Finite Group Databasebeta

← #13508266

Subgroups — #13508266

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89683C211yesmaximal, characteristic, complemented
2C22×#6372C211yesmaximal, characteristic, complemented
3#11738159C211yesmaximal, characteristic, complemented
4D7×#466C211yesmaximal, characteristic, complemented
5C7×#2215C211yesmaximal, characteristic, complemented
6#11729224C211yesmaximal, characteristic, complemented
7#11729168C211yesmaximal, characteristic, complemented
8#11728826C211yesmaximal, characteristic, complemented
9#11728825C211yesmaximal, characteristic, complemented
10#11727083C211yesmaximal, characteristic, complemented
11#11727080C211yesmaximal, characteristic, complemented
12#11726952C211yesmaximal, characteristic, complemented
13#11726951C211yesmaximal, characteristic, complemented
14#11726944C211yesmaximal, characteristic, complemented
15#11726943C211yesmaximal, characteristic, complemented
16C2×#6475C2211yescharacteristic
17C2×D7×D8C2211yescharacteristic
18C22×C8×D7C2211yescharacteristic
19#90297—21no—
20D7×#121—21no—
21C2×C7×D16C2211yescharacteristic, complemented
22C22×C7×C16C2211yescharacteristic, complemented
23C7×#466C2211yescharacteristic
24C7×#466C2211yescharacteristic, complemented
25#89929C2211yescharacteristic
26#89929C2211yescharacteristic, complemented
27C2×#6401C2211yescharacteristic, complemented
28C2×#6400C2211yescharacteristic, complemented
29#89888C2211yescharacteristic, complemented
30#89887C2211yescharacteristic
31C22×C7⋊C16C2211yescharacteristic, complemented
32#89813—21no—
33#89810—21no—
34#89683—24no—
35C2×#6374C2211yescharacteristic, complemented
36C2×#6372—24no—
37C2×#6372C2211yescharacteristic
38C2×#6372C2211yescharacteristic
39C2×#6372C2211yescharacteristic, complemented
40C2×#6372C2211yescharacteristic, complemented
41#89659C2211yescharacteristic, complemented
42#89655C2211yescharacteristic
43#89652C2211yescharacteristic
44D7×#107C2211yescharacteristic
45#89550—21no—
46#89547—21no—
47#89544—21no—
48D7×#102—21no—
49C7×#366C2211yescharacteristic, complemented
50C7×#357C2211yescharacteristic, complemented
51C7×#357C2211yescharacteristic, complemented
52#89357C2211yescharacteristic, complemented
53#89357C2211yescharacteristic, complemented
54#89304C2211yescharacteristic, complemented
55#89300C2211yescharacteristic, complemented
56#89298C2211yescharacteristic, complemented
57#89287C2211yescharacteristic, complemented
58#89286C2211yescharacteristic, complemented
59#89285C2211yescharacteristic, complemented
60#89284C2211yescharacteristic, complemented
61#33605—71nomaximal, Sylow
62C2×#1159—21no—
63C2×D4×D7—21no—
64C2×D4×D7—41no—
65C22×C4×D7D411yescharacteristic
66C2×C7×D8C2311yescharacteristic
67C2×C7×D8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×#121—21no—
70C7×#121—21no—
71#6502—41no—
72#6501—21no—
73#6501—21no—
74#6500—41no—
75D4×Dic7—41no—
76C2×#1142—21no—
77C2×#1141—21no—
78#6493—21no—
79#6487—21no—
80C22×C7⋊C8D411yescharacteristic
81#6475—22no—
82#6475—22no—
83D7×D8—22no—
84D7×D8—22no—
85C2×Dic28C2311yescharacteristic
86C2×D56C2311yescharacteristic
87C2×C8×D7—22no—
88C2×C8×D7—22no—
89C2×C8×D7D411yescharacteristic
90C2×C8×D7D411yescharacteristic
91C2×C8×D7C2311yescharacteristic
92C2×C8×D7C2311yescharacteristic
93#6460—21no—
94#6458—21no—
95#6455—21no—
96D7×C4⋊C4—21no—
97#6447—41no—
98#6446—41no—
99(C22⋊D7)⋊C4—41no—
100D7×C22⋊C4—41no—
101C7×D16—22no—
102C2×C7×C16—22no—
103C2×C7×C16C2311yescharacteristic
