Finite Group Databasebeta

← #13408107

Subgroups — #13408107

Order 1792.

1318 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#532C211yesmaximal, characteristic
2C2×#89852C211yesmaximal, characteristic, complemented
3C2×#89843C211yesmaximal, characteristic, complemented
4#11739519C211yesmaximal, characteristic, complemented
5#11729542C211yesmaximal, characteristic, complemented
6#11729519C211yesmaximal, characteristic, complemented
7#11729434C212yesmaximal, complemented
8#11729433C212yesmaximal, complemented
9#11729361C212yesmaximal, complemented
10#11729360C212yesmaximal, complemented
11#11729283C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C7×#532—24no—
14C7×#532—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#127C2211yescharacteristic
17C2×C7×#124—21no—
18C2×C7×#121—21no—
19C2×C7×#121C2212yes—
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2212yes—
22C22×C42×C7C2211yescharacteristic
23C2×#6433C2211yescharacteristic
24C2×#6406C2212yes—
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28#89984C2211yescharacteristic
29#89984C2211yescharacteristic
30#89982—21no—
31#89982—21no—
32C2×C22⋊(C7⋊C8)C2212yes—
33#89852—24no—
34#89843—24no—
35C2×#6382C2212yes—
36C2×#6381C2212yes—
37#89708C2211yescharacteristic
38#89705—21no—
39#89705—21no—
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×#6377C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#25676—71nomaximal, Sylow
48C2×C7×#41—28no—
49C2×C7×#41—44no—
50C2×C7×#41—44no—
51C22×C7×Q8—21no—
52C22×C7×D4—22no—
53C22×C7×D4—21no—
54C22×C7×D4—21no—
55C22×C7×D4C2312yes—
56C23×C4×C7—22no—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8D414yes—
62C7×#127—22no—
63C7×#127D414yes—
64C7×#124—28no—
65C7×#121—28no—
66C7×#121—24no—
67C7×#121—24no—
68C4×C7×D4—28no—
69C4×C7×D4—24no—
70C4×C7×D4—24no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C42×C7—22no—
76C2×C42×C7—22no—
77C2×C42×C7D411yescharacteristic
78C2×C42×C7D411yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C22×C7⋊C8C2312yes—
92C22×C7⋊C8C2312yes—
93#6433—22no—
94#6433—22no—
95#6406—24no—
96#6406—24no—
97#6406—24no—
98#6406—24no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C22⋊(C7⋊C8)—24no—
106#6382—24no—
107#6381—24no—
108#6377—22no—
109#6377—22no—
110C2×#532D711yescharacteristic
111C2×#501—71no—
112C2×#490—71no—
113#2851—71no—
114#2034—72no—
115#2033—72no—
116#1993—71no—
117#1961—71no—
118#1864—72no—
119#1862—72no—
120#1749—71no—
121C24×C7—22no—
122C24×C7C2×D411yescharacteristic
123C7×#41—432no—
124C2×C7×Q8—24no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×D4—28no—
128C2×C7×D4—24no—
129C2×C7×D4—24no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—42no—
136C2×C7×D4—42no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
139C2×C7×D4—44no—
140C2×C7×D4—44no—
141C2×C7×D4C2×D414yes—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—28no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—41no—
153C22×C4×C7—41no—
154C22×C4×C7—44no—
155C22×C4×C7—44no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4C2×D414yes—
169C7×C22⋊C4—28no—
170C7×C22⋊C4—28no—
171C42×C7—22no—
172C42×C7—22no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7C2×D412yes—
176C42×C7C2×D412yes—
177C22⋊Dic7—42no—
178C22⋊Dic7—42no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185Dic7⋊C4—42no—
186Dic7⋊C4—42no—
187Dic7⋊C4—42no—
188Dic7⋊C4—42no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8#4114yescomplemented
195C22×#41—21no—
196#532—24no—
197#532—24no—
198C2×C4⋊Q8C2×D711yescharacteristic
199C2×#127C2×D711yescharacteristic
200C2×#125—71no—
201C2×#124—21no—
202C2×#121—21no—
203C2×#121C2×D712yes—
204C2×C4×D4—21no—
205C2×C4×D4C2×D712yes—
206C22×C42C2×D711yescharacteristic
207#501—144no—
208#490—144no—
209C2×#107—72no—
210C2×#106—72no—
211C2×#102—72no—
212C2×#102—72no—
213C2×#102—72no—
214C2×#102—72no—
215C2×C22⋊C8—72no—
216C2×#97—71no—
217C2×#97—71no—
218#389—71no—
219#389—71no—
220#387—141no—
221#387—141no—
222#385—141no—
223#385—141no—
224#382—71no—
225#380—71no—
226#380—71no—
227#340—72no—
228#340—72no—
229#336—72no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—24no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7AGL(1,ℤ/8)12yes—
244C23×C7C22×D411yescharacteristic
245C7×Q8—48no—
246C7×D4—24no—
247C7×D4—24no—
248C7×D4—48no—
249C7×D4—48no—
250C7×D4—416no—
251C7×D4—42no—
252C7×D4—42no—
253C2×C4×C7—28no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—24no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—44no—
268C2×C4×C7—416no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7#12012yes—
274C2×C4×C7#12012yes—
275C2×C4×C7AGL(1,ℤ/8)12yes—
276C2×C4×C7C22×D411yescharacteristic
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C2×#4112yes—
