Finite Group Databasebeta

← #13510285

Subgroups — #13510285

Order 1792.

686 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
1C7×D4×Q16C211yesmaximal, characteristic, complemented
2C2×#89951C211yesmaximal, characteristic, complemented
3C22×#6402C211yesmaximal, characteristic, complemented
4#11736878C211yesmaximal, characteristic
5#11729818C211yesmaximal, characteristic, complemented
6#11729817C211yesmaximal, characteristic, complemented
7#11729069C211yesmaximal, characteristic, complemented
8#11729068C211yesmaximal, characteristic, complemented
9#11728840C211yesmaximal, characteristic, complemented
10#11728839C211yesmaximal, characteristic, complemented
11#11728838C211yesmaximal, characteristic, complemented
12#11728835C211yesmaximal, characteristic, complemented
13#11728669C211yesmaximal, characteristic, complemented
14#11728667C211yesmaximal, characteristic, complemented
15#11728611C211yesmaximal, characteristic, complemented
16C22×C7×Q16C2211yescharacteristic
17C22×C7×Q16C2211yescharacteristic, complemented
18C7×D4×Q8—21no—
19C2×#6467C2211yescharacteristic
20C22×D56C2211yescharacteristic
21#90241—21no—
22C7×#494C2211yescharacteristic, complemented
23C7×#467C2211yescharacteristic
24C7×#467C2211yescharacteristic, complemented
25C7×#462—21no—
26C7×#451—21no—
27C7×#451—21no—
28C4×C7×Q16C2211yescharacteristic, complemented
29C7×C8×D4C2211yescharacteristic
30C2×#6403C2211yescharacteristic, complemented
31C2×#6403C2211yescharacteristic, complemented
32C2×#6403C2211yescharacteristic, complemented
33#89951—24no—
34C2×#6402—24no—
35C2×#6402C2211yescharacteristic
36C2×#6402C2211yescharacteristic
37C2×#6402C2211yescharacteristic
38C2×#6402C2211yescharacteristic
39C2×#6402C2211yescharacteristic
40C2×#6402C2211yescharacteristic
41#89888C2211yescharacteristic
42#89887C2211yescharacteristic
43C2×#6386C2211yescharacteristic, complemented
44C22×C7⋊C16C2211yescharacteristic
45#89616—21no—
46#89506—21no—
47#89503—21no—
48#89468C2211yescharacteristic
49C4×D56C2211yescharacteristic
50#89359C2211yescharacteristic, complemented
51#89359C2211yescharacteristic, complemented
52#89359C2211yescharacteristic, complemented
53C22⋊(C7⋊C16)C2211yescharacteristic
54C22⋊(C7⋊C16)C2211yescharacteristic, complemented
55#89287C2211yescharacteristic, complemented
56#89286C2211yescharacteristic
57#89286C2211yescharacteristic
58#89284C2211yescharacteristic, complemented
59C4⋊(C7⋊C16)C2211yescharacteristic, complemented
60#89256C2211yescharacteristic, complemented
61#33832—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C7×Q8—21no—
64C2×#1157—21no—
65C22×D28—21no—
66C2×C7×Q16—22no—
67C2×C7×Q16—22no—
68C2×C7×Q16—21no—
69C2×C7×Q16—21no—
70C2×C7×Q16—21no—
71C2×C7×Q16—41no—
72C2×C7×Q16—41no—
73C2×C7×Q16C2311yescharacteristic
74C2×C7×Q16C2311yescharacteristic
75C2×C7×Q16C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C22×C7×C8C2311yescharacteristic
78C7×C4⋊Q8—21no—
79C7×C4⋊Q8—41no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—21no—
82C7×C22⋊Q8—41no—
83C7×C22⋊Q8—41no—
84C4×C7×Q8—21no—
85C4×C7×D4—41no—
86C4×C7×D4D411yescharacteristic
87#6500—41no—
88#6493—21no—
89#6487—21no—
90C2×Dic28C2311yescharacteristic
91#6467—22no—
92#6467—41no—
93C2×D56—22no—
94C2×D56—21no—
95C2×D56—41no—
96C2×D56C2311yescharacteristic
97C2×D56C2311yescharacteristic
98C2×#1132—21no—
99C2×#1132—21no—
100#6457—41no—
101#6447—41no—
