Finite Group Databasebeta

← #13364969

Subgroups — #13364969

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
1Dic7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#89818C211yesmaximal, characteristic
3C2×#89760C211yesmaximal, characteristic
4C7×#1872C211yesmaximal, characteristic, complemented
5#11729310C211yesmaximal, characteristic, complemented
6#11728011C212yesmaximal, complemented
7#11727994C212yesmaximal, complemented
8#11727713C211yesmaximal, characteristic
9Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
10#11726607C212yesmaximal, complemented
11#11726546C212yesmaximal, complemented
12C23×D28C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×Q8×Dic7C2211yescharacteristic, complemented
16C22×#1143C2212yes—
17C22×(C2×D7)⋊C4—21no—
18C22×(C2×D7)⋊C4—21no—
19C22×C4×Dic7C2211yescharacteristic
20C2×C7×#103C2212yescomplemented
21C2×C7×C22⋊C8C2211yescharacteristic
22#90002C2211yescharacteristic, complemented
23#89984C2211yescharacteristic
24C2×C22⋊(C7⋊C8)C2211yescharacteristic
25C2×#6395C2212yes—
26C2×#6395C2212yes—
27C2×C8×Dic7C2212yescomplemented
28#89818C418yescomplemented
29#89760—21no—
30#89760—21no—
31#89760C2212yes—
32#89760C2212yes—
33#89757C2211yescharacteristic, complemented
34#89753—21no—
35C4⋊C4×Dic7—21no—
36C4⋊C4×Dic7C2211yescharacteristic, complemented
37C2×#6383C2212yes—
38C2×#6382C2212yescomplemented
39#89722—21no—
40C22⋊C4×Dic7—21no—
41C7×#340C2212yescomplemented
42#89345C2212yescomplemented
43#89324C2212yescomplemented
44#22043—71nomaximal, Sylow
45C24×D7—21no—
46C22×C7×Q8C2311yescharacteristic
47C23×C4×C7—21no—
48C23×C4×C7C2311yescharacteristic
49C23×Dic7D411yescharacteristic
50C23×Dic7D411yescharacteristic
51C22×D28—21no—
52C22×D28—22no—
53C22×D28—42no—
54C22×D28—42no—
55C22×D28C2×C414yes—
56C22×D28C2312yes—
57C22×C7×C8C2312yes—
58C7×C22⋊Q8—22no—
59C7×C22⋊Q8C2×C414yes—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—42no—
63C2×C7×C4⋊C4C2312yes—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C22⋊Dic7—21no—
67Q8×Dic7—42no—
68C2×#1143—42no—
69C2×#1143—42no—
70C2×#1143C2×C418yescomplemented
71C2×(C2×D7)⋊C4—22no—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4—22no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—44no—
76C2×(C2×D7)⋊C4—44no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—42no—
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C7×#103—42no—
90C7×C22⋊C8—22no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C22⋊(C7⋊C8)C2×C414yes—
97#6395—42no—
98#6395—42no—
99C8×Dic7—42no—
100#6383C2×C418yescomplemented
101#6382—42no—
102C2×#450—71no—
103C4×C22⋊Q8—71no—
104C2×#390—71no—
105#1968—72no—
106#1934—72no—
107#1926—72no—
108#1880—72no—
109#1872D711yescharacteristic, complemented
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—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—42no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7#4112yes—
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—41no—
143C22×Dic7—41no—
144C22×Dic7—41no—
145C22×Dic7SD1614yescomplemented
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
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—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—42no—
169C7×C4⋊C4C22×C412yes—
170C7×C22⋊C4—24no—
171C22⋊Dic7—42no—
172#1143—48no—
173(C2×D7)⋊C4—44no—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—44no—
176(C2×D7)⋊C4—44no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—42no—
180C4⋊Dic7—42no—
181C4×Dic7—22no—
182C4×Dic7—22no—
183C4×Dic7—41no—
184C4×Dic7—41no—
185C4×Dic7—42no—
186C4×Dic7—42no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8C22×C414yes—
190C23×D4—71no—
191C22×SD16—72no—
192C2×C22⋊Q8C2×D711yescharacteristic
193C2×C4×Q8—71no—
194C22×C4⋊C4C2×D711yescharacteristic
195C22×C22⋊C4—141no—
196C22×C22⋊C4—141no—
197C22×C42—71no—
198#450—78no—
199C2×#106—72no—
200C2×#103C2×D712yescomplemented
201C2×#102—72no—
202C2×#102—72no—
203C2×#102—72no—
204C2×C22⋊C8—71no—
205C2×C22⋊C8C2×D711yescharacteristic
206C2×C4×C8—72no—
207#391—71no—
208#390—72no—
209#390—72no—
210#390—141no—
211#390—141no—
212#389—71no—
213#387—141no—
214#385—141no—
215#380—71no—
216C4×C4⋊C4—71no—
217C4×C4⋊C4—141no—
218C4×C22⋊C4—141no—
219#340—72no—
220#340C2×D712yescomplemented
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—24no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—84no—
260C2×C4×C7C4×D412yes—
261C2×C4×C7C23×C411yescharacteristic
262C2×C4×C7C2×#4111yescharacteristic
263C2×C4×C7H(5,𝔽2)12yes—
264C2×C4×C7H(5,𝔽2)12yes—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—21no—
269C2×Dic7—21no—
270C2×Dic7—21no—
271C2×Dic7—21no—
272C2×Dic7—21no—
273C2×Dic7—21no—
274C2×Dic7—44no—
275C2×Dic7—44no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7C2×SD1612yes—
