Finite Group Databasebeta

← #13395096

Subgroups — #13395096

Order 1792.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#474C211yesmaximal, characteristic, complemented
2#11740714C211yesmaximal, characteristic, complemented
3C4×#6501C211yesmaximal, characteristic
4C2×#89607C211yesmaximal, characteristic, complemented
5#11729534C211yesmaximal, characteristic, complemented
6#11729358C211yesmaximal, characteristic, complemented
7#11728391C212yesmaximal, complemented
8#11728368C212yesmaximal, complemented
9#11728208C211yesmaximal, characteristic, complemented
10#11727942C212yesmaximal, complemented
11#11727930C212yesmaximal
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6509—21no—
15C2×#6501C2211yescharacteristic
16C2×C4×C22⋊D7—21no—
17C2×C42×D7C2211yescharacteristic
18C7×#474—24no—
19C2×C7×#107C2212yes—
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×#102C2212yes—
22#90002—21no—
23#89993C2211yescharacteristic
24C2×#6406C2212yes—
25C4×C22⋊Dic7—21no—
26C2×(C2×D7)⋊C8C2212yes—
27C2×#6391C2212yes—
28C2×#6389C2212yes—
29#89759—21no—
30#89759C2211yescharacteristic
31#89759C2212yes—
32#89745—21no—
33#89745C2212yes—
34#89745C2211yescharacteristic
35#89728—21no—
36#89716—21no—
37#89707C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40#89607—24no—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24709—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C22×C22⋊D7—21no—
50C22×C4×D7D411yescharacteristic
51C22×C4×D7D411yescharacteristic
52C22×C4×D7C2311yescharacteristic
53C22×C7×C8C2312yes—
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8—22no—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7C2311yescharacteristic
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—21no—
65C2×C22⋊Dic7—21no—
66#6509—44no—
67#6501—22no—
68#6501—22no—
69C4×C22⋊D7—44no—
70C2×(C2×D7)⋊C4—21no—
71C2×(C2×D7)⋊C4—22no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7C2311yescharacteristic
75C2×C4⋊Dic7C2312yes—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—22no—
79C2×C4×Dic7—21no—
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C22×C7⋊C8C2312yes—
84C42×D7—22no—
85C42×D7—22no—
86C7×#107—24no—
87C7×C4⋊C8D414yescomplemented
88C7×#102—24no—
89#6406—24no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95(C2×D7)⋊C8—24no—
96#6391—24no—
97#6389—24no—
98C4⋊(C7⋊C8)—22no—
99C2×#474D711yescharacteristic, complemented
100C2×#432—71no—
101#2857—71no—
102C4×#121—71no—
103#2076—72no—
104#2034—72no—
105#1988—71no—
106#1948—71no—
107#1944—72no—
108#1936—72no—
109#1802—71no—
110C24×C7—21no—
111C23×D7C2×D411yescharacteristic
112C2×C7×Q8—42no—
113C2×C7×Q8—42no—
114C2×C7×Q8—42no—
115C2×C7×D4—21no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4—21no—
120C2×C7×D4#4112yes—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—42no—
124C22×C4×C7—42no—
125C22×C4×C7—42no—
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2411yescharacteristic
129C2×C22⋊D7—24no—
130C2×C22⋊D7—42no—
131C2×C22⋊D7—42no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C22×Dic7—22no—
135C22×Dic7C2×D411yescharacteristic
136C22×Dic7C2×D411yescharacteristic
137C22×Dic7C2×D411yescharacteristic
138C2×C4×D7—22no—
139C2×C4×D7—22no—
140C2×C4×D7—22no—
141C2×C4×D7—22no—
142C2×C4×D7—22no—
143C2×C4×D7—21no—
144C2×C4×D7—21no—
145C2×C4×D7—21no—
146C2×C4×D7—21no—
147C2×C4×D7—21no—
148C2×C4×D7—21no—
149C2×C4×D7—41no—
150C2×C7×C8—22no—
151C2×C7×C8—22no—
152C2×C7×C8C2×D414yes—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—42no—
158C7×C22⋊C4—42no—
159C42×C7C2×D412yes—
160C42×C7#4112yes—
161C22⋊Dic7—24no—
162C22⋊Dic7—42no—
163C22⋊Dic7—42no—
164(C2×D7)⋊C4—42no—
165(C2×D7)⋊C4—44no—
166C4⋊Dic7—21no—
167C4⋊Dic7—22no—
168C4⋊Dic7—22no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—42no—
