Finite Group Databasebeta

← C37×#18122 #22639375

Subgroups — C37×#18122 #22639375

Order 9472.

1292 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3449C211yesmaximal, characteristic, complemented
2C37×#3100C212yesmaximal, complemented
3C37×#2392C211yesmaximal, characteristic, complemented
4C37×#1610C212yesmaximal, complemented
5C37×#1607C212yesmaximal, complemented
6C37×#1598C212yesmaximal, complemented
7C37×#1595C212yesmaximal, 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×D4×C37—21no—
21C2×C4×D4×C37C2211yescharacteristic, complemented
22C2×#117×C37C2211yescharacteristic
23C37×#480C2214yescomplemented
24C37×#474—22no—
25C37×#459—22no—
26C37×#447C2214yescomplemented
27C4×D8×C37—24no—
28C37×#431C2212yescomplemented
29C37×#398C2211yescharacteristic, complemented
30C37×#397C2211yescharacteristic
31C37×#396—21no—
32C37×#394—21no—
33C37×#394C2211yescharacteristic, complemented
34C37×#392C2211yescharacteristic
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×D4×C37—21no—
49C22×D4×C37—22no—
50C22×D4×C37—21no—
51C22×D4×C37C2311yescharacteristic
52C23×C4×C37—22no—
53C23×C4×C37—21no—
54C23×C4×C37D411yescharacteristic
55C2×D8×C37—24no—
56C4⋊Q8×C37—21no—
57C4⋊Q8×C37D412yescomplemented
58#127×C37—21no—
59#127×C37D412yescomplemented
60#124×C37—22no—
61#124×C37—22no—
62#123×C37—24no—
63#123×C37—24no—
64C22⋊Q8×C37—24no—
65C22⋊Q8×C37—24no—
66#121×C37—24no—
67#121×C37—24no—
68#121×C37—22no—
69#121×C37—22no—
70#121×C37—22no—
71#121×C37C2314yes—
72#120×C37—24no—
73#120×C37—24no—
74C4×D4×C37—22no—
75C4×D4×C37—22no—
76C4×D4×C37—22no—
77C4×D4×C37—22no—
78C4×D4×C37—22no—
79C4×D4×C37—44no—
80C4×D4×C37D414yescomplemented
81#117×C37—22no—
82#117×C37—22no—
83C2×C4⋊C4×C37—21no—
84C2×C4⋊C4×C37—21no—
85C2×C4⋊C4×C37D411yescharacteristic
86C2×C4⋊C4×C37C2311yescharacteristic
87C2×C4⋊C4×C37C2311yescharacteristic
88C2×C22⋊C4×C37—21no—
89C2×C22⋊C4×C37—21no—
90C2×C22⋊C4×C37—21no—
91C2×C22⋊C4×C37—21no—
92C2×C22⋊C4×C37—21no—
93C2×C22⋊C4×C37D411yescharacteristic
94C2×C22⋊C4×C37C2311yescharacteristic
95C2×C42×C37—21no—
96C2×C42×C37D411yescharacteristic
97C2×C42×C37C2311yescharacteristic
98#107×C37—24no—
99C4⋊C8×C37—22no—
100#102×C37—24no—
101#102×C37—24no—
102C22⋊C8×C37C2314yes—
103C4×C8×C37—22no—
104#95×C37—21no—
105#95×C37—21no—
106#95×C37C2311yescharacteristic
107#95×C37C2311yescharacteristic
108C24×C37—22no—
109C24×C37—21no—
110C24×C37C2×D411yescharacteristic
111#41×C37—44no—
112#41×C37—48no—
113#41×C37—48no—
114#41×C37—44no—
115#41×C37—44no—
116#41×C37—44no—
117C2×Q8×C37—22no—
118C2×Q8×C37—22no—
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—22no—
128C2×D4×C37—42no—
129C2×D4×C37—44no—
130C2×D4×C37—42no—
131C2×D4×C37—42no—
132C2×D4×C37—44no—
133C2×D4×C37—42no—
134C2×D4×C37—44no—
135C2×D4×C37C2×D414yes—
136C22×C4×C37—21no—
137C22×C4×C37—21no—
138C22×C4×C37—22no—
139C22×C4×C37—24no—
140C22×C4×C37—22no—
141C22×C4×C37—21no—
142C22×C4×C37—24no—
143C22×C4×C37—21no—
