Finite Group Databasebeta

← #13355487

Subgroups — #13355487

Order 1792.

1914 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90519C211yesmaximal, characteristic
2C23×#1131C211yesmaximal, characteristic, complemented
3C2×C7×#416C211yesmaximal, characteristic, complemented
4C2×#90005C211yesmaximal, characteristic, complemented
5#11739712C211yesmaximal, characteristic, complemented
6#11729723C211yesmaximal, characteristic, complemented
7#11729334C211yesmaximal, characteristic, complemented
8#11728025C214yesmaximal, complemented
9#11727957C214yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C4×D7C2211yescharacteristic
12C23×C7×C8C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14#90519—24no—
15#90519C418yescomplemented
16C2×#6509C2212yes—
17C2×#6501C2212yes—
18C2×#6500—21no—
19C2×#6498—21no—
20C2×C4×C22⋊D7—22no—
21C2×#6489C2211yescharacteristic
22C2×#6489C2211yescharacteristic
23C23×C7⋊C8C2211yescharacteristic
24C22×#1131—24no—
25C22×#1131—24no—
26C22×#1131—24no—
27C22×#1131C2214yes—
28C7×#416—24no—
29C2×C7×#103C2212yescomplemented
30C2×C7×#102C2212yescomplemented
31#90005—24no—
32C2×#6409C2212yescomplemented
33C2×#6406C2212yescomplemented
34#89759C2214yes—
35#89727—21no—
36#89715—21no—
37C4×(C2×D7)⋊C4—22no—
38#89346C2214yescomplemented
39#89324C2214yescomplemented
40#20970—71nomaximal, Sylow
41C24×D7D411yescharacteristic
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41C2×C414yes—
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—22no—
54C23×Dic7D411yescharacteristic
55C22×C4×D7—22no—
56C22×C4×D7—22no—
57C22×C4×D7—22no—
58C22×C4×D7—22no—
59C22×C4×D7C2×C414yes—
60C22×C4×D7D412yes—
61C22×C4×D7D414yes—
62C22×C4×D7C2312yes—
63C22×C7×C8—22no—
64C22×C7×C8—22no—
65C22×C7×C8D414yescomplemented
66C22×C7×C8C2312yes—
67C7×#117—22no—
68C7×#117—22no—
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—21no—
71C2×C42×C7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—21no—
75#6509—24no—
76#6509C2×C418yes—
77#6501—24no—
78#6501C2×C418yes—
79#6500—28no—
80#6498—28no—
81C4×C22⋊D7—216no—
82C2×(C2×D7)⋊C4—22no—
83C2×(C2×D7)⋊C4—24no—
84#6489—22no—
85#6489—22no—
86#6489—22no—
87#6489C2×C414yes—
88C2×C4⋊Dic7C2312yes—
89C2×C4⋊Dic7C2312yes—
90C2×Dic7⋊C4—22no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C22×C7⋊C8—22no—
94C22×C7⋊C8—22no—
95C22×C7⋊C8D414yescomplemented
96C22×C7⋊C8C2312yes—
97C2×#1131—232no—
98C2×#1131—28no—
99C2×#1131—28no—
100C2×#1131—28no—
101C7×#103—24no—
102C7×#102—24no—
103#6409—24no—
104#6406—24no—
105C7⋊#95—24no—
106C23×M16—71no—
107C2×#525—71no—
108C2×#416—71no—
109C2×#416D711yescharacteristic, complemented
110#2280—71no—
111#1895—74no—
112#1894—74no—
113#1735—71no—
114#1735—71no—
115C24×C7—21no—
116C24×C7C2×D411yescharacteristic
117C23×D7—22no—
118C23×D7—22no—
119C23×D7—44no—
120C23×D7C22⋊C414yes—
121C23×D7C2×D412yes—
122C7×#41—28no—
123C7×#41—28no—
124C7×#41—44no—
125C7×#41—44no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8C22×C412yes—
132C2×C7×D4—24no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4C22×C412yes—
141C22×C4×C7—21no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7C22×C412yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7#4112yes—
152C22×C4×C7#4112yes—
153C22×C4×C7C2411yescharacteristic
154C2×C22⋊D7—28no—
155C2×C22⋊D7—48no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7C22⋊C414yes—
161C22×Dic7C2×D412yes—
162C2×C4×D7—28no—
163C2×C4×D7—28no—
164C2×C4×D7—28no—
165C2×C4×D7—24no—
166C2×C4×D7—24no—
167C2×C4×D7—28no—
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7—22no—
171C2×C4×D7—22no—
172C2×C4×D7C22⋊C418yes—
173C2×C4×D7C22×C414yes—
174C2×C4×D7C2×D414yes—
175C2×C7×C8—28no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8C2×D414yes—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C22⋊C4—42no—
182C42×C7—42no—
183C22⋊Dic7—24no—
184C22⋊Dic7—28no—
185C22⋊Dic7—42no—
186(C2×D7)⋊C4—28no—
187(C2×D7)⋊C4—48no—
188C4⋊Dic7—22no—
