Finite Group Databasebeta

← C37×#18332 #22639585

Subgroups — C37×#18332 #22639585

Order 9472.

1230 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×#3449C211yesmaximal, characteristic, complemented
2C37×#3213C212yesmaximal, complemented
3C37×#2610C211yesmaximal, characteristic, complemented
4C37×#1604C212yesmaximal, complemented
5C37×#1601C212yesmaximal, complemented
6C37×#1598C212yesmaximal, complemented
7C37×#1595C212yesmaximal, complemented
8C37×#1505C212yesmaximal, complemented
9C37×#1464C211yesmaximal, 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—
19C2×#124×C37C2211yescharacteristic, complemented
20C37×#525—24no—
21C2×#117×C37C2211yescharacteristic
22C37×#496—22no—
23C37×#489—22no—
24C37×#463—24no—
25C37×#448C2214yescomplemented
26C37×#447C2214yescomplemented
27C37×#432C2212yescomplemented
28C37×#395C2211yescharacteristic, complemented
29C37×#394C2211yescharacteristic, complemented
30C37×#393C2211yescharacteristic
31C37×#392C2211yescharacteristic
32C37×#390—21no—
33C37×#388—21no—
34C37×#388—21no—
35C37×#386—21no—
36C37×#376C2211yescharacteristic, complemented
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—
51C22×D4×C37—21no—
52C22×D4×C37C2311yescharacteristic
53C23×C4×C37—22no—
54C23×C4×C37D411yescharacteristic
55C2×SD16×C37—24no—
56C2×D8×C37—24no—
57C4⋊Q8×C37—21no—
58C4⋊Q8×C37D412yescomplemented
59#127×C37—21no—
60#127×C37D412yescomplemented
61#124×C37—22no—
62#124×C37—22no—
63#124×C37—22no—
64#124×C37—22no—
65#123×C37—24no—
66#123×C37—24no—
67C22⋊Q8×C37—24no—
68C22⋊Q8×C37—24no—
69#121×C37—24no—
70#121×C37—24no—
71#121×C37—22no—
72#121×C37—22no—
73#121×C37—22no—
74#121×C37C2314yes—
75#120×C37—24no—
76#120×C37—24no—
77C4×D4×C37—22no—
78C4×D4×C37—22no—
79C4×D4×C37—22no—
80C4×D4×C37D414yescomplemented
81#117×C37—22no—
82#117×C37—22no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37D411yescharacteristic
85C2×C4⋊C4×C37C2311yescharacteristic
86C2×C22⋊C4×C37—21no—
87C2×C22⋊C4×C37—21no—
88C2×C22⋊C4×C37—21no—
89C2×C22⋊C4×C37—21no—
90C2×C22⋊C4×C37—21no—
91C2×C22⋊C4×C37—21no—
92C2×C22⋊C4×C37D411yescharacteristic
93C2×C22⋊C4×C37C2311yescharacteristic
94C2×C42×C37—21no—
95C2×C42×C37D411yescharacteristic
96C2×C42×C37C2311yescharacteristic
97C4⋊C8×C37—22no—
98#103×C37—24no—
99#102×C37—24no—
100C22⋊C8×C37C2314yes—
101#97×C37—22no—
102#95×C37—21no—
103#95×C37—21no—
104#95×C37—21no—
105#95×C37C2311yescharacteristic
106#95×C37C2311yescharacteristic
107C24×C37—22no—
108C24×C37—21no—
109C24×C37C2×D411yescharacteristic
110#41×C37—48no—
111#41×C37—48no—
112#41×C37—44no—
113#41×C37—44no—
114#41×C37—44no—
115#41×C37—44no—
116C2×Q8×C37—22no—
117C2×Q8×C37—22no—
118C2×Q8×C37—22no—
119C2×Q8×C37—44no—
120C2×Q8×C37—81no—
121C2×D4×C37—24no—
122C2×D4×C37—22no—
123C2×D4×C37—24no—
124C2×D4×C37—22no—
125C2×D4×C37—22no—
126C2×D4×C37—22no—
127C2×D4×C37—22no—
128C2×D4×C37—22no—
129C2×D4×C37—44no—
130C2×D4×C37—42no—
