Finite Group Databasebeta

← #13323937

Subgroups — #13323937

Order 1792.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737410C211yesmaximal, characteristic, complemented
2#11737096C211yesmaximal, characteristic
3#11736871C211yesmaximal, characteristic
4C7×#1582C211yesmaximal, characteristic, complemented
5#11728708C211yesmaximal, characteristic, complemented
6SD16×C7⋊C8C211yesmaximal, characteristic, complemented
7#11728597C211yesmaximal, characteristic, complemented
8#11726287C211yesmaximal, characteristic, complemented
9#11726270C211yesmaximal, characteristic, complemented
10#11725989C211yesmaximal, characteristic, complemented
11#11725968C211yesmaximal, characteristic, complemented
12#11725929C211yesmaximal, characteristic
13#11725904C211yesmaximal, characteristic, complemented
14#11725833C211yesmaximal, characteristic, complemented
15#11725826C211yesmaximal, characteristic, complemented
16#90269C2211yescharacteristic
17C4×D4×D7C2211yescharacteristic
18C7×#446C2211yescharacteristic, complemented
19C4×C7×SD16C2211yescharacteristic
20C7×C8×D4C2211yescharacteristic
21#89941C411yescharacteristic
22#89934C411yescharacteristic
23C8×C22⋊D7—21no—
24C2×(C2×D7)⋊C8—21no—
25C4×#1143C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic, complemented
27C4×#1142C2211yescharacteristic
28D4×C7⋊C8C2211yescharacteristic
29#89642C411yescharacteristic
30#89634C411yescharacteristic
31#89612—21no—
32#89607C2211yescharacteristic
33#89606C2211yescharacteristic
34#89605C2211yescharacteristic
35#89603—21no—
36#89602C2211yescharacteristic, complemented
37#89590C411yescharacteristic
38#89577C411yescharacteristic
39#89548C411yescharacteristic
40#89533C411yescharacteristic
41(C22⋊D7)⋊C8—21no—
42D7×C22⋊C8—21no—
43#89481—21no—
44#89473—21no—
45C4×#1132C2211yescharacteristic
46C4×#1131C2211yescharacteristic
47C8×D28C2211yescharacteristic
48#89457C2211yescharacteristic
49#89457C2211yescharacteristic
50#89451C2211yescharacteristic, complemented
51C7×#334C2211yescharacteristic, complemented
52C7×Q8⋊C8C2211yescharacteristic, complemented
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic, complemented
55Q8⋊(C7⋊C8)C2211yescharacteristic, complemented
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic, complemented
58#89278C2211yescharacteristic
59#89274C2211yescharacteristic, complemented
60#89274C2211yescharacteristic, complemented
61#89254C2211yescharacteristic
62#89252C2211yescharacteristic, complemented
63#89251C2211yescharacteristic, complemented
64#89249C2211yescharacteristic, complemented
65#89249C2211yescharacteristic, complemented
66C8×C7⋊C8C2211yescharacteristic, complemented
67#17924—71nomaximal, Sylow
68C2×D4×D7C2×C411yescharacteristic
69C22×C4×D7—21no—
70C2×C7×SD16C2×C411yescharacteristic
71C22×C7×C8—21no—
72C4×C7×Q8C2311yescharacteristic
73C4×C7×D4C2311yescharacteristic
74#6509C2×C411yescharacteristic
75#6507C2×C411yescharacteristic
76C2×#1143C2×C411yescharacteristic
77D4×Dic7C2×C411yescharacteristic
78C2×#1142C2×C411yescharacteristic
79C4×C22⋊D7—21no—
80C22×C7⋊C8—21no—
81C2×#1132C2×C411yescharacteristic
82C2×#1131—21no—
83C2×C8×D7—21no—
84C2×C8×D7—21no—
85#6460C2×C411yescharacteristic
86#6459—21no—
87#6456C2×C411yescharacteristic
88D7×C4⋊C4C2×C411yescharacteristic
89#6452—21no—
90#6451C2×C411yescharacteristic
91(C22⋊D7)⋊C4—21no—
92D7×C22⋊C4—21no—
93C4×D28C2311yescharacteristic
94#6435—21no—
95C42×D7C2311yescharacteristic
96C4×Dic14—21no—
97C4×Dic14C2311yescharacteristic
98C7×#106C2×C411yescharacteristic
99C7×C4⋊C8—21no—
100C7×C4⋊C8C2311yescharacteristic
