Finite Group Databasebeta

← #13364964

Subgroups — #13364964

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×#390C211yesmaximal, characteristic
2C2×#89990C211yesmaximal, characteristic
3C4×#6487C211yesmaximal, characteristic, complemented
4#11729882C211yesmaximal, characteristic
5#11729313C211yesmaximal, characteristic, complemented
6#11729310C211yesmaximal, characteristic, complemented
7C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8#11727414C212yesmaximal, complemented
9#11727400C212yesmaximal, complemented
10#11727398C212yesmaximal, complemented
11#11727391C212yesmaximal, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4—21no—
15C22×C7×C22⋊C4—21no—
16C22×C42×C7C2211yescharacteristic
17C22×#1142C2212yes—
18C2×#6487C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic, complemented
20C7×#390—21no—
21C7×#390—21no—
22C7×#390C2212yes—
23C7×#390C2212yes—
24#89990C418yescomplemented
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28#89984C2211yescharacteristic
29#89982—21no—
30C4×C22⋊Dic7—21no—
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6384C2212yescomplemented
34C2×#6382C2212yescomplemented
35C4×C4⋊Dic7C2211yescharacteristic, complemented
36#89705—21no—
37C4×Dic7⋊C4—21no—
38#89701C2211yescharacteristic, complemented
39#89700C2211yescharacteristic, complemented
40C2×C4×C7⋊C8C2212yescomplemented
41#89324C2212yescomplemented
42#89324C2212yescomplemented
43#89319C2212yescomplemented
44#22043—71nomaximal, Sylow
45C25×C7—21no—
46C22×C7×D4—21no—
47C22×C7×D4—22no—
48C22×C7×D4—42no—
49C22×C7×D4—42no—
50C22×C7×D4C2×C414yes—
51C22×C7×D4C2312yes—
52C23×C4×C7—21no—
53C23×C4×C7D411yescharacteristic
54C23×C4×C7D411yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C22×Dic14C2311yescharacteristic
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—42no—
59C2×C7×C4⋊C4C2312yes—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—44no—
65C2×C7×C22⋊C4—44no—
66C2×C42×C7—42no—
67C2×C42×C7D411yescharacteristic
68C2×C42×C7D411yescharacteristic
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×#1142—42no—
74C2×#1142—42no—
75C2×#1142C2×C418yescomplemented
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79#6487—22no—
80#6487C2×C414yes—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C22×C7⋊C8C2312yes—
87C22×C7⋊C8C2312yes—
88C4×Dic14—42no—
89#6406—42no—
90#6406—42no—
91#6406C2×C418yescomplemented
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C22⋊(C7⋊C8)—22no—
98C22⋊(C7⋊C8)C2×C414yes—
99#6384—42no—
100#6382—42no—
101C4×C7⋊C8—42no—
102C2×#450—71no—
103C4×C22⋊Q8—71no—
104C2×#390D711yescharacteristic
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×D4—24no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—48no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4—48no—
132C2×C7×D4C22×C414yes—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—41no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7SD1614yescomplemented
145C22×C4×C7C22×C412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7#4112yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C2×Dic14—22no—
158C2×Dic14—22no—
159C2×Dic14—21no—
160C2×Dic14C22×C412yes—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—42no—
164C7×C22⋊C4—44no—
165C7×C22⋊C4—44no—
166C7×C22⋊C4—44no—
167C7×C22⋊C4—44no—
168C42×C7—22no—
169C42×C7—22no—
170C42×C7—41no—
171C42×C7—41no—
172C22⋊Dic7—24no—
173C22⋊Dic7—42no—
174#1142—48no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7C22×C412yes—
179Dic7⋊C4—24no—
180Dic7⋊C4—42no—
181Dic7⋊C4—42no—
182C4×Dic7—42no—
183C4×Dic7—42no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8C22×C414yes—
190C23×D4C2×D711yescharacteristic
191C22×SD16—72no—
192C2×C22⋊Q8—71no—
193C2×C4×Q8—71no—
194C22×C4⋊C4C2×D711yescharacteristic
195C22×C22⋊C4—21no—
196C22×C22⋊C4—21no—
197C22×C42C2×D711yescharacteristic
198#450—78no—
199C2×#106—72no—
200C2×#103—72no—
201C2×#102—72no—
202C2×#102—72no—
203C2×#102—72no—
204C2×C22⋊C8—71no—
205C2×C22⋊C8—71no—
206C2×C4×C8—72no—
207#391—71no—
208#390—21no—
209#390—21no—
210#390C2×D712yes—
211#390C2×D712yes—
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×D4—24no—
245C7×D4—24no—
246C7×D4—48no—
247C7×D4—42no—
248C7×D4—42no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—44no—
263C2×C4×C7—44no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—84no—
268C2×C4×C7C4×D412yes—
269C2×C4×C7C2×SD1612yes—
