Finite Group Databasebeta

← #1823×D71 #31997224

Subgroups — #1823×D71 #31997224

Order 18176.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×C22⋊C8×D71C211yesmaximal, characteristic
3#380×D71C211yesmaximal, characteristic
4#22588427C211yesmaximal, characteristic, complemented
5#22588426C211yesmaximal, characteristic, complemented
6#22588425C214yesmaximal, complemented
7#22588424C211yesmaximal, characteristic, complemented
8#340×D71C214yesmaximal, complemented
9C71×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D71—21no—
11C23×C4×D71C2211yescharacteristic
12C22×C8×D71C2212yes—
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4—21no—
15C2×C4⋊C4×D71—24no—
16C2×C4⋊C4×D71—21no—
17C2×C4⋊C4×D71—21no—
18C2×C4⋊C4×D71C418yescomplemented
19C2×C4⋊C4×D71C2212yes—
20C2×C4⋊C4×D71C2212yes—
21C2×C4⋊C4×D71C2212yes—
22C2×C22⋊C4×D71—21no—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×(C2×D71)⋊C8C2212yes—
25C22⋊C8×D71—24no—
26#17985859C2211yescharacteristic
27#17985858—21no—
28#17985857—21no—
29#17985856C2212yes—
30#95×D71—21no—
31#17985762C2214yescomplemented
32#17985761C2214yescomplemented
33#17985761C2214yescomplemented
34C22×C4⋊C4×C71C2211yescharacteristic
35C2×C22⋊C8×C71C2211yescharacteristic
36#380×C71C2211yescharacteristic
37#340×C71C2214yescomplemented
38C24×D71D411yescharacteristic
39C23×Dic71—21no—
40C23×Dic71D411yescharacteristic
41C22×C4×D71—28no—
42C22×C4×D71—28no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—22no—
46C22×C4×D71—44no—
47C22×C4×D71—44no—
48C22×C4×D71C2×C414yes—
49C22×C4×D71D411yescharacteristic
50C22×C4×D71D411yescharacteristic
51C22×C4×D71D414yes—
52C22×C4×D71C2311yescharacteristic
53C22×C4×D71C2311yescharacteristic
54C22×C71⋊C8C2312yes—
55C2×C8×D71—24no—
56C2×C8×D71—28no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71C2×C414yescomplemented
60C2×C4⋊Dic71C2312yes—
61C2×C4⋊Dic71C2312yes—
62C2×Dic71⋊C4—28no—
63C2×Dic71⋊C4—22no—
64C2×Dic71⋊C4—22no—
65C2×Dic71⋊C4—22no—
66C4⋊C4×D71—28no—
67C4⋊C4×D71—28no—
68C4⋊C4×D71—24no—
69C4⋊C4×D71—24no—
70C4⋊C4×D71—28no—
71C4⋊C4×D71C2×C418yes—
72C4⋊C4×D71C2×C418yescomplemented
73C2×C22⋊Dic71—21no—
74C2×(C2×D71)⋊C4—22no—
75C22⋊C4×D71—44no—
76C22⋊(C71⋊C8)—22no—
77(C2×D71)⋊C8—24no—
78C71⋊#95—22no—
79C71⋊#95—21no—
80C23×C4×C71—21no—
81C23×C4×C71C2311yescharacteristic
82C22×C8×C71C2312yes—
83C2×C4⋊C4×C71—21no—
84C2×C4⋊C4×C71—22no—
85C2×C4⋊C4×C71C2×C414yescomplemented
86C2×C4⋊C4×C71C2312yes—
87C2×C4⋊C4×C71C2312yes—
88C2×C22⋊C4×C71—21no—
89C22⋊C8×C71—22no—
90#95×C71—21no—
91C23×D71—24no—
92C23×D71—24no—
93C23×D71C22⋊C414yes—
94C23×D71D814yes—
95C23×D71SD1614yes—
96C23×D71C2×D411yescharacteristic
97C23×D71C2×D411yescharacteristic
98C22×Dic71—24no—
99C22×Dic71—24no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—42no—
103C22×Dic71—42no—
104C22×Dic71C22⋊C414yes—
105C22×Dic71C2×D411yescharacteristic
106C22×Dic71C2×D411yescharacteristic
107C2×C4×D71—24no—
