Finite Group Databasebeta

← #31997249

Subgroups — #31997249

Order 18176.

1774 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×#17985962C211yesmaximal, characteristic
3#22590719C211yesmaximal, characteristic
4#22588429C214yesmaximal, complemented
5#22588425C214yesmaximal, complemented
6#22588061C211yesmaximal, characteristic, complemented
7#22588060C211yesmaximal, characteristic, complemented
8#22588059C211yesmaximal, characteristic, complemented
9C71×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D71—21no—
11C23×C4×D71C2211yescharacteristic
12C22×#11765849C2211yescharacteristic
13C2×#14556087C2211yescharacteristic
14C22×C4⋊Dic71C2211yescharacteristic
15C22×Dic71⋊C4—21no—
16C2×C4⋊C4×D71—24no—
17C2×C4⋊C4×D71—21no—
18C2×C4⋊C4×D71—21no—
19C2×C4⋊C4×D71C418yescomplemented
20C2×C4⋊C4×D71C2212yes—
21C2×C4⋊C4×D71C2212yes—
22#17985962—24no—
23C2×(C2×D71)⋊C8C2212yes—
24C2×(C2×D71)⋊C8C2212yes—
25#17985860—21no—
26#17985856C2212yes—
27#17985856C2212yes—
28#17985854—21no—
29C4×(C2×D71)⋊C4—21no—
30C4×(C2×D71)⋊C4—21no—
31#17985762C2214yescomplemented
32#17985761C2214yescomplemented
33#17985761C2214yescomplemented
34C22×M16×C71C2211yescharacteristic
35C2×#117×C71C2211yescharacteristic
36C22×C4⋊C4×C71C2211yescharacteristic
37#340×C71C2214yescomplemented
38C24×D71D411yescharacteristic
39C23×Dic71—21no—
40C23×Dic71D411yescharacteristic
41C22×C4×D71—28no—
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—22no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—48no—
49C22×C4×D71C2×C414yes—
50C22×C4×D71D411yescharacteristic
51C22×C4×D71D411yescharacteristic
52C22×C4×D71D414yes—
53C22×C4×D71C2311yescharacteristic
54C22×C4×D71C2311yescharacteristic
55C2×#11765849—22no—
56C2×#11765849—22no—
57C2×#11765849—22no—
58C22×C71⋊C8C2312yes—
59#14556087—22no—
60#14556087—22no—
61C2×C4⋊Dic71—21no—
62C2×C4⋊Dic71—22no—
63C2×C4⋊Dic71C2×C414yes—
64C2×C4⋊Dic71C2312yes—
65C2×C4⋊Dic71C2312yes—
66C2×Dic71⋊C4—28no—
67C2×Dic71⋊C4—22no—
68C2×Dic71⋊C4—22no—
69C4⋊C4×D71—28no—
70C4⋊C4×D71—28no—
71C4⋊C4×D71—24no—
72C4⋊C4×D71—24no—
73C4⋊C4×D71C2×C418yes—
74C4⋊C4×D71C2×C418yes—
75C2×C22⋊Dic71—21no—
76C2×(C2×D71)⋊C4—22no—
77C2×(C2×D71)⋊C4—22no—
78C2×C4×Dic71—21no—
79(C2×D71)⋊C8—24no—
80(C2×D71)⋊C8—24no—
81C71⋊#95—22no—
82C71⋊#95—22no—
83C23×C4×C71—21no—
84C23×C4×C71C2311yescharacteristic
85C2×M16×C71—22no—
86C2×M16×C71—22no—
87C2×M16×C71—22no—
88C22×C8×C71C2312yes—
89#117×C71—22no—
90#117×C71—22no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71—22no—
93C2×C4⋊C4×C71C2×C414yes—
94C2×C4⋊C4×C71C2312yes—
95C2×C4⋊C4×C71C2312yes—
96C2×C22⋊C4×C71—21no—
97C2×C42×C71—21no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—24no—
101C23×D71—24no—
102C23×D71C22⋊C414yes—
103C23×D71C2×D411yescharacteristic
104C23×D71C2×D411yescharacteristic
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—24no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—44no—
111C22×Dic71C22⋊C414yes—
112C22×Dic71C2×D411yescharacteristic
113C22×Dic71C2×D411yescharacteristic
114C2×C4×D71—28no—
115C2×C4×D71—28no—
116C2×C4×D71—28no—
117C2×C4×D71—22no—
118C2×C4×D71—22no—
119C2×C4×D71—28no—
120C2×C4×D71—22no—
121C2×C4×D71—22no—
122C2×C4×D71—21no—
