Finite Group Databasebeta

← #13354573

Subgroups — #13354573

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90518C211yesmaximal, characteristic
2C2×C7×#408C211yesmaximal, characteristic
3#11741532C211yesmaximal, characteristic
4C2×#89893C211yesmaximal, characteristic, complemented
5C2×#89878C211yesmaximal, characteristic, complemented
6#11729739C211yesmaximal, characteristic, complemented
7C2×#89346C211yesmaximal, characteristic, complemented
8#11727683C212yesmaximal, complemented
9#11727683C212yesmaximal, complemented
10#11727682C212yesmaximal, complemented
11#11727681C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15#90518—24no—
16#90518—24no—
17C2×Q8×Dic7—21no—
18C2×D4×Dic7—22no—
19C2×D4×Dic7—21no—
20C2×#6489C2211yescharacteristic
21C2×#6489C2212yes—
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×C4⋊Dic7C2212yes—
25C22×C4⋊Dic7C2211yescharacteristic
26C7×#408C418yescomplemented
27C2×C7×C22⋊C8C2212yes—
28C2×C7×C22⋊C8C2212yes—
29#90002—21no—
30#89993—21no—
31#89992—22no—
32#89984C2212yes—
33#89984C2212yes—
34#89893—24no—
35#89878—24no—
36C2×#6392C2212yes—
37C2×#6392C2211yescharacteristic
38C2×#6392C2211yescharacteristic
39C2×#6391C2212yes—
40C2×#6391C2211yescharacteristic
41C2×#6391C2211yescharacteristic
42#89346C2212yes—
43#89346C2212yes—
44#89346C2212yes—
45#89346C2212yes—
46#20849—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41C2×C414yes—
52C2×C7×#41Q814yes—
53C22×C7×Q8D411yescharacteristic
54C22×C7×D4D412yes—
55C22×C7×D4D411yescharacteristic
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×Dic7—22no—
59C23×Dic7—21no—
60C23×Dic7—21no—
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16C2×C414yescomplemented
64C22×C7×C8—22no—
65C22×C7×C8—22no—
66C22×C7×C8C2×C414yescomplemented
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2312yes—
69C22×C7×C8C2311yescharacteristic
70C2×C22⋊Dic7—22no—
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—22no—
75Q8×Dic7—28no—
76D4×Dic7—216no—
77D4×Dic7—28no—
78#6489—22no—
79#6489—24no—
80#6489—24no—
81#6489—22no—
82#6489—22no—
83#6489D414yes—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—22no—
87C2×C4⋊Dic7—24no—
88C2×C4⋊Dic7—24no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7D414yes—
93C2×C4⋊Dic7C2312yes—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4×Dic7—22no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C7×C22⋊C8C2×C418yescomplemented
103C7×C22⋊C8C2×C418yescomplemented
104C7⋊#95—22no—
105C7⋊#95—22no—
106#6392—22no—
107#6392—22no—
108#6392—24no—
109#6391—22no—
110#6391—22no—
111#6391—24no—
112C2×#520—71no—
113C2×#428—71no—
114C2×#427—71no—
115C2×#408D711yescharacteristic
116#2311—71no—
117#1854—72no—
118#1854—72no—
119#1853—72no—
120#1852—72no—
121#1736—71no—
122C2×#340—71no—
123C24×C7C2×D412yes—
124C24×C7C2×D411yescharacteristic
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41—28no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8C4⋊C414yes—
132C2×C7×D4—24no—
133C2×C7×D4—24no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4C4⋊C418yes—
137C2×C7×D4C4⋊C414yes—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7C22×C414yes—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×Q814yes—
149C22×C4×C7C2×Q812yes—
150C22×C4×C7#4112yes—
151C22×C4×C7#4112yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—22no—
155C22×Dic7—21no—
156C22×Dic7—28no—
157C22×Dic7—24no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C22×Dic7—42no—
162C22×Dic7—81no—
163C22×Dic7—82no—
164C22×Dic7—81no—
165C22×Dic7—82no—
166C22×Dic7—81no—
167C22×Dic7—81no—
168C7×M16—28no—
169C2×C7×C8—28no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—22no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—22no—
176C2×C7×C8C22×C412yes—
177C2×C7×C8C22×C412yes—
178C2×C7×C8C22×C414yes—
179C22⋊Dic7—28no—
180C22⋊Dic7—24no—
181C22⋊Dic7—42no—
182C22⋊Dic7—82no—
183C22⋊Dic7—82no—
184C4⋊Dic7—28no—
185C4⋊Dic7—24no—
186C4⋊Dic7—22no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—22no—
