Finite Group Databasebeta

← #13407458

Subgroups — #13407458

Order 1792.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#542C211yesmaximal, characteristic
2C2×#89865C211yesmaximal, characteristic, complemented
3C2×#89832C211yesmaximal, characteristic, complemented
4#11739707C211yesmaximal, characteristic, complemented
5#11729542C211yesmaximal, characteristic, complemented
6#11729479C211yesmaximal, characteristic, complemented
7#11729454C212yesmaximal, complemented
8#11729439C212yesmaximal
9#11729386C212yesmaximal, complemented
10#11729366C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic, complemented
12C7×#542—24no—
13C7×#542—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—21no—
16C2×C7×#124—21no—
17C2×C7×#124C2211yescharacteristic
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117C2211yescharacteristic
25C22×#1144C2212yes—
26C22×#1142C2212yes—
27C2×#6487C2212yes—
28C2×#6432C2211yescharacteristic
29C2×#6409C2212yes—
30C2×#6406C2212yes—
31C2×C22⋊(C7⋊C8)C2212yes—
32#89865—24no—
33#89832—24no—
34#89755—21no—
35#89755—21no—
36C2×#6384C2212yes—
37C2×#6384C2212yes—
38#89725—21no—
39#89725—21no—
40#89708C2211yescharacteristic
41#89701C2212yes—
42C2×#6377C2211yescharacteristic
43C2×C4×C7⋊C8C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#25613—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×D4—21no—
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C22×Dic14C2312yes—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8D414yes—
57C7×#126—28no—
58C7×#124—28no—
59C7×#124—22no—
60C7×#124D414yes—
61C7×#123—28no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C7×#121—22no—
66C7×#121—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C7×#117—22no—
72C7×#117—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4D411yescharacteristic
86C2×C7×C22⋊C4D411yescharacteristic
87C2×C42×C7—21no—
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—22no—
91C2×#1144—24no—
92C2×#1144—24no—
93C2×#1144—42no—
94C2×#1142—24no—
95C2×#1142—24no—
96C2×#1142—42no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7C2312yes—
100#6487—42no—
101#6487—42no—
102C2×Dic7⋊C4—22no—
103C22×C7⋊C8C2312yes—
104C22×C7⋊C8C2312yes—
105#6432—22no—
106#6432—22no—
107#6409—42no—
108#6406—42no—
109C7⋊#95—22no—
110C7⋊#95—22no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C22⋊(C7⋊C8)—24no—
114#6384—24no—
115#6384—24no—
116#6377—22no—
117C4×C7⋊C8—22no—
118C2×#542D711yescharacteristic
119C2×#497—71no—
120C2×#487—71no—
121#2621—71no—
122#2001—72no—
123#2000—72no—
124#1993—71no—
125#1958—71no—
126#1878—72no—
127#1877—72no—
128#1750—71no—
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—24no—
132C2×C7×Q8—21no—
133C2×C7×Q8—22no—
134C2×C7×Q8—22no—
135C2×C7×Q8—82no—
136C2×C7×Q8C2×D412yes—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—82no—
142C2×C7×D4C2×D412yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—82no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—22no—
159C22×Dic7—22no—
160C2×Dic14—22no—
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—22no—
165C2×Dic14—22no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4C2×D412yes—
174C7×C4⋊C4C2×D412yes—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C42×C7—24no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7C2×D412yes—
186C42×C7C2×D412yes—
187C22⋊Dic7—44no—
188#1144—48no—
189#1142—48no—
190C4⋊Dic7—22no—
191C4⋊Dic7—22no—
192C4⋊Dic7—42no—
193C4⋊Dic7—42no—
194Dic7⋊C4—44no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8#4114yes—
201C22×Q16—72no—
202C22×SD16—72no—
203#542—24no—
204#542—24no—
205C2×C4⋊Q8—71no—
206C2×C4⋊Q8C2×D711yescharacteristic
207C2×#126—21no—
208C2×#124—21no—
209C2×#124C2×D711yescharacteristic
210C2×#123—21no—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8—72no—
213C2×C22⋊Q8C2×D711yescharacteristic
214C2×#121C2×D711yescharacteristic
215C2×C4×Q8C2×D711yescharacteristic
216C2×C4×D4C2×D711yescharacteristic
217C2×#117C2×D711yescharacteristic
218#497—144no—
219#487—144no—
220C2×#103—72no—
221C2×#103—72no—
222C2×#103—72no—
223C2×#102—72no—
224C2×C22⋊C8—72no—
225C2×#97—71no—
226C2×C4×C8—71no—
227#398—141no—
228#398—141no—
229#397—141no—
230#397—141no—
231#391—72no—
232#382—71no—
233#340—72no—
234#340—72no—
235#336—72no—
236C23×C7—21no—
237C23×C7—22no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7C22×D411yescharacteristic
247C7×Q8—24no—
248C7×Q8—42no—
249C7×Q8—84no—
250C7×D4—24no—
251C7×D4—42no—
252C7×D4—84no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—21no—
256C2×C4×C7—22no—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—21no—
260C2×C4×C7—22no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—84no—
272C2×C4×C7#12012yes—
273C2×C4×C7#12012yes—
