Finite Group Databasebeta

← #13392149

Subgroups — #13392149

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
1C2×C7×#483C211yesmaximal, characteristic, complemented
2#11740206C211yesmaximal, characteristic
3#11739604C211yesmaximal, characteristic, complemented
4C22×#6383C211yesmaximal, characteristic
5#11729450C211yesmaximal, characteristic, complemented
6#11728339C212yesmaximal, complemented
7#11728323C212yesmaximal, complemented
8#11727999C212yesmaximal, complemented
9#11727984C212yesmaximal, complemented
10#11727713C211yesmaximal, characteristic
11#11727704C211yesmaximal, characteristic, complemented
12C23×D28C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6510C2212yes—
16C2×#6489—21no—
17C23×C7⋊C8C2211yescharacteristic
18C7×#483—24no—
19C2×C7×#106C2211yescharacteristic, complemented
20C2×C7×#106C2211yescharacteristic, complemented
21C2×C7×#103C2211yescharacteristic, complemented
22C2×C7×#103C2211yescharacteristic, complemented
23C2×C7×C22⋊C8C2211yescharacteristic
24C2×#6409C2212yescomplemented
25#89984C2211yescharacteristic
26#89983—21no—
27C4×C22⋊Dic7C2211yescharacteristic
28C2×#6395C2212yes—
29C2×#6395C2212yes—
30C2×Dic7⋊C8C2212yescomplemented
31#89763—22no—
32#89760C2212yes—
33#89757C2212yescomplemented
34#89753—22no—
35C4⋊C4×Dic7C2212yescomplemented
36C2×#6383—24no—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39#89727—21no—
40#89727—21no—
41#89345C2212yescomplemented
42#89324C2212yescomplemented
43#24477—71nomaximal, Sylow
44C24×D7—21no—
45C22×C7×Q8C2311yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7C2311yescharacteristic
48C22×D28—22no—
49C22×D28—24no—
50C22×D28—22no—
51C22×D28—21no—
52C22×D28—42no—
53C22×D28C2312yes—
54C22×C7×C8C2311yescharacteristic
55C22×C7×C8C2311yescharacteristic
56C7×C22⋊Q8—22no—
57C7×C22⋊Q8—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4D414yes—
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C22⋊Dic7—21no—
67C2×C22⋊Dic7D411yescharacteristic
68C2×C22⋊Dic7D411yescharacteristic
69#6510—24no—
70#6510—42no—
71C2×(C2×D7)⋊C4—22no—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4—22no—
74C2×(C2×D7)⋊C4—22no—
75#6489—44no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7C2312yes—
78C2×C4×Dic7—22no—
79C2×C4×Dic7C2312yes—
80C22×C7⋊C8—22no—
81C22×C7⋊C8—42no—
82C22×C7⋊C8C2312yes—
83C7×#106—22no—
84C7×#106—22no—
85C7×#103—22no—
86C7×#103—22no—
87C7×C22⋊C8—22no—
88#6409—42no—
89C7⋊#95—21no—
90C7⋊#95—22no—
91C7⋊#95—22no—
92C7⋊#95D412yes—
93#6395—24no—
94#6395—42no—
95Dic7⋊C8—42no—
96#6383—24no—
97#6383—24no—
98C2×#483D711yescharacteristic, complemented
99C22×#102—71no—
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×C7C2×D411yescharacteristic
110C23×D7—22no—
111C23×D7—44no—
112C23×D7—42no—
113C23×D7—42no—
114C23×D7—42no—
115C23×D7—82no—
116C2×C7×Q8—21no—
117C2×C7×Q8—22no—
118C2×C7×Q8—21no—
119C2×C7×Q8—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—24no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—42no—
128C22×C4×C7—42no—
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D412yes—
131C22×C4×C7#4112yes—
132C22×C4×C7C2411yescharacteristic
133C22×Dic7—22no—
134C22×Dic7C2×D412yes—
135C22×Dic7C2×D412yes—
136C2×D28—28no—
137C2×D28—22no—
138C2×D28—22no—
139C2×D28—22no—
140C2×D28—22no—
141C2×D28—22no—
142C2×D28—48no—
143C2×D28—44no—
144C2×D28—44no—
145C2×D28—42no—
146C2×D28#4114yes—
147C2×C7×C8—21no—
148C2×C7×C8—21no—
149C2×C7×C8—22no—
150C2×C7×C8—22no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—42no—
