Finite Group Databasebeta

← #13323955

Subgroups — #13323955

Order 1792.

620 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737392C211yesmaximal, characteristic, complemented
2#11737097C211yesmaximal, characteristic
3#11736850C211yesmaximal, characteristic, complemented
4C7×#1585C211yesmaximal, characteristic, complemented
5#11728710C211yesmaximal, characteristic, complemented
6SD16⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
7#11728595C211yesmaximal, characteristic, complemented
8#11726326C211yesmaximal, characteristic, complemented
9#11726312C211yesmaximal, characteristic, complemented
10#11725988C211yesmaximal, characteristic, complemented
11#11725967C211yesmaximal, characteristic, complemented
12#11725928C211yesmaximal, characteristic, complemented
13#11725902C211yesmaximal, characteristic, complemented
14#11725842C211yesmaximal, characteristic, complemented
15#11725827C211yesmaximal, characteristic, complemented
16#90271C2211yescharacteristic
17C4×#1159C2211yescharacteristic
18C7×#446C2211yescharacteristic
19C4×C7×SD16C2211yescharacteristic
20C7×C8×D4C2211yescharacteristic
21#89945C411yescharacteristic
22#89937C411yescharacteristic
23C8×C22⋊D7—21no—
24C2×Dic7⋊C8—21no—
25C4×#1144C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27C4×#1141C2211yescharacteristic
28D4×C7⋊C8C2211yescharacteristic
29#89638C411yescharacteristic
30#89629C411yescharacteristic
31#89611—21no—
32#89610—21no—
33#89609C2211yescharacteristic
34#89606C2211yescharacteristic
35D7×C4⋊C8C2211yescharacteristic
36#89602C2211yescharacteristic
37#89587C411yescharacteristic
38#89570C411yescharacteristic
39#89553C411yescharacteristic
40#89535C411yescharacteristic
41(C22⋊D7)⋊C8—21no—
42#89490—21no—
43#89481—21no—
44#89478—21no—
45C4×#1132C2211yescharacteristic
46#89460C2211yescharacteristic
47C4×#1131C2211yescharacteristic
48#89457C2211yescharacteristic
49#89457C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89278C2211yescharacteristic
59#89274C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89253C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89250C2211yescharacteristic
66C8×C7⋊C8C2211yescharacteristic
67#17926—71nomaximal, Sylow
68C2×#1159C2×C411yescharacteristic
69C2×C7×SD16C2×C411yescharacteristic
70C22×C7×C8—21no—
71C4×C7×Q8C2311yescharacteristic
72C4×C7×D4C2311yescharacteristic
73#6510C2×C411yescharacteristic
74#6509C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76D4×Dic7C2×C411yescharacteristic
77C2×#1141C2×C411yescharacteristic
78C4×C22⋊D7—21no—
79C2×C4×Dic7—21no—
80C22×C7⋊C8—21no—
81C2×#1132C2×C411yescharacteristic
82C2×#1131—21no—
83C2×#1131—21no—
84C2×#1131—21no—
85C2×C8×D7—21no—
86C2×C8×D7—21no—
87#6461—21no—
88#6458C2×C411yescharacteristic
89#6457—21no—
90#6455C2×C411yescharacteristic
91#6453C2×C411yescharacteristic
92#6450C2×C411yescharacteristic
93(C22⋊D7)⋊C4—21no—
94#6440—21no—
95C4×D28—21no—
96C4×D28C2311yescharacteristic
97#6435—21no—
98C42×D7C2311yescharacteristic
99C4×Dic14C2311yescharacteristic
100C7×#106C2×C411yescharacteristic
101C7×C4⋊C8—21no—