104C2×C7×C16C2311yescharacteristic
105C7×#107C2311yescharacteristic
106C7×#107C2311yescharacteristic
107C7×#102—21no—
108C7×#102—21no—
109#6406—21no—
110#6406—21no—
111#6401—22no—
112#6400—22no—
113#6395—21no—
114#6392C2311yescharacteristic
115#6392C2311yescharacteristic
116#6390—21no—
117C2×C7⋊C16—22no—
118C2×C7⋊C16C2311yescharacteristic
119C2×C7⋊C16C2311yescharacteristic
120#6384—21no—
121#6383—21no—
122#6382—21no—
123#6381—21no—
124#6374—22no—
125#6372—22no—
126#6372—22no—
127#6372—22no—
128#6372D414yescomplemented
129C2×#509—71no—
130C22×M32—71no—
131#3050—71no—
132C2×#466—71no—
133#2220—71no—
134#2220—71no—
135#2218—71no—
136#2218—71no—
137#2217—71no—
138#2217—71no—
139#2215—71no—
140#2215D711yescharacteristic, complemented
141#2159—71no—
142#2140—71no—
143#2139—71no—
144C23×D7—41no—
145C23×D7D811yescharacteristic
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—41no—
149C2×C7×D4—41no—
150C22×C4×C7C2×D411yescharacteristic
151C2×C22⋊D7—41no—
152C2×C22⋊D7—41no—
153C2×C22⋊D7—41no—
154C2×C22⋊D7—41no—
155C2×C22⋊D7—41no—
156C22×Dic7—41no—
157C22×Dic7D811yescharacteristic
158#1159—42no—
159#1159—42no—
160D4×D7—42no—
161D4×D7—42no—
162D4×D7—84no—
163C2×D28—21no—
164C2×D28—41no—
165C2×C4×D7—22no—
166C2×C4×D7—22no—
167C2×C4×D7—41no—
168C2×C4×D7D811yescharacteristic
169C2×C4×D7D811yescharacteristic
170C2×C4×D7C2×D411yescharacteristic
171C2×C4×D7C2×D411yescharacteristic
172C2×Dic14—21no—
173C7×D8—21no—
174C7×D8—21no—
175C7×D8—21no—
176C7×D8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C2×D411yescharacteristic
180C2×C7×C8C2411yescharacteristic
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C22⋊C4—41no—
184C7×C22⋊C4—41no—
185C22⋊Dic7—41no—
186C22⋊Dic7—41no—
187C22⋊Dic7—41no—
188#1142—42no—
189#1141—42no—
190(C2×D7)⋊C4—41no—
191(C2×D7)⋊C4—41no—
192(C2×D7)⋊C4—41no—
193(C2×D7)⋊C4—41no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—41no—
197Dic7⋊C4—41no—
198Dic7⋊C4—41no—
199C4×Dic7—41no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8C2×D411yescharacteristic
202C2×C7⋊C8C2×D411yescharacteristic
203Dic28—21no—
204Dic28—21no—
205D56—21no—
206D56—21no—
207C8×D7—22no—
208C8×D7—21no—
209C8×D7—21no—
210C8×D7—22no—
211C8×D7C2×D412yes—
212C8×D7#4112yes—
213C7×C16—21no—
214C7×C16C2×D412yescomplemented
215C7⋊C16—21no—
216C7⋊C16C2×D412yescomplemented
217C2×#135—71no—
218C22×D8—71no—
219C23×C8—71no—
220#525—141no—
221C2×#121—141no—
222#509—144no—
223C2×SD32—71no—
224C2×SD32—71no—
225C2×D16—71no—
226C2×D16C2×D711yescharacteristic, complemented
227C2×M32—71no—
228C2×M32—71no—
229C2×M32—71no—
230C2×M32—71no—
231C2×M32—144no—
232C22×C16—71no—
233C22×C16C2×D711yescharacteristic, complemented
234#467—71no—
235#467—71no—
236#466—71no—
237#466—71no—
238#466—71no—
239#466—71no—
240#466—141no—
241#466—141no—
242#466C2×D711yescharacteristic
243#466C2×D711yescharacteristic, complemented
244#465—141no—
245#465—141no—
246#427—71no—
247C2×#107—71no—
248#416—141no—
249C2×#102—141no—
250#367—71no—
251#367—71no—
252#366—71no—
253#366C2×D711yescharacteristic, complemented
254#358—71no—
255#358—71no—
256#357—71no—
257#357—71no—
258#357—71no—
259#357—71no—
260#357C2×D711yescharacteristic, complemented
261#357C2×D711yescharacteristic, complemented
262C23×C7—41no—
263C23×C7—41no—
264C23×C7C2×D811yescharacteristic
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—82no—
270C22×D7—82no—
271C22×D7—82no—
272C22×D7C2×D811yescharacteristic
273C22×D7C2×D811yescharacteristic
