Finite Group Databasebeta

← #13321555

Subgroups — #13321555

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
1C7×#3496C211yesmaximal, characteristic
2#11739781C211yesmaximal, characteristic, complemented
3#11739747C211yesmaximal, characteristic, complemented
4#11729567C212yesmaximal, complemented
5#11729566C212yesmaximal, complemented
6#11729429C212yesmaximal, complemented
7#11729428C212yesmaximal, complemented
8#11729427C212yesmaximal, complemented
9C2×#89332C212yesmaximal, complemented
10C2×C7×H(5,𝔽2)—21no—
11C7×#547C414yescomplemented
12C7×#546—24no—
13C7×D42C414yescomplemented
14C7×#537—21no—
15C7×#534—21no—
16C7×#534—21no—
17C7×#532C2212yes—
18C2×C7×#121C2212yes—
19C2×C7×#121C2212yes—
20C2×C4×C7×D4C2211yescharacteristic
21#89992—21no—
22#89984C2211yescharacteristic, complemented
23#89983C2211yescharacteristic, complemented
24C4×C22⋊Dic7C2211yescharacteristic, complemented
25C2×C7⋊#95C2211yescharacteristic, complemented
26#89979C2212yescomplemented
27C2×C22⋊(C7⋊C8)C2212yescomplemented
28#89783—24no—
29#89753—21no—
30#89723—21no—
31C22⋊C4×Dic7—21no—
32#89709C2211yescharacteristic, complemented
33#89704C2211yescharacteristic, complemented
34#89698C2212yescomplemented
35#89332C2214yescomplemented
36#89332C2214yescomplemented
37#89332C2214yescomplemented
38#89332C2214yescomplemented
39#17620—71nomaximal, Sylow
40C7×H(5,𝔽2)—44no—
41C7×H(5,𝔽2)—44no—
42C2×C7×#41—24no—
43C2×C7×#41—22no—
44C22×C7×D4—22no—
45C22×C7×D4—22no—
46C22×C7×D4—24no—
47C22×C7×D4C2×C412yes—
48C22×C7×D4C2×C412yescomplemented
49C22×C7×D4D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×Dic7—21no—
53C7×C4⋊Q8C2×C412yescomplemented
54C7×#127C2×C412yescomplemented
55C7×#124—22no—
56C7×#123—24no—
57C7×#123—24no—
58C7×C22⋊Q8—24no—
59C7×#121—22no—
60C7×#121—22no—
61C7×#121—22no—
62C7×#121—22no—
63C7×#121—22no—
64C7×#121—24no—
65C7×#121C2×C414yes—
66C7×#121C2×C414yescomplemented
67C7×#121C2314yes—
68C7×#120—24no—
69C7×#120—24no—
70C4×C7×D4—22no—
71C4×C7×D4C2×C414yes—
72C4×C7×D4C2×C414yescomplemented
73C4×C7×D4D414yes—
74C2×C7×C4⋊C4C2×C412yes—
75C2×C7×C4⋊C4C2×C412yescomplemented
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C22⋊Dic7—21no—
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×#1136—22no—
89C22×C7⋊C8—22no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95C2311yescharacteristic
96C7⋊#95C2311yescharacteristic
97C7⋊#95C2311yescharacteristic
98C7⋊#95C2311yescharacteristic
99C22⋊(C7⋊C8)—22no—
100C22⋊(C7⋊C8)—22no—
101C22⋊(C7⋊C8)C2314yes—
102C4⋊(C7⋊C8)—22no—
103#6377—22no—
104#3496D711yescharacteristic
105#2973—71no—
106#2353—71no—
107#1519—72no—
108#1514—72no—
109#1510—72no—
110#1507—72no—
111#1505—72no—
112C2×#327—72no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7C2×D411yescharacteristic
116C24×C7C2×D411yescharacteristic
117C7×#41—44no—
118C7×#41—44no—
119C7×#41—416no—
120C2×C7×Q8—22no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—24no—
124C2×C7×D4—24no—
125C2×C7×D4—24no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—416no—
131C2×C7×D4—44no—
132C2×C7×D4—44no—
133C2×C7×D4—44no—
134C2×C7×D4—44no—
135C2×C7×D4—82no—
136C2×C7×D4—82no—
137C2×C7×D4C22⋊C414yes—
138C2×C7×D4C22×C414yes—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—24no—
142C22×C4×C7—81no—
143C22×C4×C7—81no—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C22×C412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C22×Dic7—81no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4C22⋊C414yes—
164C7×C4⋊C4C22×C414yes—
165C7×C22⋊C4—24no—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—24no—
169C7×C22⋊C4C22⋊C414yes—
170C7×C22⋊C4C22⋊C414yes—
171C42×C7C22×C412yes—
172C42×C7C2×D412yes—
173C22⋊Dic7—42no—
174C22⋊Dic7—42no—
175C22⋊Dic7—81no—
176C4⋊Dic7—81no—
177C4×Dic7—42no—
178C4×Dic7—81no—
179#1136—82no—
180C2×C7⋊C8—24no—
181C2×C7⋊C8—82no—
182C2×H(5,𝔽2)—21no—
183#547Dic714yescomplemented
184#546—24no—
185D42Dic714yescomplemented
186#537—21no—
187#534—21no—
188#534—21no—
189#532C2×D712yes—
190C2×#121C2×D712yes—
191C2×#121C2×D712yes—
192C2×C4×D4C2×D711yescharacteristic
193#435—144no—
194#432—72no—
195#407—72no—
196C2×C22⋊C8—72no—
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—24no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7—44no—
219C23×C7—44no—
220C23×C7—82no—
221C23×C7—82no—
222C23×C7—82no—
223C23×C7—82no—
224C23×C7C2×C22⋊C412yes—
225C23×C7C2×C22⋊C412yes—
226C23×C7AGL(1,ℤ/8)12yes—
227C23×C7AGL(1,ℤ/8)12yes—
228C23×C7C22×D411yescharacteristic
229C7×Q8—44no—
230C7×D4—28no—
231C7×D4—44no—
232C7×D4—44no—
233C7×D4—48no—
234C7×D4—44no—
235C7×D4—44no—
236C7×D4—48no—
237C7×D4—82no—
238C7×D4—82no—
239C7×D4—82no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—24no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—48no—