108C2×C4×D71—24no—
109C2×C4×D71—24no—
110C2×C4×D71—216no—
111C2×C4×D71—216no—
112C2×C4×D71—28no—
113C2×C4×D71—28no—
114C2×C4×D71—432no—
115C2×C4×D71—44no—
116C2×C4×D71—44no—
117C2×C4×D71—44no—
118C2×C4×D71—44no—
119C2×C4×D71—44no—
120C2×C4×D71—44no—
121C2×C4×D71—44no—
122C2×C4×D71C22⋊C414yes—
123C2×C4×D71C22⋊C414yes—
124C2×C4×D71C22×C412yes—
125C2×C4×D71C22×C412yes—
126C2×C4×D71C2×D412yes—
127C2×C4×D71C2×D412yes—
128C2×C4×D71#4118yes—
129C2×C71⋊C8—22no—
130C2×C71⋊C8—24no—
131C8×D71—48no—
132C4⋊Dic71—24no—
133C4⋊Dic71—22no—
134C4⋊Dic71—24no—
135C4⋊Dic71C22×C414yes—
136Dic71⋊C4—28no—
137Dic71⋊C4—28no—
138Dic71⋊C4—44no—
139C22⋊Dic71—42no—
140(C2×D71)⋊C4—44no—
141C24×C71C2×D411yescharacteristic
142C22×C4×C71—24no—
143C22×C4×C71—24no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—42no—
147C22×C4×C71—42no—
148C22×C4×C71C22×C412yes—
149C22×C4×C71C2×D411yescharacteristic
150C22×C4×C71C2×D412yes—
151C22×C4×C71C2411yescharacteristic
152C2×C8×C71—22no—
153C2×C8×C71—24no—
154C4⋊C4×C71—24no—
155C4⋊C4×C71—22no—
156C4⋊C4×C71—24no—
157C4⋊C4×C71C22×C414yes—
158C22⋊C4×C71—42no—
159C23×C71—22no—
160C23×C71—22no—
161C23×C71C2×C22⋊C412yes—
162C23×C71C2×D812yes—
163C23×C71C2×SD1612yes—
164C23×C71C22×D411yescharacteristic
165C22×D71—22no—
166C22×D71—22no—
167C22×D71—28no—
168C22×D71—28no—
169C22×D71—216no—
170C22×D71—216no—
171C22×D71#102116yes—
172C22×D71C2×C22⋊C412yes—
173C22×D71C2×C22⋊C412yes—
174C22×D71#12314yes—
175C22×D71C2×D812yes—
176C22×D71C2×D812yes—
177C22×D71C2×SD1612yes—
178C22×D71C2×SD1612yes—
179C22×D71#13714yes—
180C2×C4×C71—22no—
181C2×C4×C71—24no—
182C2×C4×C71—24no—
183C2×C4×C71—24no—
184C2×C4×C71—42no—
185C2×C4×C71—42no—
186C2×C4×C71—48no—
187C2×C4×C71—42no—
188C2×C4×C71—42no—
189C2×C4×C71—42no—
190C2×C4×C71C2×C22⋊C412yes—
191C2×C4×C71C23×C411yescharacteristic
192C2×C4×C71C22×D411yescharacteristic
193C2×C4×C71C2×#4114yes—
194C2×Dic71—22no—
195C2×Dic71—28no—
196C2×Dic71—24no—
197C2×Dic71—42no—
198C2×Dic71—42no—
199C2×Dic71—48no—
200C2×Dic71—42no—
201C2×Dic71—42no—
202C2×Dic71—42no—
203C2×Dic71C2×C22⋊C412yes—
204C2×Dic71C2×C22⋊C412yes—
205C2×Dic71#12314yes—
206C4×D71—28no—
207C4×D71—24no—
208C4×D71—24no—
209C4×D71—28no—
210C4×D71—416no—
211C4×D71—416no—
212C4×D71—88no—
213C4×D71#11718yes—
214C4×D71#12318yes—
215C8×C71—42no—
216C71⋊C8—42no—
217C22×C71—24no—
218C22×C71—24no—
219C22×C71—22no—
220C22×C71—22no—
221C22×C71C2×#10214yes—
222C22×C71C22×C22⋊C411yescharacteristic
223C22×C71C2×#12312yes—
224C22×C71C22×D811yescharacteristic
225C22×C71C22×SD1611yescharacteristic
226C22×C71C2×#13712yes—
227C2×D71—24no—
228C2×D71—24no—
229C2×D71—22no—
230C2×D71—22no—
231C2×D71—232no—
232C2×D71#38114yes—
233C2×D71C2×#10214yes—
234C2×D71C2×#10214yes—
235C2×D71#41914yes—
236C2×D71#48018yes—
237C2×D71#48318yes—
238C4×C71—22no—
239C4×C71—44no—
240C4×C71—82no—
241C4×C71C2×#11712yes—
242C4×C71C2×#12312yes—
243Dic71—22no—
244Dic71—44no—
