Finite Group Databasebeta

← #13357504

Subgroups — #13357504

Order 1792.

1718 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90496C211yesmaximal, characteristic
2C22×C7×#102C211yesmaximal, characteristic, complemented
3C2×#89987C211yesmaximal, characteristic, complemented
4C2×#89885C211yesmaximal, characteristic, complemented
5C2×#89873C211yesmaximal, characteristic, complemented
6#11739602C211yesmaximal, characteristic, complemented
7C2×#89738C211yesmaximal, characteristic, complemented
8#11727914C212yesmaximal, complemented
9#11727913C212yesmaximal, complemented
10#11727912C212yesmaximal, complemented
11#11727911C212yesmaximal, complemented
12C23×C7×D4C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6516—22no—
17#90496—24no—
18#90496C418yescomplemented
19C2×#6503C2212yes—
20C2×#6501—22no—
21C2×#6499C2212yes—
22C2×#6489C2211yescharacteristic
23C2×#6489C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C2×C7×#102—24no—
26C2×C7×#102C2212yes—
27C2×C7×#102C2212yes—
28#89987—24no—
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31#89982—22no—
32#89885—24no—
33C2×#6395C2211yescharacteristic, complemented
34C2×#6395C2211yescharacteristic, complemented
35C2×#6390C2211yescharacteristic, complemented
36C2×#6390C2211yescharacteristic, complemented
37#89873—24no—
38C2×#6389C2212yescomplemented
39C2×C8×Dic7C2212yescomplemented
40#89749C2212yescomplemented
41#89748C2212yescomplemented
42#89745—22no—
43C2×#6384C2212yescomplemented
44#89738—24no—
45C2×#6383C2212yescomplemented
46#21206—71nomaximal, Sylow
47C25×C7—21no—
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4—21no—
51C22×C7×D4—22no—
52C22×C7×D4—22no—
53C22×C7×D4—22no—
54C22×C7×D4C2×C414yes—
55C22×C7×D4C2312yes—
56C23×C4×C7—21no—
57C23×C4×C7C2311yescharacteristic
58C22×C22⋊D7—21no—
59C22×C22⋊D7—24no—
60C2×#1157—22no—
61C2×#1157—22no—
62C2×#1157—22no—
63C2×#1157—22no—
64C2×#1157—22no—
65C2×#1157C2×C414yes—
66C22×D28C2311yescharacteristic
67C22×C4×D7—21no—
68C22×Dic14C2311yescharacteristic
69C22×C7×C8—22no—
70C22×C7×C8—22no—
71C22×C7×C8D414yescomplemented
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4C2312yes—
78#6516—216no—
79C2×C22⋊Dic7—24no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7D412yes—
82#6503—24no—
83#6503C2×C418yescomplemented
84#6501—216no—
85#6499—24no—
86#6499C2×C418yescomplemented
87#6489—22no—
88#6489—22no—
89#6489C2×C414yes—
90#6489D414yes—
91C2×C4⋊Dic7D412yes—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—24no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7C2312yes—
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136D414yescomplemented
100C22×C7⋊C8C2312yes—
101C7×#102—24no—
102C7×#102—24no—
103#6406—24no—
104#6406—24no—
105C7⋊#95—24no—
106#6395—22no—
107#6395—22no—
108#6390—22no—
109#6390—22no—
110#6389—24no—
111C8×Dic7—24no—
112#6384—24no—
113#6383—24no—
114C2×#535—71no—
115C2×#417—71no—
116C2×#418—71no—
117C2×#416—71no—
118C22×#102D711yescharacteristic, complemented
119C2×#405—71no—
120#2317—71no—
121#1961—72no—
122#1959—72no—
123#1958—72no—
124#1957—72no—
125C24×C7—22no—
126C24×C7—24no—
127C24×C7—42no—
128C24×C7—42no—
