Finite Group Databasebeta

← #13353616

Subgroups — #13353616

Order 1792.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#520C211yesmaximal, characteristic
2#11741532C211yesmaximal, characteristic
3C2×#90006C211yesmaximal, characteristic
4#11729335C211yesmaximal, characteristic, complemented
5#11729334C211yesmaximal, characteristic, complemented
6#11729329C212yesmaximal, complemented
7#11729329C212yesmaximal, complemented
8#11729312C211yesmaximal, characteristic, complemented
9C2×#89324C211yesmaximal, characteristic, complemented
10#11729308C212yesmaximal, complemented
11#11729307C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C7×#520—24no—
14C7×#520C418yescomplemented
15C2×C4×C7×Q8—21no—
16C2×C4×C7×D4—22no—
17C2×C4×C7×D4—21no—
18C2×C7×#117C2211yescharacteristic
19C2×C7×#117C2212yes—
20C22×C7×C4⋊C4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26#90006—24no—
27#90002—21no—
28#89993—21no—
29#89992—22no—
30#89984C2212yes—
31#89984C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34#89324C2211yescharacteristic
35#89324C2212yes—
36#89324C2212yes—
37#89324C2212yes—
38#89324C2211yescharacteristic
39#89324C2212yes—
40#89324C2212yes—
41#89324C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#20703—71nomaximal, Sylow
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41C2×C414yes—
50C2×C7×#41D414yes—
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4D412yes—
53C22×C7×D4D411yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—22no—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×Dic7—21no—
59C4×C7×Q8—28no—
60C4×C7×D4—28no—
61C4×C7×D4—216no—
62C7×#117—22no—
63C7×#117—24no—
64C7×#117C2×C414yes—
65C7×#117C2×C418yescomplemented
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—24no—
70C2×C7×C4⋊C4C2×C414yes—
71C2×C7×C4⋊C4C2×C418yescomplemented
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—22no—
77C2×C42×C7—21no—
78C2×C42×C7—22no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—22no—
81C2×C22⋊Dic7—22no—
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2312yes—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4×Dic7—21no—
91C2×#1136—22no—
92C2×#1136—22no—
93C2×#1136—22no—
94C22×C7⋊C8—22no—
95C22×C7⋊C8—22no—
96C22×C7⋊C8—22no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2312yes—
100C7⋊#95—22no—
101C7⋊#95—22no—
102C22⋊(C7⋊C8)—24no—
103C22⋊(C7⋊C8)—24no—
104C2×#520D711yescharacteristic
105C2×#408—71no—
106#2311—71no—
107#1825—72no—
108#1825—72no—
109#1824—72no—
110#1823—72no—
111#1736—71no—
112#1736—71no—
113#1735—71no—
114C2×#340—71no—
115C24×C7C2×D412yes—
116C24×C7C2×D411yescharacteristic
117C7×#41—28no—
118C7×#41—28no—
119C7×#41—28no—
120C7×#41—28no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8C22⋊C414yes—
124C2×C7×D4—24no—
125C2×C7×D4—24no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4C22⋊C418yes—
129C2×C7×D4C22⋊C414yes—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—28no—
139C22×C4×C7—81no—
140C22×C4×C7—82no—
141C22×C4×C7—81no—
142C22×C4×C7—82no—
143C22×C4×C7C22×C414yes—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D414yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D412yes—
149C22×C4×C7#4112yes—
150C22×C4×C7#4112yes—
151C22×C4×C7#4112yes—
152C22×C4×C7#4112yes—
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—22no—
157C22×Dic7—42no—
158C22×Dic7—81no—
159C22×Dic7—81no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—28no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4C22×C414yes—
166C7×C4⋊C4C22×C412yes—
167C7×C4⋊C4C22×C412yes—
168C7×C22⋊C4—24no—
169C7×C22⋊C4—28no—
170C42×C7—24no—
171C42×C7—28no—
172C22⋊Dic7—42no—
173C22⋊Dic7—82no—
