Finite Group Databasebeta

← #13355063

Subgroups — #13355063

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×C7×#535C211yesmaximal, characteristic
2C2×#90518C211yesmaximal, characteristic, complemented
3C2×#90006C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5C22×#6406C211yesmaximal, characteristic, complemented
6#11729432C214yesmaximal, complemented
7#11729431C214yesmaximal, complemented
8#11729335C211yesmaximal, characteristic, complemented
9#11729334C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×D4C2211yescharacteristic
12C7×#535—24no—
13C7×#535C418yescomplemented
14C2×C7×#127—21no—
15C2×C7×#124—21no—
16C2×C7×#121C2214yes—
17C2×C7×#120—22no—
18C2×C7×#117C2211yescharacteristic
19#90518—24no—
20#90518—24no—
21C2×Q8×Dic7—21no—
22C2×D4×Dic7—21no—
23C2×D4×Dic7—22no—
24C2×#6489C2211yescharacteristic
25C2×#6489C2212yescomplemented
26C22×C4⋊Dic7C2211yescharacteristic
27C22×C4⋊Dic7C2212yescomplemented
28C22×#1136C2211yescharacteristic
29C23×C7⋊C8C2211yescharacteristic
30#90006—24no—
31#89987—24no—
32C2×#6406—24no—
33C2×#6406C2212yes—
34C2×#6406C2214yes—
35C2×#6406C2212yes—
36C2×C22⋊(C7⋊C8)C2214yescomplemented
37#89324C2212yescomplemented
38#89324C2214yescomplemented
39#89324C2212yescomplemented
40#20914—71nomaximal, Sylow
41C25×C7—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41C2×C414yes—
47C2×C7×#41D414yes—
48C22×C7×Q8D411yescharacteristic
49C22×C7×D4—24no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—21no—
53C22×C7×D4—22no—
54C22×C7×D4—22no—
55C22×C7×D4—22no—
56C22×C7×D4C2×C414yescomplemented
57C22×C7×D4D412yes—
58C22×C7×D4D411yescharacteristic
59C22×C7×D4C2312yes—
60C23×C4×C7C2312yes—
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C23×Dic7—22no—
64C7×#127—28no—
65C7×#124—28no—
66C7×#121—28no—
67C7×#121C2×C4116yescomplemented
68C7×#120—216no—
69C7×#117—22no—
70C7×#117C2×C414yescomplemented
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—24no—
74C2×C42×C7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—22no—
77Q8×Dic7—28no—
78D4×Dic7—216no—
79D4×Dic7—28no—
80#6489—22no—
81#6489—24no—
82#6489—24no—
83#6489D414yescomplemented
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—22no—
87C2×C4⋊Dic7—24no—
88C2×C4⋊Dic7—24no—
89C2×C4⋊Dic7D414yescomplemented
90C2×C4⋊Dic7C2312yes—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4×Dic7—21no—
94C2×C4×Dic7—22no—
95C2×#1136—22no—
96C2×#1136—22no—
97C2×#1136—22no—
98C22×C7⋊C8—22no—
99C22×C7⋊C8—22no—
100C22×C7⋊C8—22no—
101C22×C7⋊C8C2312yes—
102C22×C7⋊C8C2312yes—
103#6406—28no—
104#6406—24no—
105#6406—24no—
106C22⋊(C7⋊C8)—28no—
107C2×#535D711yescharacteristic
108C2×#520—71no—
109C2×#418—71no—
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×D412yes—
123C24×C7C2×D411yescharacteristic
124C7×#41—28no—
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8C22⋊C414yes—
131C2×C7×D4—24no—
132C2×C7×D4—24no—
133C2×C7×D4—216no—
134C2×C7×D4—24no—
135C2×C7×D4—28no—
136C2×C7×D4—28no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—42no—
145C2×C7×D4—44no—
146C2×C7×D4—44no—
147C2×C7×D4—42no—
148C2×C7×D4—84no—
149C2×C7×D4—84no—
150C2×C7×D4C22⋊C418yes—
151C2×C7×D4C22⋊C414yes—
152C2×C7×D4C22×C414yes—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7C22×C414yes—
159C22×C4×C7C22×C412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D414yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7#4112yes—
167C22×C4×C7#4112yes—
168C22×C4×C7C2411yescharacteristic
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—22no—
172C22×Dic7—28no—
173C22×Dic7—24no—
174C22×Dic7—82no—
175C22×Dic7—84no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4C22×C414yes—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—216no—
180C42×C7—24no—
181C22⋊Dic7—28no—
182C22⋊Dic7—24no—
183C4⋊Dic7—28no—
184C4⋊Dic7—24no—
185C4⋊Dic7—22no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7C2×D412yes—
189C4⋊Dic7C2×D414yes—
