Finite Group Databasebeta

← C37×#18123 #22639376

Subgroups — C37×#18123 #22639376

Order 9472.

1208 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3449C211yesmaximal, characteristic
2C37×#3103C212yesmaximal, complemented
3C37×#2396C211yesmaximal, characteristic, complemented
4C37×#1616C212yesmaximal, complemented
5C37×#1613C212yesmaximal, complemented
6C37×#1604C212yesmaximal, complemented
7C37×#1601C212yesmaximal, complemented
8C37×#1502C212yesmaximal, complemented
9C37×#1458C211yesmaximal, characteristic, complemented
10C22×#41×C37—21no—
11C37×#537—21no—
12C37×#537C2212yes—
13C37×#536—22no—
14C37×#535—22no—
15C37×#534—21no—
16C37×#534C2212yes—
17C37×#532—21no—
18C37×#532C2212yes—
19C37×#525—24no—
20C2×C4×Q8×C37C2211yescharacteristic, complemented
21C2×#117×C37C2211yescharacteristic
22C37×#481C2214yescomplemented
23C37×#475—22no—
24C37×#460—22no—
25C37×#448C2214yescomplemented
26C4×SD16×C37—24no—
27C37×#431C2212yescomplemented
28C37×#400C2211yescharacteristic, complemented
29C37×#399C2211yescharacteristic
30C37×#398—21no—
31C37×#396—21no—
32C37×#395C2211yescharacteristic, complemented
33C37×#393C2211yescharacteristic
34C4×C4⋊C4×C37—21no—
35C4×C4⋊C4×C37C2211yescharacteristic, complemented
36C4×C22⋊C4×C37—21no—
37C37×#327C2214yescomplemented
38C37×#327C2214yescomplemented
39C37×#324C2212yescomplemented
40C37×#323C2212yes—
41C2×#41×C37—22no—
42C2×#41×C37—22no—
43C2×#41×C37—22no—
44C2×#41×C37—22no—
45C2×#41×C37—44no—
46C2×#41×C37—44no—
47C22×Q8×C37—21no—
48C22×Q8×C37C2311yescharacteristic
49C22×D4×C37—21no—
50C22×D4×C37—22no—
51C23×C4×C37—22no—
52C23×C4×C37D411yescharacteristic
53C2×SD16×C37—24no—
54C4⋊Q8×C37—21no—
55C4⋊Q8×C37D412yes—
56#127×C37—21no—
57#127×C37D412yes—
58#124×C37—22no—
59#124×C37—22no—
60#123×C37—24no—
61#123×C37—24no—
62C22⋊Q8×C37—24no—
63C22⋊Q8×C37—24no—
64#121×C37—24no—
65#121×C37—24no—
66#121×C37—22no—
67#121×C37—22no—
68#121×C37—22no—
69#121×C37C2314yes—
70#120×C37—24no—
71#120×C37—24no—
72C4×Q8×C37—22no—
73C4×Q8×C37—22no—
74C4×D4×C37—22no—
75C4×D4×C37—22no—
76C4×D4×C37—22no—
77C4×D4×C37D414yes—
78#117×C37—22no—
79#117×C37—22no—
80C2×C4⋊C4×C37—21no—
81C2×C4⋊C4×C37—21no—
82C2×C4⋊C4×C37—21no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37D411yescharacteristic
85C2×C4⋊C4×C37C2311yescharacteristic
86C2×C4⋊C4×C37C2311yescharacteristic
87C2×C22⋊C4×C37—21no—
88C2×C22⋊C4×C37—21no—
89C2×C22⋊C4×C37D411yescharacteristic
90C2×C22⋊C4×C37C2311yescharacteristic
91C2×C42×C37—21no—
92C2×C42×C37—21no—
93C2×C42×C37D411yescharacteristic
94C2×C42×C37C2311yescharacteristic
95#106×C37—24no—
96C4⋊C8×C37—22no—
97#103×C37—24no—
98#102×C37—24no—
99C22⋊C8×C37C2314yes—
100C4×C8×C37—22no—
101#95×C37—21no—
102#95×C37—21no—
103#95×C37—21no—
104#95×C37—21no—
105#95×C37C2311yescharacteristic
106#95×C37C2311yescharacteristic
107C24×C37—22no—
108C24×C37C2×D411yescharacteristic
109#41×C37—44no—
110#41×C37—48no—
111#41×C37—48no—
112#41×C37—44no—
113#41×C37—44no—
114#41×C37—44no—
