Finite Group Databasebeta

← C37×#28633 #22649886

Subgroups — C37×#28633 #22649886

Order 9472.

772 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3153C211yesmaximal, characteristic
2C37×#3142C211yesmaximal, characteristic, complemented
3C2×C37×#488C211yesmaximal, characteristic, complemented
4C37×#2931C211yesmaximal, characteristic, complemented
5C37×#2052C212yesmaximal, complemented
6C37×#2051C212yesmaximal, complemented
7C37×#1998C211yesmaximal, characteristic, complemented
8C37×#1967C211yesmaximal, characteristic, complemented
9C37×#1908C212yesmaximal, complemented
10C37×#1907C212yesmaximal, complemented
11C4×#108×C37C211yesmaximal, characteristic, complemented
12C2×#137×C37C2212yes—
13C2×AGL(1,ℤ/8)×C37C2212yes—
14C2×#131×C37C2212yes—
15C37×#532C2211yescharacteristic
16C2×#125×C37C2211yescharacteristic
17C2×#124×C37C2211yescharacteristic
18C37×#497—21no—
19C37×#496—21no—
20C37×#495—22no—
21C37×#491—21no—
22C37×#490—21no—
23C37×#488—22no—
24C37×#488—21no—
25C37×#488—21no—
26C37×#488—21no—
27C37×#488—21no—
28C37×#436—21no—
29C37×#436—21no—
30C8×D4×C37—21no—
31C8×D4×C37—21no—
32C2×#108×C37C2212yes—
33C37×#423C2211yescharacteristic
34C37×#417C2212yes—
35C37×#417C2212yes—
36C2×#103×C37C2211yescharacteristic
37C2×#103×C37C2211yescharacteristic
38C2×#102×C37C2211yescharacteristic
39C2×#102×C37C2211yescharacteristic
40C2×#101×C37C2212yes—
41C2×#100×C37C2212yes—
42C37×#408C2212yes—
43C4×M16×C37C2211yescharacteristic
44C4×M16×C37C2211yescharacteristic
45C2×C4×C8×C37C2211yescharacteristic
46C37×#343C2212yes—
47C37×#343C2212yes—
48C37×#341C2212yes—
49C2×#41×C37C2312yes—
50C22×Q8×C37C2311yescharacteristic
51C22×D4×C37C2311yescharacteristic
52#137×C37—24no—
53#137×C37—42no—
54AGL(1,ℤ/8)×C37—24no—
55AGL(1,ℤ/8)×C37—42no—
56C2×Q16×C37—22no—
57C2×SD16×C37—22no—
58C2×SD16×C37—22no—
59C2×D8×C37—22no—
60#131×C37—42no—
61#131×C37—42no—
62C2×M16×C37—22no—
63C2×M16×C37C2312yes—
64C2×M16×C37C2312yes—
65C2×M16×C37C2312yes—
66C22×C8×C37—22no—
67C22×C8×C37C2311yescharacteristic
68C22×C8×C37C2311yescharacteristic
69C4⋊Q8×C37D411yescharacteristic
70#127×C37D411yescharacteristic
71#125×C37—21no—
72#125×C37—21no—
73#125×C37—21no—
74#125×C37—21no—
75#124×C37—21no—
76#124×C37—21no—
77#124×C37—21no—
78#124×C37—21no—
79#124×C37D412yes—
80#121×C37—21no—
81#121×C37—21no—
82C4×D4×C37—21no—
83C4×D4×C37—21no—
84C2×C4⋊C4×C37—21no—
85C2×C4⋊C4×C37—21no—
86C2×C4⋊C4×C37C2311yescharacteristic
87C2×C4⋊C4×C37C2311yescharacteristic
88C2×C22⋊C4×C37—21no—
89C2×C22⋊C4×C37—21no—
90C2×C42×C37C2311yescharacteristic
91#108×C37—24no—
92C4⋊C8×C37—21no—
93C4⋊C8×C37—21no—
94#103×C37—22no—
95#103×C37—22no—
96#103×C37—21no—
97#103×C37—21no—
98#103×C37—21no—
99#103×C37—21no—
100#102×C37—22no—
101#102×C37—22no—
102#102×C37—21no—
103#102×C37—21no—
104#102×C37—21no—
105#102×C37—21no—
106#101×C37—42no—
107#100×C37—42no—
108C22⋊C8×C37—22no—
109C22⋊C8×C37—22no—
110#97×C37—21no—
111#97×C37—21no—
112C4×C8×C37—21no—
