Finite Group Databasebeta

← C37×#29707 #22650960

Subgroups — C37×#29707 #22650960

Order 9472.

1010 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3460C211yesmaximal, characteristic
2C37×#3080C212yesmaximal, complemented
3C37×#2566C211yesmaximal, characteristic, complemented
4C37×#2061C211yesmaximal, characteristic, complemented
5C37×#2060C211yesmaximal, characteristic, complemented
6C37×#2040C211yesmaximal, characteristic, complemented
7C37×#2039C211yesmaximal, characteristic, complemented
8C37×#2037C212yesmaximal, complemented
9C37×#2037C212yesmaximal, complemented
10C37×#1806C212yesmaximal, complemented
11C37×#1800C211yesmaximal, characteristic, complemented
12C37×#544C2211yescharacteristic
13C37×#542—22no—
14C37×#540—22no—
15C37×#538—22no—
16C37×#537C2211yescharacteristic
17C37×#535C2212yes—
18C37×#532—22no—
19C37×#529—22no—
20C2×#125×C37C2211yescharacteristic
21C37×#525—22no—
22C4×#41×C37C2212yes—
23C2×C4×D4×C37C2211yescharacteristic
24C37×#479—22no—
25C37×#479—22no—
26C37×#478—22no—
27C37×#478—22no—
28C37×#428C2212yes—
29C37×#428C2212yes—
30C37×#423C2212yes—
31C2×#104×C37C2212yes—
32C2×#104×C37C2212yes—
33C37×#417C2212yes—
34C37×#417C2212yes—
35C2×#100×C37C2212yes—
36C37×#396—22no—
37C37×#390C2211yescharacteristic
38C37×#390C2211yescharacteristic
39C37×#385—22no—
40C37×#384C2211yescharacteristic
41C37×#382C2211yescharacteristic
42C4×C4⋊C4×C37C2211yescharacteristic
43C37×#343C2211yescharacteristic
44C37×#343C2211yescharacteristic
45C37×#339C2212yes—
46C37×#339C2212yes—
47C2×#41×C37—22no—
48C2×#41×C37C2312yes—
49C22×D4×C37C2311yescharacteristic
50C23×C4×C37D412yes—
51C2×M16×C37C2312yes—
52C2×M16×C37C2312yes—
53C4⋊Q8×C37—22no—
54#127×C37—22no—
55#126×C37—24no—
56#125×C37—22no—
57#125×C37—22no—
58#125×C37—22no—
59#124×C37—24no—
60#124×C37—22no—
61#123×C37—24no—
62#123×C37—24no—
63C22⋊Q8×C37—22no—
64C22⋊Q8×C37—22no—
65C22⋊Q8×C37—22no—
66#121×C37—22no—
67#121×C37—24no—
68#121×C37—22no—
69#121×C37—22no—
70#121×C37—22no—
71#121×C37—22no—
72#120×C37—24no—
73C4×Q8×C37D412yes—
74C4×Q8×C37D412yes—
75C4×D4×C37—24no—
76C4×D4×C37—22no—
77C4×D4×C37—22no—
78C4×D4×C37—22no—
79C4×D4×C37—22no—
80C4×D4×C37D412yes—
81C4×D4×C37D412yes—
82#117×C37—22no—
83#117×C37—22no—
84#117×C37C2312yes—
85C2×C4⋊C4×C37—21no—
86C2×C4⋊C4×C37—21no—
87C2×C4⋊C4×C37—21no—
88C2×C4⋊C4×C37—22no—
89C2×C4⋊C4×C37C2311yescharacteristic
90C2×C4⋊C4×C37C2311yescharacteristic
91C2×C4⋊C4×C37C2311yescharacteristic
92C2×C22⋊C4×C37—22no—
93C2×C22⋊C4×C37—22no—
94C2×C22⋊C4×C37D412yes—
95C2×C42×C37—22no—
96C2×C42×C37C2311yescharacteristic
97C2×C42×C37C2311yescharacteristic
98C2×C42×C37C2311yescharacteristic
99#107×C37—22no—
100#107×C37—22no—
101#106×C37—22no—
102#106×C37—22no—
103C4⋊C8×C37—22no—
104C4⋊C8×C37—22no—
105#104×C37—42no—
106#104×C37—42no—
