Finite Group Databasebeta

← #13330929

Subgroups — #13330929

Order 1792.

1314 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×#3498C211yesmaximal, characteristic
2#11740716C211yesmaximal, characteristic, complemented
3C2×#89848C211yesmaximal, characteristic, complemented
4#11739953C211yesmaximal, characteristic, complemented
5#11729673C214yesmaximal, complemented
6#11729661C214yesmaximal, complemented
7#11729556C211yesmaximal, characteristic, complemented
8#11729513C211yesmaximal, characteristic, complemented
9#11729392C211yesmaximal, characteristic, complemented
10C2×C7×H(5,𝔽2)—21no—
11C7×#550—21no—
12C7×#550C2212yes—
13C7×D42—22no—
14C7×D42—22no—
15C7×D42—21no—
16C7×D42C2212yes—
17C7×#535—22no—
18C7×#535—21no—
19C7×#532—21no—
20C7×#532C2211yescharacteristic
21C2×C7×#127—21no—
22C2×C7×#127C2211yescharacteristic
23C4×C7×#41C2211yescharacteristic
24C22×#1142—22no—
25C2×#6432C2211yescharacteristic, complemented
26#90004—21no—
27#89851—21no—
28#89848—22no—
29#89848C2214yescomplemented
30#89842—21no—
31#89806—22no—
32#89793C2212yescomplemented
33#89792—21no—
34#89780C2214yescomplemented
35#89780C2212yescomplemented
36#89779—21no—
37#89767—21no—
38#89697C2211yescharacteristic, complemented
39#89696C2211yescharacteristic, complemented
40C2×C4×C7⋊C8C2211yescharacteristic, complemented
41#89339C2214yescomplemented
42Q8⋊(C7⋊C8)C2212yescomplemented
43D4⋊(C7⋊C8)C2214yescomplemented
44D4⋊(C7⋊C8)C2212yescomplemented
45#18621—71nomaximal, Sylow
46C7×H(5,𝔽2)—44no—
47C7×H(5,𝔽2)—44no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—21no—
51C2×C7×#41D411yescharacteristic
52C22×C7×D4—22no—
53C22×C7×D4—24no—
54C22×C7×D4—22no—
55C22×C7×D4—22no—
56C22×C7×D4—22no—
57C22×C7×D4—21no—
58C22×C7×D4—21no—
59C22×C7×D4D412yes—
60C22×Dic14—21no—
61C7×C4⋊Q8D411yescharacteristic
62C7×#127—22no—
63C7×#127—22no—
64C7×#127—22no—
65C7×#127—21no—
66C7×#127—21no—
67C7×#127D411yescharacteristic
68C7×#127D412yes—
69C7×#127C2312yes—
70C7×#124—22no—
71C7×#124—21no—
72C7×#121—22no—
73C7×#121—24no—
74C7×#121—24no—
75C7×#121—21no—
76C7×#121—22no—
77C7×#121—22no—
78C7×#121—22no—
79C7×#121D412yes—
80C7×#120—24no—
81C7×#120—24no—
82C7×#120—24no—
83C4×C7×Q8D411yescharacteristic
84C4×C7×Q8C2311yescharacteristic
85C4×C7×D4—22no—
86C4×C7×D4—21no—
87C4×C7×D4—21no—
88C4×C7×D4—22no—
89C4×C7×D4—21no—
90C4×C7×D4D411yescharacteristic
91C4×C7×D4C2311yescharacteristic
92C4×C7×D4C2312yes—
93C7×#117—21no—
94C7×#117D411yescharacteristic
95C2×C42×C7—21no—
96C2×C42×C7C2311yescharacteristic
97C2×#1142—28no—
98C2×#1142—48no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7—21no—
101C2×#1136—21no—
102C22×C7⋊C8—21no—
103#6432—21no—
104#6432—21no—
105#6432—21no—
106#6432C2312yes—
107#6409—22no—
108#6406—22no—
109#6406—24no—
110C22⋊(C7⋊C8)—21no—
111#6382—22no—
112#6382—24no—
113#6381—22no—
114C4⋊(C7⋊C8)—21no—
115C4⋊(C7⋊C8)C2312yes—
116C4⋊(C7⋊C8)C2312yes—
117C4×C7⋊C8—21no—
118C4×C7⋊C8—21no—
