Finite Group Databasebeta

← C37×#61722 #22682975

Subgroups — C37×#61722 #22682975

Order 9472.

2014 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3496C211yesmaximal, characteristic
2C2×C37×#556C211yesmaximal, characteristic, complemented
3C37×#3353C218yesmaximal, complemented
4C37×#3324C218yesmaximal, complemented
5C37×#3226C212yesmaximal, complemented
6C37×#3193C214yesmaximal, complemented
7C37×#3103C212yesmaximal, complemented
8C37×#3094C214yesmaximal, complemented
9C37×#2975C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C37—21no—
11C37×#556—24no—
12C37×#556C2218yescomplemented
13C37×#547C2214yes—
14C37×#546—24no—
15D42×C37C2214yes—
16C37×#537—21no—
17C37×#534—22no—
18C37×#532C2212yes—
19C2×C4⋊Q8×C37C2211yescharacteristic, complemented
20C2×#126×C37—22no—
21C2×#124×C37—21no—
22C2×C22⋊Q8×C37C2212yescomplemented
23C2×#121×C37C2214yes—
24C2×C4×Q8×C37—21no—
25C2×C4×Q8×C37C2211yescharacteristic, complemented
26C2×C4×D4×C37C2211yescharacteristic
27C2×#117×C37C2212yescomplemented
28C37×#500C2214yescomplemented
29C37×#486C2214yescomplemented
30C37×#484C2218yescomplemented
31C37×#482C22116yescomplemented
32C37×#481C2214yescomplemented
33C37×#480C2218yescomplemented
34C37×#475C2214yescomplemented
35C37×#474C2218yescomplemented
36C37×#468C22116yescomplemented
37C37×#460C2214yescomplemented
38C37×#450C2218yescomplemented
39C37×#448C2214yescomplemented
40C4×SD16×C37C2218yescomplemented
41C37×#436C2218yescomplemented
42C37×#431C2212yescomplemented
43C37×#428C2218yescomplemented
44C37×#418C2218yescomplemented
45C37×#407C2214yescomplemented
46H(5,𝔽2)×C37—48no—
47C2×#41×C37—24no—
48C2×#41×C37—22no—
49C22×Q8×C37—21no—
50C22×Q8×C37C2311yescharacteristic
51C22×D4×C37—24no—
52C22×D4×C37—24no—
53C22×D4×C37D411yescharacteristic
54C22×D4×C37C2314yes—
55C23×C4×C37C2312yes—
56C2×SD16×C37C2318yes—
57C2×M16×C37C2318yes—
58C4⋊Q8×C37—22no—
59C4⋊Q8×C37C2314yes—
60C4⋊Q8×C37C2312yes—
61#127×C37C2312yes—
62#126×C37—216no—
63#124×C37—28no—
64#124×C37—22no—
65#123×C37—28no—
66C22⋊Q8×C37—24no—
67C22⋊Q8×C37—24no—
68C22⋊Q8×C37C2318yes—
69#121×C37—24no—
70#121×C37—24no—
71#121×C37—22no—
72#121×C37—24no—
73#121×C37C2314yes—
74#121×C37C2318yes—
75#120×C37—28no—
76C4×Q8×C37—28no—
77C4×Q8×C37—22no—
78C4×Q8×C37C2314yes—
79C4×D4×C37—22no—
80C4×D4×C37D414yes—
81C4×D4×C37C2314yes—
82C4×D4×C37C2314yes—
83#117×C37—24no—
84#117×C37C2318yes—
85C2×C4⋊C4×C37—22no—
86C2×C4⋊C4×C37—21no—
87C2×C4⋊C4×C37—21no—
88C2×C4⋊C4×C37D411yescharacteristic
89C2×C4⋊C4×C37C2314yes—
90C2×C4⋊C4×C37C2312yes—
91C2×C4⋊C4×C37C2311yescharacteristic
92C2×C22⋊C4×C37—22no—
93C2×C22⋊C4×C37—22no—
94C2×C22⋊C4×C37—24no—
95C2×C22⋊C4×C37D412yes—
96C2×C42×C37—21no—
97C2×C42×C37—22no—
98C2×C42×C37C2311yescharacteristic
99#107×C37C23116yes—
100#106×C37C2318yes—
101C4⋊C8×C37C2314yes—
102#103×C37C2318yes—
103#102×C37C2318yes—
104#102×C37C23116yes—
105C22⋊C8×C37C2314yes—
106C22⋊C8×C37C2318yes—
107C4×C8×C37C2314yes—
108C24×C37—24no—
109C24×C37C2×D412yes—
110#41×C37—416no—
111#41×C37—48no—
112C2×Q8×C37—21no—
113C2×Q8×C37—24no—
114C2×Q8×C37—22no—
