—21no—
15C37×#539C2212yes—
16C37×#534C2211yescharacteristic
17C2×#127×C37C2211yescharacteristic, complemented
18C37×#529—21no—
19C37×#529C2212yes—
20C2×#123×C37—21no—
21C2×#121×C37—21no—
22C37×#518—21no—
23C2×#117×C37C2211yescharacteristic
24C37×#496—22no—
25C37×#490—22no—
26C37×#460—22no—
27C37×#459—22no—
28C37×#450C2214yescomplemented
29C37×#447C2214yes—
30C37×#432C2212yescomplemented
31C37×#394—21no—
32C37×#394C2212yescomplemented
33C37×#392—21no—
34C37×#392C2212yes—
35C37×#390—21no—
36C37×#382C2211yescharacteristic, complemented
37C37×#328C2214yescomplemented
38C37×#327C2214yes—
39C37×#324C2212yescomplemented
40C37×#323C2212yes—
41C22×D4×C37—21no—
42C22×D4×C37—22no—
43C22×D4×C37—21no—
44C22×D4×C37—21no—
45C22×D4×C37C2312yes—
46C23×C4×C37D411yescharacteristic
47C2×SD16×C37—24no—
48C2×D8×C37—24no—
49#127×C37—22no—
50#127×C37—22no—
51#126×C37—22no—
52#126×C37—22no—
53#126×C37—22no—
54#126×C37—22no—
55#125×C37—21no—
56#125×C37—22no—
57#125×C37—22no—
58#125×C37D412yescomplemented
59#124×C37—21no—
60#124×C37D412yescomplemented
61#123×C37—22no—
62#123×C37—22no—
63#123×C37—22no—
64#123×C37—22no—
65#123×C37—22no—
66#123×C37—22no—
67#123×C37—22no—
68C22⋊Q8×C37—22no—
69C22⋊Q8×C37—22no—
70C22⋊Q8×C37—22no—
71C22⋊Q8×C37—21no—
72C22⋊Q8×C37—22no—
73C22⋊Q8×C37C2312yes—
74#121×C37—21no—
75#121×C37—44no—
76#121×C37C2312yes—
77#120×C37—22no—
78C4×Q8×C37—21no—
79C4×Q8×C37D412yescomplemented
80C4×D4×C37—21no—
81C4×D4×C37—22no—
82C4×D4×C37—22no—
83C4×D4×C37—22no—
84C4×D4×C37—22no—
85C4×D4×C37D412yescomplemented
86#117×C37—22no—
87#117×C37—22no—
88#117×C37—22no—
89#117×C37—22no—
90#117×C37—22no—
91C2×C4⋊C4×C37—21no—
92C2×C4⋊C4×C37—21no—
93C2×C4⋊C4×C37—21no—
94C2×C4⋊C4×C37D411yescharacteristic
95C2×C4⋊C4×C37C2311yescharacteristic
96C2×C22⋊C4×C37—21no—
97C2×C22⋊C4×C37—21no—
98C2×C22⋊C4×C37—22no—
99C2×C22⋊C4×C37—21no—
100C2×C22⋊C4×C37—22no—
101C2×C22⋊C4×C37—22no—
102C2×C22⋊C4×C37—22no—
103C2×C22⋊C4×C37D411yescharacteristic
104C2×C22⋊C4×C37C2311yescharacteristic
105C2×C42×C37D411yescharacteristic
106C2×C42×C37C2311yescharacteristic
107C4⋊C8×C37—22no—
108#102×C37—24no—
109#102×C37—24no—
110C22⋊C8×C37C2314yes—
111#97×C37—22no—
112#95×C37—21no—
113#95×C37C2312yes—
114C24×C37—21no—
115C24×C37—22no—
116C24×C37C2×D411yescharacteristic
117C2×Q8×C37—21no—
118C2×Q8×C37C2×D412yes—
119C2×D4×C37—22no—
120C2×D4×C37—22no—
121C2×D4×C37—21no—
122C2×D4×C37—24no—
123C2×D4×C37—42no—
124C2×D4×C37—42no—
125C2×D4×C37—42no—
126C2×D4×C37—42no—
127C2×D4×C37—44no—
128C2×D4×C37—41no—
129C2×D4×C37—41no—
130C2×D4×C37—42no—
131C2×D4×C37—82no—
132C2×D4×C37—82no—
133C2×D4×C37—82no—
134C2×D4×C37—82no—
135C2×D4×C37C2×D412yes—
136C22×C4×C37—21no—
137C22×C4×C37—22no—
138C22×C4×C37—21no—
139C22×C4×C37—21no—
140C22×C4×C37—22no—
141C22×C4×C37—21no—
142C22×C4×C37—21no—
143C22×C4×C37—22no—
144C22×C4×C37—22no—
145C22×C4×C37—41no—
146C22×C4×C37C2×D411yescharacteristic
147C22×C4×C37C2×D411yescharacteristic
148C22×C4×C37C2×D411yescharacteristic
149C22×C4×C37C2×D411yescharacteristic
150C22×C4×C37C2×D411yescharacteristic
151C22×C4×C37C2411yescharacteristic
152SD16×C37—84no—
153D8×C37—84no—
154C2×C8×C37—24no—
155C4⋊C4×C37—22no—
156C4⋊C4×C37—22no—
157C4⋊C4×C37—22no—
158C4⋊C4×C37—22no—
159C4⋊C4×C37—21no—
160C4⋊C4×C37—21no—
161C4⋊C4×C37—22no—
162C4⋊C4×C37—22no—
163C4⋊C4×C37—22no—
164C4⋊C4×C37—22no—
165C4⋊C4×C37—22no—
166C4⋊C4×C37—22no—
167C4⋊C4×C37—42no—
168C4⋊C4×C37C2×D412yes—
169C4⋊C4×C37C2×D412yes—
170C22⋊C4×C37—22no—
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—44no—
183C22⋊C4×C37—82no—
184C22⋊C4×C37—82no—