102C7×C4⋊C8C2311yescharacteristic
103C7×C4⋊C8C2311yescharacteristic
104C7×#103C2×C411yescharacteristic
105C7×#102C2×C411yescharacteristic
106C7×C22⋊C8—21no—
107C7×#97—21no—
108C4×C7×C8C2311yescharacteristic
109C4×C7×C8C2311yescharacteristic
110C4×C7×C8C2311yescharacteristic
111#6409C2×C411yescharacteristic
112#6406C2×C411yescharacteristic
113C22⋊(C7⋊C8)—21no—
114#6395C2×C411yescharacteristic
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—
124Dic7⋊C8—21no—
125Dic7⋊C8—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#6381C2×C411yescharacteristic
137#6381C2×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#3102—71no—
146#2993—71no—
147#2988—71no—
148#1588—71no—
149#1585D711yescharacteristic, complemented
150#1584—71no—
151#1581—71no—
152SD16⋊C8—71no—
153SD16⋊C8—71no—
154C8×Q16—71no—
155C8×D8—71no—
156#1571—71no—
157#1489—71no—
158#1483—71no—
159#1453—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×D4C22×C411yescharacteristic
162C22×C4×C7—21no—
163C2×C22⋊D7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166#1159—22no—
167#1159—22no—
168C2×D28—21no—
169C2×D28C22×C411yescharacteristic
170C2×C4×D7—21no—
171C2×C4×D7C22×C411yescharacteristic
172C2×C4×D7C2×D411yescharacteristic
173C2×C4×D7#4111yescharacteristic
174C2×Dic14C22×C411yescharacteristic
175C7×SD16—22no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—81no—
180C2×C7×C8C22×C411yescharacteristic
181C2×C7×C8C22×C411yescharacteristic
182C2×C7×C8C2×D411yescharacteristic
183C2×C7×C8C2×D411yescharacteristic
184C7×C4⋊C4—21no—
185C7×C4⋊C4C22×C411yescharacteristic
186C7×C4⋊C4C22×C411yescharacteristic
187C7×C22⋊C4—21no—
188C42×C7—21no—
189C42×C7C2411yescharacteristic
190C22⋊Dic7—21no—
191#1144—22no—
192#1141—22no—
193(C2×D7)⋊C4—21no—
194(C2×D7)⋊C4—21no—
195(C2×D7)⋊C4—21no—
196(C2×D7)⋊C4—21no—
197(C2×D7)⋊C4—21no—
198(C2×D7)⋊C4—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7C22×C411yescharacteristic
201C4⋊Dic7C22×C411yescharacteristic
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204Dic7⋊C4—21no—
205Dic7⋊C4—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7C22×C411yescharacteristic
210C4×Dic7C2×D411yescharacteristic
211C4×Dic7#4111yescharacteristic
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—81no—
216C2×C7⋊C8C22×C411yescharacteristic
217C2×C7⋊C8C22×C411yescharacteristic
218C2×C7⋊C8C2×D411yescharacteristic
219C2×C7⋊C8C2×D411yescharacteristic
220#1132—22no—
221#1131—42no—
222#1131—81no—
223#1131—81no—
224C8×D7—81no—
225C8×D7—81no—
226#529—71no—
227C4×#41—71no—
228#485—71no—
229#482—71no—
230#479—71no—
231#478—71no—
232#464—71no—
233#463—71no—
234#452—71no—
235#449—71no—
236#446C2×D711yescharacteristic
237C8×Q8—71no—
238C8×Q8—71no—
239C8×Q8—71no—
240C4×Q16—71no—
241C4×SD16—71no—
242C4×SD16C2×D711yescharacteristic
243C4×D8—71no—
244#436—71no—
245#436—141no—
246#435—141no—
247C8×D4—71no—
248C8×D4—71no—
249C8×D4—141no—
250C8×D4—141no—
251C8×D4C2×D711yescharacteristic
252#433—141no—
253#431—71no—
254#431—71no—
255#431—141no—
256#424—141no—
257#423—71no—
258C2×C4⋊C8—71no—
259C2×C4⋊C8—141no—