119C4×C7⋊C8C2312yes—
120#3498D711yescharacteristic
121#3227—71no—
122C2×#492—71no—
123#3079—71no—
124#1631—74no—
125#1629—74no—
126#1545—71no—
127#1492—71no—
128#1476—71no—
129C24×C7—24no—
130C24×C7—22no—
131C7×#41—22no—
132C7×#41—48no—
133C7×#41—44no—
134C7×#41—44no—
135C7×#41—42no—
136C7×#41—42no—
137C7×#41SD1614yescomplemented
138C2×C7×Q8—21no—
139C2×C7×Q8C2×D411yescharacteristic
140C2×C7×D4—21no—
141C2×C7×D4—24no—
142C2×C7×D4—24no—
143C2×C7×D4—22no—
144C2×C7×D4—24no—
145C2×C7×D4—24no—
146C2×C7×D4—22no—
147C2×C7×D4—21no—
148C2×C7×D4—22no—
149C2×C7×D4—21no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—48no—
154C2×C7×D4—44no—
155C2×C7×D4—416no—
156C2×C7×D4—44no—
157C2×C7×D4—44no—
158C2×C7×D4—42no—
159C2×C7×D4—42no—
160C2×C7×D4—82no—
161C2×C7×D4—84no—
162C2×C7×D4—84no—
163C2×C7×D4C2×D412yes—
164C2×C7×D4C2×D411yescharacteristic
165C2×C7×D4C2×D414yes—
166C22×C4×C7—21no—
167C22×C4×C7—24no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×Dic7—41no—
175C2×Dic14—22no—
176C2×Dic14—44no—
177C2×Dic14—41no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4C2×D411yescharacteristic
182C7×C4⋊C4C2×D411yescharacteristic
183C7×C4⋊C4C2×D412yes—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—21no—
187C7×C22⋊C4—21no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—21no—
190C7×C22⋊C4—22no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7C2×D411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2×D411yescharacteristic
197C42×C7C2411yescharacteristic
198#1142—416no—
199C4⋊Dic7—22no—
200C4⋊Dic7—22no—
201C4⋊Dic7—41no—
202C4⋊Dic7—41no—
203#1136—42no—
204C2×C7⋊C8—24no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×H(5,𝔽2)—21no—
209C22×SD16—142no—
210#550—21no—
211#550C2×D712yes—
212D42—22no—
213D42—22no—
214D42—21no—
215D42C2×D712yes—
216#535—22no—
217#535—21no—
218#532—21no—
219#532C2×D711yescharacteristic
220C2×C4⋊Q8—71no—
221C2×#127—21no—
222C2×#127C2×D711yescharacteristic
223C4×#41C2×D711yescharacteristic
224#498—141no—
225#492—74no—
226#492—142no—
227#486—141no—
228#481—142no—
229#477—72no—
230#476—72no—
231#476—74no—
232#475—141no—
233#474—141no—
234#431—71no—
235#428—141no—
236#423—71no—
237#417—141no—
238C2×C4×C8—71no—
239#331—74no—
240Q8⋊C8—72no—
241D4⋊C8—72no—
242D4⋊C8—74no—
243C23×C7—21no—
244C23×C7—22no—
245C23×C7—24no—
246C23×C7—48no—
247C23×C7—44no—
248C23×C7—82no—
249C23×C7—82no—
250C23×C7—84no—
251C23×C7—84no—
252C23×C7#12011yescharacteristic
253C7×Q8—21no—
254C7×Q8—42no—
255C7×Q8C2×SD1612yescomplemented
256C7×D4—24no—
257C7×D4—21no—
258C7×D4—24no—
259C7×D4—44no—
260C7×D4—48no—
261C7×D4—48no—
262C7×D4—44no—
263C7×D4—42no—
264C7×D4—42no—
265C7×D4—44no—
266C7×D4—42no—
267C7×D4—44no—
268C7×D4—44no—
269C7×D4—42no—
270C7×D4—82no—
271C7×D4—84no—
272C7×D4—84no—
273C7×D4C2×SD1612yescomplemented
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—48no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7#12011yescharacteristic