270C2×C4×C7C2×SD1612yes—
271C2×C4×C7AGL(1,ℤ/8)12yes—
272C2×C4×C7C23×C411yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7H(5,𝔽2)12yes—
275C2×C4×C7H(5,𝔽2)12yes—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288Dic14—24no—
289Dic14—42no—
290C7⋊C8—24no—
291C7⋊C8—42no—
292C25—21no—
293C22×Q8—71no—
294C22×D4—22no—
295C22×D4—21no—
296C22×D4—42no—
297C22×D4—42no—
298C22×D4C4×D714yes—
299C22×D4C22×D712yes—
300C23×C4—21no—
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22×D711yescharacteristic
304C2×SD16—78no—
305C2×SD16—282no—
306C2×SD16—282no—
307C22×C8—72no—
308C22×C8—72no—
309C22⋊Q8—74no—
310C22⋊Q8—142no—
311C4×Q8—282no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4C22×D712yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4—141no—
329C2×C42—42no—
330C2×C42—141no—
331C2×C42—141no—
332C2×C42C22⋊D711yescharacteristic
333C2×C42C22⋊D711yescharacteristic
334C2×C42C22×D711yescharacteristic
335C2×C42C22×D711yescharacteristic
336#106—282no—
337#103—282no—
338#102—78no—
339#102—282no—
340#102—282no—
341C22⋊C8—74no—
342C22⋊C8—142no—
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—22no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—41no—
376C4×C7—82no—
377C4×C7#51812yes—
378C4×C7#53812yes—
379Dic7—42no—
380Dic7—42no—
381Dic7—81no—
382Dic7—81no—
383C24—24no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C2×C22⋊D711yescharacteristic
392C2×Q8—72no—
393C2×Q8—141no—
394C2×Q8—142no—
395C2×Q8—142no—
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×D4C2×C4×D714yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4#114214yescomplemented
423C22×C4C2×C4×D712yes—
424C22×C4#115712yes—
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431SD16—288no—
432C2×C8—74no—
433C2×C8—142no—
434C2×C8—142no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—72no—
442C4⋊C4—144no—
443C4⋊C4—141no—
444C4⋊C4—142no—
445C4⋊C4—142no—
446C4⋊C4—282no—
447C4⋊C4—282no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—144no—
453C22⋊C4—282no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
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—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—48no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23C4×C22⋊D712yes—
488C23C2×#114212yes—
489C23#651312yes—
490C23C22×C22⋊D711yescharacteristic
491Q8—144no—
492Q8—282no—
493D4—24no—
494D4—24no—
495D4—48no—
496D4—42no—
497D4—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4C4×C22⋊D712yes—
530C2×C4#649512yes—
531C2×C4C2×#114212yes—
532C2×C4C2×#114212yes—
533C2×C4C22×C4×D711yescharacteristic
534C2×C4C2×#115711yescharacteristic
535C2×C4#654812yes—
536C2×C4#654812yes—
537C8—144no—
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—
554C22C4×#114214yes—
555C22C2×C4×C22⋊D711yescharacteristic
556C22C2×#649511yescharacteristic
557C22C22×#114211yescharacteristic
558C22#9049511yescharacteristic
559C22#9049611yescharacteristic
560C22C2×#651311yescharacteristic
561C22#9051311yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—82no—
568C4—282no—
569C4—282no—
570C4—561no—
571C4—561no—
572C4#9017712yes—
573C4#9018512yes—
574C2—24no—
575C2—84no—
576C2#1173954111yescharacteristic
577C2C4×#649511yescharacteristic
578C2C2×C4×#114211yescharacteristic
579C2#1174007111yescharacteristic
580C2#1174007511yescharacteristic
581C2#1174039111yescharacteristic
582C2#1174061411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#390C2
socleC23×C7C22×D4
prime radicalC23×C7×D4C22
cosocleC22×C4×C7C24
hypercenterC2×#390D7
hypocenterC7#22043
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
3. C4×#6487C2semidirect
5. #11729313C2semidirect
6. #11729310C2semidirect
7. C4×C22⋊(C7⋊C8)C2semidirect
8. #11727414C2semidirect
9. #11727400C2semidirect
10. #11727398C2semidirect
11. #11727391C2semidirect
19. C2×C4×Dic14C22semidirect
24. #89990C4semidirect
33. C2×#6384C22semidirect
34. C2×#6382C22semidirect
35. C4×C4⋊Dic7C22semidirect
38. #89701C22semidirect
39. #89700C22semidirect
40. C2×C4×C7⋊C8C22semidirect
41. #89324C22semidirect
42. #89324C22semidirect
43. #89319C22semidirect
75. C2×#1142C2×C4semidirect
91. #6406C2×C4semidirect
144. C22×C4×C7SD16semidirect
422. C22×C4#1142semidirect
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×#3902
C2×#899902
C4×#64872
#117298822
#117293132
#117293102
C4×C22⋊(C7⋊C8)2
#117274142
#117274002
#117273982
#117273912
#220437