131C2×D4×C37—44no—
132C2×D4×C37—44no—
133C2×D4×C37—42no—
134C2×D4×C37—81no—
135C2×D4×C37—81no—
136C2×D4×C37C2×D414yes—
137C22×C4×C37—22no—
138C22×C4×C37—21no—
139C22×C4×C37—21no—
140C22×C4×C37—22no—
141C22×C4×C37—24no—
142C22×C4×C37—22no—
143C22×C4×C37—21no—
144C22×C4×C37—24no—
145C22×C4×C37—21no—
146C22×C4×C37—21no—
147C22×C4×C37—48no—
148C22×C4×C37C2×D411yescharacteristic
149C22×C4×C37C2×D411yescharacteristic
150C22×C4×C37C2×D411yescharacteristic
151C22×C4×C37C2×D411yescharacteristic
152C22×C4×C37C2×D411yescharacteristic
153C22×C4×C37C2411yescharacteristic
154SD16×C37—84no—
155D8×C37—84no—
156C2×C8×C37—24no—
157C4⋊C4×C37—22no—
158C4⋊C4×C37—22no—
159C4⋊C4×C37—22no—
160C4⋊C4×C37—22no—
161C4⋊C4×C37—22no—
162C4⋊C4×C37—81no—
163C4⋊C4×C37C2×D414yes—
164C22⋊C4×C37—24no—
165C22⋊C4×C37—24no—
166C22⋊C4×C37—22no—
167C22⋊C4×C37—24no—
168C22⋊C4×C37—22no—
169C22⋊C4×C37—24no—
170C22⋊C4×C37—42no—
171C22⋊C4×C37—41no—
172C22⋊C4×C37—41no—
173C22⋊C4×C37—42no—
174C22⋊C4×C37—42no—
175C22⋊C4×C37—81no—
176C22⋊C4×C37—81no—
177C42×C37—21no—
178C42×C37—21no—
179C42×C37—42no—
180C42×C37C2×D412yes—
181C42×C37C2×D412yes—
182C23×C37—21no—
183C23×C37—21no—
184C23×C37—24no—
185C23×C37—41no—
186C23×C37—41no—
187C23×C37—44no—
188C23×C37—41no—
189C23×C37—41no—
190C23×C37—41no—
191C23×C37—41no—
192C23×C37—41no—
193C23×C37—41no—
194C23×C37—41no—
195C23×C37—82no—
196C23×C37—82no—
197C23×C37AGL(1,ℤ/8)12yes—
198C23×C37C22×D411yescharacteristic
199Q8×C37—44no—
200Q8×C37—44no—
201Q8×C37—44no—
202D4×C37—44no—
203D4×C37—44no—
204D4×C37—44no—
205D4×C37—44no—
206D4×C37—48no—
207D4×C37—44no—
208D4×C37—44no—
209D4×C37—82no—
210D4×C37—82no—
211D4×C37—82no—
212D4×C37—84no—
213C2×C4×C37—24no—
214C2×C4×C37—24no—
215C2×C4×C37—21no—
216C2×C4×C37—21no—
217C2×C4×C37—21no—
218C2×C4×C37—21no—
219C2×C4×C37—22no—
220C2×C4×C37—21no—
221C2×C4×C37—21no—
222C2×C4×C37—24no—
223C2×C4×C37—41no—
224C2×C4×C37—41no—
225C2×C4×C37—48no—
226C2×C4×C37—48no—
227C2×C4×C37—41no—
228C2×C4×C37—44no—
229C2×C4×C37—41no—
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#12011yescharacteristic
239C2×C4×C37#12011yescharacteristic
240C2×C4×C37#12012yes—
241C2×C4×C37AGL(1,ℤ/8)12yes—
242C2×C4×C37C22×D412yes—
243C8×C37—44no—
244#18332C3711yesmaximal, characteristic, direct product factor, Sylow
245C22×C37—21no—
246C22×C37—21no—
247C22×C37—22no—
248C22×C37—42no—
249C22×C37—41no—
250C22×C37—41no—
251C22×C37—44no—
252C22×C37—44no—
253C22×C37—44no—
254C22×C37—81no—
255C22×C37—81no—
256C22×C37—81no—
257C22×C37—81no—
258C22×C37—81no—
259C22×C37—88no—
260C22×C37—81no—
261C22×C37—81no—
262C22×C37—81no—
263C22×C37C2×#12011yescharacteristic
264C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
266C4×C37—21no—
267C4×C37—22no—
268C4×C37—21no—
269C4×C37—22no—
270C4×C37—22no—
271C4×C37—42no—
272C4×C37—44no—
273C4×C37—41no—
274C4×C37—41no—
275C4×C37—81no—
276C4×C37—81no—
