Finite Group Databasebeta

← #13357512

Subgroups — #13357512

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×C7×#535C211yesmaximal, characteristic
2C2×#90004C211yesmaximal, characteristic, complemented
3C2×#90005C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5C22×#6406C211yesmaximal, characteristic, complemented
6#11739678C211yesmaximal, characteristic, complemented
7C2×#89769C211yesmaximal, characteristic, complemented
8#11729519C212yesmaximal, complemented
9#11729518C212yesmaximal, complemented
10#11729480C212yesmaximal, complemented
11#11729479C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×D4C2211yescharacteristic
14C7×#535—24no—
15C7×#535C418yescomplemented
16C2×C7×#127C2212yes—
17C2×C7×#124C2212yes—
18C2×C7×#121—22no—
19C2×C7×#120—22no—
20C2×C7×#117C2211yescharacteristic
21C2×#6489C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C22×#1136C2211yescharacteristic
24C23×C7⋊C8C2211yescharacteristic
25#90005—24no—
26#90004—24no—
27C2×#6409C2211yescharacteristic, complemented
28C2×#6409C2212yescomplemented
29C2×#6409C2211yescharacteristic, complemented
30#89987—24no—
31C2×#6406—24no—
32C2×#6406C2212yes—
33C2×#6406C2212yes—
34C2×#6406C2212yes—
35C2×#6406C2212yes—
36C2×#6406C2211yescharacteristic, complemented
37C2×#6406C2211yescharacteristic, complemented
38C2×#6406C2212yescomplemented
39#89769—24no—
40#89757—22no—
41#89722—22no—
42#89708C2212yescomplemented
43#89707C2212yescomplemented
44C2×#6377C2212yescomplemented
45C2×C4×C7⋊C8C2212yescomplemented
46#21206—71nomaximal, Sylow
47C25×C7—21no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41C2×C414yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4—21no—
56C22×C7×D4—22no—
57C22×C7×D4—22no—
58C22×C7×D4—21no—
59C22×C7×D4—22no—
60C22×C7×D4—24no—
61C22×C7×D4—22no—
62C22×C7×D4—22no—
63C22×C7×D4C2×C414yes—
64C22×C7×D4C2312yes—
65C22×C7×D4C2311yescharacteristic
66C23×C4×C7—21no—
67C23×C4×C7C2311yescharacteristic
68C23×Dic7—21no—
69C7×#127—24no—
70C7×#127C2×C418yescomplemented
71C7×#124—24no—
72C7×#124C2×C418yescomplemented
73C7×#121—216no—
74C7×#120—216no—
75C7×#117C2×C414yes—
76C7×#117D414yes—
77C2×C7×C4⋊C4D412yes—
78C2×C7×C22⋊C4—24no—
79C2×C7×C22⋊C4D412yes—
80C2×C42×C7C2312yes—
81C2×C22⋊Dic7—21no—
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—24no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2312yes—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4×Dic7—21no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136D414yescomplemented
95C22×C7⋊C8—22no—
96C22×C7⋊C8—22no—
97C22×C7⋊C8D414yescomplemented
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2312yes—
100C22×C7⋊C8C2311yescharacteristic
101#6409—24no—
102#6409—22no—
103#6409—22no—
104#6406—24no—
105#6406—24no—
106#6406—24no—
107#6406—24no—
108#6406—24no—
109#6406—22no—
110#6406—22no—
111C7⋊#95—24no—
112#6377—24no—
113C4×C7⋊C8—24no—
114C2×#535D711yescharacteristic
115C2×#417—71no—
116C2×#418—71no—
117C2×#416—71no—
118C22×#102—71no—
119C2×#405—71no—
120#2317—71no—
121#1961—72no—
122#1959—72no—
123#1958—72no—
124#1957—72no—