115C2×Q8×C37—42no—
116C2×Q8×C37—44no—
117C2×Q8×C37C2412yes—
118C2×D4×C37—28no—
119C2×D4×C37—24no—
120C2×D4×C37—24no—
121C2×D4×C37—22no—
122C2×D4×C37—22no—
123C2×D4×C37—24no—
124C2×D4×C37—48no—
125C2×D4×C37—416no—
126C2×D4×C37—416no—
127C2×D4×C37C2×D414yes—
128C2×D4×C37C2414yes—
129C22×C4×C37—21no—
130C22×C4×C37—21no—
131C22×C4×C37—22no—
132C22×C4×C37—24no—
133C22×C4×C37—24no—
134C22×C4×C37—44no—
135C22×C4×C37C2×D412yes—
136C22×C4×C37C2×D411yescharacteristic
137C22×C4×C37C2×D414yes—
138C22×C4×C37C2×D411yescharacteristic
139C22×C4×C37C2411yescharacteristic
140C22×C4×C37C2414yes—
141SD16×C37—48no—
142M16×C37—48no—
143C2×C8×C37C2418yes—
144C4⋊C4×C37—28no—
145C4⋊C4×C37—24no—
146C4⋊C4×C37—24no—
147C4⋊C4×C37—21no—
148C4⋊C4×C37—22no—
149C4⋊C4×C37—24no—
150C4⋊C4×C37—22no—
151C4⋊C4×C37C2×D414yes—
152C4⋊C4×C37C2412yes—
153C4⋊C4×C37C2414yes—
154C4⋊C4×C37C2414yes—
155C22⋊C4×C37—28no—
156C22⋊C4×C37—28no—
157C22⋊C4×C37—24no—
158C22⋊C4×C37—28no—
159C22⋊C4×C37—24no—
160C22⋊C4×C37C2×D418yes—
161C42×C37—28no—
162C42×C37—24no—
163C42×C37C2×D412yes—
164C42×C37C2412yes—
165C23×C37—24no—
166C23×C37—24no—
167C23×C37—42no—
168C23×C37—42no—
169C23×C37—48no—
170C23×C37—84no—
171C23×C37—84no—
172C23×C37C22×D414yes—
173C23×C37C22×D411yescharacteristic
174Q8×C37—24no—
175Q8×C37—42no—
176Q8×C37—48no—
177Q8×C37—44no—
178D4×C37—28no—
179D4×C37—28no—
180D4×C37—416no—
181D4×C37—48no—
182D4×C37—44no—
183D4×C37—82no—
184D4×C37—84no—
185C2×C4×C37—28no—
186C2×C4×C37—22no—
187C2×C4×C37—22no—
188C2×C4×C37—22no—
189C2×C4×C37—24no—
190C2×C4×C37—24no—
191C2×C4×C37—44no—
192C2×C4×C37—42no—
193C2×C4×C37—41no—
194C2×C4×C37—42no—
195C2×C4×C37—48no—
196C2×C4×C37—42no—
197C2×C4×C37—44no—
198C2×C4×C37—48no—
199C2×C4×C37AGL(1,ℤ/8)14yes—
200C2×C4×C37C22×D414yes—
201C2×C4×C37C22×D411yescharacteristic
202C2×C4×C37C22×D412yes—
203C2×C4×C37C22×D412yes—
204C2×C4×C37C2511yescharacteristic
205C8×C37—28no—
206#61722C3711yesmaximal, characteristic, direct product factor, Sylow
207C22×C37—21no—
208C22×C37—21no—
209C22×C37—24no—
210C22×C37—24no—
211C22×C37—24no—
212C22×C37—44no—
213C22×C37—44no—
214C22×C37—44no—
215C22×C37—84no—
216C22×C37—168no—
217C22×C37C2×AGL(1,ℤ/8)12yes—
218C22×C37C23×D411yescharacteristic
219C4×C37—22no—
220C4×C37—21no—
221C4×C37—24no—
222C4×C37—42no—
223C4×C37—42no—
224C4×C37—44no—
225C4×C37—44no—
226C4×C37C2×AGL(1,ℤ/8)14yes—
227C4×C37#58512yes—
228#3496C2×C3711yescharacteristic
229C2×#556C2×C3711yescharacteristic, complemented
230#3353C2×C3718yescomplemented
231#3324C2×C3718yescomplemented
232#3226C2×C3712yescomplemented
233#3193C2×C3714yescomplemented
234#3103C2×C3712yescomplemented
235#3094C2×C3714yescomplemented
236#2975C2×C3711yescharacteristic, complemented
237C2×C37—22no—
238C2×C37—44no—
239C2×C37—84no—
240C2×C37#348511yescharacteristic
241C2×C37C22×AGL(1,ℤ/8)11yescharacteristic
242C2×C37#358611yescharacteristic
243C2×H(5,𝔽2)—21no—
244#556—24no—
245#556C22×C3718yescomplemented
246#547C22×C3714yes—
247#546—24no—
248D42C22×C3714yes—
249#537—21no—
250#534—22no—
251#532C22×C3712yes—