144C22×C4×C37—21no—
145C22×C4×C37—48no—
146C22×C4×C37—41no—
147C22×C4×C37—41no—
148C22×C4×C37—42no—
149C22×C4×C37—42no—
150C22×C4×C37D814yescomplemented
151C22×C4×C37C2×D411yescharacteristic
152C22×C4×C37C2×D411yescharacteristic
153C22×C4×C37C2×D411yescharacteristic
154C22×C4×C37C2×D411yescharacteristic
155C22×C4×C37C2×D411yescharacteristic
156C22×C4×C37C2411yescharacteristic
157D8×C37—48no—
158C2×C8×C37—24no—
159C4⋊C4×C37—22no—
160C4⋊C4×C37—22no—
161C4⋊C4×C37—22no—
162C4⋊C4×C37—22no—
163C4⋊C4×C37—22no—
164C4⋊C4×C37—22no—
165C4⋊C4×C37—42no—
166C4⋊C4×C37—42no—
167C4⋊C4×C37C2×D414yes—
168C22⋊C4×C37—24no—
169C22⋊C4×C37—24no—
170C22⋊C4×C37—22no—
171C22⋊C4×C37—24no—
172C22⋊C4×C37—22no—
173C22⋊C4×C37—24no—
174C22⋊C4×C37—42no—
175C22⋊C4×C37—42no—
176C22⋊C4×C37—42no—
177C22⋊C4×C37—42no—
178C22⋊C4×C37—42no—
179C42×C37—21no—
180C42×C37—21no—
181C42×C37—42no—
182C42×C37C2×D412yes—
183C42×C37C2×D412yes—
184C23×C37—21no—
185C23×C37—21no—
186C23×C37—22no—
187C23×C37—24no—
188C23×C37—41no—
189C23×C37—41no—
190C23×C37—41no—
191C23×C37—44no—
192C23×C37—41no—
193C23×C37—41no—
194C23×C37—41no—
195C23×C37—41no—
196C23×C37—41no—
197C23×C37—82no—
198C23×C37—82no—
199C23×C37C2×D812yes—
200C23×C37C22×D411yescharacteristic
201Q8×C37—44no—
202Q8×C37—44no—
203D4×C37—24no—
204D4×C37—44no—
205D4×C37—42no—
206D4×C37—44no—
207D4×C37—48no—
208D4×C37—48no—
209D4×C37—44no—
210D4×C37—44no—
211D4×C37—44no—
212D4×C37—82no—
213D4×C37—82no—
214D4×C37—82no—
215C2×C4×C37—21no—
216C2×C4×C37—21no—
217C2×C4×C37—24no—
218C2×C4×C37—24no—
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—44no—
227C2×C4×C37—41no—
228C2×C4×C37—41no—
229C2×C4×C37—48no—
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—82no—
239C2×C4×C37—82no—
240C2×C4×C37#12012yes—
241C2×C4×C37#12011yescharacteristic
242C2×C4×C37#12011yescharacteristic
243C2×C4×C37C2×D812yes—
244C2×C4×C37C2×D812yes—
245C2×C4×C37AGL(1,ℤ/8)12yes—
246C2×C4×C37C22×D412yes—
247C8×C37—44no—
248#18122C3711yesmaximal, characteristic, direct product factor, Sylow
249C22×C37—21no—
250C22×C37—21no—
251C22×C37—22no—
252C22×C37—42no—
253C22×C37—42no—
254C22×C37—42no—
255C22×C37—41no—
256C22×C37—41no—
257C22×C37—44no—
258C22×C37—44no—
259C22×C37—42no—
260C22×C37—44no—
261C22×C37—81no—
262C22×C37—81no—
263C22×C37—81no—
264C22×C37—88no—
265C22×C37—81no—
266C22×C37—81no—
267C22×C37C2×#12011yescharacteristic
268C22×C37C22×D811yescharacteristic
269C22×C37C2×AGL(1,ℤ/8)11yescharacteristic
270C4×C37—22no—
271C4×C37—22no—
272C4×C37—22no—
273C4×C37—42no—
274C4×C37—44no—
275C4×C37—41no—
276C4×C37—41no—
277C4×C37—81no—
278C4×C37—81no—
279C4×C37#44712yes—
280C4×C37#44712yes—
281#3449C2×C3711yescharacteristic, complemented
282#3100C2×C3712yescomplemented
283#2392C2×C3711yescharacteristic, complemented
284#1610C2×C3712yescomplemented
285#1607C2×C3712yescomplemented
286#1598C2×C3712yescomplemented
287#1595C2×C3712yescomplemented
288#1502C2×C3712yescomplemented