113C4×C8×C37—21no—
114C4×C8×C37D411yescharacteristic
115C4×C8×C37D411yescharacteristic
116C4×C8×C37D411yescharacteristic
117C4×C8×C37D411yescharacteristic
118C24×C37—21no—
119#41×C37—22no—
120#41×C37—22no—
121#41×C37—22no—
122#41×C37—22no—
123C2×Q8×C37—21no—
124C2×Q8×C37—21no—
125C2×Q8×C37—22no—
126C2×Q8×C37—22no—
127C2×Q8×C37C2×D412yes—
128C2×D4×C37—22no—
129C2×D4×C37—21no—
130C2×D4×C37—21no—
131C2×D4×C37—21no—
132C2×D4×C37—21no—
133C2×D4×C37—22no—
134C2×D4×C37—22no—
135C2×D4×C37C2×D412yes—
136C22×C4×C37—22no—
137C22×C4×C37—21no—
138C22×C4×C37—21no—
139C22×C4×C37—21no—
140C22×C4×C37#4111yescharacteristic
141C22×C4×C37#4111yescharacteristic
142C22×C4×C37C2411yescharacteristic
143Q16×C37—44no—
144SD16×C37—44no—
145SD16×C37—44no—
146D8×C37—44no—
147M16×C37—22no—
148M16×C37—22no—
149M16×C37—22no—
150M16×C37—22no—
151M16×C37—22no—
152M16×C37—44no—
153M16×C37#4114yes—
154C2×C8×C37—22no—
155C2×C8×C37—22no—
156C2×C8×C37—22no—
157C2×C8×C37—22no—
158C2×C8×C37—22no—
159C2×C8×C37—44no—
160C2×C8×C37C2×D411yescharacteristic
161C2×C8×C37C2×D411yescharacteristic
162C2×C8×C37C2×D411yescharacteristic
163C2×C8×C37C2×D411yescharacteristic
164C4⋊C4×C37—21no—
165C4⋊C4×C37—21no—
166C4⋊C4×C37—21no—
167C4⋊C4×C37—21no—
168C4⋊C4×C37—21no—
169C4⋊C4×C37—21no—
170C4⋊C4×C37—41no—
171C4⋊C4×C37—41no—
172C4⋊C4×C37—41no—
173C4⋊C4×C37—41no—
174C22⋊C4×C37—22no—
175C22⋊C4×C37—22no—
176C22⋊C4×C37—41no—
177C22⋊C4×C37—41no—
178C22⋊C4×C37—41no—
179C22⋊C4×C37—41no—
180C42×C37C2×D411yescharacteristic
181C42×C37C2×D411yescharacteristic
182C42×C37C2×D411yescharacteristic
183C42×C37C2×D411yescharacteristic
184C23×C37—22no—
185C23×C37—41no—
186C23×C37—42no—
187C23×C37—81no—
188C23×C37—81no—
189C23×C37C2×#4111yescharacteristic
190Q8×C37—22no—
191Q8×C37—22no—
192Q8×C37—41no—
193Q8×C37—41no—
194Q8×C37—42no—
195D4×C37—22no—
196D4×C37—22no—
197D4×C37—42no—
198D4×C37—42no—
199D4×C37—42no—
200D4×C37—42no—
201D4×C37—41no—
202D4×C37—41no—
203D4×C37—42no—
204C2×C4×C37—22no—
205C2×C4×C37—41no—
206C2×C4×C37—41no—
207C2×C4×C37—42no—
208C2×C4×C37—42no—
209C2×C4×C37—41no—
210C2×C4×C37—41no—
211C2×C4×C37—41no—
212C2×C4×C37—42no—
213C2×C4×C37—81no—
214C2×C4×C37—81no—
215C2×C4×C37#12111yescharacteristic
216C2×C4×C37#12111yescharacteristic
217C2×C4×C37#12111yescharacteristic
218C2×C4×C37#12111yescharacteristic
219C2×C4×C37C22×D411yescharacteristic
220C2×C4×C37C22×D411yescharacteristic
221C2×C4×C37C2×#4112yes—
222C2×C4×C37H(5,𝔽2)12yes—
223C8×C37—21no—
224C8×C37—21no—
225C8×C37—21no—
226C8×C37—21no—
227C8×C37—24no—
228C8×C37—42no—
229C8×C37—42no—
230#28633C3711yesmaximal, characteristic, direct product factor, Sylow
231C22×C37—21no—
232C22×C37—21no—
233C22×C37—42no—
234C22×C37—42no—
235C22×C37—42no—
236C22×C37—81no—
237C22×C37—81no—
238C22×C37—81no—
239C22×C37—162no—
240C22×C37C2×#12111yescharacteristic
241C22×C37#55912yes—
242C4×C37—21no—
243C4×C37—21no—
244C4×C37—21no—
245C4×C37—21no—
246C4×C37—42no—
247C4×C37—81no—
