Finite Group Databasebeta

← #13369462

Subgroups — #13369462

Order 1792.

1582 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90409C211yesmaximal, characteristic
2C2×#89610C211yesmaximal, characteristic, complemented
3C2×#89543C212yesmaximal, complemented
4C4⋊C4×D28C211yesmaximal, characteristic, complemented
5C7×#1982C211yesmaximal, characteristic, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11728239C211yesmaximal, characteristic, complemented
8#11727957C212yesmaximal, complemented
9#11727954C212yesmaximal, complemented
10#11727953C212yesmaximal, complemented
11C4×#6384C211yesmaximal, characteristic, complemented
12C22×#1159C2212yes—
13C2×C7×#127C2211yescharacteristic
14C2×#6503—21no—
15C2×#6502—22no—
16C2×#6501C2212yes—
17C2×D4×Dic7—21no—
18C22×#1131C2212yes—
19#90409—24no—
20#90409C418yescomplemented
21C2×D7×C4⋊C4C2212yes—
22C2×C4×D28—21no—
23C2×C4×D28C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#102C2212yes—
27C2×C7×#102C2212yes—
28C4×C7×C4⋊C4C2211yescharacteristic
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31C2×(C2×D7)⋊C8C2212yes—
32C2×#6390C2212yes—
33#89759C2212yes—
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36#89707C2211yescharacteristic
37#89700—21no—
38C2×C4×C7⋊C8C2211yescharacteristic
39#89610—24no—
40#89543—28no—
41#89440—22no—
42#89346C2212yes—
43#89324C2211yescharacteristic
44#89324C2211yescharacteristic
45#22432—71nomaximal, Sylow
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4C2312yes—
49C22×C22⋊D7—22no—
50C22×C22⋊D7—22no—
51C23×Dic7—22no—
52C2×#1159—24no—
53C2×#1159—24no—
54C2×#1159—24no—
55C2×#1159—42no—
56C2×#1159—42no—
57C2×#1159C2×C418yescomplemented
58C22×D28D411yescharacteristic
59C22×C4×D7—22no—
60C22×C4×D7C2312yes—
61C22×Dic14C2311yescharacteristic
62C22×C7×C8C2312yes—
63C7×#127—22no—
64C7×#127C2×C414yescomplemented
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70#6503—28no—
71#6502—216no—
72#6501—24no—
73#6501C2×C418yescomplemented
74D4×Dic7—28no—
75C2×(C2×D7)⋊C4—22no—
76C2×(C2×D7)⋊C4D412yes—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7D411yescharacteristic
80C2×C4⋊Dic7C2312yes—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—22no—
84C2×C4×Dic7—21no—
85C22×C7⋊C8C2311yescharacteristic
86C22×C7⋊C8C2311yescharacteristic
87C2×#1131—24no—
88C2×#1131—24no—
89C2×#1131—42no—
90D7×C4⋊C4—24no—
91D7×C4⋊C4—42no—
92C4×D28—44no—
93C4×D28C2×C414yes—
94C4×D28D414yes—
95C4×Dic14—22no—
96C4×Dic14C2×C414yescomplemented
97C7×C4⋊C8—22no—
98C7×#102—24no—
99C7×#102—42no—
100C7×#95—21no—
101#6406—24no—
102#6406—42no—
103C7⋊#95—22no—
104(C2×D7)⋊C8—24no—
105#6390—24no—
106#6384—22no—
107#6384D414yescomplemented
108C4×C7⋊C8—22no—
109C2×#550—71no—
110C2×#436—71no—
111C2×#417—72no—
112D4×C4⋊C4—71no—
113#1982D711yescharacteristic, complemented
114#1957—71no—
115#1895—72no—
116#1874—72no—
117#1803—71no—
118#1788—72no—
119C4×#103—71no—
120C24×C7—22no—
121C23×D7C2×D412yes—
122C2×C7×D4—24no—
123C2×C7×D4—24no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4—81no—