107#103×C37—22no—
108#103×C37—22no—
109#102×C37—22no—
110#102×C37—22no—
111#100×C37—42no—
112#95×C37—21no—
113#95×C37—22no—
114C24×C37C2×D412yes—
115#41×C37—22no—
116#41×C37—22no—
117#41×C37—22no—
118#41×C37—22no—
119#41×C37—84no—
120C2×Q8×C37—22no—
121C2×Q8×C37C2×D412yes—
122C2×D4×C37—22no—
123C2×D4×C37—24no—
124C2×D4×C37—24no—
125C2×D4×C37—22no—
126C2×D4×C37—22no—
127C2×D4×C37—21no—
128C2×D4×C37—21no—
129C2×D4×C37—22no—
130C2×D4×C37C2×D412yes—
131C22×C4×C37—21no—
132C22×C4×C37—22no—
133C22×C4×C37—22no—
134C22×C4×C37—22no—
135C22×C4×C37—22no—
136C22×C4×C37—22no—
137C22×C4×C37—21no—
138C22×C4×C37—21no—
139C22×C4×C37—21no—
140C22×C4×C37—82no—
141C22×C4×C37C2×D412yes—
142C22×C4×C37C2×D411yescharacteristic
143C22×C4×C37C2×D411yescharacteristic
144C22×C4×C37C2411yescharacteristic
145M16×C37—22no—
146M16×C37—22no—
147M16×C37—22no—
148M16×C37—22no—
149C2×C8×C37—22no—
150C2×C8×C37—22no—
151C4⋊C4×C37—22no—
152C4⋊C4×C37—22no—
153C4⋊C4×C37—24no—
154C4⋊C4×C37—22no—
155C4⋊C4×C37—22no—
156C4⋊C4×C37—22no—
157C4⋊C4×C37—22no—
158C4⋊C4×C37—22no—
159C4⋊C4×C37—42no—
160C4⋊C4×C37—41no—
161C4⋊C4×C37—42no—
162C4⋊C4×C37—41no—
163C4⋊C4×C37—82no—
164C4⋊C4×C37C2×D412yes—
165C4⋊C4×C37C2×D412yes—
166C22⋊C4×C37—24no—
167C22⋊C4×C37—24no—
168C22⋊C4×C37—24no—
169C22⋊C4×C37—22no—
170C22⋊C4×C37—24no—
171C22⋊C4×C37—22no—
172C22⋊C4×C37—22no—
173C22⋊C4×C37—44no—
174C22⋊C4×C37—82no—
175C42×C37—22no—
176C42×C37—22no—
177C42×C37—22no—
178C42×C37—21no—
179C42×C37—21no—
180C42×C37—21no—
181C42×C37—21no—
182C42×C37C2×D412yes—
183C42×C37C2×D412yes—
184C23×C37—22no—
185C23×C37—22no—
186C23×C37—22no—
187C23×C37—22no—
188C23×C37—42no—
189C23×C37—42no—
190C23×C37C22×D411yescharacteristic
191Q8×C37—22no—
192Q8×C37—22no—
193Q8×C37—82no—
194D4×C37—22no—
195D4×C37—22no—
196D4×C37—42no—
197D4×C37—42no—
198D4×C37—42no—
199D4×C37—42no—
200D4×C37—84no—
201D4×C37—82no—
202D4×C37—84no—
203C2×C4×C37—22no—
204C2×C4×C37—22no—
205C2×C4×C37—22no—
206C2×C4×C37—22no—
207C2×C4×C37—22no—
208C2×C4×C37—41no—
209C2×C4×C37—41no—
210C2×C4×C37—44no—
211C2×C4×C37—44no—
212C2×C4×C37—41no—
213C2×C4×C37—41no—
214C2×C4×C37—42no—
215C2×C4×C37—42no—
216C2×C4×C37—42no—
217C2×C4×C37—42no—
218C2×C4×C37—41no—
219C2×C4×C37—42no—
220C2×C4×C37—41no—
221C2×C4×C37—41no—
222C2×C4×C37—41no—
223C2×C4×C37—41no—
224C2×C4×C37—82no—
225C2×C4×C37—82no—
226C2×C4×C37#12011yescharacteristic
227C2×C4×C37#12012yes—
228C2×C4×C37#12011yescharacteristic
229C2×C4×C37C22×D412yes—
230C2×C4×C37H(5,𝔽2)11yescharacteristic
231C2×C4×C372−1+411yescharacteristic
232C2×C4×C372−1+412yes—
233C8×C37—42no—
234C8×C37—42no—
235#29707C3711yesmaximal, characteristic, direct product factor, Sylow
236C22×C37—21no—
237C22×C37—21no—
238C22×C37—44no—
239C22×C37—42no—
240C22×C37—42no—
241C22×C37—42no—
242C22×C37—42no—
243C22×C37—42no—