248C4×C37—81no—
249C4×C37—81no—
250C4×C37#53211yescharacteristic
251C4×C37#53511yescharacteristic
252C4×C37#54612yes—
253#3153C2×C3711yescharacteristic
254#3142C2×C3711yescharacteristic, complemented
255C2×#488C2×C3711yescharacteristic, complemented
256#2931C2×C3711yescharacteristic, complemented
257#2052C2×C3712yescomplemented
258#2051C2×C3712yescomplemented
259#1998C2×C3711yescharacteristic, complemented
260#1967C2×C3711yescharacteristic, complemented
261#1908C2×C3712yescomplemented
262#1907C2×C3712yescomplemented
263C4×#108C2×C3711yescharacteristic, complemented
264C2×C37—22no—
265C2×C37—82no—
266C2×C37—161no—
267C2×C37#254411yescharacteristic
268C2×C37#306711yescharacteristic
269C2×C37#306711yescharacteristic
270C2×#137C22×C3712yes—
271C2×AGL(1,ℤ/8)C22×C3712yes—
272C2×#131C22×C3712yes—
273#532C22×C3711yescharacteristic
274C2×#125C22×C3711yescharacteristic
275C2×#124C22×C3711yescharacteristic
276#497—21no—
277#496—21no—
278#495—22no—
279#491—21no—
280#490—21no—
281#488—22no—
282#488—21no—
283#488—21no—
284#488—21no—
285#488—21no—
286#436—21no—
287#436—21no—
288C8×D4—21no—
289C8×D4—21no—
290C2×#108C22×C3712yes—
291#423C22×C3711yescharacteristic
292#417C22×C3712yes—
293#417C22×C3712yes—
294C2×#103C22×C3711yescharacteristic
295C2×#103C22×C3711yescharacteristic
296C2×#102C22×C3711yescharacteristic
297C2×#102C22×C3711yescharacteristic
298C2×#101C22×C3712yes—
299C2×#100C22×C3712yes—
300#408C22×C3712yes—
301C4×M16C22×C3711yescharacteristic
302C4×M16C22×C3711yescharacteristic
303C2×C4×C8C22×C3711yescharacteristic
304#343C22×C3712yes—
305#343C22×C3712yes—
306#341C22×C3712yes—
307C37#2863311yescharacteristic, direct product factor, Sylow
308C2×#41C23×C3712yes—
309C22×Q8C23×C3711yescharacteristic
310C22×D4C23×C3711yescharacteristic
311#137—24no—
312#137—42no—
313AGL(1,ℤ/8)—24no—
314AGL(1,ℤ/8)—42no—
315C2×Q16—22no—
316C2×SD16—22no—
317C2×SD16—22no—
318C2×D8—22no—
319#131—42no—
320#131—42no—
321C2×M16—22no—
322C2×M16C23×C3712yes—
323C2×M16C23×C3712yes—
324C2×M16C23×C3712yes—
325C22×C8—22no—
326C22×C8C23×C3711yescharacteristic
327C22×C8C23×C3711yescharacteristic
328C4⋊Q8D4×C3711yescharacteristic
329#127D4×C3711yescharacteristic
330#125—21no—
331#125—21no—
332#125—21no—
333#125—21no—
334#124—21no—
335#124—21no—
336#124—21no—
337#124—21no—
338#124D4×C3712yes—
339#121—21no—
340#121—21no—
341C4×D4—21no—
342C4×D4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C23×C3711yescharacteristic
346C2×C4⋊C4C23×C3711yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C42C23×C3711yescharacteristic
350#108—24no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353#103—22no—
354#103—22no—
355#103—21no—
356#103—21no—
357#103—21no—
358#103—21no—
359#102—22no—
360#102—22no—
361#102—21no—
362#102—21no—
363#102—21no—
364#102—21no—
365#101—42no—
366#100—42no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369#97—21no—
370#97—21no—
371C4×C8—21no—
372C4×C8—21no—
373C4×C8D4×C3711yescharacteristic