132C2×C7×D4—81no—
133C2×C7×D4C22×C414yes—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2411yescharacteristic
139C2×C22⋊D7—28no—
140C2×C22⋊D7—28no—
141C2×C22⋊D7—44no—
142C2×C22⋊D7—44no—
143C2×C22⋊D7—82no—
144C2×C22⋊D7—82no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—28no—
148C22×Dic7—22no—
149C22×Dic7—44no—
150C22×Dic7—44no—
151C22×Dic7C2×D412yes—
152#1159—216no—
153#1159—48no—
154#1159—48no—
155#1159—48no—
156C2×D28—22no—
157C2×D28—22no—
158C2×D28C22⋊C414yes—
159C2×C4×D7—22no—
160C2×C4×D7—22no—
161C2×C4×D7—22no—
162C2×C4×D7—22no—
163C2×C4×D7—44no—
164C2×C4×D7—44no—
165C2×C4×D7—82no—
166C2×C4×D7C22×C414yes—
167C2×C4×D7C2×D414yes—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14C22×C412yes—
172C2×Dic14C2×D412yes—
173C2×Dic14#4112yes—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4C2×D412yes—
181C42×C7—42no—
182C42×C7C22×C412yes—
183C42×C7C2×D412yes—
184C22⋊Dic7—28no—
185(C2×D7)⋊C4—44no—
186(C2×D7)⋊C4C22⋊C418yes—
187C4⋊Dic7—21no—
188C4⋊Dic7—22no—
189C4⋊Dic7—22no—
190C4⋊Dic7—42no—
191C4⋊Dic7—81no—
192C4⋊Dic7C22⋊C414yes—
193C4⋊Dic7C22×C412yes—
194Dic7⋊C4—24no—
195Dic7⋊C4—44no—
196C4×Dic7—24no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8C2×D412yes—
203#1131—48no—
204C22×#41—72no—
205C22×M16—72no—
206#550—78no—
207#550—144no—
208C2×#127—141no—
209C2×#127C2×D711yescharacteristic
210C2×#121—72no—
211C2×#121—142no—
212C2×C4×Q8—71no—
213C2×C4×D4—71no—
214C2×C4×D4—141no—
215C2×C4×D4—141no—
216C22×C4⋊C4—72no—
217#436—144no—
218C2×C4⋊C8C2×D711yescharacteristic
219#417—148no—
220C2×#103—71no—
221C2×#103—72no—
222C2×#103—71no—
223C2×#102—72no—
224C2×#102—72no—
225C2×#102C2×D712yes—
226C2×#102C2×D712yes—
227C2×C22⋊C8—72no—
228C2×C4×C8—71no—
229#389—141no—
230#385—142no—
231#384—71no—
232#380—72no—
233C4×C4⋊C4C2×D711yescharacteristic
234#340—72no—
235#340—71no—
236#340—71no—
237C23×C7—24no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7C22×D411yescharacteristic
245C22×D7—22no—
246C22×D7—22no—
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—22no—
250C22×D7—22no—
251C22×D7C2×C22⋊C414yes—
252C7×D4—24no—
253C7×D4—24no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—48no—
257C7×D4—84no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—22no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7C2×C22⋊C412yes—
267C2×C4×C7C2×C22⋊C412yes—
268C2×C4×C7AGL(1,ℤ/8)12yes—
269C2×C4×C7#13712yes—
270C2×C4×C7C23×C411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7H(5,𝔽2)11yescharacteristic
275C2×C4×C7H(5,𝔽2)11yescharacteristic
276C22⋊D7—416no—
277C22⋊D7—88no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—22no—
284C2×Dic7—416no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7C2×C22⋊C414yes—
291D28—28no—
292C4×D7—24no—
293C4×D7—24no—
294C4×D7—24no—
295C4×D7—24no—
296C4×D7—88no—
297Dic14—22no—
298Dic14—22no—
299Dic14—22no—
300Dic14C4×D414yescomplemented
301C7×C8—42no—
302C7⋊C8—22no—
303C7⋊C8—41no—
304C2×#41—78no—
305C2×#41—144no—
306C2×#41—144no—
307C2×#41—144no—
308C2×#41—282no—
