Finite Group Databasebeta

← #13407100

Subgroups — #13407100

Order 1792.

1206 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2C2×#89861C211yesmaximal, characteristic, complemented
3C2×#89800C211yesmaximal, characteristic, complemented
4#11739822C211yesmaximal, characteristic
5#11729534C211yesmaximal, characteristic, complemented
6#11729454C212yesmaximal, complemented
7#11729453C212yesmaximal, complemented
8#11729385C211yesmaximal, characteristic, complemented
9#11729369C211yesmaximal, characteristic, complemented
10#11729314C212yesmaximal, complemented
11#11729313C212yesmaximal, complemented
12C7×#554—24no—
13C7×#554—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—22no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C22×#1144C2211yescharacteristic
23C22×#1143C2211yescharacteristic
24C2×#6493C2212yes—
25C2×#6487C2212yes—
26C2×#6438C2211yescharacteristic
27C2×#6409C2212yes—
28#89995—21no—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89861—24no—
32#89800—24no—
33#89764—21no—
34C2×#6384C2212yes—
35C2×#6384C2211yescharacteristic
36C2×#6383C2212yes—
37C2×#6383C2211yescharacteristic
38#89729—22no—
39C4×C4⋊Dic7C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#25579—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4D411yescharacteristic
49C23×C4×C7—22no—
50C23×C4×C7C2312yes—
51C22×C22⋊D7—22no—
52C22×D28C2311yescharacteristic
53C22×Dic14C2311yescharacteristic
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8D414yescomplemented
56C7×#125—28no—
57C7×C22⋊Q8—216no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—24no—
60C4×C7×Q8—22no—
61C4×C7×Q8D414yescomplemented
62C4×C7×D4—28no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—24no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4D412yes—
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—22no—
79C2×#1144—22no—
80C2×#1144—42no—
81C2×#1143—22no—
82C2×#1143—42no—
83#6493—24no—
84#6493—42no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87C2×(C2×D7)⋊C4—22no—
88C2×C4⋊Dic7C2312yes—
89#6487—24no—
90#6487—42no—
91C2×Dic7⋊C4—22no—
92C2×C4×Dic7—21no—
93C22×C7⋊C8C2312yes—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6438—22no—
97#6438—22no—
98#6409—24no—
99C7⋊#95—21no—
100C22⋊(C7⋊C8)—42no—
101C22⋊(C7⋊C8)—42no—
102#6384—24no—
103#6384—22no—
104#6383—24no—
105#6383—22no—
106C4⋊(C7⋊C8)—22no—
107#6377—22no—
108C2×#554D711yescharacteristic, complemented
109C2×#489—71no—
110C2×#464—71no—
111#2540—71no—
112#2001—72no—
113#2000—72no—
114#1988—71no—
115#1955—71no—
116#1874—72no—
117#1872—72no—
118#1763—71no—
119C24×C7C2×D412yes—
120C23×D7—21no—
121C2×C7×Q8—24no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—81no—
126C2×C7×Q8—81no—
127C2×C7×Q8C2×D412yes—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—28no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—82no—
141C22×C4×C7—82no—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2411yescharacteristic
146C2×C22⋊D7—44no—
147C2×C22⋊D7—44no—
148C2×C22⋊D7—82no—
149C22×Dic7—22no—
150C22×Dic7—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—22no—
154C2×D28—22no—
155C2×Dic14—21no—
156C2×Dic14—21no—
157C2×Dic14—22no—
158C2×Dic14—22no—
159C7×C4⋊C4—28no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4C2×D412yes—
167C7×C4⋊C4C2×D414yes—