129C24×C7—82no—
130C24×C7—82no—
131C24×C7C2×D411yescharacteristic
132C23×D7—21no—
133C2×C7×D4—24no—
134C2×C7×D4—28no—
135C2×C7×D4—28no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—44no—
142C2×C7×D4—44no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4C22×C414yes—
146C22×C4×C7—22no—
147C22×C4×C7—42no—
148C22×C4×C7—82no—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7#4112yes—
155C22×C4×C7#4112yes—
156C22×C4×C7C2411yescharacteristic
157C2×C22⋊D7—24no—
158C2×C22⋊D7—216no—
159C2×C22⋊D7—42no—
160C2×C22⋊D7—42no—
161C2×C22⋊D7—84no—
162C2×C22⋊D7—84no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7C2×D414yes—
167#1157—28no—
168#1157—28no—
169#1157—44no—
170#1157—44no—
171C2×D28—21no—
172C2×D28—21no—
173C2×D28—21no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28C22×C412yes—
177C2×C4×D7—24no—
178C2×C4×D7—42no—
179C2×C4×D7—42no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14C22×C412yes—
186C2×C7×C8—28no—
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8C2×D412yes—
192C2×C7×C8C2×D412yes—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—42no—
196C22⋊Dic7—216no—
197C22⋊Dic7—42no—
198C22⋊Dic7C22⋊C418yes—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7C22⋊C418yes—
204Dic7⋊C4—84no—
205C4×Dic7—42no—
206C4×Dic7C22×C414yes—
207C4×Dic7C2×D414yes—
208#1136—28no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8C2×D414yes—
212C22×#41—71no—
213C23×D4C2×D711yescharacteristic
214C22×M16—71no—
215C23×C8C2×D711yescharacteristic
216#535—78no—
217#535—144no—
218C2×#127—72no—
219C2×#124—72no—
220C2×#121—142no—
221C2×#120—142no—
222C2×#117—71no—
223C2×#117—71no—
224C22×C4⋊C4C2×D711yescharacteristic
225#418—144no—
226#417—144no—
227#416—144no—
228C2×#103—71no—
229C2×#103—71no—
230C2×#103—72no—
231C2×#102—24no—
232C2×#102—72no—
233C2×#102—72no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—72no—
237C2×#102C2×D712yes—
238C2×#102C2×D712yes—
239#405—144no—
240C2×#97—72no—
241C2×C4×C8—72no—
242#389—142no—
243#385—142no—
244#384—72no—
245#382—72no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—24no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—44no—
252C23×C7—44no—
253C23×C7—48no—
254C23×C7—88no—
255C23×C7—88no—
256C23×C7—84no—
257C23×C7—84no—
258C23×C7C2×C22⋊C412yes—
259C23×C7#12412yes—
260C23×C7#12712yes—
261C23×C7C22×D411yescharacteristic
262C22×D7—22no—
263C22×D7—41no—
264C22×D7—41no—
265C22×D7—42no—
266C22×D7—42no—
267C7×D4—24no—
268C7×D4—24no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—48no—
272C2×C4×C7—42no—
273C2×C4×C7—44no—
274C2×C4×C7—84no—
275C2×C4×C7C2×C22⋊C412yes—
276C2×C4×C7C4×D414yes—
277C2×C4×C7C4×D414yes—
278C2×C4×C7#12114yes—
279C2×C4×C7#12114yes—
280C2×C4×C7#12412yes—
281C2×C4×C7#12712yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C22⋊D7—48no—
289C22⋊D7—816no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7C2×C22⋊C418yes—
300D28—22no—
301D28—22no—
302D28—41no—
303D28—41no—
304C4×D7—48no—
305Dic14—22no—
306Dic14—22no—
307Dic14—41no—
308Dic14—41no—
309C7×C8—22no—
310C7×C8—22no—
311C7⋊C8—24no—
312C25—21no—
313C2×#41—74no—
314C2×#41—142no—
315C2×#41—142no—
316C2×#41—142no—
317C2×#41—142no—
