Finite Group Databasebeta

← #13245489

Subgroups — #13245489

Order 1792.

1200 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#546C211yesmaximal, characteristic
2#11729882C212yesmaximal, complemented
3#11729444C212yesmaximal, complemented
4#11729373C211yesmaximal, characteristic, complemented
5#11729354C211yesmaximal, characteristic, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C23×C7×D4C2211yescharacteristic
8C7×#546—24no—
9C7×#546—24no—
10C2×C7×#124C411yescharacteristic, complemented
11C2×C7×#123—21no—
12C2×C7×C22⋊Q8C2211yescharacteristic
13C2×C7×#121—21no—
14C2×C7×#121C411yescharacteristic, complemented
15C2×C7×#120—21no—
16C2×C4×C7×D4C411yescharacteristic, complemented
17C2×C4×C7×D4C2211yescharacteristic
18C22×C7×C22⋊C4—21no—
19C2×#6408—21no—
20C2×#6406—24no—
21#89984C2212yes—
22C2×C22⋊(C7⋊C8)C2212yes—
23#89333—24no—
24#89323—21no—
25#89321—21no—
26#89320—21no—
27#8381—71nomaximal, Sylow
28C25×C7—21no—
29C2×C7×#41—22no—
30C2×C7×#41—22no—
31C2×C7×#41—44no—
32C22×C7×Q8C2×C411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4—21no—
36C22×C7×D4—22no—
37C22×C7×D4—44no—
38C22×C7×D4C2×C411yescharacteristic
39C22×C7×D4C2×C411yescharacteristic
40C22×C7×D4D411yescharacteristic
41C22×C7×D4D414yes—
42C23×C4×C7—21no—
43C23×C4×C7—21no—
44C23×C4×C7C2×C411yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C7×#124—22no—
47C7×#124—22no—
48C7×#123—28no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8D414yes—
51C7×#121—28no—
52C7×#121—22no—
53C7×#121—22no—
54C7×#120—28no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C4×C7×D4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2×C411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C22⋊C4—28no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C7×C22⋊C4D411yescharacteristic
70C2×C42×C7C2×C411yescharacteristic
71C2×C22⋊Dic7—22no—
72C2×C4⋊Dic7—22no—
73C22×C7⋊C8—22no—
74#6408—42no—
75#6406—48no—
76C7⋊#95—22no—
77C22⋊(C7⋊C8)—22no—
78C22⋊(C7⋊C8)—22no—
79C2×#546D711yescharacteristic
80#1799—71no—
81#1786—72no—
82#1520—72no—
83#1459—71no—
84C24×C7—21no—
85C24×C7—21no—
86C24×C7—24no—
87C24×C7—44no—
88C24×C7—42no—
89C24×C7—41no—
90C24×C7—41no—
91C24×C7—82no—
92C24×C7C22⋊C411yescharacteristic
93C24×C7C2×D411yescharacteristic
94C7×#41—48no—
95C7×#41—44no—
96C7×#41—44no—
97C2×C7×Q8—21no—
98C2×C7×Q8—42no—
99C2×C7×Q8C22⋊C412yes—
100C2×C7×D4—24no—
101C2×C7×D4—28no—
102C2×C7×D4—24no—
103C2×C7×D4—22no—
104C2×C7×D4—22no—
105C2×C7×D4—21no—
106C2×C7×D4—21no—
107C2×C7×D4—24no—
108C2×C7×D4—21no—
109C2×C7×D4—21no—
110C2×C7×D4—24no—
111C2×C7×D4—42no—
112C2×C7×D4—44no—
113C2×C7×D4—48no—
114C2×C7×D4—44no—
115C2×C7×D4—44no—
116C2×C7×D4—82no—
117C2×C7×D4C22⋊C412yes—
118C2×C7×D4C22⋊C412yes—
119C22×C4×C7—24no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—24no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—42no—
126C22×C4×C7—42no—
127C22×C4×C7—44no—
128C22×C4×C7—82no—
129C22×C4×C7C22⋊C411yescharacteristic
130C22×C4×C7C22⋊C411yescharacteristic
131C22×C4×C7C22⋊C411yescharacteristic
132C22×C4×C7C22×C411yescharacteristic
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×Dic7—42no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—24no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4C22⋊C412yes—
141C7×C22⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—22no—
146C7×C22⋊C4—24no—
147C7×C22⋊C4—22no—
148C42×C7—21no—
149C42×C7—21no—
150C22⋊Dic7—42no—
