Finite Group Databasebeta

← #13321559

Subgroups — #13321559

Order 1792.

1270 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742854C211yesmaximal, characteristic
2C7×#2973C211yesmaximal, characteristic, complemented
3#11739781C211yesmaximal, characteristic, complemented
4#11728589C212yesmaximal, complemented
5#11728586C212yesmaximal, complemented
6#11727732C212yesmaximal, complemented
7#11727731C212yesmaximal, complemented
8#11727698C212yesmaximal, complemented
9C2×#89265C212yesmaximal, complemented
10C2×#6548—21no—
11C2×C4×C7×D4C2211yescharacteristic
12C2×#6493C2212yes—
13C2×#6493C2212yes—
14#90246C414yescomplemented
15#90245—24no—
16D4×D28C414yescomplemented
17#90201—21no—
18#90184—21no—
19#90184—21no—
20#90168C2212yes—
21C7×#435—24no—
22C7×#432C2212yescomplemented
23C7×#407C2212yescomplemented
24C2×C7×C22⋊C8C2212yescomplemented
25#89992—21no—
26#89984C2211yescharacteristic, complemented
27#89983C2211yescharacteristic, complemented
28C4×C22⋊Dic7C2211yescharacteristic, complemented
29C2×C7⋊#95C2211yescharacteristic, complemented
30#89753—21no—
31#89723—21no—
32C22⋊C4×Dic7—21no—
33#89709C2211yescharacteristic, complemented
34#89704C2211yescharacteristic, complemented
35#89265C2214yescomplemented
36#89265C2214yescomplemented
37#89265C2214yescomplemented
38#89265C2214yescomplemented
39#17620—71nomaximal, Sylow
40C22×C7×D4D411yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C22×C22⋊D7—22no—
44C22×C22⋊D7—22no—
45C23×Dic7—21no—
46#6548—44no—
47#6548—44no—
48C2×#1159—24no—
49C2×D4×D7—24no—
50C2×#1157—22no—
51C22×D28C2×C412yes—
52C22×D28C2×C412yescomplemented
53C2×C7×M16—22no—
54C22×C7×C8—22no—
55C4×C7×D4C2×C414yes—
56C4×C7×D4D414yes—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4D411yescharacteristic
59C2×C7×C22⋊C4D411yescharacteristic
60C2×C42×C7C2311yescharacteristic
61C2×C22⋊Dic7—21no—
62C2×C22⋊Dic7—21no—
63C2×C22⋊Dic7—21no—
64#6493—22no—
65#6493—22no—
66#6493—22no—
67#6493—22no—
68#6493—22no—
69#6493C2×C414yes—
70#6493C2×C414yescomplemented
71#6493C2314yes—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4—22no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2×C412yes—
76C2×C4⋊Dic7C2×C412yescomplemented
77C2×C4×Dic7—21no—
78C2×C4×Dic7—21no—
79#6460—24no—
80#6458—24no—
81#6449—24no—
82#6449—24no—
83#6445—24no—
84#6445—24no—
85#6438—22no—
86#6437C2×C412yescomplemented
87C4×D28—22no—
88C4×D28C2×C414yescomplemented
89#6432C2×C412yescomplemented
90C7×C4⋊C8—22no—
91C7×C22⋊C8—22no—
92C7×C22⋊C8—22no—
93C7×C22⋊C8C2314yes—
94C7×#97—22no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95C2311yescharacteristic
101C7⋊#95C2311yescharacteristic
102C7⋊#95C2311yescharacteristic
103C7⋊#95C2311yescharacteristic
104#3496—71no—
105#2973D711yescharacteristic, complemented
106#2353—71no—
107#1519—72no—
108#1514—72no—
109#1510—72no—
110#1507—72no—
111#1505—72no—
112C2×#327—72no—