252C2×C4⋊Q8C22×C3711yescharacteristic, complemented
253C2×#126—22no—
254C2×#124—21no—
255C2×C22⋊Q8C22×C3712yescomplemented
256C2×#121C22×C3714yes—
257C2×C4×Q8—21no—
258C2×C4×Q8C22×C3711yescharacteristic, complemented
259C2×C4×D4C22×C3711yescharacteristic
260C2×#117C22×C3712yescomplemented
261#500C22×C3714yescomplemented
262#486C22×C3714yescomplemented
263#484C22×C3718yescomplemented
264#482C22×C37116yescomplemented
265#481C22×C3714yescomplemented
266#480C22×C3718yescomplemented
267#475C22×C3714yescomplemented
268#474C22×C3718yescomplemented
269#468C22×C37116yescomplemented
270#460C22×C3714yescomplemented
271#450C22×C3718yescomplemented
272#448C22×C3714yescomplemented
273C4×SD16C22×C3718yescomplemented
274#436C22×C3718yescomplemented
275#431C22×C3712yescomplemented
276#428C22×C3718yescomplemented
277#418C22×C3718yescomplemented
278#407C22×C3714yescomplemented
279C37#6172211yescharacteristic, direct product factor, Sylow
280H(5,𝔽2)—48no—
281C2×#41—24no—
282C2×#41—22no—
283C22×Q8—21no—
284C22×Q8C23×C3711yescharacteristic
285C22×D4—24no—
286C22×D4—24no—
287C22×D4D4×C3711yescharacteristic
288C22×D4C23×C3714yes—
289C23×C4C23×C3712yes—
290C2×SD16C23×C3718yes—
291C2×M16C23×C3718yes—
292C4⋊Q8—22no—
293C4⋊Q8C23×C3714yes—
294C4⋊Q8C23×C3712yes—
295#127C23×C3712yes—
296#126—216no—
297#124—28no—
298#124—22no—
299#123—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8C23×C3718yes—
303#121—24no—
304#121—24no—
305#121—22no—
306#121—24no—
307#121C23×C3714yes—
308#121C23×C3718yes—
309#120—28no—
310C4×Q8—28no—
311C4×Q8—22no—
312C4×Q8C23×C3714yes—
313C4×D4—22no—
314C4×D4D4×C3714yes—
315C4×D4C23×C3714yes—
316C4×D4C23×C3714yes—
317#117—24no—
318#117C23×C3718yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4D4×C3711yescharacteristic
323C2×C4⋊C4C23×C3711yescharacteristic
324C2×C4⋊C4C23×C3712yes—
325C2×C4⋊C4C23×C3714yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4D4×C3712yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42C23×C3711yescharacteristic
333#107C23×C37116yes—
334#106C23×C3718yes—
335C4⋊C8C23×C3714yes—
336#103C23×C3718yes—
337#102C23×C3718yes—
338#102C23×C37116yes—
339C22⋊C8C23×C3714yes—
340C22⋊C8C23×C3718yes—
341C4×C8C23×C3714yes—
342C24—24no—
343C24C2×D4×C3712yes—
344#41—416no—
345#41—48no—
346C2×Q8—21no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—44no—
351C2×Q8C24×C3712yes—
352C2×D4—28no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4C2×D4×C3714yes—
362C2×D4C24×C3714yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4C2×D4×C3712yes—
370C22×C4C2×D4×C3711yescharacteristic
371C22×C4C2×D4×C3714yes—
372C22×C4C2×D4×C3711yescharacteristic
373C22×C4C24×C3711yescharacteristic
374C22×C4C24×C3714yes—
375SD16—48no—
376M16—48no—
377C2×C8C24×C3718yes—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4C2×D4×C3714yes—
386C4⋊C4C24×C3712yes—
387C4⋊C4C24×C3714yes—
388C4⋊C4C24×C3714yes—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4—24no—
394C22⋊C4C2×D4×C3718yes—
395C42—28no—
396C42—24no—
397C42C2×D4×C3712yes—
398C42C24×C3712yes—