151C22⋊Dic7—42no—
152C4⋊Dic7—44no—
153C2×C7⋊C8—44no—
154C2×C7⋊C8—42no—
155C22×#41Dic711yescharacteristic, complemented
156C23×D4C2×D711yescharacteristic
157#546—24no—
158#546—24no—
159C2×#124Dic711yescharacteristic, complemented
160C2×#123—21no—
161C2×C22⋊Q8C2×D711yescharacteristic
162C2×#121—21no—
163C2×#121Dic711yescharacteristic, complemented
164C2×#120—21no—
165C2×C4×D4Dic711yescharacteristic, complemented
166C2×C4×D4C2×D711yescharacteristic
167C22×C22⋊C4—21no—
168C2×#102—144no—
169C2×#99—141no—
170C2×C22⋊C8—72no—
171#380—72no—
172#342—141no—
173#328—144no—
174#324—141no—
175#323—141no—
176C23×C7—24no—
177C23×C7—22no—
178C23×C7—22no—
179C23×C7—21no—
180C23×C7—21no—
181C23×C7—21no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—48no—
185C23×C7—42no—
186C23×C7—42no—
187C23×C7—42no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7—41no—
191C23×C7—41no—
192C23×C7—41no—
193C23×C7—41no—
194C23×C7—44no—
195C23×C7—44no—
196C23×C7—84no—
197C23×C7—88no—
198C23×C7—84no—
199C23×C7—84no—
200C23×C7C2×C22⋊C411yescharacteristic
201C23×C7#12012yes—
202C23×C7#13512yes—
203C23×C7AGL(1,ℤ/8)11yescharacteristic
204C23×C7#13711yescharacteristic
205C7×Q8—42no—
206C7×Q8—42no—
207C7×D4—24no—
208C7×D4—22no—
209C7×D4—24no—
210C7×D4—48no—
211C7×D4—42no—
212C7×D4—48no—
213C7×D4—42no—
214C7×D4—42no—
215C7×D4—84no—
216C7×D4#9914yescomplemented
217C2×C4×C7—24no—
218C2×C4×C7—22no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—22no—
222C2×C4×C7—21no—
223C2×C4×C7—22no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—48no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—84no—
236C2×C4×C7C2×C22⋊C411yescharacteristic
237C2×C4×C7C2×C22⋊C411yescharacteristic
238C2×C4×C7#12012yes—
239C2×Dic7—82no—
240C2×Dic7—82no—
241C2×Dic7—82no—
242C7⋊C8—82no—
243C25—21no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—44no—
247C22×Q8C2×Dic711yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—44no—
253C22×D4C2×Dic711yescharacteristic
254C22×D4C2×Dic711yescharacteristic
255C22×D4C22⋊D711yescharacteristic
256C22×D4C22⋊D714yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×Dic711yescharacteristic
260C23×C4C22×D711yescharacteristic
261C22×C8—142no—
262#124—22no—
263#124—22no—
264#123—28no—
265C22⋊Q8—22no—
266C22⋊Q8C22⋊D714yes—
267#121—28no—
268#121—22no—
269#121—22no—
270#120—28no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4C2×Dic711yescharacteristic
278C2×C4⋊C4C22⋊D711yescharacteristic
279C2×C22⋊C4—28no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—142no—
286C2×C22⋊C4C22⋊D711yescharacteristic
287C2×C22⋊C4C22⋊D711yescharacteristic
288C2×C42C2×Dic711yescharacteristic
289#102—288no—
290#99—282no—
291C22⋊C8—142no—
292C22⋊C8—142no—
293#95—142no—
294C22×C7—22no—
295C22×C7—21no—
296C22×C7—22no—
297C22×C7—21no—
298C22×C7—24no—
299C22×C7—42no—
300C22×C7—41no—
301C22×C7—41no—
302C22×C7—44no—
303C22×C7—44no—
304C22×C7—42no—
305C22×C7—42no—
306C22×C7—41no—
307C22×C7—42no—
308C22×C7—41no—
309C22×C7—42no—
310C22×C7—41no—
311C22×C7—42no—
312C22×C7—88no—
313C22×C7—88no—
314C22×C7—81no—
315C22×C7—81no—
316C22×C7—82no—
317C22×C7—82no—
318C22×C7#37911yescharacteristic
319C22×C7#41611yescharacteristic
320C22×C7#41711yescharacteristic
321C22×C7#44912yes—
322C22×C7#45212yes—
323C4×C7—22no—
324C4×C7—21no—
325C4×C7—42no—
326C4×C7—41no—
327C4×C7—41no—
328C4×C7—42no—
329C4×C7—41no—
330C4×C7C2×#9912yes—
331Dic7—162no—
332C24—21no—
333C24—21no—