185C42×C37—22no—
186C42×C37—21no—
187C42×C37—21no—
188C42×C37—22no—
189C42×C37C2×D412yes—
190C42×C37C2×D412yes—
191C23×C37—21no—
192C23×C37—21no—
193C23×C37—22no—
194C23×C37—41no—
195C23×C37—41no—
196C23×C37—42no—
197C23×C37—42no—
198C23×C37—42no—
199C23×C37—42no—
200C23×C37—42no—
201C23×C37—42no—
202C23×C37—42no—
203C23×C37—42no—
204C23×C37—81no—
205C23×C37—81no—
206C23×C37AGL(1,ℤ/8)12yes—
207C23×C37C22×D411yescharacteristic
208Q8×C37—42no—
209Q8×C37—81no—
210D4×C37—42no—
211D4×C37—44no—
212D4×C37—44no—
213D4×C37—44no—
214D4×C37—42no—
215D4×C37—42no—
216D4×C37—82no—
217D4×C37—82no—
218D4×C37—84no—
219D4×C37—81no—
220D4×C37—88no—
221C2×C4×C37—21no—
222C2×C4×C37—21no—
223C2×C4×C37—21no—
224C2×C4×C37—21no—
225C2×C4×C37—22no—
226C2×C4×C37—22no—
227C2×C4×C37—22no—
228C2×C4×C37—21no—
229C2×C4×C37—21no—
230C2×C4×C37—21no—
231C2×C4×C37—41no—
232C2×C4×C37—41no—
233C2×C4×C37—42no—
234C2×C4×C37—42no—
235C2×C4×C37—42no—
236C2×C4×C37—41no—
237C2×C4×C37—41no—
238C2×C4×C37—42no—
239C2×C4×C37—42no—
240C2×C4×C37—42no—
241C2×C4×C37—82no—
242C2×C4×C37—81no—
243C2×C4×C37—82no—
244C2×C4×C37—81no—
245C2×C4×C37#12011yescharacteristic
246C2×C4×C37#12011yescharacteristic
247C2×C4×C37#12012yes—
248C2×C4×C37C22×D412yes—
249C8×C37—44no—
250#18356C3711yesmaximal, characteristic, direct product factor, Sylow
251C22×C37—21no—
252C22×C37—21no—
253C22×C37—22no—
254C22×C37—42no—
255C22×C37—41no—
256C22×C37—41no—
257C22×C37—42no—
258C22×C37—42no—
259C22×C37—42no—
260C22×C37—81no—
261C22×C37—82no—
262C22×C37—82no—
263C22×C37—82no—
264C22×C37—82no—
265C22×C37—82no—
266C22×C37—82no—
267C22×C37—82no—
268C22×C37—162no—
269C22×C37C2×#12011yescharacteristic
270C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C37#57511yescharacteristic
272C4×C37—21no—
273C4×C37—21no—
274C4×C37—22no—
275C4×C37—22no—
276C4×C37—41no—
277C4×C37—42no—
278C4×C37—41no—
279C4×C37—42no—
280C4×C37—41no—
281C4×C37—42no—
282C4×C37—82no—
283#3491C2×C3711yescharacteristic, complemented
284#3217C2×C3712yescomplemented
285#2619C2×C3711yescharacteristic, complemented
286#1597C2×C3714yescomplemented
287#1596C2×C3712yescomplemented
288#1595C2×C3712yescomplemented
289#1512C2×C3712yescomplemented
290#1464C2×C3711yescharacteristic, complemented
291C2×C37—21no—
292C2×C37—21no—
293C2×C37—42no—
294C2×C37—82no—
295C2×C37—162no—
296C2×C37#300611yescharacteristic
297C2×C37#302111yescharacteristic
298C2×C37#302711yescharacteristic
299#552—22no—
300#552—22no—
301#551—22no—
302#551—22no—
303#543C22×C3711yescharacteristic
304#539—21no—
305#539C22×C3712yes—
306#534C22×C3711yescharacteristic
307C2×#127C22×C3711yescharacteristic, complemented
308#529—21no—
309#529C22×C3712yes—
310C2×#123—21no—
311C2×#121—21no—
312#518—21no—
313C2×#117C22×C3711yescharacteristic
314#496—22no—
315#490—22no—
316#460—22no—
317#459—22no—
318#450C22×C3714yescomplemented
319#447C22×C3714yes—
320#432C22×C3712yescomplemented
321#394—21no—
322#394C22×C3712yescomplemented
323#392—21no—
324#392C22×C3712yes—
325#390—21no—
326#382C22×C3711yescharacteristic, complemented
327#328C22×C3714yescomplemented
328#327C22×C3714yes—
329#324C22×C3712yescomplemented
330#323C22×C3712yes—
331C37#1835611yescharacteristic, direct product factor, Sylow
332C22×D4—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C23×C3712yes—
337C23×C4D4×C3711yescharacteristic
338C2×SD16—24no—
339C2×D8—24no—
340#127—22no—
341#127—22no—
342#126—22no—
343#126—22no—
344#126—22no—
345#126—22no—
346#125—21no—
347#125—22no—
348#125—22no—
349#125D4×C3712yescomplemented
350#124—21no—