309C2×#41—282no—
310C22×Q8—71no—
311C22×D4—21no—
312C22×D4—21no—
313C22×D4—71no—
314C22×D4—142no—
315C22×D4—142no—
316C22×D4C22×D712yes—
317C23×C4—72no—
318C23×C4—142no—
319C23×C4—142no—
320C2×M16—144no—
321C2×M16—144no—
322C2×M16—282no—
323C22×C8—71no—
324C22×C8—71no—
325C22×C8C22×D712yes—
326#127—22no—
327#127—148no—
328#127C4×D714yescomplemented
329#121—78no—
330#121—1416no—
331#121—144no—
332C4×Q8—74no—
333C4×Q8—142no—
334C4×D4—74no—
335C4×D4—74no—
336C4×D4—148no—
337C4×D4—284no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—144no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—282no—
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C22⋊C4—72no—
350C2×C22⋊C4—142no—
351C2×C22⋊C4—142no—
352C2×C42—21no—
353C2×C42—141no—
354C2×C42C22×D711yescharacteristic
355C4⋊C8—22no—
356#103—74no—
357#103—142no—
358#103—144no—
359#102—24no—
360#102—42no—
361#102—144no—
362#102—282no—
363C22⋊C8—144no—
364C4×C8—142no—
365#95—21no—
366#95—142no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—82no—
371C22×C7—82no—
372C22×C7—82no—
373C22×C7—82no—
374C22×C7C22×C22⋊C411yescharacteristic
375C22×C7D4211yescharacteristic
376C22×C7#54611yescharacteristic
377C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
378C22×C7C2×#13711yescharacteristic
379C22×C7#57611yescharacteristic
380C22×C7#57711yescharacteristic
381C2×D7—24no—
382C2×D7—24no—
383C2×D7—24no—
384C2×D7—42no—
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—42no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—81no—
391C4×C7#41714yescomplemented
392C4×C7C2×C4×D411yescharacteristic
393C4×C7#51811yescharacteristic
394C4×C7D4211yescharacteristic
395C4×C7#54611yescharacteristic
396Dic7—24no—
397Dic7—24no—
398Dic7—42no—
399Dic7—82no—
400C24—22no—
401C24—72no—
402#41—1416no—
403#41—288no—
404#41—288no—
405#41—288no—
406C2×Q8—72no—
407C2×Q8—72no—
408C2×Q8—72no—
409C2×Q8—141no—
410C2×Q8—141no—
411C2×Q8—141no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—74no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—148no—
425C2×D4—148no—
426C2×D4—142no—
427C2×D4—142no—
428C2×D4—284no—
429C2×D4—284no—
430C2×D4—562no—
431C2×D4—562no—
432C2×D4C2×C4×D714yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—72no—
436C22×C4—74no—
437C22×C4—74no—
438C22×C4—148no—
439C22×C4—141no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—141no—
445C22×C4—142no—
446C22×C4—284no—
447C22×C4—284no—
448C22×C4—284no—
449C22×C4—284no—
450C22×C4—562no—
451C22×C4D4×D711yescharacteristic
452C22×C4D4×D711yescharacteristic
453C22×C4C23×D711yescharacteristic
454M16—288no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—72no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—74no—
468C4⋊C4—72no—
469C4⋊C4—144no—
470C4⋊C4—141no—
471C4⋊C4—142no—
472C4⋊C4—142no—
473C4⋊C4—284no—
474C4⋊C4—282no—
475C4⋊C4—561no—
476C4⋊C4D4×D712yes—
477C22⋊C4—78no—
478C22⋊C4—148no—
479C22⋊C4—284no—
480C42—42no—
481C42—144no—
482C42C2×C4×D712yes—
483C42D4×D712yes—
484C2×C7—84no—
485C2×C7D4×C22⋊C411yescharacteristic
