Finite Group Databasebeta

← C37×#18926 #22640179

Subgroups — C37×#18926 #22640179

Order 9472.

966 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3506C211yesmaximal, characteristic
2C37×#3229C211yesmaximal, characteristic, complemented
3C37×#3226C211yesmaximal, characteristic, complemented
4C2×C37×#461C211yesmaximal, characteristic, complemented
5C37×#1650C214yesmaximal, complemented
6C37×#1642C214yesmaximal, complemented
7C37×#1540C211yesmaximal, characteristic, complemented
8C37×#1481C211yesmaximal, characteristic, complemented
9C37×#1478C211yesmaximal, characteristic, complemented
10C22×SD16×C37—21no—
11C37×#559—22no—
12C37×#556C2212yes—
13C37×#555—21no—
14C37×#553—22no—
15C37×#550C2212yes—
16C37×#547—21no—
17C37×#538—22no—
18C37×#535—21no—
19C37×#533C2211yescharacteristic
20C37×#532C2211yescharacteristic
21C2×C4⋊Q8×C37C2211yescharacteristic
22C37×#525—22no—
23C37×#520—21no—
24C2×C4×D4×C37C2211yescharacteristic
25C37×#500C2212yescomplemented
26C37×#498—21no—
27C37×#487—21no—
28C37×#486C2212yescomplemented
29C37×#483—22no—
30C37×#480—22no—
31C37×#475C2214yescomplemented
32C37×#474—22no—
33C37×#461—22no—
34C37×#461C2214yes—
35C37×#431C2211yescharacteristic
36C37×#431C2211yescharacteristic
37C2×C4⋊C8×C37C2211yescharacteristic
38C2×#103×C37—21no—
39C37×#332C2212yescomplemented
40C37×#331C2212yescomplemented
41C37×Q8⋊C8C2214yescomplemented
42C37×D4⋊C8C2214yes—
43C2×#41×C37—22no—
44C2×#41×C37—21no—
45C22×Q8×C37—21no—
46C22×D4×C37D411yescharacteristic
47C23×C4×C37—21no—
48C2×SD16×C37—24no—
49C2×SD16×C37—44no—
50C22×C8×C37—21no—
51C4⋊Q8×C37—21no—
52C4⋊Q8×C37—21no—
53C4⋊Q8×C37—21no—
54C4⋊Q8×C37D411yescharacteristic
55C4⋊Q8×C37D411yescharacteristic
56C4⋊Q8×C37C2312yes—
57C4⋊Q8×C37C2311yescharacteristic
58#127×C37—22no—
59#127×C37C2311yescharacteristic
60#126×C37—24no—
61#125×C37—21no—
62#124×C37—24no—
63#124×C37—22no—
64#124×C37—21no—
65#123×C37—24no—
66#123×C37—24no—
67C22⋊Q8×C37—22no—
68C22⋊Q8×C37—21no—
69C22⋊Q8×C37D412yes—
70#121×C37—22no—
71#121×C37—22no—
72#121×C37—24no—
73#121×C37—21no—
74#121×C37—22no—
75#121×C37D412yes—
76#120×C37—24no—
77C4×Q8×C37—22no—
78C4×Q8×C37—21no—
79C4×Q8×C37C2312yes—
80C4×D4×C37—24no—
81C4×D4×C37—24no—
82C4×D4×C37—22no—
83C4×D4×C37—21no—
84C4×D4×C37—21no—
85C4×D4×C37—21no—
86C4×D4×C37D412yes—
87C4×D4×C37D412yes—
88C4×D4×C37C2312yes—
89#117×C37—22no—
90#117×C37—21no—
91#117×C37—22no—
92C2×C4⋊C4×C37—22no—
93C2×C4⋊C4×C37—21no—
94C2×C4⋊C4×C37—21no—
95C2×C4⋊C4×C37D411yescharacteristic
96C2×C22⋊C4×C37—21no—
97C2×C42×C37C2311yescharacteristic
98#107×C37—24no—
99#106×C37—24no—
100C4⋊C8×C37—21no—
101C4⋊C8×C37—21no—
102C4⋊C8×C37C2312yes—
103C4⋊C8×C37C2312yes—
104#103×C37—24no—
105#103×C37—24no—
106#102×C37—24no—
107C22⋊C8×C37—21no—
108C22⋊C8×C37—21no—
109C4×C8×C37—21no—
110C4×C8×C37C2312yes—
111C24×C37—21no—
112#41×C37—42no—