289#1458C2×C3711yescharacteristic, complemented
290C2×C37—21no—
291C2×C37—21no—
292C2×C37—42no—
293C2×C37—82no—
294C2×C37—84no—
295C2×C37C2×#44711yescharacteristic
296C2×C37#301811yescharacteristic
297C2×C37#302511yescharacteristic
298C22×#41—21no—
299#537—21no—
300#537C22×C3712yes—
301#536—22no—
302#535—22no—
303#534—21no—
304#534C22×C3712yes—
305#532—21no—
306#532C22×C3712yes—
307#525—24no—
308C2×C4×D4—21no—
309C2×C4×D4C22×C3711yescharacteristic, complemented
310C2×#117C22×C3711yescharacteristic
311#480C22×C3714yescomplemented
312#474—22no—
313#459—22no—
314#447C22×C3714yescomplemented
315C4×D8—24no—
316#431C22×C3712yescomplemented
317#398C22×C3711yescharacteristic, complemented
318#397C22×C3711yescharacteristic
319#396—21no—
320#394—21no—
321#394C22×C3711yescharacteristic, complemented
322#392C22×C3711yescharacteristic
323C4×C4⋊C4C22×C3711yescharacteristic, complemented
324C4×C22⋊C4—21no—
325#327C22×C3714yescomplemented
326#327C22×C3714yescomplemented
327#324C22×C3712yescomplemented
328#323C22×C3712yes—
329C37#1812211yescharacteristic, direct product factor, Sylow
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—44no—
335C2×#41—44no—
336C22×Q8—21no—
337C22×D4—21no—
338C22×D4—22no—
339C22×D4—21no—
340C22×D4C23×C3711yescharacteristic
341C23×C4—22no—
342C23×C4—21no—
343C23×C4D4×C3711yescharacteristic
344C2×D8—24no—
345C4⋊Q8—21no—
346C4⋊Q8D4×C3712yescomplemented
347#127—21no—
348#127D4×C3712yescomplemented
349#124—22no—
350#124—22no—
351#123—24no—
352#123—24no—
353C22⋊Q8—24no—
354C22⋊Q8—24no—
355#121—24no—
356#121—24no—
357#121—22no—
358#121—22no—
359#121—22no—
360#121C23×C3714yes—
361#120—24no—
362#120—24no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—44no—
369C4×D4D4×C3714yescomplemented
370#117—22no—
371#117—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4D4×C3711yescharacteristic
375C2×C4⋊C4C23×C3711yescharacteristic
376C2×C4⋊C4C23×C3711yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4D4×C3711yescharacteristic
383C2×C22⋊C4C23×C3711yescharacteristic
384C2×C42—21no—
385C2×C42D4×C3711yescharacteristic
386C2×C42C23×C3711yescharacteristic
387#107—24no—
388C4⋊C8—22no—
389#102—24no—
390#102—24no—
391C22⋊C8C23×C3714yes—
392C4×C8—22no—
393#95—21no—
394#95—21no—
395#95C23×C3711yescharacteristic
396#95C23×C3711yescharacteristic
397C24—22no—
398C24—21no—
399C24C2×D4×C3711yescharacteristic
400#41—44no—
401#41—48no—
402#41—48no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4C2×D4×C3714yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—48no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4D8×C3714yescomplemented
440C22×C4C2×D4×C3711yescharacteristic
441C22×C4C2×D4×C3711yescharacteristic
442C22×C4C2×D4×C3711yescharacteristic
443C22×C4C2×D4×C3711yescharacteristic
444C22×C4C2×D4×C3711yescharacteristic
445C22×C4C24×C3711yescharacteristic
446D8—48no—
447C2×C8—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4C2×D4×C3714yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C42—21no—