283C2×Dic7C2×SD1612yes—
284C2×Dic7AGL(1,ℤ/8)12yes—
285D28—24no—
286D28—24no—
287D28—48no—
288D28—42no—
289D28—42no—
290C7×C8—42no—
291C7⋊C8—24no—
292C25—141no—
293C22×Q8C22×D711yescharacteristic
294C22×D4—74no—
295C22×D4—72no—
296C22×D4—141no—
297C22×D4—142no—
298C22×D4—282no—
299C22×D4—282no—
300C23×C4—21no—
301C23×C4—71no—
302C23×C4—71no—
303C23×C4C22×D711yescharacteristic
304C2×SD16—78no—
305C2×SD16—282no—
306C2×SD16—282no—
307C22×C8—72no—
308C22×C8C22×D712yes—
309C22⋊Q8—22no—
310C22⋊Q8C4×D714yes—
311C4×Q8—282no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4C22×D712yes—
320C2×C4⋊C4C22×D711yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—142no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—284no—
328C2×C22⋊C4—284no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42—71no—
332C2×C42—71no—
333C2×C42—141no—
334C2×C42—141no—
335C2×C42—282no—
336#106—282no—
337#103—42no—
338#102—78no—
339#102—282no—
340#102—282no—
341C22⋊C8—22no—
342C22⋊C8—74no—
343C4×C8—282no—
344#95—141no—
345#95—141no—
346#95—141no—
347#95—141no—
348#95—141no—
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—42no—
373C4×C7—42no—
374C4×C7—82no—
375C4×C7#51812yes—
376C4×C7#53812yes—
377Dic7—22no—
378Dic7—22no—
379Dic7—22no—
380Dic7—41no—
381Dic7—81no—
382Dic7—81no—
383C24—144no—
384C24—142no—
385C24—282no—
386C24—282no—
387C24—282no—
388C24—282no—
389C24—282no—
390C24—282no—
391C24D4×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—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—74no—
419C22×C4—71no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—281no—
425C22×C4—281no—
426C22×C4—281no—
427C22×C4C2×C4×D712yes—
428C22×C4D4×D711yescharacteristic
429C22×C4#115912yes—
430C22×C4C23×D711yescharacteristic
431SD16—288no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—74no—
436C2×C8—142no—
437C2×C8—142no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—142no—
444C4⋊C4—142no—
445C4⋊C4—282no—
446C4⋊C4—282no—
447C4⋊C4C2×C4×D712yes—
448C22⋊C4—24no—
449C22⋊C4—284no—
450C22⋊C4—284no—
451C22⋊C4—284no—
452C22⋊C4—284no—
453C22⋊C4—282no—
454C42—142no—
455C42—142no—
456C42—281no—
457C42—281no—
458C42—282no—
459C42—282no—
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—288no—
482C23—282no—
483C23—564no—
484C23—564no—
485C23—564no—
486C23—564no—
487C23(C22⋊D7)⋊C412yes—
488C23D7×SD1612yes—
489C23#648212yes—
490C23C2×D4×D711yescharacteristic
491Q8—24no—
492Q8—42no—
493D4—144no—
494D4—144no—
495D4—288no—
496D4—282no—
497D4—282no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—72no—
512C2×C4—72no—
513C2×C4—72no—
514C2×C4—84no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—284no—
525C2×C4—284no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4(C22⋊D7)⋊C412yes—
533C2×C4C22×C4×D711yescharacteristic
534C2×C4C2×#115911yescharacteristic
535C2×C4#654812yes—
536C2×C4#655312yes—
537C8—42no—
538C8—144no—
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—
554C22#8956014yes—
555C22C2×(C22⋊D7)⋊C411yescharacteristic
556C22#9028311yescharacteristic
557C22#9034411yescharacteristic
558C22C2×D7×SD1611yescharacteristic
559C22C2×#648211yescharacteristic
560C22D7×AGL(1,ℤ/8)11yescharacteristic
561C22#9046811yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—142no—
567C4—142no—
568C4—142no—
569C4—281no—
570C4—561no—
571C4—561no—
572C4#9023612yes—
573C4#9030812yes—
574C2—24no—
575C2—564no—
576C2#1173232011yescharacteristic
577C2C2×#8956011yescharacteristic
578C2#1173631111yescharacteristic
579C2#1173758311yescharacteristic
580C2#1173764611yescharacteristic
581C2#1173888711yescharacteristic
582C2#1173892711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1872C2
socleC23×C7C22×D4
prime radicalC23×D28C22
cosocleC22×C4×C7C24
hypercenter#1872D7
hypocenterC7#22043
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
1. Dic7×C22⋊Q8C2semidirect
4. C7×#1872C2semidirect
5. #11729310C2semidirect
6. #11728011C2semidirect
7. #11727994C2semidirect
9. Dic7×C22⋊C8C2semidirect
10. #11726607C2semidirect
11. #11726546C2semidirect
15. C2×Q8×Dic7C22semidirect
20. C2×C7×#103C22semidirect
22. #90002C22semidirect
27. C2×C8×Dic7C22semidirect
28. #89818C4semidirect
33. #89757C22semidirect
36. C4⋊C4×Dic7C22semidirect
38. C2×#6382C22semidirect
41. C7×#340C22semidirect
42. #89345C22semidirect
43. #89324C22semidirect
70. C2×#1143C2×C4semidirect
100. #6383C2×C4semidirect
109. #1872D7semidirect
145. C22×Dic7SD16semidirect
200. C2×#103C2×D7semidirect
220. #340C2×D7semidirect
539. C7#22043semidirect

Maximal subgroups

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

subgroupindex
Dic7×C22⋊Q82
C2×#898182
C2×#897602
C7×#18722
#117293102
#117280112
#117279942
#117277132
Dic7×C22⋊C82
#117266072
#117265462
#220437