102D7×C22⋊C4—41no—
103#6438—21no—
104C4×D28—21no—
105C7×#107C2311yescharacteristic
106C7×C4⋊C8D411yescharacteristic
107C7×#103—21no—
108C7×#103—21no—
109C7×#103—21no—
110C7×C22⋊C8D411yescharacteristic
111C7×C22⋊C8D411yescharacteristic
112C4×C7×C8C2311yescharacteristic
113#6403—22no—
114#6403—22no—
115#6403—41no—
116#6402—22no—
117#6402—22no—
118#6402—22no—
119#6402—22no—
120#6402—41no—
121#6402D414yescomplemented
122#6395—21no—
123#6395—21no—
124#6392C2311yescharacteristic
125#6390—21no—
126#6386—22no—
127C2×C7⋊C16—22no—
128C2×C7⋊C16C2311yescharacteristic
129C2×C7⋊C16C2311yescharacteristic
130C2×C7⋊C16C2311yescharacteristic
131C2×C7⋊C16C2311yescharacteristic
132C2×#510—71no—
133C22×SD32—71no—
134D4×Q16D711yescharacteristic, complemented
135#3280—71no—
136#2251—71no—
137#2220—71no—
138#2216—71no—
139#2211—71no—
140#2208—71no—
141#2189—71no—
142#2188—71no—
143#2187—71no—
144#2185—71no—
145#2175—71no—
146#2168—71no—
147C23×D7—41no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—42no—
152C2×C7×Q8—42no—
153C2×C7×Q8—41no—
154C2×C7×Q8—41no—
155C2×C7×Q8—81no—
156C2×C7×Q8—81no—
157C2×C7×D4D811yescharacteristic
158C22×C4×C7—41no—
159C22×C4×C7—41no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C2×C22⋊D7—41no—
163C2×C22⋊D7—41no—
164#1157—42no—
165#1157—81no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28—42no—
169C2×D28—41no—
170C2×D28—81no—
171C2×C4×D7—41no—
172C2×C4×D7—41no—
173C2×Dic14—21no—
174C7×Q16—21no—
175C7×Q16—21no—
176C7×Q16—21no—
177C7×Q16—21no—
178C7×Q16—21no—
179C7×Q16—42no—
180C7×Q16—42no—
181C7×Q16—81no—
182C7×Q16C2×D412yes—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8C2×D411yescharacteristic
188C2×C7×C8C2×D411yescharacteristic
189C2×C7×C8C2×D411yescharacteristic
190C2×C7×C8C2411yescharacteristic
191C7×C4⋊C4—21no—
192C7×C4⋊C4—41no—
193C7×C4⋊C4—41no—
194C7×C4⋊C4—41no—
195C7×C4⋊C4—41no—
196C7×C4⋊C4—41no—
197C7×C4⋊C4D811yescharacteristic
198C7×C22⋊C4—41no—
199C7×C22⋊C4—41no—
200C7×C22⋊C4D811yescharacteristic
201C7×C22⋊C4D811yescharacteristic
202C42×C7—41no—
203C42×C7C2×D411yescharacteristic
204C22⋊Dic7—41no—
205(C2×D7)⋊C4—41no—
206(C2×D7)⋊C4—41no—
207(C2×D7)⋊C4—41no—
208(C2×D7)⋊C4—41no—
209C4⋊Dic7—21no—
210Dic7⋊C4—41no—
211Dic28—21no—
212Dic28—21no—
213D56—21no—
214D56—21no—
215D56—21no—
216D56—42no—
217D56C2×D412yes—
218#1132—42no—
219#1132—81no—
220C7⋊C16—21no—
221C7⋊C16—21no—
222C7⋊C16—21no—
223C7⋊C16C2×D412yes—
224C2×#135—71no—
225C22×Q16C2×D711yescharacteristic
226C22×Q16C2×D711yescharacteristic, complemented
227C22×D8—71no—
228D4×Q8—21no—
229#546—141no—
230#510—144no—
231C2×Q32—71no—
232C2×Q32—71no—
233C2×Q32—71no—
234C2×SD32—71no—
235C2×SD32—71no—
236C2×SD32—71no—
237C2×SD32—71no—
238C2×SD32—71no—
239C2×SD32—71no—
240C2×SD32—144no—
241C2×M32—71no—
242C22×C16—71no—
243#495—71no—
244#494C2×D711yescharacteristic, complemented
245#467—71no—
246#467C2×D711yescharacteristic
247#467C2×D711yescharacteristic, complemented
248#466—71no—
249#463—141no—
250#462—21no—
251#451—21no—
252#451—21no—
253#450—141no—
254#449—141no—
255C4×Q16C2×D711yescharacteristic, complemented
256C4×D8—71no—
257C8×D4C2×D711yescharacteristic
258#366—71no—
259C4⋊C16—71no—
260#358—71no—
261#358—71no—
262#358—71no—
263#358—71no—
264#357—71no—
265#357—71no—
266C22⋊C16—71no—
267C22⋊C16—71no—
268#346—71no—
269C23×C7C2×D811yescharacteristic
270C23×C7C2×D811yescharacteristic