245Dic71—82no—
246Dic71#38114yes—
247C2×#1823—711nomaximal, Sylow
248C2×C71—24no—
249C2×C71C2×#38111yescharacteristic
250C2×C71C22×#10211yescharacteristic
251C2×C71C2×#41911yescharacteristic
252C2×C71C2×#48012yes—
253C2×C71C2×#48312yes—
254D71—24no—
255D71#182318yesdirect product factor
256C23×C4⋊C4—711no—
257C22×C22⋊C8—711no—
258C2×#380—711no—
259#1823—711no—
260#1823—711no—
261#1823—714no—
262#1823—711no—
263#1823D7111yescharacteristic, direct product factor
264C2×#340—714no—
265C71C2×#182311yescharacteristic, complemented, Sylow
266C24×C4—711no—
267C24×C4—1421no—
268C23×C8—712no—
269C22×C4⋊C4—711no—
270C22×C4⋊C4—718no—
271C22×C4⋊C4—712no—
272C22×C4⋊C4—712no—
273C22×C4⋊C4—712no—
274C22×C4⋊C4—1421no—
275C22×C4⋊C4—1421no—
276C22×C4⋊C4—1424no—
277C22×C4⋊C4—1421no—
278C22×C4⋊C4C2×D7111yescharacteristic
279C22×C22⋊C4—1421no—
280C2×C22⋊C8—711no—
281C2×C22⋊C8—712no—
282C2×C22⋊C8—1424no—
283C2×C22⋊C8C2×D7111yescharacteristic
284#380—711no—
285#380—712no—
286#380—1421no—
287#380—1421no—
288#380C2×D7111yescharacteristic
289C2×#95—1421no—
290#340—714no—
291#340—714no—
292#340—714no—
293#340C2×D7114yescomplemented
294C25—711no—
295C23×C4—21no—
296C23×C4—2844no—
297C23×C4—2844no—
298C23×C4—711no—
299C23×C4—711no—
300C23×C4—714no—
301C23×C4—711no—
302C23×C4—711no—
303C23×C4—714no—
304C23×C4—711no—
305C23×C4—1422no—
306C23×C4—1422no—
307C23×C4—1422no—
308C23×C4—1428no—
309C23×C4—1428no—
310C23×C4—1421no—
311C23×C4C22×D7111yescharacteristic
312C22×C8—712no—
313C22×C8—1424no—
314C22×C8—1428no—
315C22×C8C22×D7112yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—714no—
319C2×C4⋊C4—712no—
320C2×C4⋊C4—718no—
321C2×C4⋊C4—718no—
322C2×C4⋊C4—712no—
323C2×C4⋊C4—1428no—
324C2×C4⋊C4—1424no—
325C2×C4⋊C4—1424no—
326C2×C4⋊C4—1428no—
327C2×C4⋊C4—1428no—
328C2×C4⋊C4—1422no—
329C2×C4⋊C4—1422no—
330C2×C4⋊C4—1422no—
331C2×C4⋊C4—1428no—
332C2×C4⋊C4—1422no—
333C2×C4⋊C4—1421no—
334C2×C4⋊C4C4×D7114yescomplemented
335C2×C4⋊C4C22×D7112yes—
336C2×C4⋊C4C22×D7112yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—2844no—
339C2×C22⋊C4—1422no—
340C2×C22⋊C4—1421no—
341C22⋊C8—22no—
342C22⋊C8—1424no—
343C22⋊C8—1422no—
344#95—21no—
345#95—1422no—
346#95—1421no—
347C24—711no—
348C24—711no—
349C24—714no—
350C24—714no—
351C24—714no—
352C24—1424no—
353C24—1424no—
354C24D4×D7111yescharacteristic
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—28432no—
362C22×C4—2844no—
363C22×C4—2844no—
364C22×C4—2844no—
365C22×C4—2844no—
366C22×C4—2844no—
367C22×C4—2844no—
368C22×C4—2844no—
369C22×C4—2842no—
370C22×C4—2842no—
371C22×C4—711no—
372C22×C4—712no—
373C22×C4—712no—
374C22×C4—718no—
375C22×C4—711no—
376C22×C4—714no—
377C22×C4—712no—
378C22×C4—712no—
379C22×C4—714no—
380C22×C4—714no—
381C22×C4—1428no—
382C22×C4—1428no—
383C22×C4—1424no—
384C22×C4—14216no—
385C22×C4—14216no—
386C22×C4—1424no—
387C22×C4—1422no—
388C22×C4—1422no—
389C22×C4—1424no—
390C22×C4—1424no—