244C22×C37—42no—
245C22×C37—82no—
246C22×C37C2×#12011yescharacteristic
247C22×C37#53611yescharacteristic
248C22×C37#53711yescharacteristic
249C4×C37—22no—
250C4×C37—21no—
251C4×C37—21no—
252C4×C37—42no—
253C4×C37—42no—
254C4×C37—42no—
255C4×C37—42no—
256C4×C37—42no—
257C4×C37—81no—
258C4×C37—81no—
259C4×C37—81no—
260C4×C37#53612yes—
261C4×C37#53712yes—
262#3460C2×C3711yescharacteristic
263#3080C2×C3712yescomplemented
264#2566C2×C3711yescharacteristic, complemented
265#2061C2×C3711yescharacteristic, complemented
266#2060C2×C3711yescharacteristic, complemented
267#2040C2×C3711yescharacteristic, complemented
268#2039C2×C3711yescharacteristic, complemented
269#2037C2×C3712yescomplemented
270#2037C2×C3712yescomplemented
271#1806C2×C3712yescomplemented
272#1800C2×C3711yescharacteristic, complemented
273C2×C37—21no—
274C2×C37—21no—
275C2×C37—82no—
276C2×C37—82no—
277C2×C37#261611yescharacteristic
278C2×C37#301811yescharacteristic
279C2×C37#301911yescharacteristic
280#544C22×C3711yescharacteristic
281#542—22no—
282#540—22no—
283#538—22no—
284#537C22×C3711yescharacteristic
285#535C22×C3712yes—
286#532—22no—
287#529—22no—
288C2×#125C22×C3711yescharacteristic
289#525—22no—
290C4×#41C22×C3712yes—
291C2×C4×D4C22×C3711yescharacteristic
292#479—22no—
293#479—22no—
294#478—22no—
295#478—22no—
296#428C22×C3712yes—
297#428C22×C3712yes—
298#423C22×C3712yes—
299C2×#104C22×C3712yes—
300C2×#104C22×C3712yes—
301#417C22×C3712yes—
302#417C22×C3712yes—
303C2×#100C22×C3712yes—
304#396—22no—
305#390C22×C3711yescharacteristic
306#390C22×C3711yescharacteristic
307#385—22no—
308#384C22×C3711yescharacteristic
309#382C22×C3711yescharacteristic
310C4×C4⋊C4C22×C3711yescharacteristic
311#343C22×C3711yescharacteristic
312#343C22×C3711yescharacteristic
313#339C22×C3712yes—
314#339C22×C3712yes—
315C37#2970711yescharacteristic, direct product factor, Sylow
316C2×#41—22no—
317C2×#41C23×C3712yes—
318C22×D4C23×C3711yescharacteristic
319C23×C4D4×C3712yes—
320C2×M16C23×C3712yes—
321C2×M16C23×C3712yes—
322C4⋊Q8—22no—
323#127—22no—
324#126—24no—
325#125—22no—
326#125—22no—
327#125—22no—
328#124—24no—
329#124—22no—
330#123—24no—
331#123—24no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335#121—22no—
336#121—24no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121—22no—
341#120—24no—
342C4×Q8D4×C3712yes—
343C4×Q8D4×C3712yes—
344C4×D4—24no—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4D4×C3712yes—
350C4×D4D4×C3712yes—
351#117—22no—
352#117—22no—
353#117C23×C3712yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4C23×C3711yescharacteristic
359C2×C4⋊C4C23×C3711yescharacteristic
360C2×C4⋊C4C23×C3711yescharacteristic
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4D4×C3712yes—
364C2×C42—22no—
365C2×C42C23×C3711yescharacteristic
