Finite Group Databasebeta

← #13364967

Subgroups — #13364967

Order 1792.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#450C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic
3Dic7×C22⋊Q8C211yesmaximal, characteristic, complemented
4#11729882C211yesmaximal, characteristic
5#11729803C212yesmaximal, complemented
6#11729802C212yesmaximal, complemented
7#11729446C211yesmaximal, characteristic, complemented
8#11727939C212yesmaximal, complemented
9#11727910C212yesmaximal, complemented
10#11727682C211yesmaximal, characteristic, complemented
11Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Dic7—21no—
16C22×C22⋊Dic7—21no—
17C2×Q8×Dic7C2211yescharacteristic, complemented
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4×Dic7C2211yescharacteristic
20C7×#450C418yescomplemented
21C2×C7×#102C2212yes—
22C2×C7×C22⋊C8C2211yescharacteristic
23#90002C2211yescharacteristic, complemented
24C2×#6409C2212yescomplemented
25#89993—21no—
26#89993—21no—
27#89993C2212yes—
28#89993C2212yes—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6391C2212yescomplemented
34C2×C8×Dic7C2212yescomplemented
35#89757C2211yescharacteristic, complemented
36#89753—21no—
37C4⋊C4×Dic7—21no—
38C4⋊C4×Dic7C2211yescharacteristic, complemented
39#89722—21no—
40C22⋊C4×Dic7—21no—
41#89346C2212yescomplemented
42#89345C2212yescomplemented
43#89324C2212yescomplemented
44#22043—71nomaximal, Sylow
45C25×C7—21no—
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4—22no—
49C22×C7×D4—42no—
50C22×C7×D4—42no—
51C22×C7×D4C2×C414yes—
52C22×C7×D4C2312yes—
53C23×C4×C7C2311yescharacteristic
54C23×Dic7—21no—
55C23×Dic7D411yescharacteristic
56C23×Dic7D411yescharacteristic
57C2×C7×SD16—42no—
58C2×C7×SD16—42no—
59C2×C7×SD16C2×C418yescomplemented
60C22×C7×C8C2312yes—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8C2×C414yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C22⋊Dic7—21no—
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—22no—
71C2×C22⋊Dic7—44no—
72C2×C22⋊Dic7—44no—
73Q8×Dic7—42no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—42no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2312yes—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—42no—
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C7×#102C2×C418yescomplemented
90C7×C22⋊C8C2×C414yes—
91#6409—42no—
92#6406—42no—
93#6406—42no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C22⋊(C7⋊C8)—22no—
100#6391—42no—
101C8×Dic7—42no—
102C2×#450D711yescharacteristic
103C4×C22⋊Q8—71no—
104C2×#390—71no—
105#1968—72no—
106#1934—72no—
107#1926—72no—
108#1880—72no—
109#1872—71no—
110#1853—71no—
111#1786—71no—
112C4×C22⋊C8—71no—
113C24×C7—24no—
114C24×C7—22no—
115C24×C7—42no—
116C24×C7—42no—
117C24×C7—42no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—42no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22×C412yes—
126C2×C7×D4—24no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—48no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4—48no—
136C2×C7×D4C22×C414yes—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7C22×C412yes—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7#4112yes—
144C22×C4×C7C2411yescharacteristic
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—41no—
151C22×Dic7—41no—
152C22×Dic7—41no—
153C22×Dic7—42no—
154C22×Dic7—42no—
155C22×Dic7—42no—
156C22×Dic7SD1614yescomplemented
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C7×SD16—48no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8C22×C414yes—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4C22×C412yes—
168C7×C22⋊C4—24no—
169C22⋊Dic7—44no—
170C22⋊Dic7—42no—
171C22⋊Dic7—44no—
172C22⋊Dic7—44no—
173C22⋊Dic7—44no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—42no—
179C4⋊Dic7—42no—
180C4⋊Dic7—42no—
