Finite Group Databasebeta

← #13364960

Subgroups — #13364960

Order 1792.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#89818C211yesmaximal, characteristic
3C2×#89711C211yesmaximal, characteristic
4#11729446C211yesmaximal, characteristic, complemented
5#11729387C212yesmaximal, complemented
6C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
7#11727430C212yesmaximal, complemented
8#11727369C212yesmaximal, complemented
9#11727275C211yesmaximal, characteristic
10#11727259C211yesmaximal, characteristic, complemented
11#11727217C212yesmaximal, complemented
12C23×D28C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic, complemented
15C22×C42×C7C2211yescharacteristic
16C22×#1143C2212yes—
17C22×(C2×D7)⋊C4—21no—
18C22×(C2×D7)⋊C4—21no—
19C22×C4⋊Dic7C2211yescharacteristic
20C7×#391C2211yescharacteristic, complemented
21C7×#389C2211yescharacteristic, complemented
22C7×#387—21no—
23C7×#385—21no—
24C7×#380C2211yescharacteristic
25C4×C7×C4⋊C4—21no—
26C4×C7×C4⋊C4C2211yescharacteristic, complemented
27C4×C7×C22⋊C4—21no—
28C2×#6409C2212yescomplemented
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31#89818C418yescomplemented
32C2×#6383C2212yes—
33C2×#6383C2212yes—
34C2×#6383C2212yes—
35C2×#6382C2212yescomplemented
36#89711—21no—
37#89711—21no—
38#89711C2212yes—
39#89711C2212yes—
40C2×C4×C7⋊C8C2212yescomplemented
41#89324C2212yescomplemented
42#89324C2212yescomplemented
43#89319C2212yescomplemented
44#22043—71nomaximal, Sylow
45C24×D7—21no—
46C22×C7×Q8C2311yescharacteristic
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×Dic7—21no—
51C22×D28—21no—
52C22×D28—22no—
53C22×D28—42no—
54C22×D28—42no—
55C22×D28C2×C414yes—
56C22×D28C2312yes—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8C2×C414yes—
59C4×C7×Q8—42no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C42×C7—21no—
69C2×C42×C7—21no—
70C2×C42×C7—42no—
71C2×C42×C7D411yescharacteristic
72C2×C42×C7D411yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×#1143—42no—
76C2×#1143—42no—
77C2×#1143C2×C418yescomplemented
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—22no—
81C2×(C2×D7)⋊C4—22no—
82C2×(C2×D7)⋊C4—44no—
83C2×(C2×D7)⋊C4—44no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—42no—
86C2×C4⋊Dic7C2312yes—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2312yes—
89C7×#95—21no—
90C7×#95—21no—
91C7×#95—21no—
92C7×#95—21no—
93C7×#95—21no—
94#6409—42no—
95C22⋊(C7⋊C8)—22no—
96C22⋊(C7⋊C8)C2×C414yes—
97#6383—42no—
98#6383—42no—
99#6383C2×C418yescomplemented
100#6382—42no—
101C4×C7⋊C8—42no—
102C2×#450—71no—
103C4×C22⋊Q8D711yescharacteristic, complemented
104C2×#390—71no—
105#1968—72no—
106#1934—72no—
107#1926—72no—
108#1880—72no—
109#1872—71no—
110#1853—71no—
111#1786—71no—
112C4×C22⋊C8—71no—
113C24×C7C2×D411yescharacteristic
114C23×D7—24no—
115C23×D7—22no—
116C23×D7—42no—
117C23×D7—42no—
118C23×D7—42no—
119C23×D7—42no—
120C23×D7—42no—
121C23×D7—42no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22×C412yes—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—41no—
135C22×C4×C7—41no—
136C22×C4×C7—41no—
137C22×C4×C7SD1614yescomplemented
138C22×C4×C7C22×C412yes—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7#4112yes—
145C22×C4×C7C2411yescharacteristic
146C22×Dic7—22no—
147C22×Dic7—42no—
148C22×Dic7—42no—
149C22×Dic7—42no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—22no—
153C2×D28—22no—
154C2×D28—24no—
155C2×D28—22no—
156C2×D28—48no—
157C2×D28—42no—
158C2×D28—42no—
159C2×D28—48no—
160C2×D28C22×C414yes—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—42no—
166C7×C4⋊C4—42no—
167C7×C4⋊C4C22×C412yes—
168C7×C22⋊C4—24no—
169C7×C22⋊C4—42no—
170C42×C7—22no—
171C42×C7—22no—
172C42×C7—41no—
173C42×C7—41no—
174C42×C7—42no—
175C42×C7—42no—
176#1143—48no—
