Finite Group Databasebeta

← #13387864

Subgroups — #13387864

Order 1792.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic, complemented
3C2×#89798C211yesmaximal, characteristic, complemented
4#11739860C211yesmaximal, characteristic
5#11729542C211yesmaximal, characteristic, complemented
6#11729453C212yesmaximal, complemented
7#11729448C212yesmaximal, complemented
8#11729381C211yesmaximal, characteristic, complemented
9#11729369C211yesmaximal, characteristic, complemented
10#11729340C212yesmaximal, complemented
11#11729330C212yesmaximal, complemented
12C7×#556—24no—
13C7×#556—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—22no—
16C2×C7×#124—21no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117C2212yes—
22C22×#1143C2211yescharacteristic
23C2×#6493C2212yes—
24C2×C4×D28C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2212yes—
27#89995—21no—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89856—24no—
32#89798—24no—
33#89765—21no—
34C2×#6383C2212yes—
35C2×#6383C2211yescharacteristic
36C2×#6382C2212yes—
37C2×#6381C2211yescharacteristic
38#89728—22no—
39#89708C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24131—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4D411yescharacteristic
49C23×C4×C7C2312yes—
50C22×C22⋊D7—22no—
51C22×D28C2311yescharacteristic
52C22×C4×D7—21no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8D414yescomplemented
55C7×#126—216no—
56C7×#124—28no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—24no—
59C4×C7×Q8—28no—
60C4×C7×Q8—22no—
61C4×C7×Q8D414yescomplemented
62C4×C7×D4—22no—
63C4×C7×D4—22no—
64C7×#117—24no—
65C7×#117—24no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4D412yes—
75C2×C42×C7—22no—
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—22no—
79C2×#1143—22no—
80C2×#1143—22no—
81C2×#1143—22no—
82#6493—24no—
83#6493—42no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—22no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2312yes—
90C22×C7⋊C8C2311yescharacteristic
91C4×D28—22no—
92C4×D28—22no—
93#6409—22no—
94#6409—42no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)—42no—
100#6383—24no—
101#6383—22no—
102#6382—24no—
103#6381—22no—
104C4⋊(C7⋊C8)—22no—
105#6377—22no—
106C2×#556D711yescharacteristic, complemented
107C2×#476—71no—
108C2×#440—71no—
109#2535—71no—
110#2035—72no—
111#2000—72no—
112#1993—71no—
113#1955—71no—
114#1875—72no—
115#1854—72no—
116#1805—71no—
117C24×C7C2×D412yes—
118C23×D7—21no—
119C2×C7×Q8—24no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—44no—
126C2×C7×Q8C2×D412yes—
127C2×C7×D4—22no—
128C2×C7×D4—42no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C2×C22⋊D7—44no—
143C2×C22⋊D7—44no—
144C2×C22⋊D7—82no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28#4112yes—
153C2×C4×D7—42no—
154C2×C4×D7—82no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4C2×D412yes—
162C7×C4⋊C4C2×D414yes—
163C7×C22⋊C4—28no—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—24no—
166C42×C7—28no—
167C42×C7—24no—
168C42×C7C2×D412yes—
169C42×C7C2×D412yes—
170C22⋊Dic7—82no—
171#1143—28no—
172(C2×D7)⋊C4—44no—
173(C2×D7)⋊C4—42no—
174C4⋊Dic7—21no—
175C4⋊Dic7—21no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8#4112yes—
187C22×SD16—71no—
188#556—24no—
189#556—24no—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×#126—22no—
192C2×#124—21no—
193C2×C22⋊Q8C2×D712yes—
194C2×#121—72no—
195C2×C4×Q8—21no—
196C2×C4×Q8C2×D711yescharacteristic
197C2×C4×D4—71no—
198C2×C4×D4C2×D711yescharacteristic
199C2×#117C2×D712yes—
200#476—144no—
201#440—144no—
202C2×#107—71no—
203C2×#106—72no—
204C2×C4⋊C8—71no—
205C2×#103—71no—
206C2×#103—72no—
207C2×#102—71no—
208C2×#102—72no—
209C2×C22⋊C8—72no—
210C2×C22⋊C8—72no—
211C2×#97—71no—
212#397—141no—
213#392—141no—
214#386—142no—
215#382—71no—
216#380—72no—
217#340—72no—
218#340—72no—
219#340—72no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7C22×D411yescharacteristic
228C22×D7—22no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—42no—
232C22×D7—42no—
233C7×Q8—22no—
234C7×Q8—22no—
235C7×Q8—48no—
236C7×Q8—41no—
237C7×Q8—41no—
238C7×D4—44no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—24no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—44no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7#12012yes—
256C2×C4×C7#12012yes—
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C2×#4111yescharacteristic
260C2×C4×C7C2×#4111yescharacteristic
261C2×C4×C7H(5,𝔽2)11yescharacteristic
262C2×C4×C72−1+411yescharacteristic
263C22⋊D7—88no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269D28—22no—
270D28—22no—
271D28—41no—
272D28—41no—
273C4×D7—84no—
274C7⋊C8—22no—
275C7⋊C8—41no—
276C7⋊C8—42no—
277C22×Q8—21no—
278C22×Q8C22×D711yescharacteristic
279C22×D4—71no—
280C22×D4—142no—