274C7×D4—41no—
275C7×D4—41no—
276C7×D4—41no—
277C7×D4—41no—
278C7×D4—82no—
279C7×D4—82no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7C2×D811yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C22⋊D7—82no—
287C22⋊D7—82no—
288C22⋊D7—82no—
289C22⋊D7—82no—
290C22⋊D7—82no—
291C2×Dic7—22no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—82no—
296C2×Dic7C2×D811yescharacteristic
297C2×Dic7C2×D811yescharacteristic
298D28—41no—
299D28—41no—
300D28—82no—
301C4×D7—22no—
302C4×D7—21no—
303C4×D7—21no—
304C4×D7—22no—
305C4×D7—82no—
306C4×D7#12112yes—
307C4×D7#13512yes—
308Dic14—41no—
309Dic14—41no—
310C7×C8—21no—
311C7×C8C22×D411yescharacteristic
312C7×C8C2×#4111yescharacteristic
313C7⋊C8—21no—
314C7⋊C8#12112yes—
315C2×#41—141no—
316C22×D4—141no—
317C22×D4—281no—
318C23×C4—71no—
319#135—142no—
320#135—142no—
321C2×Q16—71no—
322C2×SD16—141no—
323C2×D8—71no—
324C2×D8—142no—
325C2×D8—142no—
326C2×D8—141no—
327C2×D8C22×D711yescharacteristic
328C2×D8C22×D711yescharacteristic
329C22×C8—71no—
330C22×C8—71no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8—71no—
334C22×C8—142no—
335C22×C8—142no—
336C22×C8C22×D711yescharacteristic
337#123—281no—
338C22⋊Q8—141no—
339C22⋊Q8—141no—
340#121—21no—
341#121—21no—
342#121—141no—
343#121—141no—
344#121—141no—
345#121—141no—
346#121—281no—
347#121—281no—
348#120—281no—
349C4×D4—281no—
350C4×D4—281no—
351#117—141no—
352C2×C4⋊C4—141no—
353C2×C22⋊C4—281no—
354SD32—142no—
355SD32—142no—
356D16—22no—
357D16—142no—
358M32—74no—
359M32—142no—
360M32—142no—
361M32—142no—
362C2×C16—22no—
363C2×C16—71no—
364C2×C16—71no—
365C2×C16—142no—
366C2×C16C22×D711yescharacteristic
367C2×C16C22×D711yescharacteristic
368#107—71no—
369#107—71no—
370#107—141no—
371#107C22×D711yescharacteristic
372#107C22×D711yescharacteristic
373#106—141no—
374#103—141no—
375#103—141no—
376#102—21no—
377#102—21no—
378#102—141no—
379#102—141no—
380#102—141no—
381#102—141no—
382C22×C7—21no—
383C22×C7—21no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7—81no—
390C22×C7#50912yescomplemented
391C22×C7C22×D811yescharacteristic
392C2×D7—21no—
393C2×D7—21no—
394C2×D7—21no—
395C2×D7—21no—
396C2×D7—21no—
397C2×D7—44no—
398C2×D7—81no—
399C2×D7—81no—
400C2×D7—81no—
401C2×D7—164no—
402C2×D7#46612yes—
403C4×C7—21no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7C2×#12111yescharacteristic
407C4×C7C2×#13511yescharacteristic
408Dic7—21no—
409Dic7—81no—
410Dic7—81no—
411Dic7—81no—
412Dic7#46612yes—
413C24—71no—
414C24—281no—
415#41—282no—
416#41—282no—
417C2×Q8—141no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—141no—
423C2×D4—281no—
424C2×D4—282no—
425C2×D4—282no—
426C2×D4—281no—
427C2×D4—281no—
428C2×D4—281no—
429C2×D4—281no—
430C2×D4—281no—
431C2×D4—564no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—281no—
440C22×C4—281no—
441C22×C4D4×D711yescharacteristic
442Q16—141no—
443Q16—141no—
444SD16—282no—
445D8—21no—
446D8—21no—
447D8—21no—
448D8—21no—
449D8—141no—
450D8—141no—
451D8—282no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—71no—
455C2×C8—72no—
456C2×C8—72no—
457C2×C8—71no—
458C2×C8—142no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8D4×D711yescharacteristic
464C2×C8C23×D711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—281no—
470C4⋊C4—281no—
471C4⋊C4—281no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—281no—
475C22⋊C4—281no—
476C22⋊C4—281no—
477C22⋊C4—281no—
478C22⋊C4—281no—
479C22⋊C4—281no—
480C22⋊C4—281no—
