Finite Group Databasebeta

← #13392148

Subgroups — #13392148

Order 1792.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#102C211yesmaximal, characteristic
2#11741495C211yesmaximal, characteristic
3#11739604C211yesmaximal, characteristic, complemented
4C2×#89743C211yesmaximal, characteristic, complemented
5#11729882C211yesmaximal, characteristic
6#11729728C211yesmaximal, characteristic, complemented
7#11729726C211yesmaximal, characteristic, complemented
8#11727947C212yesmaximal, complemented
9#11727932C212yesmaximal, complemented
10#11727928C212yesmaximal, complemented
11#11727916C212yesmaximal, complemented
12C23×C7×D4C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6499C2212yes—
16C2×#6489—21no—
17C2×#6487C2211yescharacteristic
18C2×C7×#102—24no—
19C2×C7×#102C2212yes—
20C2×C7×#102C2212yes—
21#90019—21no—
22#90019—21no—
23#89994—22no—
24#89993C2212yes—
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27#89984C2211yescharacteristic
28#89983—21no—
29C4×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×#6390C2212yescomplemented
32C2×Dic7⋊C8C2212yescomplemented
33#89757C2212yescomplemented
34#89753—22no—
35C4⋊C4×Dic7C2212yescomplemented
36#89743—24no—
37C2×#6384C2211yescharacteristic, complemented
38C2×#6384C2211yescharacteristic, complemented
39C2×#6382C2211yescharacteristic, complemented
40C2×#6382C2211yescharacteristic, complemented
41#89346C2212yescomplemented
42#89345C2212yescomplemented
43#24477—71nomaximal, Sylow
44C25×C7—21no—
45C22×C7×D4—24no—
46C22×C7×D4—22no—
47C22×C7×D4—21no—
48C22×C7×D4—22no—
49C22×C7×D4—42no—
50C22×C7×D4C2312yes—
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C22×Dic14C2311yescharacteristic
54C22×C7×C8—22no—
55C22×C7×C8—42no—
56C22×C7×C8C2312yes—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4D414yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C22⋊Dic7—21no—
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—22no—
65C2×C22⋊Dic7—22no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7D411yescharacteristic
69C2×C22⋊Dic7D411yescharacteristic
70#6499—24no—
71#6499—42no—
72#6489—44no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7C2311yescharacteristic
75C2×C4⋊Dic7C2312yes—
76#6487—22no—
77#6487—22no—
78C2×Dic7⋊C4—21no—
79C2×C4×Dic7—22no—
80C2×C4×Dic7C2312yes—
81C22×C7⋊C8C2311yescharacteristic
82C22×C7⋊C8C2311yescharacteristic
83C7×#102—24no—
84C7×#102—24no—
85#6406—24no—
86#6406—42no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95D412yes—
91C22⋊(C7⋊C8)—22no—
92#6390—42no—
93Dic7⋊C8—42no—
94#6384—22no—
95#6384—22no—
96#6382—22no—
97#6382—22no—
98C2×#483—71no—
99C22×#102D711yescharacteristic
100#2435—71no—
101#2344—71no—
102#2065—72no—
103#2042—72no—
104#1983—72no—
105#1926—72no—
106#1892—71no—
107#1850—71no—
108#1786—71no—
109C24×C7—22no—
110C24×C7—44no—
111C24×C7—42no—
112C24×C7—42no—
113C24×C7—42no—
114C24×C7—82no—
115C24×C7C2×D411yescharacteristic
116C2×C7×D4—28no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4—48no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
125C2×C7×D4—42no—
126C2×C7×D4#4114yes—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D412yes—
136C22×C4×C7#4112yes—
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—22no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7C2×D412yes—
142C22×Dic7C2×D412yes—
143C2×Dic14—21no—
144C2×Dic14—21no—
145C2×Dic14—22no—
146C2×Dic14—22no—