189C4⋊Dic7—22no—
190C4⋊Dic7—22no—
191C4⋊Dic7C22×C414yes—
192Dic7⋊C4—28no—
193C4×Dic7—24no—
194C4×Dic7—42no—
195C2×C7⋊C8—28no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8C2×D414yes—
199#1131—232no—
200C22×#41C2×D711yescharacteristic
201C24×C4—71no—
202C22×M16—74no—
203C22×M16—144no—
204C22×M16—144no—
205C22×M16—144no—
206C23×C8—71no—
207C23×C8C2×D711yescharacteristic
208#525—78no—
209#525—144no—
210C2×#123—141no—
211C2×C22⋊Q8—72no—
212C2×#121—72no—
213C2×#120—141no—
214C2×C4×D4—142no—
215C2×#117—71no—
216C2×#117—71no—
217C2×#117C2×D711yescharacteristic
218#416—24no—
219#416—144no—
220C2×#103—72no—
221C2×#103C2×D712yescomplemented
222C2×#102—72no—
223C2×#102C2×D712yescomplemented
224#381—141no—
225#380—74no—
226#379—141no—
227C4×C22⋊C4—142no—
228#340—74no—
229#340—74no—
230C23×C7—24no—
231C23×C7—22no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—44no—
235C23×C7C2×C22⋊C412yes—
236C23×C7#12112yes—
237C23×C7#12112yes—
238C23×C7C22×D411yescharacteristic
239C22×D7—22no—
240C22×D7—22no—
241C22×D7—24no—
242C22×D7—24no—
243C22×D7—28no—
244C22×D7—48no—
245C22×D7—48no—
246C22×D7—48no—
247C22×D7C2×C22⋊C414yes—
248C7×Q8—22no—
249C7×Q8—22no—
250C7×Q8—41no—
251C7×Q8—41no—
252C7×D4—22no—
253C7×D4—22no—
254C7×D4—48no—
255C7×D4—41no—
256C7×D4—41no—
257C2×C4×C7—22no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—48no—
261C2×C4×C7—42no—
262C2×C4×C7—44no—
263C2×C4×C7C2×C22⋊C412yes—
264C2×C4×C7C4×D414yes—
265C2×C4×C7C4×D414yes—
266C2×C4×C7#12014yes—
267C2×C4×C7#12112yes—
268C2×C4×C7#12112yes—
269C2×C4×C7#12314yes—
270C2×C4×C7C23×C411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C22⋊D7—416no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—28no—
280C2×Dic7—24no—
281C2×Dic7—44no—
282C2×Dic7—42no—
283C2×Dic7—44no—
284C2×Dic7C2×C22⋊C414yes—
285C4×D7—24no—
286C4×D7—24no—
287C4×D7—24no—
288C4×D7—24no—
289C4×D7—216no—
290C7×C8—24no—
291C7⋊C8—24no—
292C25—71no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41C4×D714yes—
299C22×Q8C22×D711yescharacteristic
300C22×D4—21no—
301C22×D4—142no—
302C22×D4C22×D711yescharacteristic
303C23×C4—21no—
304C23×C4—71no—
305C23×C4—74no—
306C23×C4—74no—
307C23×C4—72no—
308C23×C4—72no—
309C23×C4—142no—
310C23×C4—142no—
311C23×C4—142no—
312C23×C4—142no—
313C23×C4C22×D711yescharacteristic
314C2×M16—148no—
315C2×M16—148no—
316C2×M16—1432no—
317C2×M16—148no—
318C22×C8—22no—
319C22×C8—22no—
320C22×C8—74no—
321C22×C8—72no—
322C22×C8—142no—
323C22×C8—142no—
324C22×C8C22⋊D714yescomplemented
325C22×C8C22×D712yes—
326#123—148no—
327C22⋊Q8—78no—
328C22⋊Q8—144no—
329#121—78no—
330#121—144no—
331#120—148no—
332C4×D4—1416no—
333#117—22no—
334#117—22no—
335#117—74no—
336#117—142no—
337#117—142no—
338#117—142no—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4C22×D712yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—142no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—141no—
347C2×C22⋊C4—144no—
348C2×C22⋊C4—141no—
349C2×C42—21no—
350C2×C42—141no—
351C2×C42—141no—
352#103—24no—
353#103—144no—
354#102—24no—
355#102—144no—
356#95—144no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—44no—
361C22×C7—42no—
362C22×C7—82no—
363C22×C7—82no—
364C22×C7#38614yes—
365C22×C7C22×C22⋊C411yescharacteristic
366C22×C7C2×#12111yescharacteristic
367C22×C7C2×#12111yescharacteristic
368C22×C7#57514yes—
369C2×D7—24no—
370C2×D7—24no—
371C2×D7—24no—
372C2×D7—416no—
373C2×D7—44no—
374C2×D7—44no—
375C4×C7—42no—
376C4×C7—82no—
377C4×C7#38614yes—
378C4×C7C2×C4×D411yescharacteristic
379C4×C7C2×C4×D411yescharacteristic
380C4×C7C2×#12011yescharacteristic
381C4×C7C2×#12311yescharacteristic
382Dic7—24no—
383Dic7—44no—
384Dic7—82no—
385C24—21no—
386C24—72no—
387C24—74no—
388C24—142no—
389C24—142no—
390C24—284no—
391C24D4×D711yescharacteristic
392#41—28no—
393#41—28no—
394#41—44no—
395#41—44no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C2×C4×D712yes—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—148no—