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—82no—
274C22×D7—82no—
275C22×D7—161no—
276C7×Q8—22no—
277C7×Q8—41no—
278C7×Q8—41no—
279C7×Q8—41no—
280C7×Q8—41no—
281C7×Q8—41no—
282C7×Q8—82no—
283C7×Q8—82no—
284C7×D4—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—82no—
294C2×C4×C7—82no—
295C2×C4×C7C2×D811yescharacteristic
296C2×C4×C7C2×D811yescharacteristic
297C2×C4×C7C2×D811yescharacteristic
298C2×C4×C7C2×D811yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C22⋊D7—82no—
301C22⋊D7—161no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304D28—22no—
305D28—41no—
306D28—41no—
307D28—41no—
308D28—82no—
309C4×D7—82no—
310C4×D7—161no—
311Dic14—41no—
312Dic14—41no—
313C7×C8—21no—
314C7×C8—21no—
315C7×C8—21no—
316C7×C8C22×D411yescharacteristic
317C7×C8H(5,𝔽2)11yescharacteristic
318C2×#41—141no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×D4—141no—
322#135—142no—
323#135—281no—
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—21no—
327C2×Q16—21no—
328C2×Q16—21no—
329C2×Q16—41no—
330C2×Q16—41no—
331C2×Q16—71no—
332C2×Q16C22×D711yescharacteristic
333C2×Q16C22×D711yescharacteristic
334C2×Q16C22×D711yescharacteristic
335C2×SD16—141no—
336C2×SD16—141no—
337C2×D8—71no—
338C2×D8—71no—
339C2×D8—142no—
340C2×D8—141no—
341C2×D8—281no—
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D711yescharacteristic
344C4⋊Q8—21no—
345C4⋊Q8—41no—
346#124—141no—
347#123—281no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—41no—
351C22⋊Q8—41no—
352C22⋊Q8—141no—
353#121—141no—
354#121—281no—
355#120—281no—
356C4×Q8—21no—
357C4×D4—41no—
358C4×D4—141no—
359C4×D4C22⋊D711yescharacteristic
360C2×C22⋊C4—281no—
361Q32—142no—
362Q32—142no—
363Q32—281no—
364SD32—74no—
365SD32—142no—
366SD32—142no—
367SD32—142no—
368SD32—142no—
369SD32—281no—
370M32—142no—
371C2×C16—71no—
372C2×C16—71no—
373C2×C16—71no—
374C2×C16—71no—
375C2×C16—142no—
376#107—71no—
377#107C22×D711yescharacteristic
378C4⋊C8C22⋊D711yescharacteristic
379#103—21no—
380#103—21no—
381#103—21no—
382#103—141no—
383#102—141no—
384#102—141no—
385C22⋊C8C22⋊D711yescharacteristic
386C22⋊C8C22⋊D711yescharacteristic
387C4×C8C22×D711yescharacteristic
388C22×C7—21no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7#51012yescomplemented
394C22×C7C22×D811yescharacteristic
395C2×D7—42no—
396C2×D7—81no—
397C2×D7—81no—
398C2×D7—81no—
399C2×D7—81no—
400C2×D7—81no—
401C2×D7—164no—
402C4×C7—21no—
403C4×C7—21no—
404C4×C7—21no—
405C4×C7—21no—
406C4×C7—42no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7—81no—
410C4×C7D4211yescharacteristic
411C4×C7#57611yescharacteristic
412Dic7—81no—
413Dic7—81no—
414C24—281no—
415#41—282no—
416#41—561no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—41no—
423C2×Q8—41no—
424C2×Q8—81no—
425C2×Q8—81no—
426C2×Q8—141no—
427C2×D4—141no—
428C2×D4—141no—
429C2×D4—282no—
430C2×D4—281no—
431C2×D4—281no—
432C2×D4—281no—
433C2×D4—561no—
434C2×D4#114111yescharacteristic
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—281no—
438C22×C4—281no—
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441Q16—21no—
442Q16—21no—
443Q16—21no—
444Q16—21no—
445Q16—21no—
446Q16—42no—
447Q16—42no—
448Q16—81no—
449Q16—141no—
450Q16—141no—
451Q16D4×D712yes—
452SD16—282no—
453SD16—561no—
454D8—72no—
455D8—141no—
456D8—141no—
457D8—141no—
458D8—282no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8C2×C22⋊D711yescharacteristic
