Finite Group Databasebeta

← #13381286

Subgroups — #13381286

Order 1792.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6501C211yesmaximal, characteristic
2C2×C7×#418C211yesmaximal, characteristic, complemented
3C2×#89987C211yesmaximal, characteristic, complemented
4C2×#89898C211yesmaximal, characteristic, complemented
5#11739712C211yesmaximal, characteristic, complemented
6C2×#89767C211yesmaximal, characteristic, complemented
7#11729739C211yesmaximal, characteristic, complemented
8#11727961C214yesmaximal, complemented
9#11727960C214yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C23×C22⋊D7—21no—
12C23×C4×D7C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C22×C22⋊Dic7—21no—
16C2×#6501—24no—
17C2×#6501—22no—
18C2×#6501—24no—
19C2×#6501C2214yes—
20C2×#6489C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23C7×#418—24no—
24C2×C7×#102C2212yescomplemented
25C2×C7×#102C2212yescomplemented
26#89987—24no—
27C2×#6406C2214yescomplemented
28#89898—24no—
29C2×(C2×D7)⋊C8C2214yescomplemented
30#89767—24no—
31#89759C2212yescomplemented
32#89759C2212yescomplemented
33C2×#6382C2212yescomplemented
34C2×#6381C2212yescomplemented
35#89727—21no—
36#89715—21no—
37C4×(C2×D7)⋊C4—22no—
38#89346C2214yescomplemented
39#23480—71nomaximal, Sylow
40C25×C7—21no—
41C24×D7D411yescharacteristic
42C22×C7×D4—22no—
43C22×C7×D4—21no—
44C22×C7×D4—22no—
45C22×C7×D4—22no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4D414yes—
49C22×C7×D4C2312yes—
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—28no—
52C22×C22⋊D7—24no—
53C22×C22⋊D7—48no—
54C23×Dic7—21no—
55C23×Dic7D411yescharacteristic
56C22×C4×D7—22no—
57C22×C4×D7—22no—
58C22×C4×D7—22no—
59C22×C4×D7—22no—
60C22×C4×D7—22no—
61C22×C4×D7—22no—
62C22×C4×D7D412yes—
63C22×C4×D7C2312yes—
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C22×C7×C8C2312yes—
68C7×#117—22no—
69C7×#117—22no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C42×C7—21no—
74C2×C22⋊Dic7—28no—
75C2×C22⋊Dic7—24no—
76C2×C22⋊Dic7—21no—
77#6501—216no—
78#6501—28no—
79#6501—28no—
80C2×(C2×D7)⋊C4—22no—
81C2×(C2×D7)⋊C4—22no—
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7D414yes—
87C2×C4⋊Dic7C2312yes—
88C2×C4⋊Dic7C2312yes—
89C2×C4×Dic7—21no—
90C2×#1136—22no—
91C2×#1136—22no—
92C2×#1136D414yescomplemented
93C22×C7⋊C8C2312yes—
94C7×#102—24no—
95C7×#102—24no—
96#6406—28no—
97C7⋊#95—22no—
98C7⋊#95—22no—
99(C2×D7)⋊C8—28no—
100#6382—24no—
101#6381—24no—
102C22×#121—71no—
103C2×#428—71no—
104C2×#418—71no—
105C2×#418D711yescharacteristic, complemented
106C2×#407—71no—
107#2280—71no—
108#2034—74no—
109#2033—74no—
110#1736—71no—
111C24×C7—24no—
112C24×C7—22no—
113C24×C7—44no—
114C24×C7—44no—
115C24×C7—42no—
116C24×C7—42no—
117C24×C7C2×D411yescharacteristic
118C23×D7—22no—
119C23×D7—24no—
120C23×D7—22no—
121C23×D7—22no—
122C23×D7—42no—
123C23×D7C2×D412yes—
124C2×C7×D4—24no—
125C2×C7×D4—28no—
126C2×C7×D4—28no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—44no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4—44no—
136C2×C7×D4C2×D414yes—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7#4112yes—
144C22×C4×C7#4112yes—
145C22×C4×C7#4112yes—
146C22×C4×C7C2411yescharacteristic
147C2×C22⋊D7—216no—
148C2×C22⋊D7—432no—
149C2×C22⋊D7—48no—
150C2×C22⋊D7—48no—
151C22×Dic7—24no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—44no—
158C22×Dic7C2×D412yes—
159C2×C4×D7—24no—
160C2×C4×D7—24no—
161C2×C4×D7—24no—
162C2×C4×D7—22no—
163C2×C4×D7—22no—
164C2×C4×D7—22no—
165C2×C4×D7—22no—
166C2×C4×D7—22no—
167C2×C4×D7—22no—
168C2×C4×D7—44no—
169C2×C4×D7—44no—
170C2×C4×D7—44no—
171C2×C4×D7—44no—
172C7×M16—44no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C22⋊C4—42no—
181C42×C7—42no—
182C22⋊Dic7—216no—
183C22⋊Dic7—42no—
184(C2×D7)⋊C4—44no—
185(C2×D7)⋊C4—44no—
186C4⋊Dic7—24no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7—22no—
190C4⋊Dic7C2×D414yes—