113#41×C37—42no—
114#41×C37—84no—
115C2×Q8×C37—21no—
116C2×Q8×C37—21no—
117C2×Q8×C37—22no—
118C2×Q8×C37—41no—
119C2×Q8×C37—42no—
120C2×Q8×C37—42no—
121C2×Q8×C37C2×D412yes—
122C2×D4×C37—22no—
123C2×D4×C37—24no—
124C2×D4×C37—24no—
125C2×D4×C37—21no—
126C2×D4×C37—22no—
127C2×D4×C37—22no—
128C2×D4×C37—21no—
129C2×D4×C37—21no—
130C2×D4×C37C2×D412yes—
131C2×D4×C37C2×D412yes—
132C22×C4×C37—24no—
133C22×C4×C37—21no—
134C22×C4×C37—22no—
135C22×C4×C37—22no—
136C22×C4×C37—22no—
137C22×C4×C37—41no—
138C22×C4×C37—82no—
139C22×C4×C37C2×D411yescharacteristic
140C22×C4×C37C2×D411yescharacteristic
141C22×C4×C37C2×D411yescharacteristic
142SD16×C37—48no—
143C2×C8×C37—22no—
144C2×C8×C37—22no—
145C2×C8×C37—22no—
146C2×C8×C37—42no—
147C4⋊C4×C37—24no—
148C4⋊C4×C37—21no—
149C4⋊C4×C37—21no—
150C4⋊C4×C37—21no—
151C4⋊C4×C37—22no—
152C4⋊C4×C37—21no—
153C4⋊C4×C37—21no—
154C4⋊C4×C37—22no—
155C4⋊C4×C37—22no—
156C4⋊C4×C37—22no—
157C4⋊C4×C37—41no—
158C4⋊C4×C37—41no—
159C4⋊C4×C37C2×D412yes—
160C4⋊C4×C37C2×D412yes—
161C4⋊C4×C37C2×D412yes—
162C22⋊C4×C37—24no—
163C22⋊C4×C37—24no—
164C22⋊C4×C37—24no—
165C22⋊C4×C37—22no—
166C22⋊C4×C37—22no—
167C22⋊C4×C37—22no—
168C22⋊C4×C37—22no—
169C22⋊C4×C37—22no—
170C42×C37—24no—
171C42×C37—22no—
172C42×C37C2×D411yescharacteristic
173C42×C37C2×D411yescharacteristic
174C42×C37C2×D411yescharacteristic
175C42×C37C2411yescharacteristic
176C23×C37—22no—
177C23×C37—22no—
178C23×C37—42no—
179C23×C37—81no—
180C23×C37—81no—
181C23×C37#12011yescharacteristic
182Q8×C37—24no—
183Q8×C37—41no—
184Q8×C37—41no—
185Q8×C37—42no—
186Q8×C37—42no—
187Q8×C37—81no—
188Q8×C37—82no—
189D4×C37—22no—
190D4×C37—22no—
191D4×C37—42no—
192D4×C37—44no—
193D4×C37—41no—
194D4×C37—41no—
195D4×C37—42no—
196D4×C37—42no—
197D4×C37—84no—
198D4×C37—84no—
199C2×C4×C37—22no—
200C2×C4×C37—22no—
201C2×C4×C37—22no—
202C2×C4×C37—22no—
203C2×C4×C37—22no—
204C2×C4×C37—41no—
205C2×C4×C37—44no—
206C2×C4×C37—44no—
207C2×C4×C37—42no—
208C2×C4×C37—82no—
209C2×C4×C37—81no—
210C2×C4×C37—84no—
211C2×C4×C37—82no—
212C2×C4×C37#12011yescharacteristic
213C2×C4×C37#12011yescharacteristic
214C2×C4×C37#12011yescharacteristic
215C2×C4×C37#13512yes—
216C2×C4×C37AGL(1,ℤ/8)12yes—
217C2×C4×C37C22×D411yescharacteristic
218C2×C4×C37C22×D411yescharacteristic
219C2×C4×C37C22×D411yescharacteristic
220C8×C37—42no—
221C8×C37—42no—
222C8×C37—42no—
223#18926C3711yesmaximal, characteristic, direct product factor, Sylow
224C22×C37—21no—
225C22×C37—21no—
226C22×C37—42no—
227C22×C37—42no—
228C22×C37—42no—
229C22×C37—42no—
230C22×C37—42no—
231C22×C37—42no—
232C22×C37—84no—
233C22×C37#44912yes—
234C22×C37C2×#12011yescharacteristic
235C4×C37—21no—
236C4×C37—21no—
237C4×C37—21no—
238C4×C37—42no—
239C4×C37—42no—
240C4×C37—42no—
241C4×C37—42no—
242C4×C37—82no—
243C4×C37#44912yes—
244C4×C37C2×#13511yescharacteristic