469C42—21no—
470C42—42no—
471C42C2×D4×C3712yes—
472C42C2×D4×C3712yes—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—24no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—44no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—82no—
487C23—82no—
488C23C2×D8×C3712yes—
489C23C22×D4×C3711yescharacteristic
490Q8—44no—
491Q8—44no—
492D4—24no—
493D4—44no—
494D4—44no—
495D4—42no—
496D4—48no—
497D4—48no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—82no—
502D4—82no—
503D4—82no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#120×C3712yes—
530C2×C4#120×C3711yescharacteristic
531C2×C4#120×C3711yescharacteristic
532C2×C4C2×D8×C3712yes—
533C2×C4C2×D8×C3712yes—
534C2×C4AGL(1,ℤ/8)×C3712yes—
535C2×C4C22×D4×C3712yes—
536C8—44no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—44no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—88no—
553C22—81no—
554C22—81no—
555C22C2×#120×C3711yescharacteristic
556C22C22×D8×C3711yescharacteristic
557C22C2×AGL(1,ℤ/8)×C3711yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—44no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4C37×#44712yes—
568C4C37×#44712yes—
569C2—21no—
570C2—21no—
571C2—42no—
572C2—82no—
573C2—84no—
574C2C2×C37×#44711yescharacteristic
575C2C37×#301811yescharacteristic
576C2C37×#302511yescharacteristic
subgroupG/H
centerC2×C4×C37#120
commutator, Frattini, cosocleC22×C4C24×C37
socleC22×C37C2×#120
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
1. C37×#3449C2semidirect
2. C37×#3100C2semidirect
3. C37×#2392C2semidirect
4. C37×#1610C2semidirect
5. C37×#1607C2semidirect
6. C37×#1598C2semidirect
7. C37×#1595C2semidirect
8. C37×#1502C2semidirect
9. C37×#1458C2semidirect
21. C2×C4×D4×C37C22semidirect
23. C37×#480C22semidirect
26. C37×#447C22semidirect
28. C37×#431C22semidirect
29. C37×#398C22semidirect
33. C37×#394C22semidirect
35. C4×C4⋊C4×C37C22semidirect
37. C37×#327C22semidirect
38. C37×#327C22semidirect
39. C37×#324C22semidirect
57. C4⋊Q8×C37D4semidirect
59. #127×C37D4semidirect
80. C4×D4×C37D4semidirect
150. C22×C4×C37D8semidirect
248. #18122C37direct (N × H)
281. #3449C2×C37semidirect
282. #3100C2×C37semidirect
283. #2392C2×C37semidirect
284. #1610C2×C37semidirect
285. #1607C2×C37semidirect
286. #1598C2×C37semidirect
287. #1595C2×C37semidirect
288. #1502C2×C37semidirect
289. #1458C2×C37semidirect
309. C2×C4×D4C22×C37semidirect
311. #480C22×C37semidirect
314. #447C22×C37semidirect
316. #431C22×C37semidirect
317. #398C22×C37semidirect
321. #394C22×C37semidirect
323. C4×C4⋊C4C22×C37semidirect
325. #327C22×C37semidirect
326. #327C22×C37semidirect
327. #324C22×C37semidirect
329. C37#18122direct (N × H)
346. C4⋊Q8D4×C37semidirect
348. #127D4×C37semidirect
369. C4×D4D4×C37semidirect
439. C22×C4D8×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#34492
C37×#31002
C37×#23922
C37×#16102
C37×#16072
C37×#15982
C37×#15952
C37×#15022
C37×#14582
#1812237