277#3449C2×C3711yescharacteristic, complemented
278#3213C2×C3712yescomplemented
279#2610C2×C3711yescharacteristic, complemented
280#1604C2×C3712yescomplemented
281#1601C2×C3712yescomplemented
282#1598C2×C3712yescomplemented
283#1595C2×C3712yescomplemented
284#1505C2×C3712yescomplemented
285#1464C2×C3711yescharacteristic, complemented
286C2×C37—21no—
287C2×C37—21no—
288C2×C37—42no—
289C2×C37—84no—
290C2×C37—161no—
291C2×C37#300311yescharacteristic
292C2×C37#301811yescharacteristic
293C2×C37#302711yescharacteristic
294C22×#41—21no—
295#537—21no—
296#537C22×C3712yes—
297#536—22no—
298#535—22no—
299#534—21no—
300#534C22×C3712yes—
301#532—21no—
302#532C22×C3712yes—
303C2×#124C22×C3711yescharacteristic, complemented
304#525—24no—
305C2×#117C22×C3711yescharacteristic
306#496—22no—
307#489—22no—
308#463—24no—
309#448C22×C3714yescomplemented
310#447C22×C3714yescomplemented
311#432C22×C3712yescomplemented
312#395C22×C3711yescharacteristic, complemented
313#394C22×C3711yescharacteristic, complemented
314#393C22×C3711yescharacteristic
315#392C22×C3711yescharacteristic
316#390—21no—
317#388—21no—
318#388—21no—
319#386—21no—
320#376C22×C3711yescharacteristic, complemented
321#327C22×C3714yescomplemented
322#327C22×C3714yescomplemented
323#324C22×C3712yescomplemented
324#323C22×C3712yes—
325C37#1833211yescharacteristic, direct product factor, Sylow
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—44no—
331C2×#41—44no—
332C22×Q8—21no—
333C22×Q8C23×C3711yescharacteristic
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—21no—
337C22×D4C23×C3711yescharacteristic
338C23×C4—22no—
339C23×C4D4×C3711yescharacteristic
340C2×SD16—24no—
341C2×D8—24no—
342C4⋊Q8—21no—
343C4⋊Q8D4×C3712yescomplemented
344#127—21no—
345#127D4×C3712yescomplemented
346#124—22no—
347#124—22no—
348#124—22no—
349#124—22no—
350#123—24no—
351#123—24no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354#121—24no—
355#121—24no—
356#121—22no—
357#121—22no—
358#121—22no—
359#121C23×C3714yes—
360#120—24no—
361#120—24no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4D4×C3714yescomplemented
366#117—22no—
367#117—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4D4×C3711yescharacteristic
370C2×C4⋊C4C23×C3711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4D4×C3711yescharacteristic
378C2×C22⋊C4C23×C3711yescharacteristic
379C2×C42—21no—
380C2×C42D4×C3711yescharacteristic
381C2×C42C23×C3711yescharacteristic
382C4⋊C8—22no—
383#103—24no—
384#102—24no—
385C22⋊C8C23×C3714yes—
386#97—22no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95C23×C3711yescharacteristic
391#95C23×C3711yescharacteristic
392C24—22no—
393C24—21no—
394C24C2×D4×C3711yescharacteristic
395#41—48no—
396#41—48no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—81no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4C2×D4×C3714yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—48no—