177(C2×D7)⋊C4—44no—
178(C2×D7)⋊C4—44no—
179(C2×D7)⋊C4—44no—
180(C2×D7)⋊C4—44no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7—42no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8C22×C414yes—
190C23×D4—71no—
191C22×SD16—72no—
192C2×C22⋊Q8C2×D711yescharacteristic
193C2×C4×Q8C2×D711yescharacteristic, complemented
194C22×C4⋊C4—71no—
195C22×C22⋊C4—141no—
196C22×C22⋊C4—141no—
197C22×C42C2×D711yescharacteristic
198#450—78no—
199C2×#106—72no—
200C2×#103—72no—
201C2×#102—72no—
202C2×#102—72no—
203C2×#102—72no—
204C2×C22⋊C8—71no—
205C2×C22⋊C8—71no—
206C2×C4×C8—72no—
207#391C2×D711yescharacteristic, complemented
208#390—72no—
209#390—72no—
210#390—141no—
211#390—141no—
212#389C2×D711yescharacteristic, complemented
213#387—21no—
214#385—21no—
215#380C2×D711yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2×D711yescharacteristic, complemented
218C4×C22⋊C4—21no—
219#340—72no—
220#340—72no—
221#336—72no—
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7C4×D412yes—
227C23×C7C2×SD1612yes—
228C23×C7#13512yes—
229C23×C7C22×D411yescharacteristic
230C22×D7—24no—
231C22×D7—48no—
232C22×D7—48no—
233C22×D7—42no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C22×D7—48no—
240C22×D7—84no—
241C22×D7—84no—
242C22×D7—84no—
243C22×D7—84no—
244C7×Q8—24no—
245C7×Q8—42no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—44no—
262C2×C4×C7—44no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7C4×D412yes—
274C2×C4×C7C2×SD1612yes—
275C2×C4×C7C2×SD1612yes—
276C2×C4×C7AGL(1,ℤ/8)12yes—
277C2×C4×C7C23×C411yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7H(5,𝔽2)12yes—
280C2×C4×C7H(5,𝔽2)12yes—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—84no—
285D28—24no—
286D28—24no—
287D28—48no—
288D28—42no—
289D28—42no—
290C7⋊C8—24no—
291C7⋊C8—42no—
292C25—141no—
293C22×Q8C22×D711yescharacteristic
294C22×D4—74no—
295C22×D4—72no—
296C22×D4—142no—
297C22×D4—141no—
298C22×D4—282no—
299C22×D4—282no—
300C23×C4—141no—
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22×D711yescharacteristic
304C2×SD16—78no—
305C2×SD16—282no—
306C2×SD16—282no—
307C22×C8—72no—
308C22×C8—72no—
309C22⋊Q8—22no—
310C22⋊Q8C4×D714yes—
311C4×Q8—42no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—72no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—282no—
318C2×C4⋊C4C22×D711yescharacteristic
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C4⋊C4C22×D711yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—284no—
328C2×C22⋊C4—284no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—42no—
332C2×C42C22⋊D711yescharacteristic
333C2×C42C22⋊D711yescharacteristic
334C2×C42C22×D711yescharacteristic
335C2×C42C22×D711yescharacteristic
336#106—282no—
337#103—282no—
338#102—78no—
339#102—282no—
340#102—282no—
341C22⋊C8—74no—
342C22⋊C8—142no—
343C4×C8—282no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7C4×SD1614yes—
356C22×C7C2×C4×D411yescharacteristic
357C22×C7#53411yescharacteristic
358C22×C7#53511yescharacteristic
359C22×C7C22×SD1611yescharacteristic
360C22×C7C2×#13511yescharacteristic
361C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
362C22×C7#57511yescharacteristic
363C2×D7—44no—
364C2×D7—44no—
365C2×D7—44no—
366C2×D7—816no—
367C2×D7—82no—
368C2×D7—82no—
369C2×D7—82no—
370C2×D7—82no—
371C4×C7—22no—
372C4×C7—22no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—42no—
376C4×C7—41no—
377C4×C7—42no—
378C4×C7—81no—
379C4×C7—81no—
380C4×C7#51812yes—
381C4×C7#53812yes—
382Dic7—82no—
383C24—144no—
384C24—142no—
385C24—282no—
386C24—282no—
387C24—282no—
388C24—282no—
389C24—282no—
390C24—282no—
391C24C2×C22⋊D711yescharacteristic
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C2×C4×D712yes—
396C2×D4—74no—
397C2×D4—144no—
398C2×D4—142no—
399C2×D4—142no—