113C24×C7C2×D411yescharacteristic
114C24×C7C2×D411yescharacteristic
115C23×D7—22no—
116C23×D7—22no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4C22⋊C414yes—
120C22×C4×C7—81no—
121C22×C4×C7—81no—
122C22×C4×C7C22×C412yes—
123C22×C4×C7C22×C412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2411yescharacteristic
131C2×C22⋊D7—24no—
132C2×C22⋊D7—24no—
133C2×C22⋊D7—24no—
134C2×C22⋊D7—44no—
135C2×C22⋊D7—44no—
136C2×C22⋊D7—44no—
137C2×C22⋊D7—44no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—42no—
145C22×Dic7—42no—
146C22×Dic7—81no—
147#1159—416no—
148D4×D7—416no—
149#1157—44no—
150#1157—44no—
151C2×D28—22no—
152C2×D28—22no—
153C2×D28—22no—
154C2×D28—22no—
155C2×D28—82no—
156C2×D28—82no—
157C2×D28C22×C414yes—
158C2×C4×D7—24no—
159C2×Dic14—22no—
160C7×M16—82no—
161C2×C7×C8—24no—
162C2×C7×C8—82no—
163C7×C4⋊C4C22⋊C414yes—
164C7×C22⋊C4C22⋊C414yes—
165C7×C22⋊C4C22⋊C414yes—
166C42×C7C22×C412yes—
167C42×C7C2×D412yes—
168C22⋊Dic7—42no—
169C22⋊Dic7—42no—
170C22⋊Dic7—81no—
171(C2×D7)⋊C4—24no—
172(C2×D7)⋊C4—24no—
173(C2×D7)⋊C4—24no—
174(C2×D7)⋊C4—24no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—81no—
179C4⋊Dic7C22×C414yes—
180C4×Dic7—42no—
181C4×Dic7—81no—
182C2×H(5,𝔽2)—141no—
183#547—74no—
184#546—144no—
185D42—74no—
186#537—141no—
187#534—141no—
188#534—141no—
189#532—72no—
190C2×#121—72no—
191C2×#121—72no—
192C2×C4×D4C2×D711yescharacteristic
193#435—24no—
194#432C2×D712yescomplemented
195#407C2×D712yescomplemented
196C2×C22⋊C8C2×D712yescomplemented
197#388—141no—
198#387—141no—
199#386—141no—
200#383—71no—
201#381—71no—
202#380—71no—
203C4×C22⋊C4—71no—
204C4×C22⋊C4—141no—
205#376—71no—
206C2×#95—71no—
207#327—74no—
208#327—74no—
209#327—74no—
210#327—74no—
211C23×C7—21no—
212C23×C7—21no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7C2×C22⋊C412yes—
218C23×C7C2×C22⋊C412yes—
219C23×C7AGL(1,ℤ/8)12yes—
220C23×C7AGL(1,ℤ/8)12yes—
221C23×C7C22×D411yescharacteristic
222C22×D7—24no—
223C22×D7—44no—
224C22×D7—44no—
225C22×D7—82no—
226C22×D7—82no—
227C22×D7—82no—
228C22×D7—82no—
229C7×D4—28no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—41no—
240C2×C4×C7C2×C22⋊C412yes—
241C2×C4×C7C2×C22⋊C412yes—
242C2×C4×C7C2×C22⋊C412yes—
243C2×C4×C7C2×C22⋊C412yes—
244C2×C4×C7C23×C411yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C22⋊D7—48no—
247C22⋊D7—44no—
248C22⋊D7—44no—
249C22⋊D7—48no—
250C2×Dic7—22no—
251C2×Dic7—24no—
252C2×Dic7—41no—
253C2×Dic7—44no—
254C2×Dic7—41no—
255C2×Dic7—44no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—84no—
266D28—44no—
267D28—44no—
268D28—82no—
269D28—82no—
270D28—82no—
271C4×D7—48no—
272Dic14—44no—
273C7×C8—44no—
274H(5,𝔽2)—284no—
275H(5,𝔽2)—284no—
276C2×#41—144no—
277C2×#41—142no—
278C22×D4—72no—
279C22×D4—72no—
280C22×D4—142no—
281C22×D4—142no—
282C22×D4—144no—
283C22×D4D2811yescharacteristic
284C23×C4—141no—
285C23×C4C22×D711yescharacteristic
286C23×C4C22×D711yescharacteristic