172C4⋊Dic7—42no—
173C4⋊Dic7#4112yes—
174Dic7⋊C4—42no—
175Dic7⋊C4—44no—
176C4×Dic7—22no—
177C4×Dic7—22no—
178C4×Dic7—21no—
179C4×Dic7—21no—
180C4×Dic7—42no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C4⋊Q8C2×D711yescharacteristic
185C2×C22⋊Q8—141no—
186C2×#121—71no—
187C2×C4×D4—141no—
188C2×C4×D4C2×D711yescharacteristic
189C22×C42—71no—
190#474—24no—
191#432—144no—
192C2×#107C2×D712yes—
193C2×#106—72no—
194C2×C4⋊C8—71no—
195C2×C4⋊C8C2×D711yescharacteristic
196C2×#102—72no—
197C2×#102C2×D712yes—
198C2×C22⋊C8—72no—
199C2×#97—72no—
200#391—141no—
201#390—71no—
202#389—72no—
203#389—71no—
204#389—141no—
205#388—141no—
206#386—141no—
207#384—71no—
208#380—71no—
209#380—72no—
210#380—141no—
211C4×C4⋊C4—71no—
212C4×C22⋊C4—141no—
213#340—72no—
214#340—72no—
215#340—72no—
216#340—72no—
217C23×C7—22no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7C22×D411yescharacteristic
225C22×D7—21no—
226C22×D7—21no—
227C22×D7—21no—
228C22×D7—21no—
229C22×D7—21no—
230C22×D7—21no—
231C22×D7#13712yes—
232C7×Q8—84no—
233C7×D4—22no—
234C7×D4—22no—
235C7×D4—41no—
236C7×D4—41no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—41no—
240C2×C4×C7—42no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—84no—
246C2×C4×C7#12112yes—
247C2×C4×C7#12112yes—
248C2×C4×C7AGL(1,ℤ/8)12yes—
249C2×C4×C7AGL(1,ℤ/8)12yes—
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C72−1+411yescharacteristic
255C2×C4×C72−1+411yescharacteristic
256C22⋊D7—48no—
257C2×Dic7—21no—
258C2×Dic7—21no—
259C2×Dic7—21no—
260C2×Dic7—22no—
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—21no—
264C2×Dic7—21no—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—41no—
275C4×D7—44no—
276C4×D7—41no—
277C4×D7—41no—
278C4×D7—41no—
279C4×D7—41no—
280C4×D7—44no—
281C7×C8—24no—
282C7⋊C8—42no—
283C22×Q8—21no—
284C22×D4—141no—
285C22×D4C22×D711yescharacteristic
286C23×C4—21no—
287C23×C4—71no—
288C23×C4—71no—
289C23×C4—71no—
290C22×C8—72no—
291C22×C8C22×D712yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C22⋊Q8—284no—
295#121—142no—
296#121—142no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—284no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—71no—
302C2×C4⋊C4—71no—
303C2×C4⋊C4—72no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—142no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—141no—
308C2×C4⋊C4C22×D711yescharacteristic
309C2×C4⋊C4C22×D712yes—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—141no—
312C2×C22⋊C4—141no—
313C2×C22⋊C4—142no—
314C2×C22⋊C4—141no—
315C2×C22⋊C4—141no—
316C2×C42—71no—
317C2×C42—71no—
318C2×C42—71no—
319C2×C42—142no—
320C2×C42—142no—
321C2×C42—141no—
322C2×C42C22×D711yescharacteristic
323#107—24no—
324#106—144no—
325C4⋊C8—142no—
326C4⋊C8C22⋊D714yescomplemented
327#102—24no—
328#102—144no—
329C22⋊C8—144no—
330#97—144no—
331#95—141no—
332#95—141no—
333#95—142no—
334#95—142no—
335#95—141no—
336C22×C7—42no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—81no—
340C22×C7—81no—
341C22×C7—81no—
342C22×C7—81no—
343C22×C7C2×#12111yescharacteristic
344C22×C7#52511yescharacteristic
345C22×C7#53611yescharacteristic
346C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C7C2×#13711yescharacteristic
349C22×C7#57511yescharacteristic
350C2×D7—22no—
351C2×D7—22no—
352C2×D7—22no—
353C2×D7—41no—
354C2×D7—41no—
355C2×D7—41no—
356C2×D7—41no—
357C4×C7—22no—
358C4×C7—81no—
359C4×C7—82no—
360C4×C7#46914yes—
361C4×C7#54711yescharacteristic
362C4×C7#54811yescharacteristic
363C4×C7#55211yescharacteristic
364C4×C7#55611yescharacteristic
365Dic7—42no—
366Dic7—41no—
367Dic7—41no—
368Dic7—41no—
369Dic7—82no—
370Dic7—81no—
371C24—21no—
372C24—71no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—144no—
382C2×D4—282no—
383C2×D4—282no—