411C2×D4—288no—
412C2×D4C2×C4×D712yes—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—74no—
419C22×C4—72no—
420C22×C4—78no—
421C22×C4—74no—
422C22×C4—74no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—148no—
426C22×C4—148no—
427C22×C4—144no—
428C22×C4—148no—
429C22×C4—148no—
430C22×C4—144no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4C2×C4×D712yes—
438C22×C4#115712yes—
439C22×C4D4×D712yes—
440C22×C4D4×D712yes—
441C22×C4D4×D711yescharacteristic
442C22×C4#115912yes—
443C22×C4C2×C22⋊D712yes—
444C22×C4C23×D711yescharacteristic
445M16—1432no—
446C2×C8—28no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—74no—
450C2×C8—148no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8C2×C22⋊D714yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—74no—
457C4⋊C4—148no—
458C4⋊C4—142no—
459C4⋊C4—142no—
460C4⋊C4—142no—
461C22⋊C4—42no—
462C22⋊C4—148no—
463C22⋊C4—148no—
464C22⋊C4—144no—
465C22⋊C4—288no—
466C22⋊C4—282no—
467C42—42no—
468C42—144no—
469C42—282no—
470C2×C7—24no—
471C2×C7—82no—
472C2×C7C2×#38611yescharacteristic
473C2×C7#289612yes—
474C2×C7#305412yes—
475C2×C7#305412yes—
476D7—44no—
477C23—24no—
478C23—22no—
479C23—41no—
480C23—44no—
481C23—41no—
482C23—74no—
483C23—142no—
484C23—144no—
485C23—144no—
486C23—142no—
487C23—148no—
488C23—288no—
489C23—288no—
490C23—288no—
491C23D7×C22⋊C412yes—
492C23#644712yes—
493C23#650112yes—
494C23C2×D4×D711yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499D4—22no—
500D4—22no—
501D4—48no—
502D4—41no—
503D4—41no—
504D4—2816no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—74no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—148no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—144no—
518C2×C4—144no—
519C2×C4—1416no—
520C2×C4—144no—
521C2×C4—284no—
522C2×C4—282no—
523C2×C4—284no—
524C2×C4D7×C22⋊C412yes—
525C2×C4(C22⋊D7)⋊C414yes—
526C2×C4#644614yes—
527C2×C4#644712yes—
528C2×C4C4×C22⋊D714yes—
529C2×C4#650014yes—
530C2×C4#650112yes—
531C2×C4C22×C4×D711yescharacteristic
532C2×C4C2×#115711yescharacteristic
533C2×C4C2×D4×D711yescharacteristic
534C2×C4C2×D4×D711yescharacteristic
535C2×C4C2×#115911yescharacteristic
536C2×C4C22×C22⋊D711yescharacteristic
537C8—24no—
538C8—144no—
539C7#2097011yescharacteristic, complemented, Sylow
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—144no—
548C22—144no—
549C22—144no—
550C22—2816no—
551C22—284no—
552C22—284no—
553C22#8972814yes—
554C22C2×D7×C22⋊C411yescharacteristic
555C22C2×#644711yescharacteristic
556C22#9045914yes—
557C22C2×#650111yescharacteristic
558C4—42no—
559C4—82no—
560C4—144no—
561C4—284no—
562C4—562no—
563C4#8972814yes—
564C4C2×(C22⋊D7)⋊C411yescharacteristic
565C4C2×#644611yescharacteristic
566C4C2×C4×C22⋊D711yescharacteristic
567C4C2×#650011yescharacteristic
568C2—24no—
569C2—82no—
570C2—284no—
571C2#1173627512yes—
572C2#1173628412yes—
573C2C2×#8972811yescharacteristic
574C2#1174136012yes—
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#416C2
socleC23×C7C22×D4
prime radicalC2×#90519C2
cosocleC22×C4×C7C24
hypercenterC2×#416D7
hypocenterC7#20970
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C23×#1131C2semidirect
3. C2×C7×#416C2semidirect
4. C2×#90005C2semidirect
5. #11739712C2semidirect
6. #11729723C2semidirect
7. #11729334C2semidirect
8. #11728025C2semidirect
9. #11727957C2semidirect
15. #90519C4semidirect
29. C2×C7×#103C22semidirect
30. C2×C7×#102C22semidirect
32. C2×#6409C22semidirect
33. C2×#6406C22semidirect
38. #89346C22semidirect
39. #89324C22semidirect
65. C22×C7×C8D4semidirect
95. C22×C7⋊C8D4semidirect
109. C2×#416D7semidirect
221. C2×#103C2×D7semidirect
223. C2×#102C2×D7semidirect
324. C22×C8C22⋊D7semidirect
539. C7#20970semidirect

Maximal subgroups

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

subgroupindex
C2×#905192
C23×#11312
C2×C7×#4162
C2×#900052
#117397122
#117297232
#117293342
#117280252
#117279572
#209707