318C2×#41—142no—
319C22×Q8—71no—
320C22×D4—22no—
321C22×D4—22no—
322C22×D4—21no—
323C22×D4—22no—
324C22×D4—22no—
325C22×D4—22no—
326C22×D4—71no—
327C22×D4—144no—
328C22×D4—141no—
329C22×D4C4×D714yes—
330C22×D4C22×D712yes—
331C23×C4—21no—
332C23×C4—141no—
333C23×C4C22×D711yescharacteristic
334C2×M16—74no—
335C2×M16—142no—
336C2×M16—142no—
337C22×C8—22no—
338C22×C8—22no—
339C22×C8—72no—
340C22×C8C22⋊D714yescomplemented
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343#127—78no—
344#127—144no—
345#124—78no—
346#124—144no—
347#121—1416no—
348#120—1416no—
349#117—74no—
350#117—74no—
351#117—142no—
352#117—142no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—72no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—144no—
360C2×C4⋊C4C22×D712yes—
361C2×C22⋊C4—72no—
362C2×C22⋊C4—144no—
363C2×C22⋊C4—141no—
364C2×C42—72no—
365C2×C42—141no—
366#103—144no—
367#103—142no—
368#103—142no—
369#102—24no—
370#102—24no—
371#102—144no—
372#102—144no—
373#102—144no—
374#102—142no—
375#102—142no—
376#97—144no—
377C4×C8—144no—
378#95—144no—
379C22×C7—24no—
380C22×C7—24no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—48no—
384C22×C7—44no—
385C22×C7—816no—
386C22×C7—84no—
387C22×C7—82no—
388C22×C7—82no—
389C22×C7#39014yes—
390C22×C7C22×C22⋊C411yescharacteristic
391C22×C7C2×#12411yescharacteristic
392C22×C7C2×#12711yescharacteristic
393C22×C7#57612yes—
394C22×C7#57712yes—
395C2×D7—44no—
396C2×D7—42no—
397C2×D7—82no—
398C2×D7—81no—
399C2×D7—81no—
400C4×C7—82no—
401C4×C7#39014yes—
402C4×C7C2×C4×D411yescharacteristic
403C4×C7C2×C4×D411yescharacteristic
404C4×C7C2×#12111yescharacteristic
405C4×C7C2×#12111yescharacteristic
406Dic7—28no—
407Dic7—42no—
408Dic7—81no—
409Dic7—81no—
410C24—22no—
411C24—24no—
412C24—42no—
413C24—42no—
414C24—82no—
415C24—82no—
416C24—141no—
417C24D4×D711yescharacteristic
418#41—148no—
419#41—148no—
420#41—284no—
421#41—284no—
422C2×Q8—72no—
423C2×Q8—141no—
424C2×Q8—141no—
425C2×Q8—141no—
426C2×Q8—141no—
427C2×Q8—141no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—72no—
441C2×D4—1416no—
442C2×D4—144no—
443C2×D4—141no—
444C2×D4—141no—
445C2×D4—141no—
446C2×D4—141no—
447C2×D4—141no—
448C2×D4—282no—
449C2×D4—282no—
450C2×D4—564no—
451C2×D4—564no—
452C2×D4C2×C4×D714yes—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—74no—
456C22×C4—82no—
457C22×C4—141no—
458C22×C4—144no—
459C22×C4—141no—
460C22×C4—141no—
461C22×C4—282no—
462C22×C4—282no—
463C22×C4C2×C4×D712yes—
464C22×C4#115712yes—
465C22×C4D4×D711yescharacteristic
466C22×C4D4×D712yes—
467C22×C4D4×D712yes—
468C22×C4#115912yes—
469C22×C4C2×C22⋊D712yes—
470C22×C4C23×D711yescharacteristic
471M16—148no—
472C2×C8—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—74no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8C2×C22⋊D712yes—
481C2×C8C2×C22⋊D712yes—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—78no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—564no—
491C22⋊C4—78no—
492C22⋊C4—1416no—
493C22⋊C4—282no—
494C42—74no—
495C42—74no—
496C42—282no—
497C2×C7—24no—
498C2×C7—84no—
499C2×C7C2×#39011yescharacteristic
500C2×C7#289712yes—
501C2×C7#313612yes—