288C2×C4×C7#12011yescharacteristic
289C2×C4×C7C2×SD1612yes—
290C2×C4×C7AGL(1,ℤ/8)12yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×Dic7—42no—
295C2×Dic7—82no—
296Dic14—44no—
297Dic14—82no—
298C7⋊C8—24no—
299C7⋊C8—42no—
300C7⋊C8—42no—
301H(5,𝔽2)—44no—
302H(5,𝔽2)—44no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—21no—
306C2×#41C22⋊D711yescharacteristic
307C22×Q8—141no—
308C22×D4—22no—
309C22×D4—24no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4C22⋊D712yes—
316C2×SD16—148no—
317C2×SD16—288no—
318C2×M16—141no—
319C22×C8—141no—
320C4⋊Q8—72no—
321C4⋊Q8—141no—
322C4⋊Q8—141no—
323C4⋊Q8—141no—
324C4⋊Q8C22⋊D711yescharacteristic
325#127—22no—
326#127—22no—
327#127—22no—
328#127—21no—
329#127—21no—
330#127C22⋊D711yescharacteristic
331#127C22⋊D712yes—
332#127C22×D712yes—
333#124—22no—
334#124—21no—
335#121—22no—
336#121—24no—
337#121—24no—
338#121—21no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121C22⋊D712yes—
343#120—24no—
344#120—24no—
345#120—24no—
346C4×Q8C22⋊D711yescharacteristic
347C4×Q8C22×D711yescharacteristic
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4C22⋊D711yescharacteristic
354C4×D4C22×D711yescharacteristic
355C4×D4C22×D712yes—
356#117—21no—
357#117C22⋊D711yescharacteristic
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C42—21no—
361C2×C42C22×D711yescharacteristic
362#107—142no—
363#106—142no—
364#106—144no—
365C4⋊C8—72no—
366C4⋊C8—72no—
367C4⋊C8—141no—
368#103—142no—
369#102—142no—
370#102—144no—
371C22⋊C8—141no—
372C4×C8—72no—
373C4×C8—141no—
374C4×C8—141no—
375C22×C7—21no—
376C22×C7—21no—
377C22×C7—22no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—44no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—44no—
384C22×C7—44no—
385C22×C7—42no—
386C22×C7—164no—
387C22×C7—168no—
388C22×C7#45012yes—
389C22×C7C2×#12011yescharacteristic
390C4×C7—22no—
391C4×C7—21no—
392C4×C7—21no—
393C4×C7—42no—
394C4×C7—41no—
395C4×C7—41no—
396C4×C7#45012yes—
397C4×C7#45312yes—
398C4×C7C22×SD1611yescharacteristic
399C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
400C4×C7#57712yes—
401Dic7—82no—
402C24—24no—
403C24—22no—
404#41—22no—
405#41—48no—
406#41—44no—
407#41—44no—
408#41—42no—
409#41—42no—
410#41#114214yescomplemented
411C2×Q8—21no—
412C2×Q8—142no—
413C2×Q8—284no—
414C2×Q8—281no—
415C2×Q8C2×C22⋊D711yescharacteristic
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—48no—
430C2×D4—44no—
431C2×D4—416no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—82no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4C2×C22⋊D712yes—
440C2×D4C2×C22⋊D711yescharacteristic
441C2×D4C2×C22⋊D714yes—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—281no—
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451SD16—2816no—
452M16—282no—
453C2×C8—144no—
454C2×C8—142no—
455C2×C8—142no—
456C2×C8—142no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—142no—
461C4⋊C4—142no—
462C4⋊C4—281no—