181C4×Dic7—22no—
182C4×Dic7—22no—
183C4×Dic7—41no—
184C4×Dic7—41no—
185C4×Dic7—42no—
186C4×Dic7—42no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C23×D4C2×D711yescharacteristic
191C22×SD16C2×D712yes—
192C2×C22⋊Q8C2×D711yescharacteristic
193C2×C4×Q8—71no—
194C22×C4⋊C4—71no—
195C22×C22⋊C4—141no—
196C22×C22⋊C4—141no—
197C22×C42—71no—
198#450Dic718yescomplemented
199C2×#106—72no—
200C2×#103—72no—
201C2×#102—72no—
202C2×#102—72no—
203C2×#102C2×D712yes—
204C2×C22⋊C8—71no—
205C2×C22⋊C8C2×D711yescharacteristic
206C2×C4×C8—72no—
207#391—71no—
208#390—72no—
209#390—72no—
210#390—141no—
211#390—141no—
212#389—71no—
213#387—141no—
214#385—141no—
215#380—71no—
216C4×C4⋊C4—71no—
217C4×C4⋊C4—141no—
218C4×C22⋊C4—141no—
219#340—72no—
220#340—72no—
221#336—72no—
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—24no—
227C23×C7—48no—
228C23×C7—48no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—48no—
236C23×C7—84no—
237C23×C7—84no—
238C23×C7—84no—
239C23×C7—84no—
240C23×C7C4×D412yes—
241C23×C7C2×SD1612yes—
242C23×C7#13512yes—
243C23×C7C22×D411yescharacteristic
244C7×Q8—24no—
245C7×Q8—42no—
246C7×D4—24no—
247C7×D4—24no—
248C7×D4—48no—
249C7×D4—42no—
250C7×D4—42no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7C4×D412yes—
262C2×C4×C7C23×C411yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C2×C4×C7H(5,𝔽2)12yes—
265C2×C4×C7H(5,𝔽2)12yes—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—21no—
269C2×Dic7—21no—
270C2×Dic7—21no—
271C2×Dic7—21no—
272C2×Dic7—21no—
273C2×Dic7—21no—
274C2×Dic7—21no—
275C2×Dic7—44no—
276C2×Dic7—44no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—41no—
280C2×Dic7—42no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—42no—
285C2×Dic7—41no—
286C2×Dic7—84no—
287C2×Dic7C2×SD1612yes—
288C2×Dic7C2×SD1612yes—
289C2×Dic7AGL(1,ℤ/8)12yes—
290C7×C8—24no—
291C7⋊C8—42no—
292C25—21no—
293C22×Q8C22×D711yescharacteristic
294C22×D4—21no—
295C22×D4—22no—
296C22×D4—42no—
297C22×D4—42no—
298C22×D4C2×Dic714yes—
299C22×D4C22×D712yes—
300C23×C4—71no—
301C23×C4—71no—
302C23×C4—141no—
303C23×C4C22×D711yescharacteristic
304C2×SD16—42no—
305C2×SD16—42no—
306C2×SD16C2×Dic718yescomplemented
307C22×C8—72no—
308C22×C8C22×D712yes—
309C22⋊Q8—22no—
310C22⋊Q8C2×Dic714yes—
311C4×Q8—282no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—282no—
320C2×C4⋊C4C22×D711yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—142no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—284no—
328C2×C22⋊C4—284no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42—71no—
332C2×C42—71no—
333C2×C42—141no—
334C2×C42—141no—
335C2×C42—282no—
336#106—282no—
337#103—282no—
338#102—282no—
339#102—282no—
340#102C2×Dic718yescomplemented
341C22⋊C8—142no—
342C22⋊C8C2×Dic714yes—
343C4×C8—282no—
344#95—141no—
345#95—141no—
346#95—141no—
347#95—141no—
348#95—141no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—44no—
356C22×C7—44no—
357C22×C7—44no—
358C22×C7—816no—
359C22×C7—82no—
360C22×C7—82no—
361C22×C7—82no—
362C22×C7—82no—
363C22×C7C4×SD1614yes—
364C22×C7C2×C4×D411yescharacteristic
365C22×C7#53411yescharacteristic
366C22×C7#53511yescharacteristic
367C22×C7C22×SD1611yescharacteristic
368C22×C7C2×#13511yescharacteristic
369C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C7#57511yescharacteristic
371C4×C7—22no—
372C4×C7—42no—
373C4×C7—42no—
374C4×C7#51812yes—
375C4×C7#53812yes—
376Dic7—22no—
377Dic7—22no—
378Dic7—22no—
379Dic7—41no—
380Dic7—81no—
381Dic7—82no—
382Dic7—81no—
383C24—24no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24D4×D711yescharacteristic
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22×Dic712yes—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4C22×Dic714yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—71no—