252C2×C4×C7—44no—
253C2×C4×C7—44no—
254C2×C4×C7C2×C22⋊C412yes—
255C2×C4×C7C2×C22⋊C412yes—
256C2×C4×C7C2×C22⋊C412yes—
257C2×C4×C7C2×C22⋊C412yes—
258C2×C4×C7C23×C411yescharacteristic
259C2×C4×C7C22×D411yescharacteristic
260C2×Dic7—22no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—84no—
273C7⋊C8—44no—
274H(5,𝔽2)—44no—
275H(5,𝔽2)—44no—
276C2×#41—24no—
277C2×#41—22no—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—24no—
281C22×D4C2×Dic712yes—
282C22×D4C2×Dic712yescomplemented
283C22×D4C22⋊D711yescharacteristic
284C23×C4—141no—
285C23×C4C22×D711yescharacteristic
286C23×C4C22×D711yescharacteristic
287C2×M16—142no—
288C22×C8—142no—
289C4⋊Q8C2×Dic712yescomplemented
290#127C2×Dic712yescomplemented
291#124—22no—
292#123—24no—
293#123—24no—
294C22⋊Q8—24no—
295#121—22no—
296#121—22no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—24no—
301#121C2×Dic714yes—
302#121C2×Dic714yescomplemented
303#121C22×D714yes—
304#120—24no—
305#120—24no—
306C4×D4—22no—
307C4×D4C2×Dic714yes—
308C4×D4C2×Dic714yescomplemented
309C4×D4C22⋊D714yes—
310C2×C4⋊C4—141no—
311C2×C4⋊C4C2×Dic712yes—
312C2×C4⋊C4C2×Dic712yescomplemented
313C2×C4⋊C4C22⋊D711yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—141no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—141no—
319C2×C22⋊C4C22⋊D711yescharacteristic
320C2×C22⋊C4C22⋊D711yescharacteristic
321C2×C42—141no—
322C2×C42—141no—
323C2×C42C22×D711yescharacteristic
324C4⋊C8—142no—
325C22⋊C8—74no—
326C22⋊C8—142no—
327C22⋊C8—142no—
328#97—142no—
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—44no—
348C22×C7—44no—
349C22×C7—44no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—164no—
353C22×C7—164no—
354C22×C7#41814yes—
355C22×C7C22×C22⋊C411yescharacteristic
356C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
357C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—22no—
361C4×C7—22no—
362C4×C7—22no—
363C4×C7—22no—
364C4×C7—44no—
365Dic7—42no—
366Dic7—81no—
367Dic7—81no—
368Dic7—81no—
369C24—22no—
370C24—22no—
371C24C2×C22⋊D711yescharacteristic
372C24C2×C22⋊D711yescharacteristic
373#41—44no—
374#41—44no—
375#41—416no—
376C2×Q8—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—416no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4C22⋊Dic714yes—
394C2×D4C22×Dic714yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—141no—
404C22×C4—282no—
405C22×C4—282no—
406C22×C4—561no—
407C22×C4C22×Dic712yes—
408C22×C4C22×Dic712yes—
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C2×C22⋊D711yescharacteristic
411C22×C4C2×C22⋊D711yescharacteristic
412C22×C4C2×C22⋊D711yescharacteristic
413C22×C4C2×C22⋊D712yes—
414C22×C4C2×C22⋊D712yes—
415C22×C4C23×D711yescharacteristic
416M16—562no—
417C2×C8—144no—
418C2×C8—562no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—561no—
423C4⋊C4C22⋊Dic714yes—
424C4⋊C4C22×Dic714yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—282no—
430C22⋊C4—282no—
431C22⋊C4—561no—
432C22⋊C4C22⋊Dic714yes—
433C22⋊C4C22⋊Dic714yes—
434C42—282no—
435C42—561no—
436C42C22×Dic712yes—
437C42C2×C22⋊D712yes—
438C2×C7—21no—
439C2×C7—21no—
440C2×C7—24no—
441C2×C7—42no—
442C2×C7—84no—
443C2×C7#288311yescharacteristic
444C2×C7C2×#41811yescharacteristic
445C2×C7#290111yescharacteristic
446C23—21no—
447C23—21no—
448C23—24no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—44no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23#649512yes—
460C23#649512yes—
461C23C2×C22⋊Dic712yes—
462C23C2×C22⋊Dic712yes—
463C23C22×C22⋊D711yescharacteristic
464Q8—44no—
465D4—28no—
466D4—44no—
467D4—44no—
468D4—48no—
469D4—44no—
470D4—44no—
471D4—48no—
472D4—82no—
473D4—82no—
474D4—82no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—142no—
490C2×C4—281no—
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×C22⋊Dic712yes—
503C2×C4C2×C22⋊Dic712yes—
504C2×C4C2×C22⋊Dic712yes—
505C2×C4C2×C22⋊Dic712yes—
506C2×C4C23×Dic711yescharacteristic
507C2×C4C22×C22⋊D711yescharacteristic
508C8—284no—
509C7#1762011yescharacteristic, complemented, Sylow
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—81no—
523C22—81no—
524C22—164no—
525C22—164no—
526C22#8998714yes—
527C22C2×#649511yescharacteristic
528C22C2×#649511yescharacteristic
529C22C22×C22⋊Dic711yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—44no—
537C4—282no—
538C4—561no—
539C4—561no—
540C4—561no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—42no—
545C2—84no—
546C2C2×#8998711yescharacteristic
547C2#1174155211yescharacteristic
548C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#3496C2
socleC22×C7C22×C22⋊C4
cosocleC22×C4×C7C24
hypercenter#3496D7
hypocenterC7#17620
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3496