245C4×C37C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C37#57712yes—
247#3506C2×C3711yescharacteristic
248#3229C2×C3711yescharacteristic, complemented
249#3226C2×C3711yescharacteristic, complemented
250C2×#461C2×C3711yescharacteristic, complemented
251#1650C2×C3714yescomplemented
252#1642C2×C3714yescomplemented
253#1540C2×C3711yescharacteristic, complemented
254#1481C2×C3711yescharacteristic, complemented
255#1478C2×C3711yescharacteristic, complemented
256C2×C37—22no—
257C2×C37—82no—
258C2×C37—82no—
259C2×C37C2×#44911yescharacteristic
260C2×C37#300911yescharacteristic
261C2×C37#302211yescharacteristic
262C22×SD16—21no—
263#559—22no—
264#556C22×C3712yes—
265#555—21no—
266#553—22no—
267#550C22×C3712yes—
268#547—21no—
269#538—22no—
270#535—21no—
271#533C22×C3711yescharacteristic
272#532C22×C3711yescharacteristic
273C2×C4⋊Q8C22×C3711yescharacteristic
274#525—22no—
275#520—21no—
276C2×C4×D4C22×C3711yescharacteristic
277#500C22×C3712yescomplemented
278#498—21no—
279#487—21no—
280#486C22×C3712yescomplemented
281#483—22no—
282#480—22no—
283#475C22×C3714yescomplemented
284#474—22no—
285#461—22no—
286#461C22×C3714yes—
287#431C22×C3711yescharacteristic
288#431C22×C3711yescharacteristic
289C2×C4⋊C8C22×C3711yescharacteristic
290C2×#103—21no—
291#332C22×C3712yescomplemented
292#331C22×C3712yescomplemented
293Q8⋊C8C22×C3714yescomplemented
294D4⋊C8C22×C3714yes—
295C37#1892611yescharacteristic, direct product factor, Sylow
296C2×#41—22no—
297C2×#41—21no—
298C22×Q8—21no—
299C22×D4D4×C3711yescharacteristic
300C23×C4—21no—
301C2×SD16—24no—
302C2×SD16—44no—
303C22×C8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8D4×C3711yescharacteristic
308C4⋊Q8D4×C3711yescharacteristic
309C4⋊Q8C23×C3712yes—
310C4⋊Q8C23×C3711yescharacteristic
311#127—22no—
312#127C23×C3711yescharacteristic
313#126—24no—
314#125—21no—
315#124—24no—
316#124—22no—
317#124—21no—
318#123—24no—
319#123—24no—
320C22⋊Q8—22no—
321C22⋊Q8—21no—
322C22⋊Q8D4×C3712yes—
323#121—22no—
324#121—22no—
325#121—24no—
326#121—21no—
327#121—22no—
328#121D4×C3712yes—
329#120—24no—
330C4×Q8—22no—
331C4×Q8—21no—
332C4×Q8C23×C3712yes—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4D4×C3712yes—
340C4×D4D4×C3712yes—
341C4×D4C23×C3712yes—
342#117—22no—
343#117—21no—
344#117—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4D4×C3711yescharacteristic
349C2×C22⋊C4—21no—
350C2×C42C23×C3711yescharacteristic
351#107—24no—
352#106—24no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8C23×C3712yes—
356C4⋊C8C23×C3712yes—
357#103—24no—
358#103—24no—
359#102—24no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C4×C8—21no—
363C4×C8C23×C3712yes—
364C24—21no—
365#41—42no—
366#41—42no—
367#41—84no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—41no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8C2×D4×C3712yes—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4C2×D4×C3712yes—