260C4×M16—71no—
261#335—71no—
262#335—71no—
263#335—71no—
264#335C2×D711yescharacteristic
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8—71no—
268Q8⋊C8C2×D711yescharacteristic
269D4⋊C8—71no—
270D4⋊C8—71no—
271D4⋊C8—71no—
272D4⋊C8C2×D711yescharacteristic
273#322—71no—
274#322—71no—
275#322C2×D711yescharacteristic
276C82—71no—
277C23×C7—21no—
278C22×D7—21no—
279C22×D7C4×D411yescharacteristic
280C7×Q8—21no—
281C7×Q8—21no—
282C7×D4—21no—
283C7×D4#13112yes—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7C23×C411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C22⋊D7—24no—
293C2×Dic7—24no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—81no—
298C2×Dic7C4×D411yescharacteristic
299C2×Dic7C4×D411yescharacteristic
300C2×Dic7C4×D411yescharacteristic
301D28—21no—
302D28—21no—
303D28—81no—
304C4×D7—21no—
305C4×D7—21no—
306C4×D7—21no—
307C4×D7—21no—
308C4×D7—41no—
309C4×D7—41no—
310C4×D7—41no—
311Dic14—21no—
312Dic14#13112yes—
313C7×C8—22no—
314C7×C8—21no—
315C7×C8—21no—
316C7×C8—21no—
317C7×C8—21no—
318C7×C8—41no—
319C7×C8—41no—
320C7⋊C8—21no—
321C7⋊C8—21no—
322C7⋊C8—21no—
323C7⋊C8—21no—
324C7⋊C8—41no—
325C7⋊C8—41no—
326C7⋊C8—41no—
327C2×#41—71no—
328C2×Q16—71no—
329C2×SD16—71no—
330C2×SD16C4×D711yescharacteristic
331C2×D8—71no—
332C2×M16—141no—
333C2×M16—141no—
334C2×M16—141no—
335C22×C8—21no—
336C22×C8—141no—
337C22×C8—141no—
338C22×C8—141no—
339#126—141no—
340#125—71no—
341#124—71no—
342#123—141no—
343C22⋊Q8—71no—
344#121—71no—
345C4×Q8—71no—
346C4×Q8—71no—
347C4×Q8C22×D711yescharacteristic
348C4×D4—71no—
349C4×D4—71no—
350C4×D4—141no—
351C4×D4—141no—
352C4×D4—141no—
353C4×D4C22×D711yescharacteristic
354#117—71no—
355#117—141no—
356#117—141no—
357C2×C42—71no—
358C2×C42—141no—
359#107—71no—
360#107—71no—
361#106—71no—
362#106C4×D711yescharacteristic
363C4⋊C8—21no—
364C4⋊C8—71no—
365C4⋊C8—71no—
366C4⋊C8—71no—
367C4⋊C8—71no—
368C4⋊C8—71no—
369C4⋊C8—71no—
370C4⋊C8—141no—
371C4⋊C8—141no—
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—
393#97—21no—
394#97—141no—
395C4×C8—71no—
396C4×C8—71no—
397C4×C8—71no—
398C4×C8—141no—
399C4×C8—141no—
400C4×C8—141no—
401C4×C8C22×D711yescharacteristic
402C4×C8C22×D711yescharacteristic
403C4×C8C22×D711yescharacteristic
404C22×C7—22no—
405C22×C7—41no—
406C22×C7—41no—
407C22×C7C2×C4×D411yescharacteristic
408C2×D7—21no—
409C2×D7—21no—
410C2×D7—21no—
411C2×D7—41no—
412C2×D7—41no—
413C2×D7—41no—
414C4×C7—21no—
415C4×C7—41no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7C2×#13111yescharacteristic
419C4×C7C2×#13111yescharacteristic
420C4×C7#57611yescharacteristic
421C4×C7#57811yescharacteristic
422Dic7—22no—
423Dic7—21no—
424Dic7—21no—
425Dic7—21no—
426Dic7—41no—
427Dic7—41no—
428#41—142no—
429#41—142no—
430C2×Q8—71no—
431C2×Q8C2×C4×D711yescharacteristic
432C2×D4—71no—
433C2×D4—141no—
434C2×D4—141no—
435C2×D4C2×C4×D711yescharacteristic
436C22×C4—21no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443Q16—142no—
444SD16—22no—
445SD16—142no—
446D8—142no—
447M16—282no—
448M16—561no—