Complements

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

NH ≅ G/Ntype
2. #11739781C2semidirect
3. #11739747C2semidirect
4. #11729567C2semidirect
5. #11729566C2semidirect
6. #11729429C2semidirect
7. #11729428C2semidirect
8. #11729427C2semidirect
9. C2×#89332C2semidirect
11. C7×#547C4semidirect
13. C7×D42C4semidirect
22. #89984C22semidirect
23. #89983C22semidirect
24. C4×C22⋊Dic7C22semidirect
25. C2×C7⋊#95C22semidirect
26. #89979C22semidirect
27. C2×C22⋊(C7⋊C8)C22semidirect
32. #89709C22semidirect
33. #89704C22semidirect
34. #89698C22semidirect
35. #89332C22semidirect
36. #89332C22semidirect
37. #89332C22semidirect
38. #89332C22semidirect
48. C22×C7×D4C2×C4semidirect
53. C7×C4⋊Q8C2×C4semidirect
54. C7×#127C2×C4semidirect
66. C7×#121C2×C4semidirect
72. C4×C7×D4C2×C4semidirect
75. C2×C7×C4⋊C4C2×C4semidirect
183. #547Dic7semidirect
185. D42Dic7semidirect
282. C22×D4C2×Dic7semidirect
289. C4⋊Q8C2×Dic7semidirect
290. #127C2×Dic7semidirect
302. #121C2×Dic7semidirect
308. C4×D4C2×Dic7semidirect
312. C2×C4⋊C4C2×Dic7semidirect
509. C7#17620semidirect

Maximal subgroups

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

subgroupindex
C7×#34962
#117397812
#117397472
#117295672
#117295662
#117294292
#117294282
#117294272
C2×#893322
#176207