115C2×Q8×C37—22no—
116C2×Q8×C37—22no—
117C2×Q8×C37—22no—
118C2×Q8×C37—42no—
119C2×Q8×C37—44no—
120C2×D4×C37—24no—
121C2×D4×C37—22no—
122C2×D4×C37—24no—
123C2×D4×C37—22no—
124C2×D4×C37—22no—
125C2×D4×C37—22no—
126C2×D4×C37—22no—
127C2×D4×C37—44no—
128C2×D4×C37—44no—
129C2×D4×C37—44no—
130C2×D4×C37C2×D414yes—
131C22×C4×C37—21no—
132C22×C4×C37—21no—
133C22×C4×C37—22no—
134C22×C4×C37—24no—
135C22×C4×C37—22no—
136C22×C4×C37—21no—
137C22×C4×C37—21no—
138C22×C4×C37—24no—
139C22×C4×C37—21no—
140C22×C4×C37—21no—
141C22×C4×C37—48no—
142C22×C4×C37SD1614yescomplemented
143C22×C4×C37C2×D411yescharacteristic
144C22×C4×C37C2×D411yescharacteristic
145C22×C4×C37C2×D411yescharacteristic
146C22×C4×C37C2×D411yescharacteristic
147C22×C4×C37C2×D411yescharacteristic
148C22×C4×C37C2411yescharacteristic
149SD16×C37—48no—
150C2×C8×C37—24no—
151C4⋊C4×C37—22no—
152C4⋊C4×C37—22no—
153C4⋊C4×C37—22no—
154C4⋊C4×C37—22no—
155C4⋊C4×C37—22no—
156C4⋊C4×C37—22no—
157C4⋊C4×C37—42no—
158C4⋊C4×C37—42no—
159C4⋊C4×C37—42no—
160C4⋊C4×C37—42no—
161C4⋊C4×C37C2×D414yes—
162C22⋊C4×C37—24no—
163C22⋊C4×C37—24no—
164C22⋊C4×C37—22no—
165C22⋊C4×C37—24no—
166C22⋊C4×C37—22no—
167C22⋊C4×C37—24no—
168C22⋊C4×C37—42no—
169C22⋊C4×C37—42no—
170C42×C37—21no—
171C42×C37—21no—
172C42×C37—41no—
173C42×C37—41no—
174C42×C37—42no—
175C42×C37C2×D412yes—
176C42×C37C2×D412yes—
177C23×C37—21no—
178C23×C37—21no—
179C23×C37—24no—
180C23×C37—41no—
181C23×C37—41no—
182C23×C37—44no—
183C23×C37—82no—
184C23×C37—82no—
185C23×C37C2×SD1612yes—
186C23×C37C22×D411yescharacteristic
187Q8×C37—24no—
188Q8×C37—42no—
189Q8×C37—44no—
190Q8×C37—44no—
191D4×C37—44no—
192D4×C37—44no—
193D4×C37—48no—
194D4×C37—44no—
195D4×C37—44no—
196D4×C37—44no—
197D4×C37—82no—
198D4×C37—82no—
199D4×C37—82no—
200C2×C4×C37—21no—
201C2×C4×C37—21no—
202C2×C4×C37—24no—
203C2×C4×C37—24no—
204C2×C4×C37—22no—
205C2×C4×C37—22no—
206C2×C4×C37—21no—
207C2×C4×C37—21no—
208C2×C4×C37—24no—
209C2×C4×C37—41no—
210C2×C4×C37—41no—
211C2×C4×C37—48no—
212C2×C4×C37—41no—
213C2×C4×C37—44no—
214C2×C4×C37—41no—
215C2×C4×C37—41no—
216C2×C4×C37—41no—
217C2×C4×C37—48no—
218C2×C4×C37—41no—
219C2×C4×C37—41no—
220C2×C4×C37—41no—
221C2×C4×C37—41no—
222C2×C4×C37—41no—
223C2×C4×C37—41no—
224C2×C4×C37—41no—
225C2×C4×C37—41no—
226C2×C4×C37#12012yes—
227C2×C4×C37#12011yescharacteristic
228C2×C4×C37#12011yescharacteristic
229C2×C4×C37C2×SD1612yes—
230C2×C4×C37C2×SD1612yes—
231C2×C4×C37AGL(1,ℤ/8)12yes—
232C2×C4×C37C22×D412yes—
233C8×C37—44no—
234#18123C3711yesmaximal, characteristic, direct product factor, Sylow
235C22×C37—21no—
236C22×C37—21no—
237C22×C37—22no—
238C22×C37—42no—
239C22×C37—41no—
240C22×C37—41no—
241C22×C37—44no—
242C22×C37—44no—
243C22×C37—44no—
244C22×C37—81no—
245C22×C37—88no—
246C22×C37C2×#12011yescharacteristic
247C22×C37C22×SD1611yescharacteristic
248C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
249C4×C37—22no—
250C4×C37—22no—
251C4×C37—22no—
252C4×C37—42no—
253C4×C37—44no—
254C4×C37—41no—