157C7×C4⋊C4C2×D412yes—
158C7×C4⋊C4C2×D412yes—
159C7×C22⋊C4—42no—
160C22⋊Dic7—22no—
161C22⋊Dic7—22no—
162C22⋊Dic7—42no—
163(C2×D7)⋊C4—44no—
164(C2×D7)⋊C4—44no—
165(C2×D7)⋊C4—82no—
166(C2×D7)⋊C4—82no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7—44no—
170C4×Dic7—22no—
171C4×Dic7—22no—
172C4×Dic7—44no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—44no—
177C23×D4—71no—
178C23×C8—71no—
179C2×#124—72no—
180C2×C22⋊Q8C2×D711yescharacteristic
181C2×#117—141no—
182C22×C4⋊C4C2×D711yescharacteristic
183#483—24no—
184C2×#106C2×D711yescharacteristic, complemented
185C2×#106C2×D711yescharacteristic, complemented
186C2×C4⋊C8—72no—
187C2×#103—72no—
188C2×#103C2×D711yescharacteristic, complemented
189C2×#103C2×D711yescharacteristic, complemented
190C2×#102—72no—
191C2×#102—72no—
192C2×#102—72no—
193C2×#102—72no—
194C2×#102—144no—
195C2×C22⋊C8C2×D711yescharacteristic
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#12112yes—
211C23×C7#13512yes—
212C23×C7AGL(1,ℤ/8)12yes—
213C23×C7C22×D411yescharacteristic
214C22×D7—24no—
215C22×D7—48no—
216C22×D7—42no—
217C22×D7—42no—
218C22×D7—42no—
219C22×D7—42no—
220C22×D7—42no—
221C22×D7—42no—
222C22×D7—88no—
223C22×D7—88no—
224C22×D7—84no—
225C22×D7—84no—
226C7×Q8—41no—
227C7×Q8—41no—
228C7×Q8—42no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—24no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—48no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7#12112yes—
245C2×C4×C7C22×D411yescharacteristic
246C2×C4×C7C2×#4111yescharacteristic
247C2×C4×C7C2×#4112yes—
248C2×C4×C7H(5,𝔽2)12yes—
249C2×Dic7—22no—
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258D28—24no—
259D28—24no—
260D28—48no—
261D28—42no—
262D28—42no—
263C7×C8—41no—
264C7×C8—41no—
265C7⋊C8—42no—
266C25—141no—
267C22×Q8C22×D711yescharacteristic
268C22×D4—72no—
269C22×D4—144no—
270C22×D4—141no—
271C22×D4—142no—
272C22×D4—142no—
273C22×D4—282no—
274C23×C4—21no—
275C23×C4C22×D711yescharacteristic
276C22×C8—72no—
277C22×C8—142no—
278C22×C8—282no—
279C22×C8C22×D711yescharacteristic
280C22×C8C22×D711yescharacteristic
281#124—144no—
282#124—282no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285#117—284no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—72no—
290C2×C4⋊C4—142no—
291C2×C4⋊C4C22⋊D714yes—
292C2×C4⋊C4C22×D711yescharacteristic
293C2×C4⋊C4C22×D711yescharacteristic
294C2×C4⋊C4C22×D711yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—71no—
297C2×C22⋊C4—71no—
298C2×C22⋊C4—142no—
299C2×C22⋊C4—142no—
300C2×C22⋊C4—141no—
301C2×C22⋊C4—142no—
302C2×C22⋊C4—142no—
303C2×C42—72no—
304C2×C42—142no—
305#106—22no—
306#106—22no—
307C4⋊C8—282no—
308#103—22no—
309#103—22no—
310#103—282no—
311#102—144no—
312#102—144no—
313#102—144no—
314#102—282no—
315C22⋊C8—22no—
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#46314yes—
325C22×C7C2×#12111yescharacteristic
326C22×C7#54612yes—
327C22×C7C2×#13511yescharacteristic
328C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C7#57712yes—
330C2×D7—44no—
331C2×D7—44no—
332C2×D7—44no—
333C2×D7—816no—
334C2×D7—82no—
335C2×D7—82no—
336C2×D7—82no—
337C2×D7—82no—
338C4×C7—22no—
339C4×C7—42no—
340C4×C7—42no—
341C4×C7—81no—
342C4×C7—81no—
343C4×C7#52511yescharacteristic
344C4×C7#53511yescharacteristic
345C4×C7#55112yes—
346Dic7—42no—
347Dic7—42no—
348Dic7—82no—
349C24—142no—
350C24—284no—
351C24—282no—
352C24—282no—
353C24—282no—
354C24—562no—
355C24D4×D711yescharacteristic
