Finite Group Databasebeta

← #13395100

Subgroups — #13395100

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
1C4×C7×#121C211yesmaximal, characteristic
2C2×#89779C211yesmaximal, characteristic, complemented
3#11739521C211yesmaximal, characteristic, complemented
4C2×#89698C211yesmaximal, characteristic, complemented
5#11729434C212yesmaximal
6#11729358C211yesmaximal, characteristic, complemented
7#11727399C212yesmaximal, complemented
8#11727375C212yesmaximal, complemented
9#11727273C212yesmaximal, complemented
10#11727244C211yesmaximal, characteristic, complemented
11#11727221C211yesmaximal, characteristic, complemented
12C2×C7×#121C2211yescharacteristic
13C2×C4×C7×D4—21no—
14C2×C4×C7×D4C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C2×#6487—21no—
17C2×#6432C2211yescharacteristic
18C7×#390C2211yescharacteristic
19C7×#389C2211yescharacteristic
20C7×#388—21no—
21C7×#386—21no—
22C7×#380C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C4×C7×C22⋊C4—21no—
25C2×#6406C2212yes—
26C2×#6406C2212yes—
27#89984—21no—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30#89779—24no—
31C2×#6382C2212yes—
32C2×#6381C2212yes—
33#89707C2211yescharacteristic
34#89701—21no—
35#89700—21no—
36#89700C2212yes—
37#89698—24no—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×#6377C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24709—71nomaximal, Sylow
46C22×C7×D4—21no—
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C22×Dic14—21no—
53C7×#121—22no—
54C7×#121—22no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C4×C7×D4—44no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C42×C7—22no—
68C2×C42×C7—22no—
69C2×C42×C7—21no—
70C2×C42×C7D411yescharacteristic
71C2×C42×C7D411yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C22⋊Dic7—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2312yes—
79C2×C4⋊Dic7C2312yes—
80#6487—44no—
81C2×Dic7⋊C4—22no—
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2312yes—
84#6432—22no—
85#6432—22no—
86C7×#95—21no—
87C7×#95—21no—
88C7×#95—21no—
89#6406—24no—
90#6406—24no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93C22⋊(C7⋊C8)—24no—
94#6382—24no—
95#6381—24no—
96C4⋊(C7⋊C8)—22no—
97C4⋊(C7⋊C8)D414yescomplemented
98#6377—24no—
99C2×#474—71no—
100C2×#432—71no—
101#2857—71no—
102C4×#121D711yescharacteristic
103#2076—72no—
104#2034—72no—
105#1988—71no—
106#1948—71no—
107#1944—72no—
108#1936—72no—
109#1802—71no—
110C24×C7—21no—
111C24×C7C2×D411yescharacteristic
112C2×C7×D4—24no—
113C2×C7×D4—21no—
114C2×C7×D4—21no—
115C2×C7×D4—21no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4—42no—
119C2×C7×D4—42no—
120C2×C7×D4#4112yes—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—41no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2411yescharacteristic
145C22×Dic7—22no—
146C22×Dic7—22no—
147C2×Dic14—42no—
148C2×Dic14—42no—
149C2×Dic14—42no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—42no—
156C7×C4⋊C4#4112yes—
157C7×C22⋊C4—24no—
158C7×C22⋊C4—42no—
159C7×C22⋊C4—42no—
160C7×C22⋊C4—42no—
161C7×C22⋊C4—42no—
162C42×C7—22no—
163C42×C7—22no—
164C42×C7—21no—
165C42×C7—21no—
166C42×C7—42no—
167C42×C7C2×D412yes—
168C42×C7#4112yes—
169C22⋊Dic7—44no—
170C4⋊Dic7—22no—
171C4⋊Dic7—22no—
172C4⋊Dic7—22no—
173C4⋊Dic7—22no—
174C4⋊Dic7—42no—
175C4⋊Dic7—42no—
176C4⋊Dic7—42no—
177Dic7⋊C4—44no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8C2×D414yes—
184C2×C4⋊Q8—71no—
185C2×C22⋊Q8—141no—
186C2×#121C2×D711yescharacteristic
187C2×C4×D4—21no—
188C2×C4×D4C2×D711yescharacteristic
189C22×C42C2×D711yescharacteristic
190#474—144no—
191#432—144no—
192C2×#107—72no—
193C2×#106—72no—
194C2×C4⋊C8—71no—
195C2×C4⋊C8—71no—
196C2×#102—72no—
197C2×#102—72no—
198C2×C22⋊C8—72no—
199C2×#97—72no—
200#391—141no—
201#390C2×D711yescharacteristic
202#389—72no—
203#389—141no—
204#389C2×D711yescharacteristic
205#388—21no—
206#386—21no—
207#384—71no—
208#380—72no—
209#380—141no—
210#380C2×D711yescharacteristic
211C4×C4⋊C4C2×D711yescharacteristic
212C4×C22⋊C4—21no—
213#340—72no—
214#340—72no—
215#340—72no—
216#340—72no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7#13712yes—
231C23×C7C22×D411yescharacteristic
232C7×D4—22no—
233C7×D4—22no—
234C7×D4—48no—
235C7×D4—41no—
236C7×D4—41no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—22no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—44no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—44no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—84no—
264C2×C4×C7#12112yes—
265C2×C4×C7#12112yes—
266C2×C4×C7AGL(1,ℤ/8)12yes—
267C2×C4×C7AGL(1,ℤ/8)12yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C72−1+411yescharacteristic
273C2×C4×C72−1+411yescharacteristic
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280Dic14—84no—
281C7⋊C8—24no—
282C7⋊C8—42no—
283C22×Q8—141no—
284C22×D4—21no—
285C22×D4C22×D711yescharacteristic
286C23×C4—21no—