400C2×D4—142no—
401C2×D4—142no—
402C2×D4—142no—
403C2×D4—288no—
404C2×D4—282no—
405C2×D4—282no—
406C2×D4—288no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—142no—
419C22×C4—282no—
420C22×C4—282no—
421C22×C4—282no—
422C22×C4#114314yescomplemented
423C22×C4C2×C4×D712yes—
424C22×C4#115712yes—
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431SD16—288no—
432C2×C8—74no—
433C2×C8—142no—
434C2×C8—142no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—142no—
445C4⋊C4—142no—
446C4⋊C4—282no—
447C4⋊C4C2×C4×D712yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—284no—
451C22⋊C4—284no—
452C22⋊C4—284no—
453C22⋊C4—284no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—42no—
460C2×C7—24no—
461C2×C7#232111yescharacteristic
462C2×C7C2×C4×SD1611yescharacteristic
463C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
464C2×C7#318711yescharacteristic
465C2×C7#318811yescharacteristic
466C2×C7#320811yescharacteristic
467C2×C7#321111yescharacteristic
468D7—84no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—144no—
474C23—288no—
475C23—288no—
476C23—282no—
477C23—282no—
478C23—282no—
479C23—282no—
480C23—282no—
481C23—282no—
482C23—288no—
483C23—564no—
484C23—564no—
485C23—564no—
486C23—564no—
487C23C4×C22⋊D712yes—
488C23C2×#114312yes—
489C23#651312yes—
490C23C22×C22⋊D711yescharacteristic
491Q8—24no—
492Q8—42no—
493D4—144no—
494D4—144no—
495D4—288no—
496D4—282no—
497D4—282no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—564no—
529C2×C4C4×C22⋊D712yes—
530C2×C4C2×#114312yes—
531C2×C4C2×#114312yes—
532C2×C4#651212yes—
533C2×C4C22×C4×D711yescharacteristic
534C2×C4C2×#115711yescharacteristic
535C2×C4#654812yes—
536C2×C4#655312yes—
537C8—144no—
538C8—282no—
539C7#2204311yescharacteristic, complemented, Sylow
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—284no—
547C22—284no—
548C22—284no—
549C22—5616no—
550C22—562no—
551C22—562no—
552C22—562no—
553C22—562no—
554C22C4×#114314yes—
555C22C2×C4×C22⋊D711yescharacteristic
556C22#9049511yescharacteristic
557C22C22×#114311yescharacteristic
558C22C2×#651211yescharacteristic
559C22C2×#651311yescharacteristic
560C22#9051311yescharacteristic
561C22#9052111yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—562no—
572C4#9023612yes—
573C4#9025112yes—
574C2—24no—
575C2—564no—
576C2#1173984011yescharacteristic
577C2C2×C4×#114311yescharacteristic
578C2C4×#651211yescharacteristic
579C2#1174018111yescharacteristic
580C2#1174037411yescharacteristic
581C2#1174058911yescharacteristic
582C2#1174080011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×C22⋊Q8C2
socleC23×C7C22×D4
prime radicalC23×D28C22
cosocleC22×C4×C7C24
hypercenterC4×C22⋊Q8D7
hypocenterC7#22043
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C4×C7×C22⋊Q8C2semidirect
4. #11729446C2semidirect
5. #11729387C2semidirect
6. C4×C22⋊(C7⋊C8)C2semidirect
7. #11727430C2semidirect
8. #11727369C2semidirect
10. #11727259C2semidirect
11. #11727217C2semidirect
14. C2×C4×C7×Q8C22semidirect
20. C7×#391C22semidirect
21. C7×#389C22semidirect
26. C4×C7×C4⋊C4C22semidirect
28. C2×#6409C22semidirect
31. #89818C4semidirect
35. C2×#6382C22semidirect
40. C2×C4×C7⋊C8C22semidirect
41. #89324C22semidirect
42. #89324C22semidirect
43. #89319C22semidirect
77. C2×#1143C2×C4semidirect
99. #6383C2×C4semidirect
103. C4×C22⋊Q8D7semidirect
137. C22×C4×C7SD16semidirect
193. C2×C4×Q8C2×D7semidirect
207. #391C2×D7semidirect
212. #389C2×D7semidirect
217. C4×C4⋊C4C2×D7semidirect
422. C22×C4#1143semidirect
539. C7#22043semidirect

Maximal subgroups

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

subgroupindex
C4×C7×C22⋊Q82
C2×#898182
C2×#897112
#117294462
#117293872
C4×C22⋊(C7⋊C8)2
#117274302
#117273692
#117272752
#117272592
#117272172
#220437