Finite Group Databasebeta

← #31963607

Subgroups — #31963607

Order 18176.

952 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986591C211yesmaximal, characteristic
2#22597369C211yesmaximal, characteristic, complemented
3#22586707C212yesmaximal, complemented
4#22586705C212yesmaximal, complemented
5#22586695C212yesmaximal, complemented
6#22586689C212yesmaximal, complemented
7#22586676C212yesmaximal, complemented
8#22586669C212yesmaximal, complemented
9C71×#2975C211yesmaximal, characteristic, complemented
10#17986591—24no—
11#17986591—24no—
12C2×#14556122C2211yescharacteristic
13C2×#14556121C411yescharacteristic
14C2×#14556120C2211yescharacteristic
15C2×#14556118C411yescharacteristic
16C2×#14556112C411yescharacteristic
17C2×#14556111C2211yescharacteristic
18C2×#14556110C411yescharacteristic
19C2×#14556109C411yescharacteristic
20C2×#14556107C2211yescharacteristic
21C2×#14556106C411yescharacteristic
22C2×#14556105C411yescharacteristic
23C2×#14556100C411yescharacteristic
24C2×C4×C22⋊D71C2211yescharacteristic
25C22×Dic71⋊C4C2211yescharacteristic
26#17986071—24no—
27#17986056C2212yes—
28#17985963C2212yes—
29C2×C22⋊(C71⋊C8)C2212yes—
30#17985724C2214yes—
31#17985724C2214yes—
32#17985724C2214yes—
33#17985722C2214yes—
34C2×C4×D4×C71C2211yescharacteristic
35#436×C71—24no—
36#431×C71C2212yes—
37#407×C71C2212yes—
38#407×C71C2212yes—
39C22×Dic142C2×C411yescharacteristic
40C22×C22⋊D71C2×C411yescharacteristic
41C22×C22⋊D71C2×C411yescharacteristic
42C23×Dic71C2×C411yescharacteristic
43C23×Dic71C2×C411yescharacteristic
44C22×C4×D71C2×C411yescharacteristic
45C2×#11765849—22no—
46C22×C71⋊C8—22no—
47#14556122—22no—
48#14556122—22no—
49#14556121—24no—
50#14556120—22no—
51#14556120—22no—
52#14556118—24no—
53#14556112—24no—
54#14556111—22no—
55#14556111—22no—
56#14556110—24no—
57#14556109—24no—
58#14556107—22no—
59#14556107—22no—
60#14556106—24no—
61#14556105—24no—
62#14556100—24no—
63C4×C22⋊D71—22no—
64C4×C22⋊D71—22no—
65C2×C4⋊Dic71C2×C411yescharacteristic
66C2×C4⋊Dic71C2×C411yescharacteristic
67C2×C4⋊Dic71C2×C411yescharacteristic
68C2×Dic71⋊C4—22no—
69C2×Dic71⋊C4—22no—
70C2×Dic71⋊C4C2×C411yescharacteristic
71C2×Dic71⋊C4C2×C411yescharacteristic
72C2×Dic71⋊C4C2×C411yescharacteristic
73C2×Dic71⋊C4C2×C411yescharacteristic
74C2×Dic71⋊C4C2×C411yescharacteristic
75C2×Dic71⋊C4C2×C411yescharacteristic
76C2×Dic71⋊C4C2×C411yescharacteristic
77C2×Dic71⋊C4C2311yescharacteristic
78C2×Dic71⋊C4C2311yescharacteristic
79C2×Dic71⋊C4C2311yescharacteristic
80C2×C22⋊Dic71C2×C411yescharacteristic