374C4×C8D4×C3711yescharacteristic
375C4×C8D4×C3711yescharacteristic
376C4×C8D4×C3711yescharacteristic
377C24—21no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8C2×D4×C3712yes—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4C2×D4×C3712yes—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4#41×C3711yescharacteristic
400C22×C4#41×C3711yescharacteristic
401C22×C4C24×C3711yescharacteristic
402Q16—44no—
403SD16—44no—
404SD16—44no—
405D8—44no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—22no—
411M16—44no—
412M16#41×C3714yes—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—44no—
419C2×C8C2×D4×C3711yescharacteristic
420C2×C8C2×D4×C3711yescharacteristic
421C2×C8C2×D4×C3711yescharacteristic
422C2×C8C2×D4×C3711yescharacteristic
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42C2×D4×C3711yescharacteristic
440C42C2×D4×C3711yescharacteristic
441C42C2×D4×C3711yescharacteristic
442C42C2×D4×C3711yescharacteristic
443C23—22no—
444C23—41no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23C2×#41×C3711yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453Q8—42no—
454D4—22no—
455D4—22no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—41no—
461D4—41no—
462D4—42no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#121×C3711yescharacteristic
475C2×C4#121×C3711yescharacteristic
476C2×C4#121×C3711yescharacteristic
477C2×C4#121×C3711yescharacteristic
478C2×C4C22×D4×C3711yescharacteristic
479C2×C4C22×D4×C3711yescharacteristic
480C2×C4C2×#41×C3712yes—
481C2×C4H(5,𝔽2)×C3712yes—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—24no—
487C8—42no—
488C8—42no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—162no—
498C22C2×#121×C3711yescharacteristic
499C22C37×#55912yes—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4C37×#53211yescharacteristic
509C4C37×#53511yescharacteristic
510C4C37×#54612yes—
511C2—22no—
512C2—82no—
513C2—161no—
514C2C37×#254411yescharacteristic
515C2C37×#306711yescharacteristic
516C2C37×#306711yescharacteristic
subgroupG/H
center, socleC22×C37C2×#121
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC37×#3153C2
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×#3142C2semidirect
3. C2×C37×#488C2semidirect
4. C37×#2931C2semidirect
5. C37×#2052C2semidirect
6. C37×#2051C2semidirect
7. C37×#1998C2semidirect
8. C37×#1967C2semidirect
9. C37×#1908C2semidirect
10. C37×#1907C2semidirect
11. C4×#108×C37C2semidirect
230. #28633C37direct (N × H)
254. #3142C2×C37semidirect
255. C2×#488C2×C37semidirect
256. #2931C2×C37semidirect
257. #2052C2×C37semidirect
258. #2051C2×C37semidirect
259. #1998C2×C37semidirect
260. #1967C2×C37semidirect
261. #1908C2×C37semidirect
262. #1907C2×C37semidirect
263. C4×#108C2×C37semidirect
307. C37#28633direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#31532
C37×#31422
C2×C37×#4882
C37×#29312
C37×#20522
C37×#20512
C37×#19982
C37×#19672
C37×#19082
C37×#19072
C4×#108×C372
#2863337