190C4⋊Dic7C2×D412yes—
191C4×Dic7—28no—
192C4×Dic7—24no—
193#1136—44no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—44no—
201C22×#41C2×D711yescharacteristic
202C23×D4C2×D711yescharacteristic
203C22×M16—71no—
204C23×C8—71no—
205#535—24no—
206#535Dic718yescomplemented
207C2×#127—21no—
208C2×#124—21no—
209C2×#121C2×D714yes—
210C2×#120—22no—
211#520—144no—
212#520—144no—
213C2×C4×Q8—141no—
214C2×C4×D4—142no—
215C2×C4×D4—141no—
216C2×#117—71no—
217C2×#117—72no—
218C2×#117C2×D711yescharacteristic
219C22×C4⋊C4—71no—
220C22×C4⋊C4—72no—
221#418—144no—
222C2×#102—74no—
223C2×#102—72no—
224C2×#102—72no—
225C2×#102—144no—
226#408—144no—
227C2×C22⋊C8—74no—
228#340—72no—
229#340—74no—
230#340—72no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—24no—
236C23×C7—24no—
237C23×C7—24no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—44no—
241C23×C7—44no—
242C23×C7—48no—
243C23×C7—88no—
244C23×C7—88no—
245C23×C7—84no—
246C23×C7—84no—
247C23×C7C2×C22⋊C414yes—
248C23×C7C2×C22⋊C412yes—
249C23×C7#12012yes—
250C23×C7#12312yes—
251C23×C7C22×D411yescharacteristic
252C7×Q8—28no—
253C7×D4—216no—
254C7×D4—24no—
255C7×D4—24no—
256C7×D4—28no—
257C7×D4—42no—
258C7×D4—42no—
259C7×D4—48no—
260C7×D4—816no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7C2×C22⋊C414yes—
271C2×C4×C7C2×C22⋊C412yes—
272C2×C4×C7C4×D414yes—
273C2×C4×C7C4×D414yes—
274C2×C4×C7#12012yes—
275C2×C4×C7#12114yes—
276C2×C4×C7#12114yes—
277C2×C4×C7#12312yes—
278C2×C4×C7C23×C411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—24no—
286C2×Dic7—22no—
287C2×Dic7—44no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—84no—
291C2×Dic7—88no—
292C7⋊C8—42no—
293C7⋊C8—42no—
294C25—21no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41C2×Dic714yes—
300C2×#41C22⋊D714yes—
301C22×Q8C22⋊D711yescharacteristic
302C22×D4—24no—
303C22×D4—22no—
304C22×D4—22no—
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—21no—
308C22×D4—22no—
309C22×D4C2×Dic714yescomplemented
310C22×D4C22⋊D712yes—
311C22×D4C22⋊D711yescharacteristic
312C22×D4C22×D712yes—
313C23×C4—142no—
314C23×C4—141no—
315C23×C4C22×D712yes—
316C23×C4C22×D711yescharacteristic
317C2×M16—142no—
318C2×M16—142no—
319C2×M16—142no—
320C22×C8—72no—
321C22×C8—72no—
322C22×C8—142no—
323C22×C8—142no—
324C22×C8—142no—
325#127—28no—
326#124—28no—
327#121—28no—
328#121C2×Dic7116yescomplemented
329#120—216no—
330C4×Q8—148no—
331C4×D4—1416no—
332C4×D4—148no—
333#117—22no—
334#117—74no—
335#117—144no—
336#117—144no—
337#117—142no—
338#117C2×Dic714yescomplemented
339C2×C4⋊C4—74no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—144no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4C22×D712yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—142no—
352C2×C22⋊C4—141no—
353C2×C42—21no—
354C2×C42—142no—
355C2×C42—141no—
356#102—148no—
357#102—144no—
358#102—144no—
359C22⋊C8—148no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—28no—
365C22×C7—24no—
366C22×C7—44no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—48no—
370C22×C7—44no—
371C22×C7—816no—
372C22×C7—84no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7#38614yes—
376C22×C7C22×C22⋊C411yescharacteristic
377C22×C7C2×#12011yescharacteristic
378C22×C7C2×#12311yescharacteristic
379C22×C7#57612yes—
380C22×C7#57712yes—
381C4×C7—24no—
382C4×C7—44no—
383C4×C7#38614yes—
384C4×C7C2×C4×D411yescharacteristic
385C4×C7C2×C4×D411yescharacteristic
386C4×C7C2×#12111yescharacteristic
387C4×C7C2×#12111yescharacteristic
388Dic7—44no—
389Dic7—42no—
390Dic7—42no—
391C24—22no—
392C24—24no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24—82no—
397C24C2×C22⋊D712yes—
398C24C2×C22⋊D711yescharacteristic
399#41—28no—
400#41—28no—
401#41—28no—
402#41—28no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22⋊Dic714yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—216no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