125C24×C7—22no—
126C24×C7—21no—
127C24×C7—24no—
128C24×C7—42no—
129C24×C7—42no—
130C24×C7—82no—
131C24×C7—82no—
132C24×C7C2×D411yescharacteristic
133C7×#41—28no—
134C7×#41—28no—
135C7×#41—44no—
136C7×#41—44no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8C22×C412yes—
143C2×C7×D4—24no—
144C2×C7×D4—24no—
145C2×C7×D4—216no—
146C2×C7×D4—28no—
147C2×C7×D4—28no—
148C2×C7×D4—22no—
149C2×C7×D4—22no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—42no—
159C2×C7×D4—42no—
160C2×C7×D4—44no—
161C2×C7×D4—44no—
162C2×C7×D4—42no—
163C2×C7×D4—42no—
164C2×C7×D4—84no—
165C2×C7×D4—84no—
166C2×C7×D4C22×C414yes—
167C2×C7×D4C22×C412yes—
168C22×C4×C7—21no—
169C22×C4×C7—24no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7C22×C412yes—
173C22×C4×C7C2×D414yes—
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D412yes—
176C22×C4×C7C2×D412yes—
177C22×C4×C7C2×D412yes—
178C22×C4×C7#4112yes—
179C22×C4×C7#4112yes—
180C22×C4×C7C2411yescharacteristic
181C22×Dic7—21no—
182C22×Dic7—22no—
183C22×Dic7—21no—
184C22×Dic7—42no—
185C22×Dic7—82no—
186C7×C4⋊C4C22⋊C418yes—
187C7×C22⋊C4—216no—
188C7×C22⋊C4C22⋊C418yes—
189C42×C7C22×C414yes—
190C42×C7C2×D414yes—
191C22⋊Dic7—42no—
192C4⋊Dic7—22no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—22no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—42no—
199C4⋊Dic7—84no—
200C4×Dic7—42no—
201#1136—28no—
202C2×C7⋊C8—28no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8C2×D412yes—
210C2×C7⋊C8C2×D412yes—
211C2×C7⋊C8C2×D414yes—
212C22×#41C2×D711yescharacteristic
213C23×D4C2×D711yescharacteristic
214C22×M16—71no—
215C23×C8—71no—
216#535—24no—
217#535Dic718yescomplemented
218C2×#127C2×D712yes—
219C2×#124C2×D712yes—
220C2×#121—22no—
221C2×#120—22no—
222C2×#117—71no—
223C2×#117C2×D711yescharacteristic
224C22×C4⋊C4—71no—
225#418—144no—
226#417—144no—
227#416—144no—
228C2×#103—71no—
229C2×#103—71no—
230C2×#103—72no—
231C2×#102—72no—
232C2×#102—72no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#102—72no—
236C2×#102—72no—
237C2×#102—72no—
238C2×#102—144no—
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—22no—
250C23×C7—41no—
251C23×C7—42no—
252C23×C7—41no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—44no—
256C23×C7—42no—
257C23×C7—44no—
258C23×C7—48no—
259C23×C7—88no—
260C23×C7—88no—
261C23×C7—84no—
262C23×C7—84no—
263C23×C7C2×C22⋊C412yes—
264C23×C7#12412yes—
265C23×C7#12712yes—
266C23×C7C22×D411yescharacteristic
267C7×Q8—22no—
268C7×Q8—22no—
269C7×Q8—41no—
270C7×Q8—41no—
271C7×D4—24no—
272C7×D4—24no—
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—48no—
276C7×D4—42no—
277C7×D4—42no—
278C7×D4—41no—
279C7×D4—41no—
280C7×D4—48no—
281C7×D4—816no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—22no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—48no—
288C2×C4×C7C2×C22⋊C418yes—
289C2×C4×C7C2×C22⋊C412yes—
290C2×C4×C7C4×D414yes—
291C2×C4×C7C4×D414yes—
292C2×C4×C7#12114yes—
293C2×C4×C7#12114yes—
294C2×C4×C7#12412yes—
295C2×C4×C7#12712yes—
296C2×C4×C7C23×C411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—44no—
307C2×Dic7—42no—