81C2×C22⋊Dic71C2×C411yescharacteristic
82C2×C22⋊Dic71C2×C411yescharacteristic
83C2×C22⋊Dic71C2×C411yescharacteristic
84C2×(C2×D71)⋊C4C2×C411yescharacteristic
85C2×(C2×D71)⋊C4C2×C411yescharacteristic
86C2×(C2×D71)⋊C4C2×C411yescharacteristic
87C2×(C2×D71)⋊C4C2311yescharacteristic
88C2×C4×Dic71C2×C411yescharacteristic
89C4⋊(C71⋊C8)—22no—
90C22⋊(C71⋊C8)—22no—
91C22⋊(C71⋊C8)—22no—
92C22⋊(C71⋊C8)C2314yes—
93#14556039—22no—
94C22×D4×C71C2×C411yescharacteristic
95C23×C4×C71C2311yescharacteristic
96C23×C4×C71C2311yescharacteristic
97C2×M16×C71—22no—
98C2×M16×C71—22no—
99C4×D4×C71D414yes—
100C4×D4×C71D414yes—
101C2×C4⋊C4×C71C2×C411yescharacteristic
102C2×C22⋊C4×C71C2×C411yescharacteristic
103C2×C22⋊C4×C71C2×C411yescharacteristic
104C2×C42×C71C2311yescharacteristic
105C4⋊C8×C71—22no—
106C22⋊C8×C71—22no—
107C22⋊C8×C71—22no—
108C22⋊C8×C71C2314yes—
109C4×C8×C71—22no—
110C23×D71C22×C411yescharacteristic
111C2×Dic142—22no—
112C2×Dic142—81no—
113C2×C22⋊D71—22no—
114C2×C22⋊D71—22no—
115C2×C22⋊D71—42no—
116C2×C22⋊D71—81no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—42no—
120C22×Dic71—81no—
121C22×Dic71C22×C411yescharacteristic
122C22×Dic71C22×C411yescharacteristic
123C22×Dic71C22×C411yescharacteristic
124C22×Dic71C22×C411yescharacteristic
125C22×Dic71C22×C411yescharacteristic
126C22×Dic71C22×C411yescharacteristic
127C22×Dic71C22×C411yescharacteristic
128C2×C4×D71—22no—
129C2×C4×D71—81no—
130#11765849—82no—
131C2×C71⋊C8—24no—
132C2×C71⋊C8—82no—
133C4⋊Dic71—22no—
134C4⋊Dic71—22no—
135C4⋊Dic71—22no—
136Dic71⋊C4—22no—
137Dic71⋊C4—22no—
138Dic71⋊C4—22no—
139Dic71⋊C4—22no—
140Dic71⋊C4—22no—
141Dic71⋊C4—22no—
142Dic71⋊C4—22no—
143Dic71⋊C4—21no—
144Dic71⋊C4—21no—
145Dic71⋊C4—21no—
146Dic71⋊C4—21no—
147Dic71⋊C4—21no—
148Dic71⋊C4—21no—
149C22⋊Dic71—22no—
150C22⋊Dic71—22no—
151C22⋊Dic71—22no—
152C22⋊Dic71—22no—
153(C2×D71)⋊C4—22no—
154(C2×D71)⋊C4—22no—
155(C2×D71)⋊C4—22no—
156(C2×D71)⋊C4—21no—
157(C2×D71)⋊C4—21no—
158C4×Dic71—22no—
159C24×C71C22×C411yescharacteristic
160C24×C71C22×C411yescharacteristic
161C2×D4×C71—42no—
162C2×D4×C71C22⋊C414yes—
163C22×C4×C71—81no—
164C22×C4×C71—81no—
165C22×C4×C71C22×C411yescharacteristic
166C22×C4×C71C22×C411yescharacteristic
167C22×C4×C71C22×C411yescharacteristic
168C22×C4×C71C22×C411yescharacteristic
169C22×C4×C71C2×D412yes—
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×D412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2411yescharacteristic
174M16×C71—82no—
175M16×C71—82no—
176C2×C8×C71—24no—