486C2×C7C2×#41711yescharacteristic
487C2×C7#289711yescharacteristic
488C2×C7#327411yescharacteristic
489C2×C7#327811yescharacteristic
490C2×C7#333211yescharacteristic
491C2×C7#333311yescharacteristic
492D7—44no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—74no—
501C23—142no—
502C23—142no—
503C23—142no—
504C23—142no—
505C23—142no—
506C23—142no—
507C23C2×D4×D711yescharacteristic
508Q8—74no—
509Q8—142no—
510Q8—142no—
511Q8—142no—
512D4—24no—
513D4—24no—
514D4—42no—
515D4—48no—
516D4—42no—
517D4—84no—
518D4—148no—
519D4—2816no—
520D4—568no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—74no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—142no—
540C2×C4—2816no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—568no—
547C2×C4D7×C22⋊C412yes—
548C2×C4D7×C22⋊C412yes—
549C2×C4#647412yes—
550C2×C4#647812yes—
551C2×C4C22×C4×D711yescharacteristic
552C2×C4C2×D4×D711yescharacteristic
553C2×C4C2×D4×D711yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4#655311yescharacteristic
557C8—42no—
558C8—142no—
559C8—281no—
560C7#2243211yescharacteristic, complemented, Sylow
561C22—44no—
562C22—44no—
563C22—44no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—144no—
569C22—144no—
570C22—144no—
571C22—282no—
572C22—282no—
573C22—282no—
574C22—282no—
575C22C2×D7×C22⋊C411yescharacteristic
576C22#9030011yescharacteristic
577C22#9030311yescharacteristic
578C22C2×#647411yescharacteristic
579C22C2×#647811yescharacteristic
580C22#9046511yescharacteristic
581C22#9046611yescharacteristic
582C4—22no—
583C4—42no—
584C4—81no—
585C4—144no—
586C4—144no—
587C4—282no—
588C4—562no—
589C4#8954314yescomplemented
590C4C4×D4×D711yescharacteristic
591C4#9023611yescharacteristic
592C4#9036711yescharacteristic
593C4#9040811yescharacteristic
594C2—84no—
595C2—284no—
596C2#1173387611yescharacteristic
597C2C2×#8954311yescharacteristic
598C2#1173637011yescharacteristic
599C2#1173779411yescharacteristic
600C2#1173780211yescharacteristic
601C2#1173783811yescharacteristic
602C2#1173784511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1982C2
socleC23×C7C22×D4
prime radicalC2×#90409C2
cosocleC22×C4×C7C24
hypercenter#1982D7
hypocenterC7#22432
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. C2×#89610C2semidirect
3. C2×#89543C2semidirect
4. C4⋊C4×D28C2semidirect
5. C7×#1982C2semidirect
6. #11729518C2semidirect
7. #11728239C2semidirect
8. #11727957C2semidirect
9. #11727954C2semidirect
10. #11727953C2semidirect
11. C4×#6384C2semidirect
20. #90409C4semidirect
57. C2×#1159C2×C4semidirect
64. C7×#127C2×C4semidirect
73. #6501C2×C4semidirect
96. C4×Dic14C2×C4semidirect
107. #6384D4semidirect
113. #1982D7semidirect
300. Dic14C4×D4semidirect
328. #127C4×D7semidirect
391. C4×C7#417semidirect
560. C7#22432semidirect
589. C4#89543semidirect

Maximal subgroups

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

subgroupindex
C2×#904092
C2×#896102
C2×#895432
C4⋊C4×D282
C7×#19822
#117295182
#117282392
#117279572
#117279542
#117279532
C4×#63842
#224327