308C2×Dic7—84no—
309C7⋊C8—22no—
310C7⋊C8—22no—
311C7⋊C8—24no—
312C25—21no—
313C2×#41—22no—
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—22no—
317C2×#41—22no—
318C2×#41C2×Dic714yes—
319C22×Q8C22×D711yescharacteristic
320C22×D4—21no—
321C22×D4—22no—
322C22×D4—22no—
323C22×D4—22no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—22no—
327C22×D4—24no—
328C22×D4C2×Dic714yes—
329C22×D4C22×D712yes—
330C22×D4C22×D711yescharacteristic
331C23×C4—21no—
332C23×C4—141no—
333C23×C4C22×D711yescharacteristic
334C2×M16—74no—
335C2×M16—142no—
336C2×M16—142no—
337C22×C8—74no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—72no—
341C22×C8—142no—
342C22×C8—142no—
343#127—24no—
344#127C2×Dic718yescomplemented
345#124—24no—
346#124C2×Dic718yescomplemented
347#121—216no—
348#120—216no—
349#117—142no—
350#117—142no—
351#117C2×Dic714yes—
352#117C22⋊D714yes—
353C2×C4⋊C4—72no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—144no—
360C2×C4⋊C4C22⋊D712yes—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4C22⋊D712yes—
364C2×C42—141no—
365C2×C42C22×D712yes—
366#103—142no—
367#103—142no—
368#103—144no—
369#102—144no—
370#102—144no—
371#102—142no—
372#102—142no—
373#102—144no—
374#102—144no—
375#102—144no—
376#97—144no—
377C4×C8—144no—
378#95—144no—
379C22×C7—24no—
380C22×C7—24no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—44no—
384C22×C7—48no—
385C22×C7—42no—
386C22×C7—44no—
387C22×C7—816no—
388C22×C7—82no—
389C22×C7—84no—
390C22×C7—81no—
391C22×C7—82no—
392C22×C7—81no—
393C22×C7—82no—
394C22×C7#39014yes—
395C22×C7C22×C22⋊C411yescharacteristic
396C22×C7C2×#12411yescharacteristic
397C22×C7C2×#12711yescharacteristic
398C22×C7#57612yes—
399C22×C7#57712yes—
400C4×C7—28no—
401C4×C7—42no—
402C4×C7#39014yes—
403C4×C7C2×C4×D411yescharacteristic
404C4×C7C2×C4×D411yescharacteristic
405C4×C7C2×#12111yescharacteristic
406C4×C7C2×#12111yescharacteristic
407Dic7—81no—
408Dic7—81no—
409Dic7—82no—
410C24—22no—
411C24—21no—
412C24—24no—
413C24—42no—
414C24—42no—
415C24—82no—
416C24—82no—
417C24C2×C22⋊D711yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C22×Dic712yes—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—216no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4C22×Dic714yes—
452C2×D4C22×Dic712yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—141no—
458C22×C4—142no—
459C22×C4—141no—
460C22×C4—282no—
461C22×C4—562no—
462C22×C4D4×D712yes—
463C22×C4D4×D712yes—
464C22×C4#115912yes—
465C22×C4#115912yes—
466C22×C4C22×Dic712yes—
467C22×C4C2×C22⋊D714yes—
468C22×C4C2×C22⋊D711yescharacteristic
469C22×C4C2×C22⋊D712yes—
470C22×C4C23×D711yescharacteristic
471M16—148no—
472C2×C8—72no—
473C2×C8—72no—
474C2×C8—74no—
475C2×C8—148no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—142no—
479C2×C8—141no—
480C2×C8—142no—
481C2×C8—141no—
482C4⋊C4—142no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—142no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—282no—
489C4⋊C4—564no—
490C4⋊C4C22⋊Dic718yes—
491C22⋊C4—216no—
492C22⋊C4—282no—
493C22⋊C4C22⋊Dic718yes—
494C42—282no—