502C2×C7#315411yescharacteristic
503C2×C7#315511yescharacteristic
504D7—82no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—48no—
513C23—88no—
514C23—84no—
515C23—84no—
516C23—88no—
517C23—142no—
518C23—281no—
519C23—282no—
520C23—282no—
521C23—281no—
522C23D7×C22⋊C412yes—
523C23#644812yes—
524C23#650312yes—
525C23C2×D4×D711yescharacteristic
526Q8—142no—
527Q8—142no—
528Q8—281no—
529Q8—281no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—48no—
535D4—142no—
536D4—142no—
537D4—288no—
538D4—281no—
539D4—281no—
540D4—5616no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—78no—
544C2×C4—84no—
545C2×C4—144no—
546C2×C4—144no—
547C2×C4—142no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—288no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4D7×C22⋊C412yes—
556C2×C4(C22⋊D7)⋊C414yes—
557C2×C4#644714yes—
558C2×C4#644812yes—
559C2×C4C4×C22⋊D714yes—
560C2×C4#650214yes—
561C2×C4#650312yes—
562C2×C4C22×C4×D711yescharacteristic
563C2×C4C2×#115711yescharacteristic
564C2×C4C2×D4×D711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×#115911yescharacteristic
567C2×C4C22×C22⋊D711yescharacteristic
568C8—22no—
569C8—22no—
570C8—144no—
571C7#2120611yescharacteristic, complemented, Sylow
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—48no—
577C22—44no—
578C22—816no—
579C22—82no—
580C22—82no—
581C22—84no—
582C22—284no—
583C22—282no—
584C22—561no—
585C22—561no—
586C22—562no—
587C22#8972914yes—
588C22C2×D7×C22⋊C411yescharacteristic
589C22C2×#644811yescharacteristic
590C22#9044812yes—
591C22#9045312yes—
592C22C2×#650311yescharacteristic
593C4—82no—
594C4—148no—
595C4—282no—
596C4—561no—
597C4—561no—
598C4#8972914yes—
599C4C2×(C22⋊D7)⋊C411yescharacteristic
600C4C2×#644711yescharacteristic
601C4C2×C4×C22⋊D711yescharacteristic
602C4C2×#650211yescharacteristic
603C2—24no—
604C2—84no—
605C2—562no—
606C2#1173618212yes—
607C2#1173618712yes—
608C2C2×#8972911yescharacteristic
609C2#1174127511yescharacteristic
610C2#1174129911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#102C2
socleC23×C7C22×D4
prime radicalC2×#90496C2
cosocleC22×C4×C7C24
hypercenterC22×#102D7
hypocenterC7#21206
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C7×#102C2semidirect
3. C2×#89987C2semidirect
4. C2×#89885C2semidirect
5. C2×#89873C2semidirect
6. #11739602C2semidirect
7. C2×#89738C2semidirect
8. #11727914C2semidirect
9. #11727913C2semidirect
10. #11727912C2semidirect
11. #11727911C2semidirect
18. #90496C4semidirect
33. C2×#6395C22semidirect
34. C2×#6395C22semidirect
35. C2×#6390C22semidirect
36. C2×#6390C22semidirect
38. C2×#6389C22semidirect
39. C2×C8×Dic7C22semidirect
40. #89749C22semidirect
41. #89748C22semidirect
43. C2×#6384C22semidirect
45. C2×#6383C22semidirect
71. C22×C7×C8D4semidirect
83. #6503C2×C4semidirect
86. #6499C2×C4semidirect
99. C2×#1136D4semidirect
118. C22×#102D7semidirect
340. C22×C8C22⋊D7semidirect
571. C7#21206semidirect

Maximal subgroups

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

subgroupindex
C2×#904962
C22×C7×#1022
C2×#899872
C2×#898852
C2×#898732
#117396022
C2×#897382
#117279142
#117279132
#117279122
#117279112
#212067