Finite Group Databasebeta

← #13355064

Subgroups — #13355064

Order 1792.

1970 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90496C211yesmaximal, characteristic
2C2×#90214C211yesmaximal, characteristic, complemented
3C2×C7×#418C211yesmaximal, characteristic, complemented
4C22×#6406C211yesmaximal, characteristic, complemented
5C2×#89901C211yesmaximal, characteristic, complemented
6#11729739C211yesmaximal, characteristic, complemented
7#11729334C211yesmaximal, characteristic, complemented
8#11727955C214yesmaximal, complemented
9#11727954C214yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×#1157C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×#6516—22no—
15#90496—24no—
16#90496C418yescomplemented
17C2×#6503—21no—
18C2×#6501C2214yes—
19C2×#6499—21no—
20C2×#6489C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C23×C7⋊C8C2211yescharacteristic
23#90214—24no—
24#90214—24no—
25C2×#6455C2212yescomplemented
26C2×D7×C4⋊C4C2212yescomplemented
27C2×(C22⋊D7)⋊C4—22no—
28C2×C4×D28—21no—
29C2×C4×Dic14—21no—
30C7×#418—24no—
31C2×C7×#102C2212yes—
32C2×C7×#102C2212yes—
33C2×#6406—24no—
34C2×#6406C2214yes—
35#89901—24no—
36C2×(C2×D7)⋊C8C2214yescomplemented
37#89346C2214yescomplemented
38#89324C2212yescomplemented
39#89324C2212yescomplemented
40#20914—71nomaximal, Sylow
41C25×C7—21no—
42C22×C7×D4—22no—
43C22×C7×D4—22no—
44C22×C7×D4—21no—
45C22×C7×D4—22no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4C2×C414yescomplemented
49C22×C7×D4C2312yes—
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—24no—
52C22×C22⋊D7D412yes—
53C23×Dic7—21no—
54C2×#1157—22no—
55C2×#1157—22no—
56C2×#1157—22no—
57C2×#1157—22no—
58C2×#1157C2×C414yes—
59C2×#1157D414yes—
60C22×D28D411yescharacteristic
61C22×C4×D7—22no—
62C22×C4×D7C2312yes—
63C22×Dic14D411yescharacteristic
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C22×C7×C8C2312yes—
68C7×#117—22no—
69C7×#117D414yescomplemented
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C42×C7—21no—
74#6516—216no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—24no—
77#6503—28no—
78#6501—28no—
79#6501C2×C4116yescomplemented
80#6499—28no—
81C2×(C2×D7)⋊C4—22no—
82#6489—22no—
83#6489C2×C414yescomplemented
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7D414yescomplemented
87C2×C4⋊Dic7C2312yes—
88C2×C4⋊Dic7C2312yes—
89C2×Dic7⋊C4—22no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—22no—
92C22×C7⋊C8—22no—
93C22×C7⋊C8—22no—
94C22×C7⋊C8—22no—
95C22×C7⋊C8C2312yes—
96#6455—24no—
97#6455—24no—
98D7×C4⋊C4—24no—
99D7×C4⋊C4—24no—
100(C22⋊D7)⋊C4—216no—
101C4×D28—28no—
102C4×Dic14—28no—
103C7×#102—24no—
104C7×#102—24no—
105#6406—28no—
106(C2×D7)⋊C8—28no—
107C2×#535—71no—
108C2×#520—71no—
109C2×#418D711yescharacteristic, complemented
110C22×#102—71no—
111C2×#408—71no—
112#1875—74no—
113#1874—74no—
114#1736—71no—
115#1735—71no—
116C24×C7—22no—
117C24×C7—24no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—82no—
121C24×C7—82no—
122C24×C7C2×D411yescharacteristic
123C23×D7C2×D412yes—