255C4×C37—42no—
256C4×C37—41no—
257C4×C37—81no—
258C4×C37—81no—
259C4×C37#45012yes—
260C4×C37#45012yes—
261#3449C2×C3711yescharacteristic
262#3103C2×C3712yescomplemented
263#2396C2×C3711yescharacteristic, complemented
264#1616C2×C3712yescomplemented
265#1613C2×C3712yescomplemented
266#1604C2×C3712yescomplemented
267#1601C2×C3712yescomplemented
268#1502C2×C3712yescomplemented
269#1458C2×C3711yescharacteristic, complemented
270C2×C37—21no—
271C2×C37—21no—
272C2×C37—42no—
273C2×C37—84no—
274C2×C37C2×#45011yescharacteristic
275C2×C37#301811yescharacteristic
276C2×C37#302511yescharacteristic
277C22×#41—21no—
278#537—21no—
279#537C22×C3712yes—
280#536—22no—
281#535—22no—
282#534—21no—
283#534C22×C3712yes—
284#532—21no—
285#532C22×C3712yes—
286#525—24no—
287C2×C4×Q8C22×C3711yescharacteristic, complemented
288C2×#117C22×C3711yescharacteristic
289#481C22×C3714yescomplemented
290#475—22no—
291#460—22no—
292#448C22×C3714yescomplemented
293C4×SD16—24no—
294#431C22×C3712yescomplemented
295#400C22×C3711yescharacteristic, complemented
296#399C22×C3711yescharacteristic
297#398—21no—
298#396—21no—
299#395C22×C3711yescharacteristic, complemented
300#393C22×C3711yescharacteristic
301C4×C4⋊C4—21no—
302C4×C4⋊C4C22×C3711yescharacteristic, complemented
303C4×C22⋊C4—21no—
304#327C22×C3714yescomplemented
305#327C22×C3714yescomplemented
306#324C22×C3712yescomplemented
307#323C22×C3712yes—
308C37#1812311yescharacteristic, direct product factor, Sylow
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—44no—
314C2×#41—44no—
315C22×Q8—21no—
316C22×Q8C23×C3711yescharacteristic
317C22×D4—21no—
318C22×D4—22no—
319C23×C4—22no—
320C23×C4D4×C3711yescharacteristic
321C2×SD16—24no—
322C4⋊Q8—21no—
323C4⋊Q8D4×C3712yes—
324#127—21no—
325#127D4×C3712yes—
326#124—22no—
327#124—22no—
328#123—24no—
329#123—24no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332#121—24no—
333#121—24no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121C23×C3714yes—
338#120—24no—
339#120—24no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4D4×C3714yes—
346#117—22no—
347#117—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4D4×C3711yescharacteristic
353C2×C4⋊C4C23×C3711yescharacteristic
354C2×C4⋊C4C23×C3711yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4D4×C3711yescharacteristic
358C2×C22⋊C4C23×C3711yescharacteristic
359C2×C42—21no—
360C2×C42—21no—
361C2×C42D4×C3711yescharacteristic
362C2×C42C23×C3711yescharacteristic
363#106—24no—
364C4⋊C8—22no—
365#103—24no—
366#102—24no—
367C22⋊C8C23×C3714yes—
368C4×C8—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95C23×C3711yescharacteristic
374#95C23×C3711yescharacteristic
375C24—22no—
376C24C2×D4×C3711yescharacteristic
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—44no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4C2×D4×C3714yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—48no—
410C22×C4SD16×C3714yescomplemented
411C22×C4C2×D4×C3711yescharacteristic