399C23—24no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—48no—
404C23—84no—
405C23—84no—
406C23C22×D4×C3714yes—
407C23C22×D4×C3711yescharacteristic
408Q8—24no—
409Q8—42no—
410Q8—48no—
411Q8—44no—
412D4—28no—
413D4—28no—
414D4—416no—
415D4—48no—
416D4—44no—
417D4—84no—
418D4—82no—
419C2×C4—28no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4AGL(1,ℤ/8)×C3714yes—
434C2×C4C22×D4×C3714yes—
435C2×C4C22×D4×C3711yescharacteristic
436C2×C4C22×D4×C3712yes—
437C2×C4C22×D4×C3712yes—
438C2×C4C25×C3711yescharacteristic
439C8—28no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—84no—
449C22—168no—
450C22C2×AGL(1,ℤ/8)×C3712yes—
451C22C23×D4×C3711yescharacteristic
452C4—21no—
453C4—24no—
454C4—22no—
455C4—42no—
456C4—44no—
457C4—44no—
458C4—42no—
459C4C2×AGL(1,ℤ/8)×C3714yes—
460C4C37×#58512yes—
461C2—22no—
462C2—44no—
463C2—84no—
464C2C37×#348511yescharacteristic
465C2C22×AGL(1,ℤ/8)×C3711yescharacteristic
466C2C37×#358611yescharacteristic
subgroupG/H
center, socleC22×C37C23×D4
commutator, Frattini, cosocleC2×C4C25×C37
prime radicalC37×#3496C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C4×D4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C37×#556C2semidirect
3. C37×#3353C2semidirect
4. C37×#3324C2semidirect
5. C37×#3226C2semidirect
6. C37×#3193C2semidirect
7. C37×#3103C2semidirect
8. C37×#3094C2semidirect
9. C37×#2975C2semidirect
12. C37×#556C22semidirect
19. C2×C4⋊Q8×C37C22semidirect
22. C2×C22⋊Q8×C37C22semidirect
25. C2×C4×Q8×C37C22semidirect
27. C2×#117×C37C22semidirect
28. C37×#500C22semidirect
29. C37×#486C22semidirect
30. C37×#484C22semidirect
31. C37×#482C22semidirect
32. C37×#481C22semidirect
33. C37×#480C22semidirect
34. C37×#475C22semidirect
35. C37×#474C22semidirect
36. C37×#468C22semidirect
37. C37×#460C22semidirect
38. C37×#450C22semidirect
39. C37×#448C22semidirect
40. C4×SD16×C37C22semidirect
41. C37×#436C22semidirect
42. C37×#431C22semidirect
43. C37×#428C22semidirect
44. C37×#418C22semidirect
45. C37×#407C22semidirect
206. #61722C37direct (N × H)
229. C2×#556C2×C37semidirect
230. #3353C2×C37semidirect
231. #3324C2×C37semidirect
232. #3226C2×C37semidirect
233. #3193C2×C37semidirect
234. #3103C2×C37semidirect
235. #3094C2×C37semidirect
236. #2975C2×C37semidirect
245. #556C22×C37semidirect
252. C2×C4⋊Q8C22×C37semidirect
255. C2×C22⋊Q8C22×C37semidirect
258. C2×C4×Q8C22×C37semidirect
260. C2×#117C22×C37semidirect
261. #500C22×C37semidirect
262. #486C22×C37semidirect
263. #484C22×C37semidirect
264. #482C22×C37semidirect
265. #481C22×C37semidirect
266. #480C22×C37semidirect
267. #475C22×C37semidirect
268. #474C22×C37semidirect
269. #468C22×C37semidirect
270. #460C22×C37semidirect
271. #450C22×C37semidirect
272. #448C22×C37semidirect
273. C4×SD16C22×C37semidirect
274. #436C22×C37semidirect
275. #431C22×C37semidirect
276. #428C22×C37semidirect
277. #418C22×C37semidirect
278. #407C22×C37semidirect
279. C37#61722direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#34962
C2×C37×#5562
C37×#33532
C37×#33242
C37×#32262
C37×#31932
C37×#31032
C37×#30942
C37×#29752
#6172237