481C42—281no—
482C16—21no—
483C16—72no—
484C16—141no—
485C16D4×D712yescomplemented
486C2×C7—22no—
487C2×C7—161no—
488C2×C7—161no—
489C2×C7C2×#46611yescharacteristic
490C2×C7C2×#50911yescharacteristic
491C2×C7#341411yescharacteristic
492D7—22no—
493D7—41no—
494D7—161no—
495C23—41no—
496C23—41no—
497C23—71no—
498C23—71no—
499C23—142no—
500C23—142no—
501C23—281no—
502C23—281no—
503C23—562no—
504C23—562no—
505C23—562no—
506C23D7×D811yescharacteristic
507Q8—281no—
508Q8—281no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—82no—
514D4—82no—
515D4—281no—
516D4—281no—
517D4—562no—
518D4—562no—
519D4—562no—
520D4—562no—
521D4—562no—
522D4—562no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—71no—
528C2×C4—72no—
529C2×C4—72no—
530C2×C4—71no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—562no—
540C2×C4—562no—
541C2×C4D7×D811yescharacteristic
542C2×C4C2×D4×D711yescharacteristic
543C8—21no—
544C8—72no—
545C8—141no—
546C8C2×D4×D711yescharacteristic
547C8D7×#4111yescharacteristic
548C7#3360511yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—72no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—141no—
559C22—141no—
560C22—141no—
561C22—141no—
562C22—141no—
563C22—284no—
564C22—561no—
565C22—561no—
566C22—561no—
567C22—1124no—
568C22#8968312yescomplemented
569C22C2×D7×D811yescharacteristic
570C4—21no—
571C4—72no—
572C4—81no—
573C4—81no—
574C4—141no—
575C4—561no—
576C4—561no—
577C4—561no—
578C4D7×#12111yescharacteristic
579C4D7×#13511yescharacteristic
580C2—22no—
581C2—142no—
582C2—161no—
583C2—161no—
584C2—281no—
585C2—1121no—
586C2D7×#46611yescharacteristic
587C2C2×#8968311yescharacteristic
588C2#1173947511yescharacteristic
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2215C2
socleC22×C7C22×D8
hypercenter#2215D7
hypocenterC7#33605
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. C2×#89683C2semidirect
2. C22×#6372C2semidirect
3. #11738159C2semidirect
4. D7×#466C2semidirect
5. C7×#2215C2semidirect
6. #11729224C2semidirect
7. #11729168C2semidirect
8. #11728826C2semidirect
9. #11728825C2semidirect
10. #11727083C2semidirect
11. #11727080C2semidirect
12. #11726952C2semidirect
13. #11726951C2semidirect
14. #11726944C2semidirect
15. #11726943C2semidirect
21. C2×C7×D16C22semidirect
22. C22×C7×C16C22semidirect
24. C7×#466C22semidirect
26. #89929C22semidirect
27. C2×#6401C22semidirect
28. C2×#6400C22semidirect
29. #89888C22semidirect
31. C22×C7⋊C16C22semidirect
35. C2×#6374C22semidirect
39. C2×#6372C22semidirect
40. C2×#6372C22semidirect
41. #89659C22semidirect
49. C7×#366C22semidirect
50. C7×#357C22semidirect
51. C7×#357C22semidirect
52. #89357C22semidirect
53. #89357C22semidirect
54. #89304C22semidirect
55. #89300C22semidirect
56. #89298C22semidirect
57. #89287C22semidirect
58. #89286C22semidirect
59. #89285C22semidirect
60. #89284C22semidirect
128. #6372D4semidirect
140. #2215D7semidirect
214. C7×C16C2×D4semidirect
216. C7⋊C16C2×D4semidirect
226. C2×D16C2×D7semidirect
233. C22×C16C2×D7semidirect
243. #466C2×D7semidirect
253. #366C2×D7semidirect
260. #357C2×D7semidirect
261. #357C2×D7semidirect
390. C22×C7#509semidirect
485. C16D4×D7semidirect
548. C7#33605semidirect
568. C22#89683semidirect

Maximal subgroups

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

subgroupindex
C2×#896832
C22×#63722
#117381592
D7×#4662
C7×#22152
#117292242
#117291682
#117288262
#117288252
#117270832
#117270802
#117269522
#117269512
#117269442
#117269432
#336057