147C2×C7×C8—22no—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C2×C7×C8—44no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4C2×D412yes—
155C7×C4⋊C4C2×D412yes—
156C22⋊Dic7—22no—
157C22⋊Dic7—22no—
158C22⋊Dic7—44no—
159C22⋊Dic7—44no—
160C22⋊Dic7—42no—
161C22⋊Dic7—42no—
162C22⋊Dic7—82no—
163C22⋊Dic7—82no—
164C4⋊Dic7—21no—
165C4⋊Dic7—22no—
166C4⋊Dic7—21no—
167C4⋊Dic7—22no—
168C4⋊Dic7—44no—
169Dic7⋊C4—42no—
170C4×Dic7—22no—
171C4×Dic7—22no—
172C4×Dic7—44no—
173C2×C7⋊C8—21no—
174C2×C7⋊C8—21no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177C23×D4C2×D711yescharacteristic
178C23×C8C2×D711yescharacteristic
179C2×#124—72no—
180C2×C22⋊Q8—71no—
181C2×#117—141no—
182C22×C4⋊C4C2×D711yescharacteristic
183#483—144no—
184C2×#106—71no—
185C2×#106—71no—
186C2×C4⋊C8—72no—
187C2×#103—71no—
188C2×#103—71no—
189C2×#103—72no—
190C2×#102—24no—
191C2×#102—72no—
192C2×#102—72no—
193C2×#102C2×D712yes—
194C2×#102C2×D712yes—
195C2×C22⋊C8—71no—
196#394—142no—
197#390—72no—
198#389—72no—
199#387—142no—
200#381—141no—
201#380—71no—
202#379—141no—
203#379—141no—
204C4×C4⋊C4—72no—
205C4×C22⋊C4—71no—
206#340—72no—
207#336—72no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—24no—
211C23×C7—48no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—88no—
219C23×C7—88no—
220C23×C7—84no—
221C23×C7—84no—
222C23×C7#12112yes—
223C23×C7#13512yes—
224C23×C7AGL(1,ℤ/8)12yes—
225C23×C7C22×D411yescharacteristic
226C7×D4—24no—
227C7×D4—24no—
228C7×D4—48no—
229C7×D4—42no—
230C7×D4—42no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—48no—
242C2×C4×C7#12112yes—
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C2×#4112yes—
245C2×C4×C7C2×#4111yescharacteristic
246C2×C4×C7H(5,𝔽2)12yes—
247C2×Dic7—22no—
248C2×Dic7—22no—
249C2×Dic7—22no—
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260Dic14—41no—
261Dic14—41no—
262Dic14—42no—
263C7×C8—42no—
264C7⋊C8—41no—
265C7⋊C8—41no—
266C25—21no—
267C22×Q8—71no—
268C22×D4—24no—
269C22×D4—21no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—42no—
273C22×D4C22×D712yes—
274C23×C4—21no—
275C23×C4C22×D711yescharacteristic
276C22×C8—22no—
277C22×C8—42no—
278C22×C8—71no—
279C22×C8—71no—
280C22×C8C22×D712yes—
281#124—144no—
282#124—282no—
283C22⋊Q8—142no—
284C22⋊Q8—142no—
285#117—284no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—71no—
289C2×C4⋊C4—72no—
290C2×C4⋊C4—142no—
291C2×C4⋊C4—141no—
292C2×C4⋊C4C22⋊D714yes—
293C2×C4⋊C4C22×D711yescharacteristic
294C2×C4⋊C4C22×D711yescharacteristic
295C2×C22⋊C4—71no—
296C2×C22⋊C4—71no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4—142no—
299C2×C22⋊C4—141no—
300C2×C22⋊C4—141no—
301C2×C22⋊C4—142no—
302C2×C22⋊C4—142no—
303C2×C42—72no—
304C2×C42—142no—
305#106—142no—
306#106—142no—
307C4⋊C8—282no—
308#103—142no—
309#103—142no—
310#103—282no—
311#102—24no—
312#102—24no—
313#102—144no—
314#102—282no—
315C22⋊C8—142no—
316#95—72no—
317#95—142no—
318#95—142no—
319#95—141no—
320C22×C7—22no—
321C22×C7—24no—
322C22×C7—22no—
323C22×C7—24no—
324C22×C7—44no—
325C22×C7—44no—
326C22×C7—44no—
327C22×C7—816no—
328C22×C7—82no—
329C22×C7—82no—
330C22×C7—82no—
331C22×C7—82no—
332C22×C7#46314yes—
333C22×C7C2×#12111yescharacteristic
334C22×C7#54612yes—
335C22×C7C2×#13511yescharacteristic
336C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
337C22×C7#57712yes—
338C4×C7—22no—
339C4×C7—42no—
340C4×C7—42no—
341C4×C7#52511yescharacteristic
342C4×C7#53511yescharacteristic
343C4×C7#55112yes—
344Dic7—42no—
345Dic7—42no—
346Dic7—81no—
347Dic7—81no—