495C42C22×Dic714yes—
496C42C2×C22⋊D714yes—
497C2×C7—24no—
498C2×C7—82no—
499C2×C7—84no—
500C2×C7C2×#39011yescharacteristic
501C2×C7#289712yes—
502C2×C7#313612yes—
503C2×C7#315411yescharacteristic
504C2×C7#315511yescharacteristic
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—48no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—88no—
519C23—88no—
520C23—84no—
521C23—84no—
522C23#649912yes—
523C23#650312yes—
524C23C2×C22⋊Dic712yes—
525C23C22×C22⋊D711yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530D4—24no—
531D4—24no—
532D4—22no—
533D4—22no—
534D4—48no—
535D4—42no—
536D4—42no—
537D4—41no—
538D4—41no—
539D4—48no—
540D4—816no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—284no—
552C2×C4—282no—
553C2×C4—564no—
554C2×C4D4×Dic714yes—
555C2×C4D4×Dic714yes—
556C2×C4#649912yes—
557C2×C4#650114yes—
558C2×C4#650114yes—
559C2×C4#650312yes—
560C2×C4C2×C22⋊Dic718yes—
561C2×C4C2×C22⋊Dic712yes—
562C2×C4C2×D4×D711yescharacteristic
563C2×C4C2×D4×D711yescharacteristic
564C2×C4C2×#115911yescharacteristic
565C2×C4C2×#115911yescharacteristic
566C2×C4C23×Dic711yescharacteristic
567C2×C4C22×C22⋊D711yescharacteristic
568C8—142no—
569C8—142no—
570C8—144no—
571C7#2120611yescharacteristic, complemented, Sylow
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—48no—
580C22—816no—
581C22—84no—
582C22—82no—
583C22—82no—
584C22—81no—
585C22—82no—
586C22—81no—
587C22#8999314yes—
588C22C2×#649911yescharacteristic
589C22C2×#650311yescharacteristic
590C22#9052612yes—
591C22#9052712yes—
592C22C22×C22⋊Dic711yescharacteristic
593C4—28no—
594C4—42no—
595C4—561no—
596C4—562no—
597C4—561no—
598C4#8999314yes—
599C4C2×D4×Dic711yescharacteristic
600C4C2×D4×Dic711yescharacteristic
601C4C2×#650111yescharacteristic
602C4C2×#650111yescharacteristic
603C2—24no—
604C2—84no—
605C2—82no—
606C2#1174082512yes—
607C2#1174082611yescharacteristic
608C2#1174082711yescharacteristic
609C2C2×#8999311yescharacteristic
610C2#1174154412yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#535C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#535D7
hypocenterC7#21206
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. C2×#90004C2semidirect
3. C2×#90005C2semidirect
4. C2×#89987C2semidirect
5. C22×#6406C2semidirect
6. #11739678C2semidirect
7. C2×#89769C2semidirect
8. #11729519C2semidirect
9. #11729518C2semidirect
10. #11729480C2semidirect
11. #11729479C2semidirect
15. C7×#535C4semidirect
27. C2×#6409C22semidirect
28. C2×#6409C22semidirect
29. C2×#6409C22semidirect
36. C2×#6406C22semidirect
37. C2×#6406C22semidirect
38. C2×#6406C22semidirect
42. #89708C22semidirect
43. #89707C22semidirect
44. C2×#6377C22semidirect
45. C2×C4×C7⋊C8C22semidirect
70. C7×#127C2×C4semidirect
72. C7×#124C2×C4semidirect
94. C2×#1136D4semidirect
97. C22×C7⋊C8D4semidirect
217. #535Dic7semidirect
344. #127C2×Dic7semidirect
346. #124C2×Dic7semidirect
571. C7#21206semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5352
C2×#900042
C2×#900052
C2×#899872
C22×#64062
#117396782
C2×#897692
#117295192
#117295182
#117294802
#117294792
#212067