464C2×C8C2×C22⋊D711yescharacteristic
465C2×C8C2×C22⋊D711yescharacteristic
466C2×C8C23×D711yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—141no—
474C4⋊C4—281no—
475C4⋊C4#114111yescharacteristic
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—281no—
479C22⋊C4—281no—
480C22⋊C4—281no—
481C22⋊C4—281no—
482C22⋊C4—281no—
483C22⋊C4#114111yescharacteristic
484C22⋊C4#114111yescharacteristic
485C42—41no—
486C42C2×C22⋊D711yescharacteristic
487C16—72no—
488C16—141no—
489C16—141no—
490C16—141no—
491C2×C7—22no—
492C2×C7—41no—
493C2×C7D4×D811yescharacteristic
494C2×C7C2×#51011yescharacteristic
495C2×C7#341611yescharacteristic
496D7—82no—
497D7—161no—
498C23—281no—
499C23—281no—
500C23—562no—
501C23—562no—
502C23—1121no—
503C23C2×#114111yescharacteristic
504C23C2×#114111yescharacteristic
505Q8—22no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—82no—
512Q8—82no—
513Q8—281no—
514Q8—281no—
515D4—22no—
516D4—142no—
517D4—281no—
518D4—281no—
519D4—281no—
520D4—562no—
521D4—562no—
522D4—1121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—562no—
536C2×C4—1121no—
537C2×C4C2×#114111yescharacteristic
538C2×C4C2×#114111yescharacteristic
539C2×C4C2×#114111yescharacteristic
540C2×C4C2×#114111yescharacteristic
541C2×C4C22×C22⋊D711yescharacteristic
542C8—21no—
543C8—21no—
544C8—21no—
545C8C2×D4×D711yescharacteristic
546C8#654811yescharacteristic
547C7#3383211yescharacteristic, complemented, Sylow
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—282no—
554C22—561no—
555C22—561no—
556C22—561no—
557C22—561no—
558C22—561no—
559C22—1124no—
560C22#8995112yescomplemented
561C22C22×#114111yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—561no—
571C4—561no—
572C4D4×C22⋊D711yescharacteristic
573C4#9052611yescharacteristic
574C2—22no—
575C2—41no—
576C2—562no—
577C2—1121no—
578C2D4×#114111yescharacteristic
579C2C2×#8995111yescharacteristic
580C2#1174146111yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×D4×Q16C2
socleC22×C7C22×D8
prime radical#11736878C2
hypercenterD4×Q16D7
hypocenterC7#33832
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C7×D4×Q16C2semidirect
2. C2×#89951C2semidirect
3. C22×#6402C2semidirect
5. #11729818C2semidirect
6. #11729817C2semidirect
7. #11729069C2semidirect
8. #11729068C2semidirect
9. #11728840C2semidirect
10. #11728839C2semidirect
11. #11728838C2semidirect
12. #11728835C2semidirect
13. #11728669C2semidirect
14. #11728667C2semidirect
15. #11728611C2semidirect
17. C22×C7×Q16C22semidirect
22. C7×#494C22semidirect
24. C7×#467C22semidirect
28. C4×C7×Q16C22semidirect
30. C2×#6403C22semidirect
31. C2×#6403C22semidirect
32. C2×#6403C22semidirect
43. C2×#6386C22semidirect
50. #89359C22semidirect
51. #89359C22semidirect
52. #89359C22semidirect
54. C22⋊(C7⋊C16)C22semidirect
55. #89287C22semidirect
58. #89284C22semidirect
59. C4⋊(C7⋊C16)C22semidirect
60. #89256C22semidirect
121. #6402D4semidirect
134. D4×Q16D7semidirect
226. C22×Q16C2×D7semidirect
244. #494C2×D7semidirect
247. #467C2×D7semidirect
255. C4×Q16C2×D7semidirect
393. C22×C7#510semidirect
547. C7#33832semidirect
560. C22#89951semidirect

Maximal subgroups

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

subgroupindex
C7×D4×Q162
C2×#899512
C22×#64022
#117368782
#117298182
#117298172
#117290692
#117290682
#117288402
#117288392
#117288382
#117288352
#117286692
#117286672
#117286112
#338327