412C22×C4—71no—
413C22×C4—71no—
414C22×C4—74no—
415C22×C4—71no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—142no—
421C22×C4—281no—
422C22×C4—281no—
423C22×C4—281no—
424C22×C4—282no—
425C22×C4—282no—
426C22×C4—282no—
427C22×C4D4×D711yescharacteristic
428C22×C4#115912yes—
429C22×C4C22×Dic712yes—
430C22×C4C23×D711yescharacteristic
431SD16—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—142no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8C22×Dic714yes—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—142no—
441C4⋊C4—142no—
442C4⋊C4—142no—
443C4⋊C4—142no—
444C4⋊C4—282no—
445C4⋊C4—282no—
446C4⋊C4—282no—
447C4⋊C4C22×Dic712yes—
448C22⋊C4—24no—
449C22⋊C4—284no—
450C22⋊C4—284no—
451C22⋊C4—284no—
452C22⋊C4—284no—
453C22⋊C4—282no—
454C42—142no—
455C42—142no—
456C42—281no—
457C42—281no—
458C42—282no—
459C42—282no—
460C2×C7—24no—
461C2×C7—84no—
462C2×C7#232111yescharacteristic
463C2×C7C2×C4×SD1611yescharacteristic
464C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
465C2×C7#318711yescharacteristic
466C2×C7#318811yescharacteristic
467C2×C7#320811yescharacteristic
468C2×C7#321111yescharacteristic
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23D7×SD1612yes—
488C23#647912yes—
489C23D4×Dic712yes—
490C23C2×D4×D711yescharacteristic
491Q8—24no—
492Q8—42no—
493D4—24no—
494D4—24no—
495D4—48no—
496D4—42no—
497D4—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—72no—
509C2×C4—72no—
510C2×C4—72no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—141no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—284no—
521C2×C4—284no—
522C2×C4—282no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—282no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—282no—
530C2×C4—281no—
531C2×C4—564no—
532C2×C4D4×Dic712yes—
533C2×C4C2×#115911yescharacteristic
534C2×C4#654812yes—
535C2×C4#654812yes—
536C2×C4C23×Dic711yescharacteristic
537C8—24no—
538C8—282no—
539C7#2204311yescharacteristic, complemented, Sylow
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22SD16×Dic714yes—
555C22#9028011yescharacteristic
556C22#9028311yescharacteristic
557C22C2×D7×SD1611yescharacteristic
558C22C2×#647911yescharacteristic
559C22D7×AGL(1,ℤ/8)11yescharacteristic
560C22#9046411yescharacteristic
561C22C2×D4×Dic711yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—142no—
566C4—142no—
567C4—142no—
568C4—281no—
569C4—562no—
570C4—561no—
571C4—561no—
572C4#9028412yes—
573C4#9048912yes—
574C2—24no—
575C2—84no—
576C2#1173758311yescharacteristic
577C2#1173758411yescharacteristic
578C2#1173759511yescharacteristic
579C2#1173759611yescharacteristic
580C2#1174002311yescharacteristic
581C2C2×SD16×Dic711yescharacteristic
582C2Dic7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#450C2
socleC23×C7C22×D4
prime radicalC23×C7×D4C22
cosocleC22×C4×C7C24
hypercenterC2×#450D7
hypocenterC7#22043
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
3. Dic7×C22⋊Q8C2semidirect
5. #11729803C2semidirect
6. #11729802C2semidirect
7. #11729446C2semidirect
8. #11727939C2semidirect
9. #11727910C2semidirect
10. #11727682C2semidirect
11. Dic7×C22⋊C8C2semidirect
17. C2×Q8×Dic7C22semidirect
20. C7×#450C4semidirect
23. #90002C22semidirect
24. C2×#6409C22semidirect
33. C2×#6391C22semidirect
34. C2×C8×Dic7C22semidirect
35. #89757C22semidirect
38. C4⋊C4×Dic7C22semidirect
41. #89346C22semidirect
42. #89345C22semidirect
43. #89324C22semidirect
59. C2×C7×SD16C2×C4semidirect
89. C7×#102C2×C4semidirect
156. C22×Dic7SD16semidirect
198. #450Dic7semidirect
306. C2×SD16C2×Dic7semidirect
340. #102C2×Dic7semidirect
539. C7#22043semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4502
C2×#899932
Dic7×C22⋊Q82
#117298822
#117298032
#117298022
#117294462
#117279392
#117279102
#117276822
Dic7×C22⋊C82
#220437