168C7×C22⋊C4—28no—
169C7×C22⋊C4—24no—
170C42×C7—24no—
171C42×C7C2×D412yes—
172C42×C7C2×D412yes—
173C22⋊Dic7—44no—
174#1144—44no—
175#1143—44no—
176(C2×D7)⋊C4—44no—
177(C2×D7)⋊C4—42no—
178(C2×D7)⋊C4—42no—
179C4⋊Dic7—22no—
180C4⋊Dic7#4114yes—
181Dic7⋊C4—44no—
182C4×Dic7—81no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C22×Q16—71no—
191C22×SD16—71no—
192#554—24no—
193#554—24no—
194C2×C4⋊Q8C2×D711yescharacteristic
195C2×#125—21no—
196C2×#124—71no—
197C2×C22⋊Q8—22no—
198C2×C22⋊Q8—72no—
199C2×C22⋊Q8C2×D712yes—
200C2×#121—72no—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4—21no—
203C2×C4×D4C2×D711yescharacteristic
204C22×C4⋊C4C2×D712yes—
205#489—144no—
206#464—144no—
207C2×C4⋊C8—71no—
208C2×#103—72no—
209C2×#103—72no—
210C2×#103—71no—
211C2×#102—72no—
212C2×#102—71no—
213C2×C22⋊C8—72no—
214C2×C22⋊C8—72no—
215C2×#97—71no—
216#399—141no—
217#392—141no—
218#390—142no—
219C4×C4⋊C4—71no—
220#340—72no—
221#340—72no—
222#340—72no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7#13714yes—
230C23×C7C22×D411yescharacteristic
231C22×D7—41no—
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—41no—
235C22×D7—41no—
236C7×Q8—22no—
237C7×Q8—22no—
238C7×Q8—42no—
239C7×Q8—84no—
240C7×D4—28no—
241C2×C4×C7—22no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—24no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—88no—
260C2×C4×C7#12012yes—
261C2×C4×C7#12012yes—
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C2×#4112yes—
265C2×C4×C7H(5,𝔽2)12yes—
266C22⋊D7—88no—
267C2×Dic7—24no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—41no—
271C2×Dic7—42no—
272D28—41no—
273D28—41no—
274D28—42no—
275Dic14—41no—
276Dic14—41no—
277Dic14—42no—
278C7⋊C8—41no—
279C7⋊C8—41no—
280C7⋊C8—42no—
281C22×Q8—21no—
282C22×Q8—71no—
283C22×Q8C22×D711yescharacteristic
284C22×D4—71no—
285C22×D4—142no—
286C22×D4C22⋊D711yescharacteristic
287C23×C4—22no—
288C23×C4C22×D712yes—
289C2×Q16—142no—
290C2×Q16—282no—
291C2×SD16—142no—
292C2×SD16—282no—
293C22×C8—71no—
294C22×C8—71no—
295C22×C8—72no—
296C4⋊Q8—22no—
297C4⋊Q8C22⋊D714yescomplemented
298#125—28no—
299#124—142no—
300#124—142no—
301C22⋊Q8—216no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—144no—
305C22⋊Q8—282no—
306#121—144no—
307#121—282no—
308C4×Q8—22no—
309C4×Q8C22⋊D714yescomplemented
310C4×D4—28no—
311C4×D4—22no—
312C4×D4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—72no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4C22⋊D711yescharacteristic
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C4⋊C4C22×D712yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4—141no—
329C2×C22⋊C4C22⋊D712yes—
330C2×C42—21no—
331C2×C42—141no—
332C2×C42C22×D711yescharacteristic
333C4⋊C8—142no—
334#103—144no—
335#103—142no—
336#103—144no—
337#102—142no—
338#102—144no—
339C22⋊C8—282no—
340C22⋊C8—282no—
341#97—142no—
342#95—141no—
343C22×C7—24no—
344C22×C7—24no—
345C22×C7—24no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7C2×#12011yescharacteristic
351C22×C7#54612yes—
352C22×C7C2×#13712yes—
353C22×C7#57611yescharacteristic
354C22×C7#57711yescharacteristic
355C2×D7—81no—
356C2×D7—81no—
357C2×D7—81no—
358C2×D7—82no—
359C2×D7—82no—
360C4×C7—22no—
361C4×C7—22no—
362C4×C7—42no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—44no—