433C22×C4C2×D4×C3711yescharacteristic
434C22×C4C2×D4×C3711yescharacteristic
435C22×C4C2×D4×C3711yescharacteristic
436C22×C4C2×D4×C3711yescharacteristic
437C22×C4C2×D4×C3711yescharacteristic
438C22×C4C24×C3711yescharacteristic
439SD16—84no—
440D8—84no—
441C2×C8—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—81no—
448C4⋊C4C2×D4×C3714yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C42—21no—
463C42—21no—
464C42—42no—
465C42C2×D4×C3712yes—
466C42C2×D4×C3712yes—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—82no—
481C23—82no—
482C23AGL(1,ℤ/8)×C3712yes—
483C23C22×D4×C3711yescharacteristic
484Q8—44no—
485Q8—44no—
486Q8—44no—
487D4—44no—
488D4—44no—
489D4—44no—
490D4—44no—
491D4—48no—
492D4—44no—
493D4—44no—
494D4—82no—
495D4—82no—
496D4—82no—
497D4—84no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4#120×C3711yescharacteristic
524C2×C4#120×C3711yescharacteristic
525C2×C4#120×C3712yes—
526C2×C4AGL(1,ℤ/8)×C3712yes—
527C2×C4C22×D4×C3712yes—
528C8—44no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—88no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#120×C3711yescharacteristic
548C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
549C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
550C4—21no—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—44no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—84no—
565C2—161no—
566C2C37×#300311yescharacteristic
567C2C37×#301811yescharacteristic
568C2C37×#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×#3449C2semidirect
2. C37×#3213C2semidirect
3. C37×#2610C2semidirect
4. C37×#1604C2semidirect
5. C37×#1601C2semidirect
6. C37×#1598C2semidirect
7. C37×#1595C2semidirect
8. C37×#1505C2semidirect
9. C37×#1464C2semidirect
19. C2×#124×C37C22semidirect
25. C37×#448C22semidirect
26. C37×#447C22semidirect
27. C37×#432C22semidirect
28. C37×#395C22semidirect
29. C37×#394C22semidirect
36. C37×#376C22semidirect
37. C37×#327C22semidirect
38. C37×#327C22semidirect
39. C37×#324C22semidirect
58. C4⋊Q8×C37D4semidirect
60. #127×C37D4semidirect
80. C4×D4×C37D4semidirect
244. #18332C37direct (N × H)
277. #3449C2×C37semidirect
278. #3213C2×C37semidirect
279. #2610C2×C37semidirect
280. #1604C2×C37semidirect
281. #1601C2×C37semidirect
282. #1598C2×C37semidirect
283. #1595C2×C37semidirect
284. #1505C2×C37semidirect
285. #1464C2×C37semidirect
303. C2×#124C22×C37semidirect
309. #448C22×C37semidirect
310. #447C22×C37semidirect
311. #432C22×C37semidirect
312. #395C22×C37semidirect
313. #394C22×C37semidirect
320. #376C22×C37semidirect
321. #327C22×C37semidirect
322. #327C22×C37semidirect
323. #324C22×C37semidirect
325. C37#18332direct (N × H)
343. C4⋊Q8D4×C37semidirect
345. #127D4×C37semidirect
365. C4×D4D4×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34492
C37×#32132
C37×#26102
C37×#16042
C37×#16012
C37×#15982
C37×#15952
C37×#15052
C37×#14642
#1833237