194C4⋊Dic7—22no—
195C4⋊Dic7—42no—
196C4⋊Dic7C2×D412yes—
197C4⋊Dic7C2×D414yes—
198C4⋊Dic7C2×D412yes—
199C4×Dic7—28no—
200C4×Dic7—24no—
201C4×Dic7—42no—
202C22×#41C2×D711yescharacteristic
203C22×M16C2×D711yescharacteristic
204C23×C8C2×D711yescharacteristic
205#520—144no—
206#520—144no—
207C2×C4×Q8—141no—
208C2×C4×D4—142no—
209C2×C4×D4—141no—
210C2×#117—71no—
211C2×#117—71no—
212C2×#117—72no—
213C22×C4⋊C4—71no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—72no—
216#428—144no—
217#427—144no—
218C2×#107—71no—
219C2×#107—72no—
220C2×#107—71no—
221C2×#106—71no—
222C2×#106—72no—
223C2×#106—71no—
224#408Dic718yescomplemented
225C2×C22⋊C8C2×D712yes—
226C2×C22⋊C8C2×D712yes—
227#391—141no—
228#390—141no—
229#386—142no—
230#380—72no—
231#380—72no—
232#340—72no—
233#340—72no—
234#340—72no—
235#340—72no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7C2×C4⋊C414yes—
247C23×C7C2×C4⋊C412yes—
248C23×C7#12012yes—
249C23×C7#12312yes—
250C23×C7C22×D411yescharacteristic
251C7×Q8—28no—
252C7×D4—216no—
253C7×D4—28no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7C2×C4⋊C414yes—
261C2×C4×C7C2×C4⋊C412yes—
262C2×C4×C7C4×D414yes—
263C2×C4×C7C4×D414yes—
264C2×C4×C7#12012yes—
265C2×C4×C7C22⋊Q814yes—
266C2×C4×C7C22⋊Q814yes—
267C2×C4×C7#12312yes—
268C2×C4×C7C23×C411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×Q811yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—22no—
277C2×Dic7—42no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—42no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—42no—
286C2×Dic7—41no—
287C2×Dic7—42no—
288C2×Dic7—41no—
289C2×Dic7—42no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—84no—
293C2×Dic7—88no—
294C2×Dic7—84no—
295C7×C8—22no—
296C7×C8—22no—
297C7×C8—24no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41—22no—
302C2×#41Dic1414yes—
303C2×#41C2×Dic714yes—
304C22×Q8D2811yescharacteristic
305C22×D4D2812yes—
306C22×D4D2811yescharacteristic
307C23×C4—142no—
308C23×C4—141no—
309C23×C4—141no—
310C23×C4C22×D712yes—
311C23×C4C22×D711yescharacteristic
312C2×M16—22no—
313C2×M16—22no—
314C2×M16C2×Dic714yescomplemented
315C22×C8—22no—
316C22×C8—22no—
317C22×C8C2×Dic714yescomplemented
318C22×C8C22×D711yescharacteristic
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D712yes—
321C4×Q8—148no—
322C4×D4—1416no—
323C4×D4—148no—
324#117—74no—
325#117—144no—
326#117—144no—
327#117—142no—
328#117—142no—
329#117—142no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—72no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—144no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4—142no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—141no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—142no—
349C2×C22⋊C4—142no—
350C2×C42—142no—
351C2×C42—141no—
352C2×C42—141no—
353#107—142no—
354#107—142no—
355#107—144no—
356#106—142no—
357#106—142no—
358#106—144no—
359C22⋊C8C2×Dic718yescomplemented
360C22⋊C8C2×Dic718yescomplemented
361#95—142no—
362#95—142no—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—24no—
370C22×C7—24no—
371C22×C7—24no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7#38514yes—
377C22×C7C22×C4⋊C411yescharacteristic
378C22×C7C2×#12011yescharacteristic
379C22×C7C2×#12311yescharacteristic
380C22×C7#57611yescharacteristic
381C22×C7#57711yescharacteristic
382C22×C7#57711yescharacteristic
383C22×C7#57811yescharacteristic
384C4×C7—24no—
385C4×C7#38514yes—
386C4×C7C2×C4×D411yescharacteristic
387C4×C7C2×C4×D411yescharacteristic
388C4×C7C2×C22⋊Q811yescharacteristic
389C4×C7C2×C22⋊Q811yescharacteristic
390Dic7—44no—
391Dic7—42no—
392Dic7—42no—
393Dic7—82no—
394Dic7—81no—
395Dic7—81no—
396C24C2×D2812yes—
397C24C2×D2811yescharacteristic
398#41—28no—
399#41—28no—
400#41—28no—
401#41—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C4⋊Dic714yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C4⋊Dic718yes—
410C2×D4C4⋊Dic714yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—141no—