281C22×D4C22⋊D711yescharacteristic
282C23×C4—141no—
283C23×C4C22×D712yes—
284C2×SD16—142no—
285C2×SD16—142no—
286C2×SD16—142no—
287C22×C8—71no—
288C22×C8—71no—
289C22×C8—72no—
290C4⋊Q8—22no—
291C4⋊Q8C22⋊D714yescomplemented
292#126—216no—
293#124—28no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296#121—144no—
297#121—282no—
298C4×Q8—28no—
299C4×Q8—22no—
300C4×Q8C22⋊D714yescomplemented
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—142no—
304C4×D4—142no—
305#117—24no—
306#117—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4C22⋊D711yescharacteristic
313C2×C4⋊C4C22×D711yescharacteristic
314C2×C4⋊C4C22×D712yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—142no—
319C2×C22⋊C4—142no—
320C2×C22⋊C4C22⋊D712yes—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42C22×D711yescharacteristic
324#107—142no—
325#106—144no—
326C4⋊C8—142no—
327#103—142no—
328#103—282no—
329#102—142no—
330#102—144no—
331C22⋊C8—144no—
332C22⋊C8—282no—
333#97—142no—
334#95—142no—
335#95—141no—
336#95—141no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7C2×#12011yescharacteristic
345C22×C7#54611yescharacteristic
346C22×C7#54711yescharacteristic
347C22×C7#57512yes—
348C22×C7#57712yes—
349C2×D7—42no—
350C2×D7—44no—
351C2×D7—82no—
352C2×D7—81no—
353C2×D7—81no—
354C4×C7—22no—
355C4×C7—22no—
356C4×C7—42no—
357C4×C7—42no—
358C4×C7—44no—
359C4×C7—42no—
360C4×C7#52511yescharacteristic
361C4×C7#53711yescharacteristic
362C4×C7#54811yescharacteristic
363C4×C7#55011yescharacteristic
364Dic7—81no—
365Dic7—82no—
366C24—141no—
367C24C2×C22⋊D712yes—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—44no—
375C2×Q8C2×C22⋊D712yes—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—72no—
379C2×D4—141no—
380C2×D4—141no—
381C2×D4—141no—
382C2×D4—141no—
383C2×D4—141no—
384C2×D4—284no—
385C2×D4—284no—
386C2×D4—562no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—142no—
397C22×C4—141no—
398C22×C4—282no—
399C22×C4—562no—
400C22×C4C2×C22⋊D712yes—
401C22×C4C2×C22⋊D711yescharacteristic
402C22×C4C2×C22⋊D711yescharacteristic
403C22×C4C23×D711yescharacteristic
404SD16—148no—
405C2×C8—72no—
406C2×C8—141no—
407C2×C8—141no—
408C2×C8—142no—
409C2×C8—141no—
410C2×C8—142no—
411C2×C8—141no—
412C2×C8—141no—
413C2×C8—142no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—142no—
421C4⋊C4—142no—
422C4⋊C4—141no—
423C4⋊C4—141no—
424C4⋊C4C2×C22⋊D714yes—
425C4⋊C4C2×C22⋊D712yes—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—24no—
429C22⋊C4—284no—
430C22⋊C4—282no—
431C22⋊C4—562no—
432C42—28no—
433C42—24no—
434C42C2×C22⋊D712yes—
435C42C2×C22⋊D712yes—
436C2×C7—82no—
437C2×C7#240711yescharacteristic
438C2×C7#300511yescharacteristic
439C2×C7#301011yescharacteristic
440C2×C7#331112yes—
441C2×C7#335012yes—
442D7—82no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—142no—
451C23—282no—
452C23—281no—
453C23—282no—
454C23—281no—
455C23C22×C22⋊D711yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—48no—
459Q8—41no—
460Q8—41no—
461D4—44no—
462D4—142no—
463D4—142no—
464D4—281no—
465D4—281no—
466D4—568no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—281no—
488C2×C4—564no—
489C2×C4#651612yes—
490C2×C4#651612yes—
491C2×C4D7×#4111yescharacteristic
492C2×C4D7×#4111yescharacteristic
493C2×C4#655311yescharacteristic
494C2×C4#655411yescharacteristic
495C2×C4C22×C22⋊D711yescharacteristic
496C2×C4C22×C22⋊D711yescharacteristic
497C8—142no—
498C8—281no—
499C8—282no—
500C7#2413111yescharacteristic, complemented, Sylow
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—282no—
509C22—284no—
510C22—561no—
511C22—561no—
512C22—562no—
513C22#9051312yes—
514C22#9051511yescharacteristic
515C22#9052011yescharacteristic
516C22#9052812yes—
517C22C2×#651611yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—561no—
525C4—562no—
526C4#9051911yescharacteristic
527C4#9052211yescharacteristic
528C4#9052411yescharacteristic
529C4#9052511yescharacteristic
530C2—82no—
531C2—562no—
532C2#1174110612yes—
533C2#1174111012yes—
534C2#1174152611yescharacteristic
535C2#1174154211yescharacteristic
536C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radical#11739860C2
hypercenterC2×#556D7
hypocenterC7#24131
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×C7×#556C2semidirect
2. C2×#89856C2semidirect
3. C2×#89798C2semidirect
5. #11729542C2semidirect
6. #11729453C2semidirect
7. #11729448C2semidirect
8. #11729381C2semidirect
9. #11729369C2semidirect
10. #11729340C2semidirect
11. #11729330C2semidirect
54. C7×C4⋊Q8D4semidirect
61. C4×C7×Q8D4semidirect
106. C2×#556D7semidirect
291. C4⋊Q8C22⋊D7semidirect
300. C4×Q8C22⋊D7semidirect
500. C7#24131semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#898562
C2×#897982
#117398602
#117295422
#117294532
#117294482
#117293812
#117293692
#117293402
#117293302
#241317