449M16—561no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—71no—
454C2×C8—71no—
455C2×C8—71no—
456C2×C8—71no—
457C2×C8—81no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—561no—
462C2×C8—561no—
463C2×C8—561no—
464C2×C8C2×C4×D711yescharacteristic
465C2×C8C2×C4×D711yescharacteristic
466C2×C8D4×D711yescharacteristic
467C2×C8D4×D711yescharacteristic
468C4⋊C4—21no—
469C4⋊C4—71no—
470C4⋊C4—71no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4C2×C4×D711yescharacteristic
477C4⋊C4C2×C4×D711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—141no—
480C22⋊C4—141no—
481C22⋊C4—141no—
482C22⋊C4—141no—
483C22⋊C4—141no—
484C22⋊C4—141no—
485C22⋊C4—141no—
486C42—21no—
487C42—71no—
488C42—71no—
489C42—71no—
490C42—141no—
491C42—141no—
492C42—141no—
493C42C23×D711yescharacteristic
494C2×C7—42no—
495C2×C7#292911yescharacteristic
496C2×C7#295111yescharacteristic
497C2×C7#295611yescharacteristic
498D7—41no—
499D7—81no—
500C23—21no—
501C23—71no—
502C23—141no—
503Q8—21no—
504Q8—21no—
505Q8—72no—
506Q8—141no—
507D4—21no—
508D4—144no—
509D4—141no—
510D4—141no—
511D4—561no—
512D4#647012yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—71no—
519C2×C4—71no—
520C2×C4—71no—
521C2×C4—144no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—561no—
533C2×C4C22×C4×D711yescharacteristic
534C2×C4C2×D4×D711yescharacteristic
535C2×C4D7×#4111yescharacteristic
536C8—22no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—141no—
544C8—141no—
545C8—141no—
546C8—141no—
547C8—281no—
548C8—281no—
549C8—281no—
550C7#1792611yescharacteristic, complemented, Sylow
551C22—22no—
552C22—41no—
553C22—41no—
554C22—141no—
555C22—141no—
556C22—141no—
557C22—281no—
558C22—281no—
559C22—281no—
560C22C4×D4×D711yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—141no—
566C4—142no—
567C4—141no—
568C4—141no—
569C4—281no—
570C4—281no—
571C4C2×#647011yescharacteristic
572C4D7×#13111yescharacteristic
573C4#9046511yescharacteristic
574C4#9047011yescharacteristic
575C2—42no—
576C2—281no—
577C2—561no—
578C2#1173692311yescharacteristic
579C2#1173722311yescharacteristic
580C2#1173733411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1585C2
socleC22×C7C2×C4×D4
prime radical#89553C4
cosocleC42×C7C24
hypercenter#1585D7
hypocenterC7#17926
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1585

Complements

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

NH ≅ G/Ntype
1. #11737392C2semidirect
3. #11736850C2semidirect
4. C7×#1585C2semidirect
5. #11728710C2semidirect
6. SD16⋊(C7⋊C8)C2semidirect
7. #11728595C2semidirect
8. #11726326C2semidirect
9. #11726312C2semidirect
10. #11725988C2semidirect
11. #11725967C2semidirect
12. #11725928C2semidirect
13. #11725902C2semidirect
14. #11725842C2semidirect
15. #11725827C2semidirect
149. #1585D7semidirect
550. C7#17926semidirect

Maximal subgroups

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

subgroupindex
#117373922
#117370972
#117368502
C7×#15852
#117287102
SD16⋊(C7⋊C8)2
#117285952
#117263262
#117263122
#117259882
#117259672
#117259282
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