348Dic7—82no—
349C24—22no—
350C24—44no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—82no—
355C24D4×D711yescharacteristic
356C2×Q8—142no—
357C2×Q8—141no—
358C2×Q8—142no—
359C2×Q8—141no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—42no—
370C2×D4#115714yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—72no—
379C22×C4—72no—
380C22×C4—142no—
381C22×C4—141no—
382C22×C4—141no—
383C22×C4D4×D711yescharacteristic
384C22×C4#115912yes—
385C22×C4C2×C22⋊D712yes—
386C22×C4C23×D711yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—44no—
391C2×C8—142no—
392C2×C8—142no—
393C2×C8—141no—
394C2×C8—141no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—142no—
399C4⋊C4—141no—
400C4⋊C4—142no—
401C4⋊C4—141no—
402C4⋊C4—284no—
403C4⋊C4—282no—
404C4⋊C4C2×C22⋊D712yes—
405C4⋊C4C2×C22⋊D712yes—
406C22⋊C4—142no—
407C22⋊C4—142no—
408C22⋊C4—284no—
409C22⋊C4—284no—
410C22⋊C4—282no—
411C22⋊C4—282no—
412C22⋊C4—562no—
413C22⋊C4—562no—
414C42—142no—
415C42—142no—
416C42—284no—
417C2×C7—24no—
418C2×C7—84no—
419C2×C7#241111yescharacteristic
420C2×C7C2×#46311yescharacteristic
421C2×C7#304311yescharacteristic
422C2×C7#329612yes—
423C2×C7#330612yes—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—48no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—88no—
435C23—88no—
436C23—84no—
437C23—84no—
438C23#647412yes—
439C23#647512yes—
440C23#650212yes—
441C23C2×D4×D711yescharacteristic
442Q8—282no—
443Q8—281no—
444Q8—281no—
445D4—24no—
446D4—24no—
447D4—48no—
448D4—42no—
449D4—42no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—142no—
462C2×C4—142no—
463C2×C4—142no—
464C2×C4—142no—
465C2×C4—142no—
466C2×C4—142no—
467C2×C4—282no—
468C2×C4—282no—
469C2×C4—282no—
470C2×C4—282no—
471C2×C4—282no—
472C2×C4—281no—
473C2×C4—281no—
474C2×C4#650212yes—
475C2×C4C2×#115712yes—
476C2×C4C2×#115911yescharacteristic
477C2×C4#654812yes—
478C2×C4C22×C22⋊D711yescharacteristic
479C8—42no—
480C8—281no—
481C8—281no—
482C7#2447711yescharacteristic, complemented, Sylow
483C22—22no—
484C22—24no—
485C22—24no—
486C22—22no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#8992514yes—
496C22#9018112yes—
497C22C2×#647411yescharacteristic
498C22C2×#647511yescharacteristic
499C22#9045312yes—
500C22C2×#650211yescharacteristic
501C4—22no—
502C4—42no—
503C4—42no—
504C4—282no—
505C4—282no—
506C4—561no—
507C4—562no—
508C4—561no—
509C4#9018612yes—
510C4#9048211yescharacteristic
511C4#9049611yescharacteristic
512C2—24no—
513C2—84no—
514C2#1173620412yes—
515C2#1173620612yes—
516C2#1173955111yescharacteristic
517C2C2×#8992511yescharacteristic
518C2#1174129411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#102C2
socleC23×C7C22×D4
prime radicalC23×C7×D4C22
hypercenterC22×#102D7
hypocenterC7#24477
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
3. #11739604C2semidirect
4. C2×#89743C2semidirect
6. #11729728C2semidirect
7. #11729726C2semidirect
8. #11727947C2semidirect
9. #11727932C2semidirect
10. #11727928C2semidirect
11. #11727916C2semidirect
31. C2×#6390C22semidirect
32. C2×Dic7⋊C8C22semidirect
33. #89757C22semidirect
35. C4⋊C4×Dic7C22semidirect
37. C2×#6384C22semidirect
38. C2×#6384C22semidirect
39. C2×#6382C22semidirect
40. C2×#6382C22semidirect
41. #89346C22semidirect
42. #89345C22semidirect
482. C7#24477semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1022
#117414952
#117396042
C2×#897432
#117298822
#117297282
#117297262
#117279472
#117279322
#117279282
#117279162
#244777