177C4⋊C4×C71C22⋊C414yes—
178C22⋊C4×C71C22⋊C414yes—
179C22⋊C4×C71C22⋊C414yes—
180C42×C71C2×D412yes—
181C42×C71C2×D412yes—
182C23×C71—22no—
183C23×C71—22no—
184C23×C71—41no—
185C23×C71—41no—
186C23×C71—41no—
187C23×C71—41no—
188C23×C71C2×C22⋊C412yes—
189C23×C71C2×C22⋊C412yes—
190C23×C71C23×C411yescharacteristic
191C22×D71—21no—
192C22×D71—22no—
193C22×D71—41no—
194C22×D71—41no—
195D4×C71—44no—
196C2×C4×C71—22no—
197C2×C4×C71—42no—
198C2×C4×C71—42no—
199C2×C4×C71—41no—
200C2×C4×C71—41no—
201C2×C4×C71—41no—
202C2×C4×C71—42no—
203C2×C4×C71—42no—
204C2×C4×C71—41no—
205C2×C4×C71C2×C22⋊C412yes—
206C2×C4×C71C2×C22⋊C412yes—
207C2×C4×C71C2×C22⋊C412yes—
208C2×C4×C71C2×C22⋊C412yes—
209C2×C4×C71C22×D411yescharacteristic
210C2×C4×C71C22×D411yescharacteristic
211C22⋊D71—44no—
212C22⋊D71—82no—
213C2×Dic71—21no—
214C2×Dic71—21no—
215C2×Dic71—21no—
216C2×Dic71—21no—
217C2×Dic71—21no—
218C2×Dic71—21no—
219C2×Dic71—21no—
220C2×Dic71—22no—
221C2×Dic71—22no—
222C2×Dic71—22no—
223C2×Dic71—44no—
224C2×Dic71—41no—
225C2×Dic71—41no—
226C2×Dic71—41no—
227C2×Dic71—41no—
228C2×Dic71—82no—
229C4×D71—82no—
230Dic142—82no—
231C8×C71—44no—
232C71⋊C8—44no—
233C22×C71—21no—
234C22×C71—21no—
235C22×C71—22no—
236C22×C71—22no—
237C22×C71—22no—
238C22×C71—22no—
239C22×C71—22no—
240C22×C71—22no—
241C22×C71—44no—
242C22×C71—82no—
243C22×C71C22×C22⋊C411yescharacteristic
244C22×C71#56812yes—
245C2×D71—41no—
246C2×D71—41no—
247C2×D71—41no—
248C2×D71—82no—
249C4×C71—22no—
250C4×C71—21no—
251C4×C71—21no—
252C4×C71—22no—
253C4×C71—22no—
254C4×C71—22no—
255Dic71—41no—
256Dic71—41no—
257Dic71—41no—
258Dic71—41no—
259Dic71—41no—
260Dic71—41no—
261Dic71—41no—
262#17406—711nomaximal, Sylow
263C2×C71—22no—
264C2×C71—42no—
265C2×C71—42no—
266C2×C71#287911yescharacteristic
267C2×C71#288311yescharacteristic
268C2×C71#288911yescharacteristic
269D71—81no—
270C2×#539—711no—
271#2975D7111yescharacteristic, complemented
272#2973—711no—
273#1473—712no—
274#1472—712no—
275#1470—712no—
276#1468—712no—
277#1465—712no—
278#1463—712no—
279C71#1740611yescharacteristic, complemented, Sylow
280#539—1424no—
281#539—1424no—
282C2×#126—711no—
283C2×#126—711no—
284C2×#125—711no—
285C2×#125—711no—
286C2×#123—711no—
287C2×#123—711no—
288C2×#123—711no—
289C2×#123—711no—
290C2×C22⋊Q8—711no—
291C2×C22⋊Q8—711no—
292C2×C22⋊Q8—711no—
293C2×#121—711no—
294C2×C4×D4—711no—
295C2×C4×D4C2×D7111yescharacteristic
296C22×C4⋊C4—711no—
297#436—24no—
298#435—1424no—
299#432—712no—
300#431C2×D7112yes—
301#407—712no—
302#407C2×D7112yes—