463C4⋊C4—281no—
464C4⋊C4C2×C22⋊D711yescharacteristic
465C4⋊C4C2×C22⋊D712yes—
466C4⋊C4C2×C22⋊D711yescharacteristic
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42C2×C22⋊D711yescharacteristic
478C42C2×C22⋊D711yescharacteristic
479C42C2×C22⋊D711yescharacteristic
480C42C23×D711yescharacteristic
481C2×C7—22no—
482C2×C7—42no—
483C2×C7—82no—
484C2×C7—84no—
485C2×C7C2×#45011yescharacteristic
486C2×C7#300911yescharacteristic
487C2×C7C2×#45311yescharacteristic
488C23—21no—
489C23—24no—
490C23—22no—
491C23—48no—
492C23—44no—
493C23—82no—
494C23—82no—
495C23—84no—
496C23—84no—
497C23#651611yescharacteristic
498Q8—21no—
499Q8—42no—
500Q8—284no—
501Q8—562no—
502Q8C2×#114212yescomplemented
503D4—24no—
504D4—21no—
505D4—24no—
506D4—44no—
507D4—48no—
508D4—48no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—44no—
514D4—44no—
515D4—42no—
516D4—44no—
517D4—82no—
518D4—84no—
519D4—84no—
520D4C2×#114212yescomplemented
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—282no—
534C2×C4—562no—
535C2×C4#649512yes—
536C2×C4C2×#114212yes—
537C2×C4#651611yescharacteristic
538C2×C4#651611yescharacteristic
539C2×C4#651611yescharacteristic
540C2×C4C22×C22⋊D711yescharacteristic
541C2×C4C22×C22⋊D711yescharacteristic
542C2×C4C22×C22⋊D711yescharacteristic
543C8—144no—
544C8—282no—
545C8—282no—
546C7#1862111yescharacteristic, complemented, Sylow
547C22—21no—
548C22—21no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—164no—
559C22—168no—
560C22#8999012yes—
561C22C2×#651611yescharacteristic
562C4—22no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—562no—
569C4#8999012yes—
570C4#9001312yes—
571C4C2×#649511yescharacteristic
572C4C22×#114211yescharacteristic
573C4#9052712yes—
574C2—22no—
575C2—42no—
576C2—84no—
577C2—82no—
578C2C2×#8999011yescharacteristic
579C2C2×#9001311yescharacteristic
580C2#1174155011yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3498C2
socleC22×C7C2×#120
hypercenter#3498D7
hypocenterC7#18621
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
2. #11740716C2semidirect
3. C2×#89848C2semidirect
4. #11739953C2semidirect
5. #11729673C2semidirect
6. #11729661C2semidirect
7. #11729556C2semidirect
8. #11729513C2semidirect
9. #11729392C2semidirect
25. C2×#6432C22semidirect
29. #89848C22semidirect
32. #89793C22semidirect
34. #89780C22semidirect
35. #89780C22semidirect
38. #89697C22semidirect
39. #89696C22semidirect
40. C2×C4×C7⋊C8C22semidirect
41. #89339C22semidirect
42. Q8⋊(C7⋊C8)C22semidirect
43. D4⋊(C7⋊C8)C22semidirect
44. D4⋊(C7⋊C8)C22semidirect
137. C7×#41SD16semidirect
255. C7×Q8C2×SD16semidirect
273. C7×D4C2×SD16semidirect
410. #41#1142semidirect
502. Q8C2×#1142semidirect
520. D4C2×#1142semidirect
546. C7#18621semidirect

Maximal subgroups

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

subgroupindex
C7×#34982
#117407162
C2×#898482
#117399532
#117296732
#117296612
#117295562
#117295132
#117293922
#186217