384C2×D4C2×D4×C3712yes—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—82no—
392C22×C4C2×D4×C3711yescharacteristic
393C22×C4C2×D4×C3711yescharacteristic
394C22×C4C2×D4×C3711yescharacteristic
395SD16—48no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—42no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4C2×D4×C3712yes—
413C4⋊C4C2×D4×C3712yes—
414C4⋊C4C2×D4×C3712yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C42—24no—
424C42—22no—
425C42C2×D4×C3711yescharacteristic
426C42C2×D4×C3711yescharacteristic
427C42C2×D4×C3711yescharacteristic
428C42C24×C3711yescharacteristic
429C23—22no—
430C23—22no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23#120×C3711yescharacteristic
435Q8—24no—
436Q8—41no—
437Q8—41no—
438Q8—42no—
439Q8—42no—
440Q8—81no—
441Q8—82no—
442D4—22no—
443D4—22no—
444D4—42no—
445D4—44no—
446D4—41no—
447D4—41no—
448D4—42no—
449D4—42no—
450D4—84no—
451D4—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—84no—
464C2×C4—82no—
465C2×C4#120×C3711yescharacteristic
466C2×C4#120×C3711yescharacteristic
467C2×C4#120×C3711yescharacteristic
468C2×C4#135×C3712yes—
469C2×C4AGL(1,ℤ/8)×C3712yes—
470C2×C4C22×D4×C3711yescharacteristic
471C2×C4C22×D4×C3711yescharacteristic
472C2×C4C22×D4×C3711yescharacteristic
473C8—42no—
474C8—42no—
475C8—42no—
476C22—21no—
477C22—21no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—84no—
485C22C37×#44912yes—
486C22C2×#120×C3711yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—82no—
495C4C37×#44912yes—
496C4C2×#135×C3711yescharacteristic
497C4C2×AGL(1,ℤ/8)×C3711yescharacteristic
498C4C37×#57712yes—
499C2—22no—
500C2—82no—
501C2—82no—
502C2C2×C37×#44911yescharacteristic
503C2C37×#300911yescharacteristic
504C2C37×#302211yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#3506C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3229C2semidirect
3. C37×#3226C2semidirect
4. C2×C37×#461C2semidirect
5. C37×#1650C2semidirect
6. C37×#1642C2semidirect
7. C37×#1540C2semidirect
8. C37×#1481C2semidirect
9. C37×#1478C2semidirect
25. C37×#500C22semidirect
28. C37×#486C22semidirect
31. C37×#475C22semidirect
39. C37×#332C22semidirect
40. C37×#331C22semidirect
41. C37×Q8⋊C8C22semidirect
223. #18926C37direct (N × H)
248. #3229C2×C37semidirect
249. #3226C2×C37semidirect
250. C2×#461C2×C37semidirect
251. #1650C2×C37semidirect
252. #1642C2×C37semidirect
253. #1540C2×C37semidirect
254. #1481C2×C37semidirect
255. #1478C2×C37semidirect
277. #500C22×C37semidirect
280. #486C22×C37semidirect
283. #475C22×C37semidirect
291. #332C22×C37semidirect
292. #331C22×C37semidirect
293. Q8⋊C8C22×C37semidirect
295. C37#18926direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#35062
C37×#32292
C37×#32262
C2×C37×#4612
C37×#16502
C37×#16422
C37×#15402
C37×#14812
C37×#14782
#1892637