101C7×C4⋊C8C2311yescharacteristic
102C7×#103C2×C411yescharacteristic
103C7×#102C2×C411yescharacteristic
104C7×C22⋊C8—21no—
105C7×#97—21no—
106C4×C7×C8C2311yescharacteristic
107C4×C7×C8C2311yescharacteristic
108C4×C7×C8C2311yescharacteristic
109#6409C2×C411yescharacteristic
110#6406C2×C411yescharacteristic
111C22⋊(C7⋊C8)—21no—
112#6395C2×C411yescharacteristic
113(C2×D7)⋊C8—21no—
114(C2×D7)⋊C8—21no—
115(C2×D7)⋊C8—21no—
116(C2×D7)⋊C8—21no—
117(C2×D7)⋊C8C2×C411yescharacteristic
118(C2×D7)⋊C8C2×C411yescharacteristic
119(C2×D7)⋊C8D411yescharacteristic
120(C2×D7)⋊C8D411yescharacteristic
121#6391C2×C411yescharacteristic
122#6390C2×C411yescharacteristic
123#6389—21no—
124#6389—21no—
125#6389—21no—
126Dic7⋊C8—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8C2×C411yescharacteristic
129Dic7⋊C8C2×C411yescharacteristic
130Dic7⋊C8D411yescharacteristic
131Dic7⋊C8D411yescharacteristic
132C8×Dic7—21no—
133C8×Dic7—21no—
134#6384C2×C411yescharacteristic
135#6383C2×C411yescharacteristic
136#6382C2×C411yescharacteristic
137#6382C2×C411yescharacteristic
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2311yescharacteristic
140C4⋊(C7⋊C8)C2311yescharacteristic
141C4×C7⋊C8—21no—
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3101—71no—
146#2985—71no—
147#2972—71no—
148#1585—71no—
149#1585—71no—
150#1582—71no—
151#1582D711yescharacteristic, complemented
152SD16⋊C8—71no—
153SD16⋊C8—71no—
154C8×SD16—71no—
155C8×SD16—71no—
156#1572—71no—
157#1490—71no—
158#1485—71no—
159#1453—71no—
160C23×D7—21no—
161C2×C7×Q8C22×C411yescharacteristic
162C2×C7×D4C22×C411yescharacteristic
163C22×C4×C7—21no—
164C2×C22⋊D7—21no—
165C22×Dic7—21no—
166D4×D7—22no—
167D4×D7—22no—
168C2×D28C22×C411yescharacteristic
169C2×C4×D7—21no—
170C2×C4×D7—21no—
171C2×C4×D7—21no—
172C2×C4×D7—81no—
173C2×C4×D7—81no—
174C2×C4×D7C22×C411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7#4111yescharacteristic
177C2×Dic14—21no—
178C2×Dic14C22×C411yescharacteristic
179C7×SD16—22no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—81no—
184C2×C7×C8C22×C411yescharacteristic
185C2×C7×C8C22×C411yescharacteristic
186C2×C7×C8C2×D411yescharacteristic
187C2×C7×C8C2×D411yescharacteristic
188C7×C4⋊C4—21no—
189C7×C4⋊C4C22×C411yescharacteristic
190C7×C4⋊C4C22×C411yescharacteristic
191C7×C22⋊C4—21no—
192C42×C7—21no—
193C42×C7C2411yescharacteristic
194C22⋊Dic7—21no—
195#1143—22no—
196#1142—22no—
197(C2×D7)⋊C4—21no—
198(C2×D7)⋊C4—21no—
199(C2×D7)⋊C4—21no—
200(C2×D7)⋊C4—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7C22×C411yescharacteristic
203C4⋊Dic7C22×C411yescharacteristic
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206Dic7⋊C4—21no—
207Dic7⋊C4—21no—
208Dic7⋊C4—21no—
209Dic7⋊C4—21no—
210C4×Dic7—21no—
211C4×Dic7C22×C411yescharacteristic
212C4×Dic7C2×D411yescharacteristic
213C4×Dic7#4111yescharacteristic
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—81no—
218C2×C7⋊C8C22×C411yescharacteristic
219C2×C7⋊C8C22×C411yescharacteristic
220C2×C7⋊C8C2×D411yescharacteristic
221C2×C7⋊C8C2×D411yescharacteristic
222#1132—22no—
223#1131—42no—
224C8×D7—81no—
225C8×D7—81no—
226#533—71no—
227C2×C4×D4—71no—
228#483—71no—
229#483—71no—
230#476—71no—
231#476—71no—
232#461—71no—
233#461—71no—
234#450—71no—
235#450—71no—
236#446—71no—
237#446—141no—
238#446—141no—
239#446C2×D711yescharacteristic, complemented
240C8×Q8—71no—
241C8×Q8—71no—
242C4×SD16—71no—
243C4×SD16—71no—
244C4×SD16—71no—
245C4×SD16C2×D711yescharacteristic
246C8×D4—71no—
247C8×D4—71no—
248C8×D4—71no—
249C8×D4—141no—
250C8×D4—141no—
251C8×D4C2×D711yescharacteristic