287C2×M16—22no—
288C22×C8—22no—
289C4⋊Q8—72no—
290#127—72no—
291#124—142no—
292#123—144no—
293#123—144no—
294C22⋊Q8—144no—
295#121—74no—
296#121—74no—
297#121—74no—
298#121—142no—
299#121—142no—
300#121—142no—
301#121—142no—
302#121—142no—
303#121—144no—
304#120—144no—
305#120—144no—
306C4×D4—74no—
307C4×D4—142no—
308C4×D4C4×D714yes—
309C4×D4C22⋊D714yes—
310C2×C4⋊C4—72no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4D2811yescharacteristic
314C2×C22⋊C4—142no—
315C2×C22⋊C4—142no—
316C2×C22⋊C4—141no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—141no—
319C2×C22⋊C4D2811yescharacteristic
320C2×C22⋊C4D2811yescharacteristic
321C2×C42—141no—
322C2×C42—141no—
323C2×C42C22×D711yescharacteristic
324C4⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8C22×D714yes—
328#97—22no—
329#95—71no—
330#95—71no—
331#95—71no—
332#95—71no—
333#95—141no—
334#95—141no—
335#95—141no—
336#95—141no—
337#95—141no—
338C22×C7—21no—
339C22×C7—22no—
340C22×C7—21no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7#41814yes—
350C22×C7C22×C22⋊C411yescharacteristic
351C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
352C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
353C2×D7—44no—
354C2×D7—44no—
355C2×D7—44no—
356C2×D7—164no—
357C2×D7—164no—
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—22no—
361C4×C7—22no—
362C4×C7—22no—
363C4×C7—22no—
364Dic7—42no—
365Dic7—44no—
366Dic7—81no—
367Dic7—81no—
368Dic7—81no—
369C24—142no—
370C24—142no—
371C24C2×D2811yescharacteristic
372C24C2×D2811yescharacteristic
373#41—284no—
374#41—284no—
375#41—2816no—
376C2×Q8—142no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—74no—
380C2×D4—144no—
381C2×D4—144no—
382C2×D4—144no—
383C2×D4—142no—
384C2×D4—142no—
385C2×D4—142no—
386C2×D4—142no—
387C2×D4—2816no—
388C2×D4—284no—
389C2×D4—284no—
390C2×D4—284no—
391C2×D4—284no—
392C2×D4—562no—
393C2×D4—562no—
394C2×D4(C2×D7)⋊C414yes—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—141no—
400C22×C4—144no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—141no—
404C22×C4—282no—
405C22×C4—282no—
406C22×C4—561no—
407C22×C4C2×C4×D712yes—
408C22×C4C2×C4×D712yes—
409C22×C4C2×D2811yescharacteristic
410C22×C4C2×D2811yescharacteristic
411C22×C4C2×D2811yescharacteristic
412C22×C4C2×D2811yescharacteristic
413C22×C4C2×C22⋊D712yes—
414C22×C4C2×C22⋊D712yes—
415C22×C4C23×D711yescharacteristic
416M16—82no—
417C2×C8—24no—
418C2×C8—82no—
419C4⋊C4—74no—
420C4⋊C4—142no—
421C4⋊C4—142no—
422C4⋊C4—142no—
423C4⋊C4—561no—
424C4⋊C4(C2×D7)⋊C414yes—
425C22⋊C4—144no—
426C22⋊C4—144no—
427C22⋊C4—144no—
428C22⋊C4—144no—
429C22⋊C4—282no—
430C22⋊C4—282no—
431C22⋊C4—561no—
432C22⋊C4(C2×D7)⋊C414yes—
433C22⋊C4(C2×D7)⋊C414yes—
434C42—282no—
435C42—561no—
436C42C2×C4×D712yes—
437C42C2×C22⋊D712yes—
438C2×C7—21no—
439C2×C7—21no—
440C2×C7—24no—
441C2×C7—42no—
442C2×C7#288311yescharacteristic