303#407C2×D7112yes—
304C2×C22⋊C8—712no—
305#324—714no—
306#324—714no—
307#324—714no—
308#323—714no—
309C22×Q8—711no—
310C22×D4—711no—
311C22×D4—711no—
312C22×D4C4×D7111yescharacteristic
313C23×C4—711no—
314C23×C4—711no—
315C23×C4—711no—
316C23×C4C22×D7111yescharacteristic
317C23×C4C22×D7111yescharacteristic
318C2×M16—22no—
319C2×M16—22no—
320C2×M16—1422no—
321C22×C8—1422no—
322#126—1424no—
323#126—1422no—
324#126—1422no—
325#125—1424no—
326#125—1422no—
327#125—1422no—
328#123—1424no—
329#123—1422no—
330#123—1422no—
331#123—1424no—
332#123—1424no—
333C22⋊Q8—1422no—
334C22⋊Q8—1422no—
335C22⋊Q8—1424no—
336C22⋊Q8—1424no—
337#121—1424no—
338C4×D4—1422no—
339C4×D4—1422no—
340C4×D4D28414yes—
341C4×D4C22⋊D7114yes—
342C2×C4⋊C4—711no—
343C2×C4⋊C4—711no—
344C2×C4⋊C4—711no—
345C2×C4⋊C4—711no—
346C2×C4⋊C4—711no—
347C2×C4⋊C4—711no—
348C2×C4⋊C4—711no—
349C2×C4⋊C4—711no—
350C2×C4⋊C4—711no—
351C2×C4⋊C4—711no—
352C2×C4⋊C4—711no—
353C2×C4⋊C4—711no—
354C2×C4⋊C4—711no—
355C2×C4⋊C4—1422no—
356C2×C4⋊C4—1422no—
357C2×C4⋊C4C4×D7111yescharacteristic
358C2×C22⋊C4—711no—
359C2×C22⋊C4—711no—
360C2×C22⋊C4—711no—
361C2×C22⋊C4—711no—
362C2×C22⋊C4—711no—
363C2×C22⋊C4—711no—
364C2×C22⋊C4—711no—
365C2×C22⋊C4—711no—
366C2×C22⋊C4C4×D7111yescharacteristic
367C2×C22⋊C4C4×D7111yescharacteristic
368C2×C42—711no—
369C2×C42C22×D7111yescharacteristic
370C4⋊C8—22no—
371C4⋊C8—1422no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—714no—
375C22⋊C8—1422no—
376C22⋊C8—1422no—
377C22⋊C8C22×D7114yes—
378#97—1422no—
379C4×C8—22no—
380C24—711no—
381C24C2×C4×D7111yescharacteristic
382C24C2×C4×D7111yescharacteristic
383C2×Q8—5681no—
384C2×Q8—1422no—
385C2×D4—42no—
386C2×D4—2842no—
387C2×D4—5681no—
388C2×D4—1422no—
389C2×D4—1422no—
390C2×D4(C2×D71)⋊C414yes—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—2842no—
394C22×C4—5681no—
395C22×C4—5681no—
396C22×C4—711no—
397C22×C4—711no—
398C22×C4—711no—
399C22×C4—711no—
400C22×C4—711no—
401C22×C4—711no—
402C22×C4—711no—
403C22×C4—1422no—
404C22×C4—1422no—
405C22×C4—1422no—
406C22×C4C2×C4×D7111yescharacteristic
407C22×C4C2×C4×D7111yescharacteristic
408C22×C4C2×C4×D7111yescharacteristic
409C22×C4C2×C4×D7111yescharacteristic
410C22×C4C2×C22⋊D7112yes—
411C22×C4C2×C22⋊D7112yes—
412C22×C4C2×D28412yes—
413C22×C4C2×D28412yes—
414C22×C4C23×D7111yescharacteristic
415M16—82no—
416M16—82no—
417M16—5682no—
418C2×C8—24no—
419C2×C8—5682no—
420C2×C8—1424no—
421C4⋊C4—1421no—
422C4⋊C4—1421no—
423C4⋊C4—1421no—
424C4⋊C4—1421no—
425C4⋊C4—1421no—
426C4⋊C4—1421no—