252#433—141no—
253#433—141no—
254#432—71no—
255#431—71no—
256#431—71no—
257#431—71no—
258C2×C22⋊C8—141no—
259C2×C22⋊C8—141no—
260C4×M16—71no—
261#335—71no—
262#335—71no—
263#334—71no—
264#334C2×D711yescharacteristic, complemented
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8—71no—
268Q8⋊C8C2×D711yescharacteristic, complemented
269D4⋊C8—71no—
270D4⋊C8—71no—
271D4⋊C8—71no—
272D4⋊C8C2×D711yescharacteristic
273#322—71no—
274#322—71no—
275#322C2×D711yescharacteristic, complemented
276C82—71no—
277C23×C7—21no—
278C22×D7—21no—
279C22×D7—24no—
280C22×D7—81no—
281C22×D7—81no—
282C22×D7C4×D411yescharacteristic
283C7×Q8—21no—
284C7×Q8—21no—
285C7×D4—21no—
286C7×D4#13112yes—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7C23×C411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C22⋊D7—24no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—81no—
300C2×Dic7C4×D411yescharacteristic
301C2×Dic7C4×D411yescharacteristic
302C2×Dic7C4×D411yescharacteristic
303D28—21no—
304D28#13112yes—
305C4×D7—21no—
306C4×D7—21no—
307C4×D7—21no—
308C4×D7—21no—
309C4×D7—41no—
310C4×D7—41no—
311C4×D7—41no—
312C4×D7—81no—
313C4×D7—81no—
314Dic14—21no—
315Dic14—21no—
316Dic14—81no—
317C7×C8—21no—
318C7×C8—21no—
319C7×C8—21no—
320C7×C8—21no—
321C7×C8—41no—
322C7×C8—41no—
323C7×C8—41no—
324C7⋊C8—22no—
325C7⋊C8—21no—
326C7⋊C8—21no—
327C7⋊C8—21no—
328C7⋊C8—21no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C22×D4—71no—
332C23×C4—141no—
333C2×SD16—71no—
334C2×SD16—71no—
335C2×SD16—71no—
336C2×SD16C4×D711yescharacteristic
337C2×M16—141no—
338C22×C8—21no—
339C22×C8—141no—
340C22×C8—141no—
341C22×C8—141no—
342C4⋊Q8—71no—
343C4⋊Q8—71no—
344#125—141no—
345C22⋊Q8—71no—
346C22⋊Q8—71no—
347C22⋊Q8—141no—
348C4×Q8—71no—
349C4×Q8—141no—
350C4×Q8C22×D711yescharacteristic
351C4×D4—71no—
352C4×D4—71no—
353C4×D4—71no—
354C4×D4—141no—
355C4×D4—141no—
356C4×D4C22×D711yescharacteristic
357#117—141no—
358C2×C4⋊C4—71no—
359C2×C22⋊C4—141no—
360C2×C42—71no—
361#106—71no—
362#106—71no—
363#106—71no—
364#106C4×D711yescharacteristic
365C4⋊C8—21no—
366C4⋊C8—71no—
367C4⋊C8—71no—
368C4⋊C8—71no—
369C4⋊C8—71no—
370C4⋊C8—71no—
371C4⋊C8—71no—
372C4⋊C8—141no—
373C4⋊C8—141no—
374C4⋊C8—141no—
375C4⋊C8C22×D711yescharacteristic
376C4⋊C8C22×D711yescharacteristic
377#103—71no—
378#103—71no—
379#103—71no—
380#103C4×D711yescharacteristic
381#102—71no—
382#102—71no—
383#102—71no—
384#102C4×D711yescharacteristic
385C22⋊C8—21no—
386C22⋊C8—71no—
387C22⋊C8—71no—
388C22⋊C8—71no—
389C22⋊C8—71no—
390C22⋊C8—141no—
391C22⋊C8—141no—
392C22⋊C8—141no—
393C22⋊C8—141no—
394C22⋊C8—141no—
395#97—21no—
396#97—141no—
397#97—141no—
398#97—141no—
399C4×C8—71no—
400C4×C8—71no—
401C4×C8—71no—
402C4×C8—141no—
403C4×C8—141no—
404C4×C8—141no—
405C4×C8C22×D711yescharacteristic
406C4×C8C22×D711yescharacteristic
407C4×C8C22×D711yescharacteristic
408C22×C7—22no—
409C22×C7—41no—
410C22×C7—41no—
411C22×C7C2×C4×D411yescharacteristic
412C2×D7—21no—
413C2×D7—22no—
414C2×D7—21no—
415C2×D7—21no—
416C2×D7—41no—
417C2×D7—41no—
418C2×D7—84no—
419C4×C7—21no—
420C4×C7—41no—
421C4×C7—41no—
422C4×C7—41no—
423C4×C7C2×#13111yescharacteristic
424C4×C7C2×#13111yescharacteristic
425C4×C7#57711yescharacteristic
426C4×C7#57711yescharacteristic
427Dic7—21no—
428Dic7—21no—
429Dic7—21no—
430Dic7—41no—
431Dic7—41no—
432Dic7—41no—
433C24—141no—
434C2×Q8—71no—
435C2×Q8—141no—
436C2×Q8C2×C4×D711yescharacteristic
437C2×D4—71no—
438C2×D4—141no—
439C2×D4—142no—
440C2×D4—142no—
441C2×D4C2×C4×D711yescharacteristic
442C22×C4—21no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—561no—