124C2×C7×D4—24no—
125C2×C7×D4—28no—
126C2×C7×D4—28no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—42no—
133C2×C7×D4—44no—
134C2×C7×D4—44no—
135C2×C7×D4—42no—
136C2×C7×D4C22×C414yes—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D412yes—
144C22×C4×C7#4112yes—
145C22×C4×C7#4112yes—
146C22×C4×C7C2411yescharacteristic
147C2×C22⋊D7—24no—
148C2×C22⋊D7—24no—
149C2×C22⋊D7—216no—
150C2×C22⋊D7—84no—
151C2×C22⋊D7—84no—
152C2×C22⋊D7C22⋊C418yes—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—24no—
156C22×Dic7—82no—
157C22×Dic7C2×D412yes—
158#1157—28no—
159#1157—28no—
160#1157—28no—
161#1157—28no—
162C2×D28—22no—
163C2×D28—22no—
164C2×D28C22⋊C414yes—
165C2×C4×D7—22no—
166C2×C4×D7—22no—
167C2×C4×D7—22no—
168C2×C4×D7—22no—
169C2×C4×D7—28no—
170C2×C4×D7—84no—
171C2×C4×D7C22×C414yes—
172C2×C4×D7C2×D414yes—
173C2×Dic14—22no—
174C2×Dic14—22no—
175C2×Dic14C22⋊C414yes—
176C7×M16—44no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4C2×D412yes—
183C7×C4⋊C4C2×D412yes—
184C7×C22⋊C4—24no—
185C42×C7—24no—
186C22⋊Dic7—24no—
187C22⋊Dic7—216no—
188(C2×D7)⋊C4—28no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7—22no—
192C4⋊Dic7C22×C414yes—
193C4⋊Dic7C2×D414yes—
194Dic7⋊C4—28no—
195C4×Dic7—28no—
196C4×Dic7—24no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—44no—
201C22×#41—71no—
202C23×D4C2×D711yescharacteristic
203C22×M16C2×D711yescharacteristic
204C23×C8—71no—
205#535—78no—
206#535—144no—
207C2×#127—141no—
208C2×#124—141no—
209C2×#121—74no—
210C2×#120—142no—
211#520—144no—
212#520—144no—
213C2×C4×Q8—141no—
214C2×C4×D4—141no—
215C2×C4×D4—142no—
216C2×#117—71no—
217C2×#117—72no—
218C2×#117C2×D711yescharacteristic
219C22×C4⋊C4—71no—
220C22×C4⋊C4—72no—
221#418—24no—
222C2×#102—74no—
223C2×#102—144no—
224C2×#102C2×D712yes—
225C2×#102C2×D712yes—
226#408—144no—
227C2×C22⋊C8—74no—
228#340—74no—
229#340—72no—
230#340—72no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—24no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—44no—
237C23×C7—44no—
238C23×C7—48no—
239C23×C7—88no—
240C23×C7—88no—
241C23×C7—84no—
242C23×C7—84no—
243C23×C7C2×C22⋊C412yes—
244C23×C7#12012yes—
245C23×C7#12312yes—
246C23×C7C22×D411yescharacteristic
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—24no—
250C22×D7—24no—
251C22×D7C2×C22⋊C414yes—
252C7×D4—24no—
253C7×D4—24no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—48no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7C2×C22⋊C412yes—
262C2×C4×C7C4×D414yes—
263C2×C4×C7C4×D414yes—
264C2×C4×C7#12012yes—
265C2×C4×C7#12114yes—
266C2×C4×C7#12114yes—
267C2×C4×C7#12312yes—
268C2×C4×C7C23×C411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C22⋊D7—216no—
275C22⋊D7—816no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7—44no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—84no—
284C2×Dic7C2×C22⋊C414yes—
285D28—28no—
286C4×D7—24no—
287C4×D7—24no—
288C4×D7—24no—
289C4×D7—24no—
290C4×D7—88no—
291Dic14—28no—
292C7×C8—42no—
293C7⋊C8—42no—
294C25—21no—
295C2×#41—74no—
296C2×#41—74no—
297C2×#41—142no—