287C23×C4C22⋊D711yescharacteristic
288C23×C4C22⋊D711yescharacteristic
289C23×C4C22×D711yescharacteristic
290C22×C8—72no—
291C22×C8—72no—
292C4⋊Q8—142no—
293C4⋊Q8—142no—
294C22⋊Q8—284no—
295#121—22no—
296#121—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—72no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—142no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4C22×D711yescharacteristic
308C2×C4⋊C4C22×D711yescharacteristic
309C2×C4⋊C4C22×D711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—142no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42C22⋊D711yescharacteristic
320C2×C42C22⋊D711yescharacteristic
321C2×C42C22×D711yescharacteristic
322C2×C42C22×D711yescharacteristic
323#107—144no—
324#106—144no—
325C4⋊C8—74no—
326C4⋊C8—142no—
327#102—144no—
328#102—144no—
329C22⋊C8—144no—
330#97—144no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—142no—
335#95—142no—
336C22×C7—22no—
337C22×C7—22no—
338C22×C7—22no—
339C22×C7—41no—
340C22×C7—41no—
341C22×C7—41no—
342C22×C7—41no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—81no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7C2×#12111yescharacteristic
351C22×C7#52511yescharacteristic
352C22×C7#53611yescharacteristic
353C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C7C2×#13711yescharacteristic
356C22×C7#57511yescharacteristic
357C4×C7—22no—
358C4×C7—42no—
359C4×C7—41no—
360C4×C7—41no—
361C4×C7—41no—
362C4×C7—81no—
363C4×C7—81no—
364C4×C7#46914yes—
365C4×C7#54711yescharacteristic
366C4×C7#54811yescharacteristic
367C4×C7#55211yescharacteristic
368C4×C7#55611yescharacteristic
369Dic7—82no—
370Dic7—82no—
371C24—21no—
372C24C2×C22⋊D711yescharacteristic
373C2×Q8—282no—
374C2×Q8—282no—
375C2×Q8—282no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4#115712yes—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C2×C22⋊D711yescharacteristic
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C2×C22⋊D711yescharacteristic
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C23×D711yescharacteristic
411C2×C8—74no—
412C2×C8—142no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—142no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—42no—
423C4⋊C4—142no—
424C4⋊C4—142no—
425C4⋊C4—142no—
426C4⋊C4—142no—
427C4⋊C4—284no—
428C4⋊C4—282no—
429C4⋊C4—282no—
430C4⋊C4—282no—
431C4⋊C4#115712yes—
432C22⋊C4—24no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—284no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—42no—
443C42D4×D712yes—
444C42#115912yes—
445C2×C7—42no—
446C2×C7—82no—
447C2×C7#241511yescharacteristic
448C2×C7C2×#46911yescharacteristic
449C2×C7#305611yescharacteristic
450C2×C7#311011yescharacteristic
451C2×C7#311511yescharacteristic
452C2×C7#323511yescharacteristic
453C2×C7#323911yescharacteristic
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23#651412yes—
468C23C22×C22⋊D711yescharacteristic
469Q8—564no—
470D4—22no—
471D4—22no—
472D4—48no—
473D4—41no—
474D4—41no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—84no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4#649512yes—
509C2×C4#650212yes—
510C2×C4#650212yes—
511C2×C4#651212yes—
512C2×C4C2×#115711yescharacteristic
513C2×C4C2×#115711yescharacteristic
514C2×C4C2×D4×D711yescharacteristic
515C2×C4C2×#115911yescharacteristic
516C2×C4#655411yescharacteristic
517C2×C4#655411yescharacteristic
518C8—144no—
519C8—282no—
520C7#2470911yescharacteristic, complemented, Sylow
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22#9048211yescharacteristic
536C22C2×#649511yescharacteristic
537C22C2×#650211yescharacteristic
538C22C2×#651211yescharacteristic
539C22#9051311yescharacteristic
540C22C2×#651411yescharacteristic
541C22#9051611yescharacteristic
542C4—22no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—81no—
548C4—81no—
549C4—562no—
550C4—562no—
551C4#8981114yes—
552C4#9023211yescharacteristic
553C4#9023311yescharacteristic
554C4#9024211yescharacteristic
555C4#9024311yescharacteristic
556C2—42no—
557C2—82no—
558C2#1173976011yescharacteristic
559C2#1173979611yescharacteristic
560C2#1173997611yescharacteristic
561C2C2×#8981111yescharacteristic
562C2#1174037011yescharacteristic
563C2#1174056911yescharacteristic
564C2#1174071911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×#121C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC4×#121D7
hypocenterC7#24709
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#89779C2semidirect
3. #11739521C2semidirect
4. C2×#89698C2semidirect
6. #11729358C2semidirect
7. #11727399C2semidirect
8. #11727375C2semidirect
9. #11727273C2semidirect
10. #11727244C2semidirect
11. #11727221C2semidirect
97. C4⋊(C7⋊C8)D4semidirect
520. C7#24709semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1212
C2×#897792
#117395212
C2×#896982
#117294342
#117293582
#117273992
#117273752
#117272732
#117272442
#117272212
#247097