416C22×C4—142no—
417C22×C4—141no—
418C22×C4—148no—
419C22×C4—144no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—142no—
423C22×C4—282no—
424C22×C4—561no—
425C22×C4—562no—
426C22×C4—561no—
427C22×C4—562no—
428C22×C4—561no—
429C22×C4—561no—
430C22×C4C2×Dic1414yes—
431C22×C4C2×Dic1412yes—
432C22×C4C2×D2812yes—
433C22×C4C2×D2811yescharacteristic
434C22×C4D4×D712yes—
435C22×C4D4×D712yes—
436C22×C4#115912yes—
437C22×C4#115912yes—
438C22×C4C22×Dic714yes—
439C22×C4C22×Dic712yes—
440C22×C4C23×D711yescharacteristic
441M16—28no—
442C2×C8—28no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8C22×Dic712yes—
450C2×C8C22×Dic712yes—
451C2×C8C22×Dic714yes—
452C4⋊C4—74no—
453C4⋊C4—72no—
454C4⋊C4—72no—
455C4⋊C4—148no—
456C4⋊C4—144no—
457C4⋊C4—141no—
458C4⋊C4—142no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—142no—
465C4⋊C4—142no—
466C4⋊C4—282no—
467C22⋊C4—148no—
468C22⋊C4—144no—
469C22⋊C4—282no—
470C22⋊C4—562no—
471C22⋊C4—562no—
472C42—148no—
473C42—144no—
474C42—282no—
475C2×C7—24no—
476C2×C7—44no—
477C2×C7C2×#38511yescharacteristic
478C2×C7#291211yescharacteristic
479C2×C7#291311yescharacteristic
480C2×C7#301211yescharacteristic
481C2×C7#301311yescharacteristic
482C2×C7#309611yescharacteristic
483C2×C7#309711yescharacteristic
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23#644512yes—
495C23#644912yes—
496C23C2×C4⋊Dic714yes—
497C23C2×C4⋊Dic712yes—
498C23C22×D2811yescharacteristic
499Q8—28no—
500D4—216no—
501D4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—142no—
509C2×C4—144no—
510C2×C4—142no—
511C2×C4—281no—
512C2×C4—282no—
513C2×C4—281no—
514C2×C4—282no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—282no—
520C2×C4—281no—
521C2×C4—282no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—282no—
525C2×C4—281no—
526C2×C4—564no—
527C2×C4—568no—
528C2×C4—564no—
529C2×C4#644114yes—
530C2×C4#644114yes—
531C2×C4#644512yes—
532C2×C4#644912yes—
533C2×C4C2×C4⋊Dic714yes—
534C2×C4C2×C4⋊Dic712yes—
535C2×C4D4×Dic714yes—
536C2×C4D4×Dic714yes—
537C2×C4C22×Dic1411yescharacteristic
538C2×C4C2×D4×D711yescharacteristic
539C2×C4C2×D4×D711yescharacteristic
540C2×C4C2×#115911yescharacteristic
541C2×C4C2×#115911yescharacteristic
542C2×C4C23×Dic711yescharacteristic
543C8—22no—
544C8—22no—
545C8—24no—
546C7#2084911yescharacteristic, complemented, Sylow
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#8972214yes—
561C22C2×#644511yescharacteristic
562C22C2×#644911yescharacteristic
563C22#9044211yescharacteristic
564C22#9044211yescharacteristic
565C22#9044311yescharacteristic
566C22#9044411yescharacteristic
567C22C22×C4⋊Dic711yescharacteristic
568C4—24no—
569C4—284no—
570C4—282no—
571C4—282no—
572C4—562no—
573C4—561no—
574C4—561no—
575C4#8972214yes—
576C4C2×#644111yescharacteristic
577C4C2×#644111yescharacteristic
578C4C2×D4×Dic711yescharacteristic
579C4C2×D4×Dic711yescharacteristic
580C2—24no—
581C2—44no—
582C2#1173609611yescharacteristic
583C2#1173609711yescharacteristic
584C2#1173611711yescharacteristic
585C2#1173612511yescharacteristic
586C2C2×#8972211yescharacteristic
587C2#1174122911yescharacteristic
588C2#1174123011yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#408C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#408D7
hypocenterC7#20849
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
4. C2×#89893C2semidirect
5. C2×#89878C2semidirect
6. #11729739C2semidirect
7. C2×#89346C2semidirect
8. #11727683C2semidirect
9. #11727683C2semidirect
10. #11727682C2semidirect
11. #11727681C2semidirect
26. C7×#408C4semidirect
63. C2×C7×M16C2×C4semidirect
66. C22×C7×C8C2×C4semidirect
102. C7×C22⋊C8C2×C4semidirect
103. C7×C22⋊C8C2×C4semidirect
224. #408Dic7semidirect
314. C2×M16C2×Dic7semidirect
317. C22×C8C2×Dic7semidirect
359. C22⋊C8C2×Dic7semidirect
360. C22⋊C8C2×Dic7semidirect
546. C7#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#905182
C2×C7×#4082
#117415322
C2×#898932
C2×#898782
#117297392
C2×#893462
#117276832
#117276822
#117276812
#208497