191C4×Dic7—42no—
192#1136—28no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8C2×D414yescomplemented
196C23×D4—141no—
197C23×D4C2×D711yescharacteristic
198C24×C4—71no—
199C22×M16—71no—
200C22×M16C2×D711yescharacteristic
201C2×#121—74no—
202C2×#121—144no—
203C2×#121—142no—
204C2×#121—144no—
205C2×#117—71no—
206C2×#117C2×D711yescharacteristic
207C22×C4⋊C4—71no—
208C22×C22⋊C4—141no—
209#428—144no—
210C2×#107—72no—
211C2×#106—72no—
212#418—24no—
213#418—144no—
214C2×#102—74no—
215C2×#102C2×D712yescomplemented
216C2×#102C2×D712yescomplemented
217#407—144no—
218C2×C22⋊C8—74no—
219#381—141no—
220#380—72no—
221#380—72no—
222#379—141no—
223C4×C22⋊C4—142no—
224#340—74no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—24no—
228C23×C7—48no—
229C23×C7—48no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—44no—
233C23×C7—44no—
234C23×C7—48no—
235C23×C7—88no—
236C23×C7—88no—
237C23×C7#12012yes—
238C23×C7#12112yes—
239C23×C7#12312yes—
240C23×C7C22×D411yescharacteristic
241C22×D7—28no—
242C22×D7—22no—
243C22×D7—22no—
244C22×D7—22no—
245C22×D7—22no—
246C22×D7—22no—
247C22×D7—22no—
248C22×D7—48no—
249C22×D7—48no—
250C22×D7—44no—
251C22×D7—44no—
252C22×D7AGL(1,ℤ/8)14yes—
253C7×D4—24no—
254C7×D4—24no—
255C7×D4—42no—
256C7×D4—42no—
257C7×D4—48no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7#12012yes—
263C2×C4×C7#12114yes—
264C2×C4×C7#12112yes—
265C2×C4×C7#12114yes—
266C2×C4×C7#12312yes—
267C2×C4×C7#12314yes—
268C2×C4×C7#12414yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C22⋊D7—432no—
276C2×Dic7—24no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—48no—
281C2×Dic7—44no—
282C2×Dic7—44no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C4×D7—42no—
287C4×D7—42no—
288C4×D7—42no—
289C4×D7—42no—
290C4×D7—48no—
291C7×C8—42no—
292C7⋊C8—24no—
293C25—21no—
294C25—71no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4—22no—
298C22×D4—21no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—144no—
302C22×D4—148no—
303C22×D4—288no—
304C22×D4C22⋊D714yes—
305C22×D4C22×D712yes—
306C23×C4—71no—
307C23×C4—72no—
308C23×C4—72no—
309C23×C4—141no—
310C23×C4—142no—
311C23×C4—142no—
312C23×C4—142no—
313C23×C4—142no—
314C23×C4—142no—
315C23×C4—142no—
316C23×C4C22×D711yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C2×M16—74no—
321C2×M16—142no—
322C2×M16—142no—
323C22×C8—72no—
324C22×C8C22×D712yes—
325#121—148no—
326#121—1416no—
327#121—148no—
328#117—22no—
329#117—22no—
330#117—142no—
331#117—142no—
332C2×C4⋊C4—74no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—72no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—144no—
341C2×C22⋊C4—148no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4—141no—
345C2×C42—21no—
346C2×C42—141no—
347#107—144no—
348#106—144no—
349#102—24no—
350#102—24no—
351#102—148no—
352C22⋊C8—148no—
353#95—142no—
354#95—142no—
355C22×C7—24no—
356C22×C7—22no—
357C22×C7—22no—
358C22×C7—24no—
359C22×C7—44no—
360C22×C7—44no—
361C22×C7—44no—
362C22×C7—816no—
363C22×C7—84no—
364C22×C7—84no—
365C22×C7#39414yes—
366C22×C7C2×#12011yescharacteristic
367C22×C7C2×#12111yescharacteristic
368C22×C7C2×#12311yescharacteristic
369C22×C7C2×AGL(1,ℤ/8)12yes—
370C22×C7#57512yes—
371C2×D7—24no—
372C2×D7—24no—
373C2×D7—24no—
374C2×D7—416no—
375C2×D7—42no—
376C2×D7—42no—
377C2×D7—42no—
378C2×D7—42no—
379C4×C7—81no—
380C4×C7—81no—
381C4×C7#39414yes—
382C4×C7C2×#12111yescharacteristic
383C4×C7C2×#12111yescharacteristic
384C4×C7C2×#12311yescharacteristic
385C4×C7C2×#12411yescharacteristic
386Dic7—42no—
387Dic7—44no—
388Dic7—82no—
389C24—24no—
390C24—22no—
391C24—44no—
392C24—44no—
393C24—42no—
394C24—42no—
395C24—72no—
396C24—144no—
397C24—142no—
398C24—142no—
399C24—142no—
400C24—282no—
401C24D4×D711yescharacteristic
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—1416no—
415C2×D4—288no—
416C2×D4—2832no—
417C2×D4—288no—