174C22⋊Dic7—82no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—42no—
182C4×Dic7—42no—
183#1136—44no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—44no—
194C22×#41C2×D711yescharacteristic
195C22×M16—71no—
196C23×C8—71no—
197#520—24no—
198#520Dic718yescomplemented
199C2×C4×Q8—21no—
200C2×C4×D4—22no—
201C2×C4×D4—21no—
202C2×#117—71no—
203C2×#117C2×D711yescharacteristic
204C2×#117C2×D712yes—
205C22×C4⋊C4—71no—
206C22×C4⋊C4C2×D711yescharacteristic
207C22×C4⋊C4C2×D712yes—
208#408—144no—
209C2×C22⋊C8—72no—
210C2×C22⋊C8—72no—
211#391—141no—
212#390—141no—
213#386—142no—
214#380—72no—
215#380—72no—
216#340—71no—
217#340—72no—
218#340—72no—
219#340—72no—
220#340—71no—
221#340—72no—
222#340—72no—
223#340—71no—
224#340—71no—
225#340—72no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7C2×C22⋊C414yes—
237C23×C7C2×C22⋊C412yes—
238C23×C7#12312yes—
239C23×C7#12312yes—
240C23×C7C22×D411yescharacteristic
241C7×Q8—28no—
242C7×D4—216no—
243C7×D4—28no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—84no—
260C2×C4×C7—88no—
261C2×C4×C7C2×C22⋊C414yes—
262C2×C4×C7C2×C22⋊C412yes—
263C2×C4×C7#11714yes—
264C2×C4×C7#11714yes—
265C2×C4×C7#12314yes—
266C2×C4×C7#12314yes—
267C2×C4×C7#12312yes—
268C2×C4×C7#12312yes—
269C2×C4×C7C23×C411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—42no—
278C2×Dic7—41no—
279C2×Dic7—42no—
280C2×Dic7—41no—
281C2×Dic7—42no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—84no—
285C7⋊C8—41no—
286C7⋊C8—41no—
287C7⋊C8—42no—
288C2×#41—22no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41C2×Dic714yes—
293C2×#41C22⋊D714yes—
294C22×Q8C22⋊D711yescharacteristic
295C22×D4C22⋊D712yes—
296C22×D4C22⋊D711yescharacteristic
297C23×C4—21no—
298C23×C4—22no—
299C23×C4—141no—
300C23×C4C22×D712yes—
301C23×C4C22×D711yescharacteristic
302C2×M16—142no—
303C2×M16—142no—
304C2×M16—142no—
305C22×C8—71no—
306C22×C8—71no—
307C22×C8—72no—
308C22×C8—142no—
309C22×C8—142no—
310C22×C8—142no—
311C4×Q8—28no—
312C4×D4—28no—
313C4×D4—216no—
314#117—22no—
315#117—24no—
316#117—142no—
317#117—142no—
318#117C2×Dic714yes—
319#117C2×Dic718yescomplemented
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—72no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—142no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4C2×Dic714yes—
331C2×C4⋊C4C2×Dic718yescomplemented
332C2×C4⋊C4C22×D712yes—
333C2×C4⋊C4C22×D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—141no—
338C2×C22⋊C4—142no—
339C2×C22⋊C4—142no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—141no—
343C22⋊C8—144no—
344C22⋊C8—144no—
345#95—142no—
346#95—142no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—24no—
354C22×C7—24no—
355C22×C7—24no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7#38114yes—
361C22×C7C22×C22⋊C411yescharacteristic
362C22×C7C2×#12311yescharacteristic
363C22×C7C2×#12311yescharacteristic
364C22×C7#57611yescharacteristic
365C22×C7#57711yescharacteristic
366C22×C7#57711yescharacteristic
367C22×C7#57811yescharacteristic
368C4×C7—24no—
369C4×C7—42no—
370C4×C7—44no—
371C4×C7—42no—
372C4×C7#38114yes—
373C4×C7C2×#11711yescharacteristic
374C4×C7C2×#11711yescharacteristic
375C4×C7C2×#12311yescharacteristic
376C4×C7C2×#12311yescharacteristic
377Dic7—82no—
378Dic7—81no—
379Dic7—81no—
380C24C2×C22⋊D712yes—
381C24C2×C22⋊D711yescharacteristic
382#41—28no—
383#41—28no—
384#41—28no—
385#41—28no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8C22⋊Dic714yes—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4C22⋊Dic718yes—
394C2×D4C22⋊Dic714yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—81no—
405C22×C4—82no—
406C22×C4—81no—
407C22×C4—82no—
408C22×C4—141no—