298C2×#41—142no—
299C2×#41—142no—
300C2×#41—142no—
301C22×Q8—71no—
302C22×D4—22no—
303C22×D4—22no—
304C22×D4—21no—
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—72no—
309C22×D4—71no—
310C22×D4—144no—
311C22×D4C4×D714yescomplemented
312C22×D4C22×D712yes—
313C23×C4—72no—
314C23×C4—142no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C22×C8—72no—
321C22×C8—142no—
322C22×C8—142no—
323C22×C8—142no—
324C22×C8C22×D712yes—
325#127—148no—
326#124—148no—
327#121—716no—
328#121—148no—
329#120—1416no—
330C4×Q8—148no—
331C4×D4—148no—
332C4×D4—1416no—
333#117—22no—
334#117—74no—
335#117—144no—
336#117—142no—
337#117—144no—
338#117C22⋊D714yescomplemented
339C2×C4⋊C4—74no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—72no—
342C2×C4⋊C4—144no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—144no—
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—142no—
351C2×C22⋊C4—144no—
352C2×C22⋊C4—141no—
353C2×C42—21no—
354C2×C42—142no—
355C2×C42—141no—
356#102—24no—
357#102—24no—
358#102—148no—
359C22⋊C8—148no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—48no—
365C22×C7—44no—
366C22×C7—816no—
367C22×C7—84no—
368C22×C7—82no—
369C22×C7—82no—
370C22×C7#38614yes—
371C22×C7C22×C22⋊C411yescharacteristic
372C22×C7C2×#12011yescharacteristic
373C22×C7C2×#12311yescharacteristic
374C22×C7#57612yes—
375C22×C7#57712yes—
376C2×D7—28no—
377C2×D7—24no—
378C2×D7—44no—
379C2×D7—42no—
380C2×D7—42no—
381C4×C7—42no—
382C4×C7—42no—
383C4×C7#38614yes—
384C4×C7C2×C4×D411yescharacteristic
385C4×C7C2×C4×D411yescharacteristic
386C4×C7C2×#12111yescharacteristic
387C4×C7C2×#12111yescharacteristic
388Dic7—24no—
389Dic7—44no—
390Dic7—44no—
391C24—22no—
392C24—24no—
393C24—42no—
394C24—42no—
395C24—72no—
396C24—82no—
397C24—82no—
398C24D4×D711yescharacteristic
399#41—148no—
400#41—148no—
401#41—148no—
402#41—148no—
403C2×Q8—74no—
404C2×Q8—142no—
405C2×Q8—142no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—78no—
419C2×D4—74no—
420C2×D4—144no—
421C2×D4—144no—
422C2×D4—1416no—
423C2×D4—142no—
424C2×D4—142no—
425C2×D4—564no—
426C2×D4—564no—
427C2×D4C2×C4×D714yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—72no—
431C22×C4—74no—
432C22×C4—74no—
433C22×C4—148no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—144no—
441C22×C4—564no—
442C22×C4—562no—
443C22×C4C2×C4×D712yes—
444C22×C4#115712yes—
445C22×C4D4×D711yescharacteristic
446C22×C4D4×D712yes—
447C22×C4D4×D712yes—
448C22×C4#115912yes—
449C22×C4C2×C22⋊D712yes—
450C22×C4C23×D711yescharacteristic
451M16—44no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—142no—
456C2×C8—142no—
457C2×C8—142no—
458C2×C8—284no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—74no—
462C4⋊C4—74no—
463C4⋊C4—148no—
464C4⋊C4—144no—
465C4⋊C4—142no—
466C4⋊C4—142no—
467C4⋊C4C2×C22⋊D712yes—
468C4⋊C4C2×C22⋊D712yes—
469C22⋊C4—24no—
470C22⋊C4—148no—
471C22⋊C4—1416no—
472C22⋊C4—144no—
473C42—24no—
474C42—148no—
475C42—144no—
476C2×C7—24no—
477C2×C7—84no—
478C2×C7C2×#38611yescharacteristic
479C2×C7#289712yes—