427C4⋊C4—1422no—
428C4⋊C4—1422no—
429C4⋊C4—1422no—
430C4⋊C4—1422no—
431C4⋊C4—1422no—
432C4⋊C4—1422no—
433C4⋊C4—1422no—
434C4⋊C4—1422no—
435C4⋊C4—1422no—
436C4⋊C4—1422no—
437C4⋊C4(C2×D71)⋊C414yes—
438C22⋊C4—1422no—
439C22⋊C4—1422no—
440C22⋊C4—1422no—
441C22⋊C4—1422no—
442C22⋊C4—1421no—
443C22⋊C4—1421no—
444C22⋊C4—1422no—
445C22⋊C4—1422no—
446C22⋊C4—1422no—
447C22⋊C4(C2×D71)⋊C414yes—
448C22⋊C4(C2×D71)⋊C414yes—
449C42—1422no—
450C42C2×C22⋊D7112yes—
451C42C2×D28412yes—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—2841no—
459C23—2841no—
460C23—1421no—
461C23—1422no—
462C23C2×(C2×D71)⋊C412yes—
463C23C2×(C2×D71)⋊C412yes—
464C23C22×C4×D7111yescharacteristic
465Q8—5682no—
466D4—44no—
467D4—2844no—
468D4—5682no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—2844no—
479C2×C4—2841no—
480C2×C4—2841no—
481C2×C4—2841no—
482C2×C4—2841no—
483C2×C4—5682no—
484C2×C4—5682no—
485C2×C4—1421no—
486C2×C4—1421no—
487C2×C4—1421no—
488C2×C4—1421no—
489C2×C4—1421no—
490C2×C4—1421no—
491C2×C4—1421no—
492C2×C4—1422no—
493C2×C4—1422no—
494C2×C4—1422no—
495C2×C4C2×(C2×D71)⋊C412yes—
496C2×C4C2×(C2×D71)⋊C412yes—
497C2×C4C2×(C2×D71)⋊C412yes—
498C2×C4C2×(C2×D71)⋊C412yes—
499C2×C4C22×C22⋊D7111yescharacteristic
500C2×C4C22×D28411yescharacteristic
501C8—44no—
502C8—2844no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—44no—
512C22—82no—
513C22—2841no—
514C22—2841no—
515C22—2841no—
516C22—5682no—
517C22C22×(C2×D71)⋊C411yescharacteristic
518C22#1798676512yes—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—2841no—
526C4—2841no—
527C4—2841no—
528C4—2841no—
529C4—2841no—
530C4—2841no—
531C4—2841no—
532C2—22no—
533C2—42no—
534C2—42no—
535C2—5681no—
536C2#2259678411yescharacteristic
537C2#2259680011yescharacteristic
538C2#2259682211yescharacteristic
subgroupG/H
centerC22C22×(C2×D71)⋊C4
commutatorC23×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2975C2
socleC22×C71C22×C22⋊C4
prime radicalC2×#14556100C4
cosocleC22×C4×C71C24
hypercenter#2975D71
hypocenterC71#17406
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #22597369C2semidirect
3. #22586707C2semidirect
4. #22586705C2semidirect
5. #22586695C2semidirect
6. #22586689C2semidirect
7. #22586676C2semidirect
8. #22586669C2semidirect
9. C71×#2975C2semidirect
271. #2975D71semidirect
279. C71#17406semidirect

Maximal subgroups

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

subgroupindex
C2×#179865912
#225973692
#225867072
#225867052
#225866952
#225866892
#225866762
#225866692
C71×#29752
#1740671