409C22×C4—141no—
410C22×C4—142no—
411C22×C4—282no—
412C22×C4—561no—
413C22×C4—561no—
414C22×C4#115712yes—
415C22×C4#115712yes—
416C22×C4#115712yes—
417C22×C4#115712yes—
418C22×C4C22×Dic714yes—
419C22×C4C22×Dic712yes—
420C22×C4C2×C22⋊D712yes—
421C22×C4C2×C22⋊D714yes—
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D712yes—
424C22×C4C23×D711yescharacteristic
425M16—284no—
426C2×C8—141no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—142no—
431C2×C8—142no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—142no—
435C2×C8—284no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—142no—
442C4⋊C4—142no—
443C4⋊C4—141no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—282no—
448C4⋊C4C22×Dic714yes—
449C4⋊C4C22×Dic712yes—
450C4⋊C4C22×Dic712yes—
451C22⋊C4—24no—
452C22⋊C4—28no—
453C22⋊C4—282no—
454C22⋊C4—562no—
455C22⋊C4—562no—
456C42—24no—
457C42—28no—
458C42—282no—
459C2×C7—24no—
460C2×C7—44no—
461C2×C7C2×#38111yescharacteristic
462C2×C7#289711yescharacteristic
463C2×C7#289811yescharacteristic
464C2×C7#309611yescharacteristic
465C2×C7#309711yescharacteristic
466C2×C7#309811yescharacteristic
467C2×C7#309911yescharacteristic
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23#649212yes—
479C23#649212yes—
480C23C2×C22⋊Dic714yes—
481C23C2×C22⋊Dic712yes—
482C23C22×C22⋊D711yescharacteristic
483Q8—28no—
484D4—216no—
485D4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—84no—
502C2×C4—88no—
503C2×C4—281no—
504C2×C4—281no—
505C2×C4—282no—
506C2×C4—281no—
507C2×C4—282no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4—282no—
511C2×C4—281no—
512C2×C4—564no—
513C2×C4#648914yes—
514C2×C4#648914yes—
515C2×C4#649214yes—
516C2×C4#649214yes—
517C2×C4#649212yes—
518C2×C4#649212yes—
519C2×C4C2×C22⋊Dic714yes—
520C2×C4C2×C22⋊Dic712yes—
521C2×C4C2×#115711yescharacteristic
522C2×C4C2×#115711yescharacteristic
523C2×C4C2×#115711yescharacteristic
524C2×C4C2×#115711yescharacteristic
525C2×C4C23×Dic711yescharacteristic
526C2×C4C22×C22⋊D711yescharacteristic
527C8—281no—
528C8—281no—
529C8—282no—
530C7#2070311yescharacteristic, complemented, Sylow
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22#8998314yes—
545C22C2×#649211yescharacteristic
546C22C2×#649211yescharacteristic
547C22#9052611yescharacteristic
548C22#9052711yescharacteristic
549C22#9052811yescharacteristic
550C22#9052911yescharacteristic
551C22C22×C22⋊Dic711yescharacteristic
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—562no—
557C4—561no—
558C4—561no—
559C4#8998314yes—
560C4C2×#648911yescharacteristic
561C4C2×#648911yescharacteristic
562C4C2×#649211yescharacteristic
563C4C2×#649211yescharacteristic
564C2—24no—
565C2—44no—
566C2#1174001111yescharacteristic
567C2#1174001211yescharacteristic
568C2#1174001811yescharacteristic
569C2#1174001911yescharacteristic
570C2C2×#8998311yescharacteristic
571C2#1174154411yescharacteristic
572C2#1174155611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#520C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#520D7
hypocenterC7#20703
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
4. #11729335C2semidirect
5. #11729334C2semidirect
6. #11729329C2semidirect
7. #11729329C2semidirect
8. #11729312C2semidirect
9. C2×#89324C2semidirect
10. #11729308C2semidirect
11. #11729307C2semidirect
14. C7×#520C4semidirect
65. C7×#117C2×C4semidirect
71. C2×C7×C4⋊C4C2×C4semidirect
198. #520Dic7semidirect
319. #117C2×Dic7semidirect
331. C2×C4⋊C4C2×Dic7semidirect
530. C7#20703semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5202
#117415322
C2×#900062
#117293352
#117293342
#117293292
#117293122
C2×#893242
#117293082
#117293072
#207037