391C22×C4—1424no—
392C22×C4C2×C4×D7112yes—
393C22×C4D4×D7112yes—
394C22×C4D4×D7111yescharacteristic
395C22×C4C23×D7111yescharacteristic
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—2848no—
399C2×C8—1422no—
400C2×C8—1424no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—2844no—
405C4⋊C4—714no—
406C4⋊C4—1428no—
407C4⋊C4—1428no—
408C4⋊C4—1424no—
409C4⋊C4—1422no—
410C4⋊C4—1424no—
411C4⋊C4C2×C4×D7114yes—
412C22⋊C4—42no—
413C22⋊C4—2844no—
414C22⋊C4—2842no—
415C23—22no—
416C23—22no—
417C23—712no—
418C23—714no—
419C23—712no—
420C23—712no—
421C23—714no—
422C23—712no—
423C23—7116no—
424C23—712no—
425C23—712no—
426C23—1428no—
427C23—14216no—
428C23—14216no—
429C23—1428no—
430C23—1422no—
431C23—1422no—
432C23C22⋊C4×D7112yes—
433C23D8×D7112yes—
434C23SD16×D7112yes—
435C23C2×D4×D7111yescharacteristic
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—28416no—
447C2×C4—28416no—
448C2×C4—2848no—
449C2×C4—2842no—
450C2×C4—2842no—
451C2×C4—2842no—
452C2×C4—2842no—
453C2×C4—2842no—
454C2×C4—5688no—
455C2×C4—712no—
456C2×C4—714no—
457C2×C4—718no—
458C2×C4—718no—
459C2×C4—712no—
460C2×C4—1428no—
461C2×C4—1428no—
462C2×C4—1424no—
463C2×C4—1428no—
464C2×C4—1422no—
465C2×C4—1424no—
466C2×C4—1424no—
467C2×C4C22⋊C4×D7112yes—
468C2×C4C22×C4×D7111yescharacteristic
469C2×C4C2×D4×D7111yescharacteristic
470C2×C4#41×D7114yes—
471C8—42no—
472C8—2842no—
473C22—24no—
474C22—24no—
475C22—22no—
476C22—22no—
477C22—718no—
478C22—718no—
479C22—714no—
480C22—714no—
481C22—714no—
482C22—714no—
483C22—14232no—
484C22—1424no—
485C22—1424no—
486C22—1422no—
487C22—1422no—
488C22#102×D7114yes—
489C22C2×C22⋊C4×D7111yescharacteristic
490C22#123×D7112yes—
491C22C2×D8×D7111yescharacteristic
492C22C2×SD16×D7111yescharacteristic
493C22#137×D7112yes—
494C4—22no—
495C4—44no—
496C4—82no—
497C4—2844no—
498C4—5682no—
499C4—714no—
500C4—1422no—
501C4#117×D7112yes—
502C4#123×D7112yes—
503C2—24no—
504C2—718no—
505C2—1424no—
506C2#381×D7111yescharacteristic
507C2C2×#102×D7111yescharacteristic
508C2#419×D7111yescharacteristic
509C2#480×D7112yes—
510C2#483×D7112yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1823C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1823D71
hypocenterC71C2×#1823
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #22588427C2semidirect
5. #22588426C2semidirect
6. #22588425C2semidirect
7. #22588424C2semidirect
8. #340×D71C2semidirect
9. C71×#1823C2semidirect
18. C2×C4⋊C4×D71C4semidirect
31. #17985762C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
37. #340×C71C22semidirect
59. C2×C4⋊Dic71C2×C4semidirect
72. C4⋊C4×D71C2×C4semidirect
85. C2×C4⋊C4×C71C2×C4semidirect
255. D71#1823direct (N × H)
263. #1823D71direct (N × H)
265. C71C2×#1823semidirect
293. #340C2×D71semidirect
334. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D712
C2×C22⋊C8×D712
#380×D712
#225884272
#225884262
#225884252
#225884242
#340×D712
C71×#18232
C2×#182371