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7C2×#4112yes—
277C2×C4×C7H(5,𝔽2)12yes—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284Dic14—42no—
285Dic14—42no—
286Dic14—42no—
287Dic14—42no—
288C7⋊C8—24no—
289C7⋊C8—42no—
290C22×Q8—21no—
291C22×Q8—72no—
292C22×Q8C22×D711yescharacteristic
293C22×D4—21no—
294C22×D4C22×D711yescharacteristic
295C23×C4—21no—
296C23×C4C22×D711yescharacteristic
297C2×Q16—144no—
298C2×Q16—144no—
299C2×Q16—282no—
300C2×SD16—144no—
301C2×SD16—144no—
302C2×SD16—282no—
303C22×C8—72no—
304C22×C8—72no—
305C4⋊Q8—22no—
306C4⋊Q8—142no—
307C4⋊Q8—142no—
308C4⋊Q8C22⋊D714yes—
309#126—28no—
310#124—28no—
311#124—22no—
312#124C22⋊D714yes—
313#123—28no—
314C22⋊Q8—28no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—282no—
318C22⋊Q8—282no—
319#121—22no—
320#121—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×D4—22no—
324C4×D4—22no—
325#117—22no—
326#117—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—72no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22⋊D711yescharacteristic
335C2×C4⋊C4C22⋊D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4C22⋊D711yescharacteristic
345C2×C22⋊C4C22⋊D711yescharacteristic
346C2×C42—21no—
347C2×C42C22×D711yescharacteristic
348C2×C42C22×D711yescharacteristic
349#103—144no—
350#103—144no—
351#103—282no—
352#102—282no—
353C22⋊C8—144no—
354#97—142no—
355C4×C8—142no—
356#95—141no—
357#95—141no—
358#95—142no—
359#95—142no—
360C22×C7—42no—
361C22×C7—41no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—42no—
368C22×C7—82no—
369C22×C7—82no—
370C22×C7C2×#12011yescharacteristic
371C22×C7#53211yescharacteristic
372C22×C7#53511yescharacteristic
373C22×C7#57712yes—
374C22×C7#57812yes—
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—41no—
378C4×C7—42no—
379C4×C7—41no—
380C4×C7—42no—
381C4×C7—42no—
382C4×C7—41no—
383C4×C7#54612yes—
384C4×C7#55012yes—
385Dic7—82no—
386Dic7—82no—
387C24—21no—
388C24C2×C22⋊D711yescharacteristic
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—82no—
394C2×Q8—142no—
395C2×Q8—142no—
396C2×Q8—142no—
397C2×Q8—142no—
398C2×Q8—142no—
399C2×Q8—142no—
400C2×Q8C2×C22⋊D712yes—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—82no—
406C2×D4C2×C22⋊D712yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—82no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4C2×C22⋊D711yescharacteristic
419C22×C4C2×C22⋊D711yescharacteristic
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C23×D711yescharacteristic
424Q16—288no—
425SD16—288no—
426C2×C8—74no—
427C2×C8—142no—
428C2×C8—142no—
429C2×C8—142no—
430C2×C8—142no—
431C2×C8—142no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—142no—
440C4⋊C4—142no—
441C4⋊C4—284no—
442C4⋊C4—282no—
443C4⋊C4—282no—
444C4⋊C4C2×C22⋊D712yes—
445C4⋊C4C2×C22⋊D712yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—284no—
454C42—24no—
455C42—21no—
456C42—21no—
457C42C2×C22⋊D712yes—
458C42C2×C22⋊D712yes—
459C2×C7—81no—
460C2×C7—82no—
461C2×C7#243111yescharacteristic
462C2×C7#300911yescharacteristic
463C2×C7#301111yescharacteristic
464C2×C7#317812yes—
465C2×C7#322512yes—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23C22×C22⋊D711yescharacteristic
477Q8—24no—
478Q8—42no—
479Q8—84no—
480Q8—282no—
481Q8—282no—
482Q8—282no—
483Q8—282no—
484D4—24no—
485D4—42no—
486D4—84no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—282no—
512C2×C4#651612yes—
513C2×C4#651612yes—
514C2×C4D7×#4112yes—
515C2×C4#655312yes—
516C2×C4C22×C22⋊D711yescharacteristic
517C2×C4C22×C22⋊D711yescharacteristic
518C8—144no—
519C8—282no—
520C7#2561311yescharacteristic, complemented, Sylow
521C22—41no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—82no—
530C22—82no—
531C22#9052111yescharacteristic
532C22#9052311yescharacteristic
533C22#9052712yes—
534C22#9052912yes—
535C22C2×#651611yescharacteristic
536C4—22no—
537C4—22no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—562no—
545C4—562no—
546C4#9052012yes—
547C4#9052512yes—
548C2—81no—
549C2—82no—
550C2#1174090312yes—
551C2#1174090912yes—
552C2#1174153711yescharacteristic
553C2#1174155011yescharacteristic
554C2#1174156211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#542C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC2×#542D7
hypocenterC7#25613
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
2. C2×#89865C2semidirect
3. C2×#89832C2semidirect
4. #11739707C2semidirect
5. #11729542C2semidirect
6. #11729479C2semidirect
7. #11729454C2semidirect
9. #11729386C2semidirect
10. #11729366C2semidirect
11. #11729336C2semidirect
520. C7#25613semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5422
C2×#898652
C2×#898322
#117397072
#117295422
#117294792
#117294542
#117294392
#117293862
#117293662
#117293362
#256137