351#124D4×C3712yescomplemented
352#123—22no—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8—22no—
364C22⋊Q8C23×C3712yes—
365#121—21no—
366#121—44no—
367#121C23×C3712yes—
368#120—22no—
369C4×Q8—21no—
370C4×Q8D4×C3712yescomplemented
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4D4×C3712yescomplemented
377#117—22no—
378#117—22no—
379#117—22no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4D4×C3711yescharacteristic
386C2×C4⋊C4C23×C3711yescharacteristic
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4D4×C3711yescharacteristic
395C2×C22⋊C4C23×C3711yescharacteristic
396C2×C42D4×C3711yescharacteristic
397C2×C42C23×C3711yescharacteristic
398C4⋊C8—22no—
399#102—24no—
400#102—24no—
401C22⋊C8C23×C3714yes—
402#97—22no—
403#95—21no—
404#95C23×C3712yes—
405C24—21no—
406C24—22no—
407C24C2×D4×C3711yescharacteristic
408C2×Q8—21no—
409C2×Q8C2×D4×C3712yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—24no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4C2×D4×C3712yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4C2×D4×C3711yescharacteristic
438C22×C4C2×D4×C3711yescharacteristic
439C22×C4C2×D4×C3711yescharacteristic
440C22×C4C2×D4×C3711yescharacteristic
441C22×C4C2×D4×C3711yescharacteristic
442C22×C4C24×C3711yescharacteristic
443SD16—84no—
444D8—84no—
445C2×C8—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4C2×D4×C3712yes—
460C4⋊C4C2×D4×C3712yes—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42C2×D4×C3712yes—
481C42C2×D4×C3712yes—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23AGL(1,ℤ/8)×C3712yes—
498C23C22×D4×C3711yescharacteristic
499Q8—42no—
500Q8—81no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—81no—
511D4—88no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4#120×C3711yescharacteristic
537C2×C4#120×C3711yescharacteristic
538C2×C4#120×C3712yes—
539C2×C4C22×D4×C3712yes—
540C8—44no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—162no—
559C22C2×#120×C3711yescharacteristic
560C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
561C22C37×#57511yescharacteristic
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—82no—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—82no—
577C2—162no—
578C2C37×#300611yescharacteristic
579C2C37×#302111yescharacteristic
580C2C37×#302711yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3491C2semidirect
2. C37×#3217C2semidirect
3. C37×#2619C2semidirect
4. C37×#1597C2semidirect
5. C37×#1596C2semidirect
6. C37×#1595C2semidirect
7. C37×#1512C2semidirect
8. C37×#1464C2semidirect
17. C2×#127×C37C22semidirect
28. C37×#450C22semidirect
30. C37×#432C22semidirect
32. C37×#394C22semidirect
36. C37×#382C22semidirect
37. C37×#328C22semidirect
39. C37×#324C22semidirect
58. #125×C37D4semidirect
60. #124×C37D4semidirect
79. C4×Q8×C37D4semidirect
85. C4×D4×C37D4semidirect
250. #18356C37direct (N × H)
283. #3491C2×C37semidirect
284. #3217C2×C37semidirect
285. #2619C2×C37semidirect
286. #1597C2×C37semidirect
287. #1596C2×C37semidirect
288. #1595C2×C37semidirect
289. #1512C2×C37semidirect
290. #1464C2×C37semidirect
307. C2×#127C22×C37semidirect
318. #450C22×C37semidirect
320. #432C22×C37semidirect
322. #394C22×C37semidirect
326. #382C22×C37semidirect
327. #328C22×C37semidirect
329. #324C22×C37semidirect
331. C37#18356direct (N × H)
349. #125D4×C37semidirect
351. #124D4×C37semidirect
370. C4×Q8D4×C37semidirect
376. C4×D4D4×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34912
C37×#32172
C37×#26192
C37×#15972
C37×#15962
C37×#15952
C37×#15122
C37×#14642
#1835637