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—44no—
422C2×D4—44no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4C22⋊Dic718yes—
426C2×D4C22⋊Dic714yes—
427C2×D4C22×Dic714yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—141no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—148no—
437C22×C4—144no—
438C22×C4—562no—
439C22×C4—564no—
440C22×C4D4×D712yes—
441C22×C4D4×D712yes—
442C22×C4#115912yes—
443C22×C4#115912yes—
444C22×C4C22×Dic714yes—
445C22×C4C22×Dic712yes—
446C22×C4C2×C22⋊D712yes—
447C22×C4C2×C22⋊D714yes—
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C2×C22⋊D712yes—
450C22×C4C23×D711yescharacteristic
451M16—284no—
452C2×C8—142no—
453C2×C8—142no—
454C2×C8—142no—
455C2×C8—142no—
456C2×C8—142no—
457C2×C8—142no—
458C2×C8—284no—
459C4⋊C4—22no—
460C4⋊C4—74no—
461C4⋊C4—72no—
462C4⋊C4—72no—
463C4⋊C4—148no—
464C4⋊C4—144no—
465C4⋊C4—141no—
466C4⋊C4—142no—
467C4⋊C4—141no—
468C4⋊C4C22×Dic714yes—
469C22⋊C4—24no—
470C22⋊C4—216no—
471C22⋊C4—148no—
472C22⋊C4—144no—
473C42—24no—
474C42—148no—
475C42—144no—
476C2×C7—24no—
477C2×C7—44no—
478C2×C7—84no—
479C2×C7C2×#38611yescharacteristic
480C2×C7#289712yes—
481C2×C7#300811yescharacteristic
482C2×C7#301011yescharacteristic
483C2×C7#309612yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—48no—
494C23—44no—
495C23—44no—
496C23—88no—
497C23—88no—
498C23—84no—
499C23—84no—
500C23#649812yes—
501C23#650012yes—
502C23C2×C22⋊Dic714yes—
503C23C2×C22⋊Dic712yes—
504C23C22×C22⋊D711yescharacteristic
505Q8—28no—
506D4—216no—
507D4—24no—
508D4—24no—
509D4—28no—
510D4—42no—
511D4—42no—
512D4—48no—
513D4—816no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—142no—
524C2×C4—144no—
525C2×C4—142no—
526C2×C4—284no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—564no—
530C2×C4—568no—
531C2×C4D4×Dic714yes—
532C2×C4D4×Dic714yes—
533C2×C4#649812yes—
534C2×C4#650012yes—
535C2×C4#650214yes—
536C2×C4#650214yes—
537C2×C4C2×C22⋊Dic714yes—
538C2×C4C2×C22⋊Dic712yes—
539C2×C4C2×D4×D711yescharacteristic
540C2×C4C2×D4×D711yescharacteristic
541C2×C4C2×#115911yescharacteristic
542C2×C4C2×#115911yescharacteristic
543C2×C4C23×Dic711yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C8—282no—
546C8—282no—
547C7#2091411yescharacteristic, complemented, Sylow
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—28no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—48no—
559C22—816no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22#8999214yes—
564C22C2×#649811yescharacteristic
565C22C2×#650011yescharacteristic
566C22#9052612yes—
567C22#9052712yes—
568C22C22×C22⋊Dic711yescharacteristic
569C4—24no—
570C4—44no—
571C4—284no—
572C4—282no—
573C4—282no—
574C4#8999214yes—
575C4C2×D4×Dic711yescharacteristic
576C4C2×D4×Dic711yescharacteristic
577C4C2×#650211yescharacteristic
578C4C2×#650211yescharacteristic
579C2—24no—
580C2—44no—
581C2—84no—
582C2#1174082112yes—
583C2#1174082811yescharacteristic
584C2#1174083511yescharacteristic
585C2C2×#8999211yescharacteristic
586C2#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#20914
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. C2×#90518C2semidirect
3. C2×#90006C2semidirect
4. C2×#89987C2semidirect
5. C22×#6406C2semidirect
6. #11729432C2semidirect
7. #11729431C2semidirect
8. #11729335C2semidirect
9. #11729334C2semidirect
13. C7×#535C4semidirect
25. C2×#6489C22semidirect
27. C22×C4⋊Dic7C22semidirect
36. C2×C22⋊(C7⋊C8)C22semidirect
37. #89324C22semidirect
38. #89324C22semidirect
39. #89324C22semidirect
56. C22×C7×D4C2×C4semidirect
67. C7×#121C2×C4semidirect
70. C7×#117C2×C4semidirect
83. #6489D4semidirect
89. C2×C4⋊Dic7D4semidirect
206. #535Dic7semidirect
309. C22×D4C2×Dic7semidirect
328. #121C2×Dic7semidirect
338. #117C2×Dic7semidirect
547. C7#20914semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5352
C2×#905182
C2×#900062
C2×#899872
C22×#64062
#117294322
#117294312
#117293352
#117293342
#209147