480C2×C7#300811yescharacteristic
481C2×C7#301011yescharacteristic
482C2×C7#309612yes—
483D7—44no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—48no—
492C23—74no—
493C23—88no—
494C23—88no—
495C23—84no—
496C23—84no—
497C23—142no—
498C23—144no—
499C23—144no—
500C23—142no—
501C23D7×C22⋊C412yes—
502C23#644612yes—
503C23#650012yes—
504C23C2×D4×D711yescharacteristic
505Q8—148no—
506D4—24no—
507D4—24no—
508D4—42no—
509D4—42no—
510D4—48no—
511D4—1416no—
512D4—148no—
513D4—5616no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—74no—
519C2×C4—142no—
520C2×C4—142no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—144no—
524C2×C4—144no—
525C2×C4—144no—
526C2×C4—144no—
527C2×C4—284no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—568no—
531C2×C4—564no—
532C2×C4D7×C22⋊C412yes—
533C2×C4(C22⋊D7)⋊C414yes—
534C2×C4#644612yes—
535C2×C4#644714yes—
536C2×C4C4×C22⋊D714yes—
537C2×C4#650012yes—
538C2×C4#650114yes—
539C2×C4C22×C4×D711yescharacteristic
540C2×C4C2×#115711yescharacteristic
541C2×C4C2×D4×D711yescharacteristic
542C2×C4C2×D4×D711yescharacteristic
543C2×C4C2×#115911yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C8—42no—
546C8—282no—
547C7#2091411yescharacteristic, complemented, Sylow
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—48no—
553C22—44no—
554C22—816no—
555C22—84no—
556C22—82no—
557C22—82no—
558C22—148no—
559C22—144no—
560C22—284no—
561C22—282no—
562C22—282no—
563C22#8972814yes—
564C22C2×D7×C22⋊C411yescharacteristic
565C22C2×#644611yescharacteristic
566C22#9046512yes—
567C22#9046612yes—
568C22C2×#650011yescharacteristic
569C4—42no—
570C4—42no—
571C4—144no—
572C4—284no—
573C4—284no—
574C4#8972814yes—
575C4C2×(C22⋊D7)⋊C411yescharacteristic
576C4C2×#644711yescharacteristic
577C4C2×C4×C22⋊D711yescharacteristic
578C4C2×#650111yescharacteristic
579C2—24no—
580C2—84no—
581C2—284no—
582C2#1173636412yes—
583C2#1173637012yes—
584C2C2×#8972811yescharacteristic
585C2#1174139211yescharacteristic
586C2#1174139311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#418C2
socleC23×C7C22×D4
prime radicalC2×#90496C2
cosocleC22×C4×C7C24
hypercenterC2×#418D7
hypocenterC7#20914
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#90214C2semidirect
3. C2×C7×#418C2semidirect
4. C22×#6406C2semidirect
5. C2×#89901C2semidirect
6. #11729739C2semidirect
7. #11729334C2semidirect
8. #11727955C2semidirect
9. #11727954C2semidirect
16. #90496C4semidirect
25. C2×#6455C22semidirect
26. C2×D7×C4⋊C4C22semidirect
36. C2×(C2×D7)⋊C8C22semidirect
37. #89346C22semidirect
38. #89324C22semidirect
39. #89324C22semidirect
48. C22×C7×D4C2×C4semidirect
69. C7×#117D4semidirect
79. #6501C2×C4semidirect
83. #6489C2×C4semidirect
86. C2×C4⋊Dic7D4semidirect
109. C2×#418D7semidirect
311. C22×D4C4×D7semidirect
338. #117C22⋊D7semidirect
547. C7#20914semidirect

Maximal subgroups

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

subgroupindex
C2×#904962
C2×#902142
C2×C7×#4182
C22×#64062
C2×#899012
#117297392
#117293342
#117279552
#117279542
#209147