412C22×C4C2×D4×C3711yescharacteristic
413C22×C4C2×D4×C3711yescharacteristic
414C22×C4C2×D4×C3711yescharacteristic
415C22×C4C2×D4×C3711yescharacteristic
416C22×C4C24×C3711yescharacteristic
417SD16—48no—
418C2×C8—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4C2×D4×C3714yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42C2×D4×C3712yes—
444C42C2×D4×C3712yes—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—41no—
449C23—41no—
450C23—44no—
451C23—82no—
452C23—82no—
453C23C2×SD16×C3712yes—
454C23C22×D4×C3711yescharacteristic
455Q8—24no—
456Q8—44no—
457Q8—42no—
458Q8—44no—
459D4—44no—
460D4—44no—
461D4—48no—
462D4—44no—
463D4—44no—
464D4—44no—
465D4—82no—
466D4—82no—
467D4—82no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—48no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—48no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4#120×C3712yes—
495C2×C4#120×C3711yescharacteristic
496C2×C4#120×C3711yescharacteristic
497C2×C4C2×SD16×C3712yes—
498C2×C4C2×SD16×C3712yes—
499C2×C4AGL(1,ℤ/8)×C3712yes—
500C2×C4C22×D4×C3712yes—
501C8—44no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—88no—
513C22C2×#120×C3711yescharacteristic
514C22C22×SD16×C3711yescharacteristic
515C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—44no—
521C4—41no—
522C4—42no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4C37×#45012yes—
527C4C37×#45012yes—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—84no—
532C2C2×C37×#45011yescharacteristic
533C2C37×#301811yescharacteristic
534C2C37×#302511yescharacteristic
subgroupG/H
centerC2×C4×C37#120
commutator, Frattini, cosocleC22×C4C24×C37
socleC22×C37C2×#120
prime radicalC37×#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3103C2semidirect
3. C37×#2396C2semidirect
4. C37×#1616C2semidirect
5. C37×#1613C2semidirect
6. C37×#1604C2semidirect
7. C37×#1601C2semidirect
8. C37×#1502C2semidirect
9. C37×#1458C2semidirect
20. C2×C4×Q8×C37C22semidirect
22. C37×#481C22semidirect
25. C37×#448C22semidirect
27. C37×#431C22semidirect
28. C37×#400C22semidirect
32. C37×#395C22semidirect
35. C4×C4⋊C4×C37C22semidirect
37. C37×#327C22semidirect
38. C37×#327C22semidirect
39. C37×#324C22semidirect
142. C22×C4×C37SD16semidirect
234. #18123C37direct (N × H)
262. #3103C2×C37semidirect
263. #2396C2×C37semidirect
264. #1616C2×C37semidirect
265. #1613C2×C37semidirect
266. #1604C2×C37semidirect
267. #1601C2×C37semidirect
268. #1502C2×C37semidirect
269. #1458C2×C37semidirect
287. C2×C4×Q8C22×C37semidirect
289. #481C22×C37semidirect
292. #448C22×C37semidirect
294. #431C22×C37semidirect
295. #400C22×C37semidirect
299. #395C22×C37semidirect
302. C4×C4⋊C4C22×C37semidirect
304. #327C22×C37semidirect
305. #327C22×C37semidirect
306. #324C22×C37semidirect
308. C37#18123direct (N × H)
410. C22×C4SD16×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34492
C37×#31032
C37×#23962
C37×#16162
C37×#16132
C37×#16042
C37×#16012
C37×#15022
C37×#14582
#1812337