366C2×C42C23×C3711yescharacteristic
367C2×C42C23×C3711yescharacteristic
368#107—22no—
369#107—22no—
370#106—22no—
371#106—22no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374#104—42no—
375#104—42no—
376#103—22no—
377#103—22no—
378#102—22no—
379#102—22no—
380#100—42no—
381#95—21no—
382#95—22no—
383C24C2×D4×C3712yes—
384#41—22no—
385#41—22no—
386#41—22no—
387#41—22no—
388#41—84no—
389C2×Q8—22no—
390C2×Q8C2×D4×C3712yes—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4C2×D4×C3712yes—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—82no—
410C22×C4C2×D4×C3712yes—
411C22×C4C2×D4×C3711yescharacteristic
412C22×C4C2×D4×C3711yescharacteristic
413C22×C4C24×C3711yescharacteristic
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418C2×C8—22no—
419C2×C8—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—82no—
433C4⋊C4C2×D4×C3712yes—
434C4⋊C4C2×D4×C3712yes—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42C2×D4×C3712yes—
452C42C2×D4×C3712yes—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23C22×D4×C3711yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—82no—
463D4—22no—
464D4—22no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—82no—
470D4—84no—
471D4—84no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#120×C3711yescharacteristic
496C2×C4#120×C3712yes—
497C2×C4#120×C3711yescharacteristic
498C2×C4C22×D4×C3712yes—
499C2×C4H(5,𝔽2)×C3711yescharacteristic
500C2×C42−1+4×C3711yescharacteristic
501C2×C42−1+4×C3712yes—
502C8—42no—
503C8—42no—
504C22—21no—
505C22—21no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—82no—
514C22C2×#120×C3711yescharacteristic
515C22C37×#53611yescharacteristic
516C22C37×#53711yescharacteristic
517C4—22no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4C37×#53612yes—
529C4C37×#53712yes—
530C2—21no—
531C2—21no—
532C2—82no—
533C2—82no—
534C2C37×#261611yescharacteristic
535C2C37×#301811yescharacteristic
536C2C37×#301911yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#3460C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#3080C2semidirect
3. C37×#2566C2semidirect
4. C37×#2061C2semidirect
5. C37×#2060C2semidirect
6. C37×#2040C2semidirect
7. C37×#2039C2semidirect
8. C37×#2037C2semidirect
9. C37×#2037C2semidirect
10. C37×#1806C2semidirect
11. C37×#1800C2semidirect
235. #29707C37direct (N × H)
263. #3080C2×C37semidirect
264. #2566C2×C37semidirect
265. #2061C2×C37semidirect
266. #2060C2×C37semidirect
267. #2040C2×C37semidirect
268. #2039C2×C37semidirect
269. #2037C2×C37semidirect
270. #2037C2×C37semidirect
271. #1806C2×C37semidirect
272. #1800C2×C37semidirect
315. C37#29707direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#34602
C37×#30802
C37×#25662
C37×#20612
C37×#20602
C37×#20402
C37×#20392
C37×#20372
C37×#18062
C37×#18002
#2970737