451C22×C4—561no—
452SD16—22no—
453SD16—142no—
454SD16—142no—
455SD16—142no—
456M16—282no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—71no—
461C2×C8—71no—
462C2×C8—71no—
463C2×C8—71no—
464C2×C8—81no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—561no—
469C2×C8—561no—
470C2×C8—561no—
471C2×C8C2×C4×D711yescharacteristic
472C2×C8C2×C4×D711yescharacteristic
473C2×C8D4×D711yescharacteristic
474C2×C8D4×D711yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—71no—
477C4⋊C4—71no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4C2×C4×D711yescharacteristic
486C4⋊C4C2×C4×D711yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—141no—
489C22⋊C4—141no—
490C22⋊C4—141no—
491C22⋊C4—141no—
492C22⋊C4—141no—
493C42—21no—
494C42—71no—
495C42—71no—
496C42—71no—
497C42—141no—
498C42C23×D711yescharacteristic
499C2×C7—42no—
500C2×C7#292911yescharacteristic
501C2×C7#294911yescharacteristic
502C2×C7#295611yescharacteristic
503D7—41no—
504D7—42no—
505C23—21no—
506C23—71no—
507C23—144no—
508C23—141no—
509C23—561no—
510C23—561no—
511Q8—21no—
512Q8—21no—
513Q8—141no—
514Q8—141no—
515Q8—561no—
516D4—21no—
517D4—72no—
518D4—144no—
519D4—141no—
520D4#647012yes—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—71no—
527C2×C4—71no—
528C2×C4—71no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—561no—
540C2×C4—561no—
541C2×C4—561no—
542C2×C4C22×C4×D711yescharacteristic
543C2×C4C2×D4×D711yescharacteristic
544C2×C4D7×#4111yescharacteristic
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—142no—
553C8—141no—
554C8—141no—
555C8—141no—
556C8—141no—
557C8—281no—
558C8—281no—
559C7#1792411yescharacteristic, complemented, Sylow
560C22—22no—
561C22—41no—
562C22—41no—
563C22—141no—
564C22—141no—
565C22—142no—
566C22—141no—
567C22—281no—
568C22—281no—
569C22—564no—
570C22C4×D4×D711yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—141no—
576C4—141no—
577C4—141no—
578C4—281no—
579C4—281no—
580C4—281no—
581C4C2×#647011yescharacteristic
582C4D7×#13111yescharacteristic
583C4#9045311yescharacteristic
584C4#9046111yescharacteristic
585C2—42no—
586C2—281no—
587C2—282no—
588C2#1173692311yescharacteristic
589C2#1173709411yescharacteristic
590C2#1173734311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1582C2
socleC22×C7C2×C4×D4
prime radicalC2×D4×D7C2×C4
cosocleC42×C7C24
hypercenter#1582D7
hypocenterC7#17924
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
1. #11737410C2semidirect
4. C7×#1582C2semidirect
5. #11728708C2semidirect
6. SD16×C7⋊C8C2semidirect
7. #11728597C2semidirect
8. #11726287C2semidirect
9. #11726270C2semidirect
10. #11725989C2semidirect
11. #11725968C2semidirect
13. #11725904C2semidirect
14. #11725833C2semidirect
15. #11725826C2semidirect
18. C7×#446C22semidirect
26. Q8×C7⋊C8C22semidirect
36. #89602C22semidirect
50. #89451C22semidirect
51. C7×#334C22semidirect
52. C7×Q8⋊C8C22semidirect
54. C7×#322C22semidirect
55. Q8⋊(C7⋊C8)C22semidirect
57. #89279C22semidirect
59. #89274C22semidirect
60. #89274C22semidirect
62. #89252C22semidirect
63. #89251C22semidirect
64. #89249C22semidirect
65. #89249C22semidirect
66. C8×C7⋊C8C22semidirect
151. #1582D7semidirect
239. #446C2×D7semidirect
264. #334C2×D7semidirect
268. Q8⋊C8C2×D7semidirect
275. #322C2×D7semidirect
559. C7#17924semidirect

Maximal subgroups

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

subgroupindex
#117374102
#117370962
#117368712
C7×#15822
#117287082
SD16×C7⋊C82
#117285972
#117262872
#117262702
#117259892
#117259682
#117259292
#117259042
#117258332
#117258262
#179247