366C4×C7#53211yescharacteristic
367C4×C7#53511yescharacteristic
368C4×C7#55012yes—
369Dic7—44no—
370Dic7—81no—
371C24—141no—
372C24C2×C22⋊D712yes—
373C2×Q8—24no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—142no—
380C2×Q8—142no—
381C2×Q8—141no—
382C2×Q8—141no—
383C2×Q8C2×C22⋊D712yes—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—142no—
387C2×D4—142no—
388C2×D4—141no—
389C2×D4—141no—
390C2×D4—284no—
391C2×D4—284no—
392C2×D4—562no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—142no—
406C22×C4—141no—
407C22×C4C2×C22⋊D712yes—
408C22×C4C2×C22⋊D711yescharacteristic
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411Q16—284no—
412SD16—284no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—142no—
417C2×C8—142no—
418C2×C8—141no—
419C2×C8—141no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—74no—
428C4⋊C4—142no—
429C4⋊C4—284no—
430C4⋊C4C2×C22⋊D712yes—
431C4⋊C4C2×C22⋊D714yes—
432C22⋊C4—28no—
433C22⋊C4—24no—
434C22⋊C4—284no—
435C22⋊C4—284no—
436C22⋊C4—282no—
437C22⋊C4—282no—
438C42—24no—
439C42—561no—
440C42C2×C22⋊D712yes—
441C42C2×C22⋊D712yes—
442C2×C7—44no—
443C2×C7#243111yescharacteristic
444C2×C7#300411yescharacteristic
445C2×C7#301211yescharacteristic
446C2×C7#331412yes—
447C2×C7#331612yes—
448D7—161no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—282no—
456C23—282no—
457C23—281no—
458C23—281no—
459C23—281no—
460C23#650514yes—
461C23C22×C22⋊D711yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—42no—
465Q8—84no—
466Q8—281no—
467Q8—281no—
468Q8—282no—
469D4—28no—
470D4—281no—
471D4—281no—
472D4—282no—
473D4—568no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—88no—
493C2×C4—144no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—281no—
498C2×C4#651612yes—
499C2×C4#651612yes—
500C2×C4D7×#4112yes—
501C2×C4#655312yes—
502C2×C4C22×C22⋊D711yescharacteristic
503C2×C4C22×C22⋊D711yescharacteristic
504C8—281no—
505C8—281no—
506C8—282no—
507C7#2557911yescharacteristic, complemented, Sylow
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—561no—
516C22—561no—
517C22—561no—
518C22—562no—
519C22—562no—
520C22C2×#650512yes—
521C22#9052012yes—
522C22#9052611yescharacteristic
523C22#9052711yescharacteristic
524C22C2×#651611yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—44no—
531C4—284no—
532C4—561no—
533C4#9052111yescharacteristic
534C4#9052311yescharacteristic
535C4#9052512yes—
536C2—44no—
537C2—1121no—
538C2#1174108012yes—
539C2#1174108612yes—
540C2#1174150411yescharacteristic
541C2#1174153711yescharacteristic
542C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radical#11739822C2
hypercenterC2×#554D7
hypocenterC7#25579
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C7×#554C2semidirect
2. C2×#89861C2semidirect
3. C2×#89800C2semidirect
5. #11729534C2semidirect
6. #11729454C2semidirect
7. #11729453C2semidirect
8. #11729385C2semidirect
9. #11729369C2semidirect
10. #11729314C2semidirect
11. #11729313C2semidirect
55. C7×C4⋊Q8D4semidirect
61. C4×C7×Q8D4semidirect
108. C2×#554D7semidirect
297. C4⋊Q8C22⋊D7semidirect
309. C4×Q8C22⋊D7semidirect
507. C7#25579semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#898612
C2×#898002
#117398222
#117295342
#117294542
#117294532
#117293852
#117293692
#117293142
#117293132
#255797