443C2×C7C2×#41811yescharacteristic
444C2×C7#290111yescharacteristic
445D7—84no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—144no—
453C23—284no—
454C23—284no—
455C23—562no—
456C23—562no—
457C23—562no—
458C23—562no—
459C23#647112yes—
460C23#647112yes—
461C23C2×(C2×D7)⋊C412yes—
462C23C2×(C2×D7)⋊C412yes—
463C23C22×D2811yescharacteristic
464Q8—284no—
465D4—28no—
466D4—284no—
467D4—284no—
468D4—288no—
469D4—288no—
470D4—284no—
471D4—284no—
472D4—562no—
473D4—562no—
474D4—562no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—142no—
486C2×C4—144no—
487C2×C4—288no—
488C2×C4—281no—
489C2×C4—284no—
490C2×C4—284no—
491C2×C4—281no—
492C2×C4—281no—
493C2×C4—281no—
494C2×C4—281no—
495C2×C4—281no—
496C2×C4—281no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—564no—
502C2×C4C2×(C2×D7)⋊C412yes—
503C2×C4C2×(C2×D7)⋊C412yes—
504C2×C4C2×(C2×D7)⋊C412yes—
505C2×C4C2×(C2×D7)⋊C412yes—
506C2×C4C22×C4×D711yescharacteristic
507C2×C4C22×C22⋊D711yescharacteristic
508C8—44no—
509C7#1762011yescharacteristic, complemented, Sylow
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—284no—
522C22—284no—
523C22—284no—
524C22—1124no—
525C22—1124no—
526C22#8990314yes—
527C22C2×#647111yescharacteristic
528C22C2×#647111yescharacteristic
529C22C22×(C2×D7)⋊C411yescharacteristic
530C4—22no—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—22no—
535C4—22no—
536C4—282no—
537C4—284no—
538C4—561no—
539C4—561no—
540C4—561no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—42no—
545C2—564no—
546C2#1173999111yescharacteristic
547C2C2×#8990311yescharacteristic
548C2#1174126211yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2973C2
socleC22×C7C22×C22⋊C4
prime radical#11742854C2
cosocleC22×C4×C7C24
hypercenter#2973D7
hypocenterC7#17620
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. C7×#2973C2semidirect
3. #11739781C2semidirect
4. #11728589C2semidirect
5. #11728586C2semidirect
6. #11727732C2semidirect
7. #11727731C2semidirect
8. #11727698C2semidirect
9. C2×#89265C2semidirect
14. #90246C4semidirect
16. D4×D28C4semidirect
22. C7×#432C22semidirect
23. C7×#407C22semidirect
24. C2×C7×C22⋊C8C22semidirect
26. #89984C22semidirect
27. #89983C22semidirect
28. C4×C22⋊Dic7C22semidirect
29. C2×C7⋊#95C22semidirect
33. #89709C22semidirect
34. #89704C22semidirect
35. #89265C22semidirect
36. #89265C22semidirect
37. #89265C22semidirect
38. #89265C22semidirect
52. C22×D28C2×C4semidirect
70. #6493C2×C4semidirect
76. C2×C4⋊Dic7C2×C4semidirect
86. #6437C2×C4semidirect
88. C4×D28C2×C4semidirect
89. #6432C2×C4semidirect
105. #2973D7semidirect
194. #432C2×D7semidirect
195. #407C2×D7semidirect
196. C2×C22⋊C8C2×D7semidirect
509. C7#17620semidirect

Maximal subgroups

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

subgroupindex
#117428542
C7×#29732
#117397812
#117285892
#117285862
#117277322
#117277312
#117276982
C2×#892652
#176207