418C2×D4C2×C22⋊D714yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—72no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—144no—
426C22×C4—142no—
427C22×C4—144no—
428C22×C4—144no—
429C22×C4—144no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—284no—
438C22×C4—284no—
439C22×C4—284no—
440C22×C4—284no—
441C22×C4—284no—
442C22×C4#115712yes—
443C22×C4#115712yes—
444C22×C4D4×D711yescharacteristic
445C22×C4D4×D712yes—
446C22×C4D4×D712yes—
447C22×C4#115912yes—
448C22×C4C2×C22⋊D712yes—
449C22×C4C23×D711yescharacteristic
450M16—44no—
451M16—148no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—74no—
456C2×C8—142no—
457C2×C8—142no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—74no—
463C4⋊C4—144no—
464C4⋊C4—142no—
465C4⋊C4—142no—
466C4⋊C4—142no—
467C22⋊C4—42no—
468C22⋊C4—1416no—
469C22⋊C4—284no—
470C22⋊C4—284no—
471C22⋊C4—282no—
472C42—42no—
473C42—282no—
474C2×C7—24no—
475C2×C7—84no—
476C2×C7C2×#39411yescharacteristic
477C2×C7#300311yescharacteristic
478C2×C7#300511yescharacteristic
479C2×C7#305512yes—
480C2×C7#309412yes—
481D7—44no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—48no—
486C23—48no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—74no—
493C23—88no—
494C23—88no—
495C23—148no—
496C23—142no—
497C23—142no—
498C23—142no—
499C23—142no—
500C23—142no—
501C23—142no—
502C23—288no—
503C23—288no—
504C23—284no—
505C23—284no—
506C23#644612yes—
507C23#644712yes—
508C23#650012yes—
509C23C2×D4×D711yescharacteristic
510D4—24no—
511D4—24no—
512D4—42no—
513D4—42no—
514D4—48no—
515D4—2832no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—144no—
524C2×C4—288no—
525C2×C4—284no—
526C2×C4—284no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—288no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4#644612yes—
536C2×C4#644714yes—
537C2×C4#644712yes—
538C2×C4#644814yes—
539C2×C4#649214yes—
540C2×C4#650012yes—
541C2×C4#650214yes—
542C2×C4C2×#115711yescharacteristic
543C2×C4C2×#115711yescharacteristic
544C2×C4C2×D4×D711yescharacteristic
545C2×C4C2×D4×D711yescharacteristic
546C2×C4C2×#115911yescharacteristic
547C2×C4C22×C22⋊D711yescharacteristic
548C8—42no—
549C8—144no—
550C7#2348011yescharacteristic, complemented, Sylow
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—816no—
559C22—84no—
560C22—84no—
561C22—144no—
562C22—144no—
563C22—144no—
564C22—2816no—
565C22—282no—
566C22—282no—
567C22—282no—
568C22—282no—
569C22#8973114yes—
570C22C2×#644611yescharacteristic
571C22C2×#644711yescharacteristic
572C22D7×AGL(1,ℤ/8)12yes—
573C22#9046412yes—
574C22C2×#650011yescharacteristic
575C4—81no—
576C4—81no—
577C4—282no—
578C4—284no—
579C4—562no—
580C4#8973114yes—
581C4C2×#644711yescharacteristic
582C4C2×#644811yescharacteristic
583C4C2×#649211yescharacteristic
584C4C2×#650211yescharacteristic
585C2—24no—
586C2—84no—
587C2—284no—
588C2#1173637512yes—
589C2#1173637812yes—
590C2C2×#8973111yescharacteristic
591C2#1174138911yescharacteristic
592C2#1174139011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#418C2
socleC23×C7C22×D4
prime radicalC22×#6501C2
hypercenterC2×#418D7
hypocenterC7#23480
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×C7×#418C2semidirect
3. C2×#89987C2semidirect
4. C2×#89898C2semidirect
5. #11739712C2semidirect
6. C2×#89767C2semidirect
7. #11729739C2semidirect
8. #11727961C2semidirect
9. #11727960C2semidirect
24. C2×C7×#102C22semidirect
25. C2×C7×#102C22semidirect
27. C2×#6406C22semidirect
29. C2×(C2×D7)⋊C8C22semidirect
31. #89759C22semidirect
32. #89759C22semidirect
33. C2×#6382C22semidirect
34. C2×#6381C22semidirect
38. #89346C22semidirect
92. C2×#1136D4semidirect
105. C2×#418D7semidirect
195. C2×C7⋊C8C2×D4semidirect
215. C2×#102C2×D7semidirect
216. C2×#102C2×D7semidirect
550. C7#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#65012
C2×C7×#4182
C2×#899872
C2×#898982
#117397122
C2×#897672
#117297392
#117279612
#117279602
#234807