384C2×D4#116112yes—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—71no—
391C22×C4—71no—
392C22×C4—71no—
393C22×C4—142no—
394C22×C4—141no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—141no—
404C22×C4—141no—
405C22×C4—141no—
406C22×C4—142no—
407C22×C4—281no—
408C22×C4D4×D711yescharacteristic
409C22×C4D4×D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8C2×C22⋊D714yes—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—42no—
422C4⋊C4—72no—
423C4⋊C4—141no—
424C4⋊C4—142no—
425C4⋊C4—141no—
426C4⋊C4—142no—
427C4⋊C4—141no—
428C4⋊C4—282no—
429C4⋊C4—282no—
430C4⋊C4—284no—
431C4⋊C4—282no—
432C22⋊C4—42no—
433C22⋊C4—144no—
434C22⋊C4—282no—
435C22⋊C4—284no—
436C22⋊C4—282no—
437C22⋊C4—282no—
438C42—142no—
439C42—142no—
440C42—141no—
441C42—141no—
442C42—282no—
443C42#115912yes—
444C42C2×C22⋊D712yes—
445C2×C7—82no—
446C2×C7#241511yescharacteristic
447C2×C7C2×#46911yescharacteristic
448C2×C7#305611yescharacteristic
449C2×C7#311011yescharacteristic
450C2×C7#311511yescharacteristic
451C2×C7#323511yescharacteristic
452C2×C7#323911yescharacteristic
453D7—42no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—72no—
462C23—141no—
463C23—141no—
464C23—141no—
465C23—141no—
466C23—141no—
467C23—141no—
468C23C2×D4×D711yescharacteristic
469Q8—84no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—288no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—84no—
484C2×C4—141no—
485C2×C4—141no—
486C2×C4—141no—
487C2×C4—141no—
488C2×C4—141no—
489C2×C4—141no—
490C2×C4—141no—
491C2×C4—141no—
492C2×C4—142no—
493C2×C4—141no—
494C2×C4—284no—
495C2×C4—281no—
496C2×C4—281no—
497C2×C4—284no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—282no—
504C2×C4—281no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—281no—
508C2×C4#647412yes—
509C2×C4#647412yes—
510C2×C4#650112yes—
511C2×C4#650112yes—
512C2×C4C2×#115911yescharacteristic
513C2×C4C2×#116111yescharacteristic
514C2×C4#655111yescharacteristic
515C2×C4D7×#4111yescharacteristic
516C2×C4#655411yescharacteristic
517C2×C4C22×C22⋊D711yescharacteristic
518C8—24no—
519C8—282no—
520C7#2470911yescharacteristic, complemented, Sylow
521C22—42no—
522C22—42no—
523C22—42no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—142no—
529C22—142no—
530C22—142no—
531C22—281no—
532C22—281no—
533C22—281no—
534C22—281no—
535C22#9031811yescharacteristic
536C22#9031911yescharacteristic
537C22C2×#647411yescharacteristic
538C22C2×#647411yescharacteristic
539C22D7×#13711yescharacteristic
540C22#9046811yescharacteristic
541C22C2×#650111yescharacteristic
542C4—22no—
543C4—81no—
544C4—82no—
545C4—282no—
546C4—281no—
547C4—281no—
548C4—281no—
549C4—562no—
550C4—561no—
551C4#8993114yes—
552C4#9033211yescharacteristic
553C4#9033311yescharacteristic
554C4#9050811yescharacteristic
555C4#9051511yescharacteristic
556C2—82no—
557C2—282no—
558C2#1173839511yescharacteristic
559C2#1173839611yescharacteristic
560C2#1173841611yescharacteristic
561C2#1173841711yescharacteristic
562C2#1174025811yescharacteristic
563C2C2×#8993111yescharacteristic
564C2#1174143011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#474C2
socleC23×C7C22×D4
prime radicalC2×#6501C22
hypercenterC2×#474D7
hypocenterC7#24709
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C7×#474C2semidirect
2. #11740714C2semidirect
4. C2×#89607C2semidirect
5. #11729534C2semidirect
6. #11729358C2semidirect
7. #11728391C2semidirect
8. #11728368C2semidirect
9. #11728208C2semidirect
10. #11727942C2semidirect
87. C7×C4⋊C8D4semidirect
99. C2×#474D7semidirect
326. C4⋊C8C22⋊D7semidirect
520. C7#24709semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4742
#117407142
C4×#65012
C2×#896072
#117295342
#117293582
#117283912
#117283682
#117282082
#117279422
#117279302
#247097