334C24—24no—
335C24—44no—
336C24—41no—
337C24—42no—
338C24—41no—
339C24—82no—
340C24C22⋊Dic711yescharacteristic
341C24C2×C22⋊D711yescharacteristic
342#41—48no—
343#41—44no—
344#41—44no—
345C2×Q8—21no—
346C2×Q8—42no—
347C2×Q8C22⋊Dic712yes—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—24no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4C22⋊Dic712yes—
366C2×D4C22⋊Dic712yes—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—82no—
377C22×C4—282no—
378C22×C4C22⋊Dic711yescharacteristic
379C22×C4C22⋊Dic711yescharacteristic
380C22×C4C22⋊Dic711yescharacteristic
381C22×C4C22×Dic711yescharacteristic
382C22×C4C2×C22⋊D712yes—
383C22×C4C2×C22⋊D712yes—
384C22×C4C2×C22⋊D711yescharacteristic
385C2×C8—284no—
386C2×C8—282no—
387C4⋊C4—21no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—284no—
391C4⋊C4C22⋊Dic712yes—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—282no—
400C22⋊C4—282no—
401C42—21no—
402C42—21no—
403C2×C7—22no—
404C2×C7—21no—
405C2×C7—21no—
406C2×C7—22no—
407C2×C7—44no—
408C2×C7—81no—
409C2×C7—82no—
410C2×C7#178111yescharacteristic
411C2×C7#178811yescharacteristic
412C2×C7#179511yescharacteristic
413C23—24no—
414C23—22no—
415C23—22no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—41no—
420C23—41no—
421C23—48no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—44no—
432C23—44no—
433C23—84no—
434C23—84no—
435C23—88no—
436C23—84no—
437C23#651211yescharacteristic
438C23#651312yes—
439C23#651411yescharacteristic
440C23C2×C22⋊Dic711yescharacteristic
441C23#651612yes—
442Q8—42no—
443Q8—42no—
444D4—24no—
445D4—22no—
446D4—24no—
447D4—48no—
448D4—42no—
449D4—48no—
450D4—42no—
451D4—42no—
452D4—84no—
453D4#640814yescomplemented
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—84no—
473C2×C4—562no—
474C2×C4—562no—
475C2×C4—562no—
476C2×C4C2×C22⋊Dic711yescharacteristic
477C2×C4C2×C22⋊Dic711yescharacteristic
478C2×C4#651612yes—
479C8—562no—
480C7#838111yescharacteristic, complemented, Sylow
481C22—21no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—24no—
486C22—41no—
487C22—42no—
488C22—41no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—42no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—42no—
499C22—88no—
500C22—88no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22#9000411yescharacteristic
506C22#9000511yescharacteristic
507C22#9001012yes—
508C22#9001112yes—
509C22#9001911yescharacteristic
510C4—22no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—1122no—
518C4C2×#640812yes—
519C2—22no—
520C2—21no—
521C2—21no—
522C2—22no—
523C2—44no—
524C2—82no—
525C2—81no—
526C2#1172988411yescharacteristic
527C2#1172988911yescharacteristic
528C2#1172989511yescharacteristic
subgroupG/H
centerC22#90019
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC2×C7×#546C2
socleC22×C7#379
cosocleC23×C4×C7C23
hypercenterC2×#546D7
hypocenterC7#8381
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
2. #11729882C2semidirect
3. #11729444C2semidirect
4. #11729373C2semidirect
5. #11729354C2semidirect
6. C22×C7×#41C4semidirect
10. C2×C7×#124C4semidirect
14. C2×C7×#121C4semidirect
16. C2×C4×C7×D4C4semidirect
155. C22×#41Dic7semidirect
159. C2×#124Dic7semidirect
163. C2×#121Dic7semidirect
165. C2×C4×D4Dic7semidirect
216. C7×D4#99semidirect
453. D4#6408semidirect
480. C7#8381semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5462
#117298822
#117294442
#117293732
#117293542
#83817