279C2×C4×C72−1+412yes—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C7⋊C8—24no—
293C7⋊C8—42no—
294C2×#41—28no—
295C2×#41—44no—
296C2×#41—44no—
297C22×Q8—21no—
298C22×D4—22no—
299C22×D4—21no—
300C22×D4—21no—
301C22×D4C22×D712yes—
302C23×C4—22no—
303C23×C4C22⋊D711yescharacteristic
304C23×C4C22⋊D711yescharacteristic
305C23×C4C22×D711yescharacteristic
306C22×C8—72no—
307C22×C8—72no—
308C4⋊Q8—22no—
309C4⋊Q8C22⋊D714yes—
310#127—22no—
311#127C22⋊D714yes—
312#125—142no—
313#125—142no—
314#124—28no—
315#121—28no—
316#121—24no—
317#121—24no—
318C4×D4—28no—
319C4×D4—24no—
320C4×D4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4C22×D712yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—141no—
334C2×C22⋊C4—141no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42C22⋊D711yescharacteristic
338C2×C42C22⋊D711yescharacteristic
339C2×C42C22×D711yescharacteristic
340C2×C42C22×D711yescharacteristic
341#107—144no—
342#106—144no—
343#102—144no—
344#102—144no—
345#102—144no—
346#102—144no—
347C22⋊C8—144no—
348#97—142no—
349#97—142no—
350#95—141no—
351#95—141no—
352#95—141no—
353#95—141no—
354#95—141no—
355#95—141no—
356C22×C7—22no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—41no—
360C22×C7—41no—
361C22×C7—41no—
362C22×C7—41no—
363C22×C7—44no—
364C22×C7—44no—
365C22×C7—44no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7—82no—
369C22×C7—82no—
370C22×C7C2×#12011yescharacteristic
371C22×C7#53211yescharacteristic
372C22×C7#53611yescharacteristic
373C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
375C22×C7#57511yescharacteristic
376C22×C7#57511yescharacteristic
377C4×C7—22no—
378C4×C7—22no—
379C4×C7—22no—
380C4×C7—41no—
381C4×C7—44no—
382C4×C7—41no—
383C4×C7#54712yes—
384C4×C7#54812yes—
385Dic7—81no—
386Dic7—81no—
387Dic7—81no—
388Dic7—81no—
389C24—22no—
390C24C2×C22⋊D711yescharacteristic
391#41—432no—
392C2×Q8—24no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4C2×C22⋊D714yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C23×D711yescharacteristic
434C2×C8—74no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C2×C8—142no—
439C2×C8—142no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—142no—
443C4⋊C4—142no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—282no—
449C4⋊C4—282no—
450C4⋊C4—282no—
451C4⋊C4—282no—
452C4⋊C4C2×C22⋊D714yes—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—282no—
456C22⋊C4—282no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42C2×C22⋊D712yes—
462C42C2×C22⋊D712yes—
463C2×C7—42no—
464C2×C7—84no—
465C2×C7#243211yescharacteristic
466C2×C7#300311yescharacteristic
467C2×C7#300511yescharacteristic
468C2×C7#317111yescharacteristic
469C2×C7#318011yescharacteristic
470C2×C7#323411yescharacteristic
471C2×C7#323511yescharacteristic
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23#649512yes—
486C23C22×C22⋊D711yescharacteristic
487Q8—48no—
488D4—24no—
489D4—24no—
490D4—48no—
491D4—48no—
492D4—416no—
493D4—42no—
494D4—42no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—44no—
510C2×C4—416no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4#649512yes—
528C2×C4#651612yes—
529C2×C4#651612yes—
530C2×C4D7×#4112yes—
531C2×C4#655412yes—
532C2×C4C22×C22⋊D711yescharacteristic
533C2×C4C22×C22⋊D711yescharacteristic
534C8—144no—
535C8—282no—
536C7#2567611yescharacteristic, complemented, Sylow
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×#649511yescharacteristic
552C22C2×#649511yescharacteristic
553C22#9051311yescharacteristic
554C22#9051311yescharacteristic
555C22#9051611yescharacteristic
556C22#9052311yescharacteristic
557C22C2×#651611yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—44no—
562C4—41no—
563C4—41no—
564C4—561no—
565C4—561no—
566C4—561no—
567C4—561no—
568C4#9051512yes—
569C4#9052212yes—
570C2—42no—
571C2—84no—
572C2#1174071811yescharacteristic
573C2#1174071911yescharacteristic
574C2#1174073711yescharacteristic
575C2#1174073811yescharacteristic
576C2#1174148311yescharacteristic
577C2#1174152611yescharacteristic
578C2#1174153811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#532C2
socleC23×C7C22×D4
hypercenterC2×#532D7
hypocenterC7#25676
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#89852C2semidirect
3. C2×#89843C2semidirect
4. #11739519C2semidirect
5. #11729542C2semidirect
6. #11729519C2semidirect
7. #11729434C2semidirect
8. #11729433C2semidirect
9. #11729361C2semidirect
10. #11729360C2semidirect
11. #11729283C2semidirect
194. C2×C7⋊C8#41semidirect
536. C7#25676semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5322
C2×#898522
C2×#898432
#117395192
#117295422
#117295192
#117294342
#117294332
#117293612
#117293602
#117292832
#256767