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×D4—74no—
361C2×D4—148no—
362C2×D4—142no—
363C2×D4—142no—
364C2×D4—142no—
365C2×D4—142no—
366C2×D4—142no—
367C2×D4—288no—
368C2×D4—284no—
369C2×D4—284no—
370C2×D4—282no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—72no—
381C22×C4—72no—
382C22×C4—142no—
383C22×C4D4×D711yescharacteristic
384C22×C4#115912yes—
385C22×C4C2×C22⋊D712yes—
386C22×C4C23×D711yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—142no—
392C2×C8—142no—
393C2×C8—142no—
394C2×C8—284no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—142no—
402C4⋊C4—142no—
403C4⋊C4—284no—
404C4⋊C4C2×C22⋊D712yes—
405C4⋊C4C2×C22⋊D712yes—
406C22⋊C4—42no—
407C22⋊C4—142no—
408C22⋊C4—142no—
409C22⋊C4—284no—
410C22⋊C4—284no—
411C22⋊C4—282no—
412C22⋊C4—562no—
413C22⋊C4—562no—
414C42—142no—
415C42—142no—
416C42—284no—
417C2×C7—24no—
418C2×C7#241111yescharacteristic
419C2×C7C2×#46311yescharacteristic
420C2×C7#304311yescharacteristic
421C2×C7#329612yes—
422C2×C7#330612yes—
423D7—84no—
424C23—22no—
425C23—22no—
426C23—144no—
427C23—288no—
428C23—282no—
429C23—282no—
430C23—282no—
431C23—282no—
432C23—282no—
433C23—282no—
434C23—568no—
435C23—568no—
436C23—564no—
437C23—564no—
438C23#647712yes—
439C23#647912yes—
440C23#650212yes—
441C23C2×D4×D711yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—42no—
445D4—144no—
446D4—144no—
447D4—288no—
448D4—282no—
449D4—282no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—142no—
466C2×C4—142no—
467C2×C4—142no—
468C2×C4—142no—
469C2×C4—142no—
470C2×C4—142no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4#650212yes—
475C2×C4C2×#115911yescharacteristic
476C2×C4D7×#4112yes—
477C2×C4#655312yes—
478C2×C4C22×C22⋊D711yescharacteristic
479C8—41no—
480C8—41no—
481C8—282no—
482C7#2447711yescharacteristic, complemented, Sylow
483C22—24no—
484C22—24no—
485C22—22no—
486C22—22no—
487C22—284no—
488C22—284no—
489C22—284no—
490C22—5616no—
491C22—562no—
492C22—562no—
493C22—562no—
494C22—562no—
495C22#8993714yes—
496C22#9034512yes—
497C22C2×#647711yescharacteristic
498C22C2×#647911yescharacteristic
499C22#9046912yes—
500C22C2×#650211yescharacteristic
501C4—22no—
502C4—42no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—282no—
507C4—282no—
508C4—562no—
509C4#9034912yes—
510C4#9051911yescharacteristic
511C4#9052111yescharacteristic
512C2—24no—
513C2—564no—
514C2#1173889012yes—
515C2#1173889112yes—
516C2#1174033911yescharacteristic
517C2C2×#8993711yescharacteristic
518C2#1174141411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#483C2
socleC23×C7C22×D4
prime radicalC23×D28C22
hypercenterC2×#483D7
hypocenterC7#24477
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
1. C2×C7×#483C2semidirect
3. #11739604C2semidirect
5. #11729450C2semidirect
6. #11728339C2semidirect
7. #11728323C2semidirect
8. #11727999C2semidirect
9. #11727984C2semidirect
11. #11727704C2semidirect
19. C2×C7×#106C22semidirect
20. C2×C7×#106C22semidirect
21. C2×C7×#103C22semidirect
22. C2×C7×#103C22semidirect
24. C2×#6409C22semidirect
30. C2×Dic7⋊C8C22semidirect
33. #89757C22semidirect
35. C4⋊C4×Dic7C22semidirect
41. #89345C22semidirect
42. #89324C22semidirect
98. C2×#483D7semidirect
184. C2×#106C2×D7semidirect
185. C2×#106C2×D7semidirect
188. C2×#103C2×D7semidirect
189. C2×#103C2×D7semidirect
482. C7#24477semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4832
#117402062
#117396042
C22×#63832
#117294502
#117283392
#117283232
#117279992
#117279842
#117277132
#117277042
#244777