123C2×C4×D71—21no—
124C2×C4×D71—21no—
125C2×C4×D71—21no—
126C2×C4×D71—21no—
127C2×C4×D71—21no—
128C2×C4×D71—21no—
129C2×C4×D71—21no—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71—48no—
133C2×C4×D71—432no—
134C2×C4×D71—48no—
135C2×C4×D71C22⋊C414yes—
136C2×C4×D71C22⋊C414yes—
137C2×C4×D71C22×C412yes—
138C2×C4×D71C22×C412yes—
139C2×C4×D71C2×D412yes—
140C2×C4×D71C2×D412yes—
141#11765849—44no—
142C2×C71⋊C8—22no—
143C2×C71⋊C8—22no—
144C2×C71⋊C8—22no—
145C4⋊Dic71—24no—
146C4⋊Dic71—22no—
147C4⋊Dic71—22no—
148C4⋊Dic71—22no—
149C4⋊Dic71C22×C414yes—
150Dic71⋊C4—28no—
151Dic71⋊C4—28no—
152C22⋊Dic71—42no—
153(C2×D71)⋊C4—44no—
154(C2×D71)⋊C4—44no—
155C4×Dic71—42no—
156C24×C71C2×D411yescharacteristic
157C22×C4×C71—24no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—44no—
161C22×C4×C71C22×C412yes—
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D412yes—
164C22×C4×C71#4112yes—
165C22×C4×C71#4112yes—
166C22×C4×C71#4112yes—
167C22×C4×C71#4112yes—
168C22×C4×C71C2411yescharacteristic
169M16×C71—44no—
170C2×C8×C71—22no—
171C2×C8×C71—22no—
172C2×C8×C71—22no—
173C4⋊C4×C71—24no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—22no—
177C4⋊C4×C71C22×C414yes—
178C22⋊C4×C71—42no—
179C42×C71—42no—
180C23×C71—22no—
181C23×C71—22no—
182C23×C71C2×C22⋊C412yes—
183C23×C71#12312yes—
184C23×C71#12312yes—
185C23×C71C22×D411yescharacteristic
186C22×D71—216no—
187C22×D71—216no—
188C22×D71—21no—
189C22×D71—21no—
190C22×D71—21no—
191C22×D71—21no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—28no—
195C22×D71—28no—
196C22×D71—28no—
197C22×D71C2×C22⋊C412yes—
198C22×D71C2×C22⋊C412yes—
199C22×D71AGL(1,ℤ/8)14yes—
200C22×D71#13714yes—
201C2×C4×C71—24no—
202C2×C4×C71—48no—
203C2×C4×C71—44no—
204C2×C4×C71—42no—
205C2×C4×C71—42no—
206C2×C4×C71—42no—
207C2×C4×C71C2×C22⋊C412yes—
208C2×C4×C71#11714yes—
209C2×C4×C71#11714yes—
210C2×C4×C71#12314yes—
211C2×C4×C71#12312yes—
212C2×C4×C71#12314yes—
213C2×C4×C71#12312yes—
214C2×C4×C71C23×C411yescharacteristic
215C2×C4×C71C22×D411yescharacteristic
216C2×C4×C71C2×#4111yescharacteristic
217C2×C4×C71C2×#4111yescharacteristic
218C2×C4×C71C2×#4111yescharacteristic
219C2×C4×C71C2×#4111yescharacteristic
220C2×Dic71—21no—
221C2×Dic71—21no—
222C2×Dic71—21no—
223C2×Dic71—21no—
224C2×Dic71—28no—
225C2×Dic71—24no—
226C2×Dic71—48no—
227C2×Dic71—44no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71C2×C22⋊C412yes—
232C2×Dic71C2×C22⋊C412yes—
233C4×D71—22no—
234C4×D71—22no—
235C4×D71—22no—
236C4×D71—22no—
237C4×D71—22no—
238C4×D71—22no—
239C4×D71—22no—
240C4×D71—22no—
241C4×D71—28no—
242C4×D71—28no—
243C4×D71—416no—
244C4×D71—416no—
245C8×C71—42no—
246C71⋊C8—42no—
247C22×C71—24no—
248C22×C71—24no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71#38114yes—
252C22×C71C22×C22⋊C411yescharacteristic
253C22×C71C2×#12311yescharacteristic
254C22×C71C2×#12311yescharacteristic
255C22×C71C2×AGL(1,ℤ/8)12yes—
256C22×C71C2×#13712yes—
257C2×D71—232no—
258C2×D71—24no—
259C2×D71—24no—
260C2×D71—24no—
261C2×D71—24no—
262C2×D71—22no—
263C2×D71—22no—
264C2×D71—22no—
265C2×D71—22no—
266C2×D71#41814yes—
267C2×D71#41914yes—
268C4×C71—44no—
269C4×C71—82no—
270C4×C71#38114yes—
271C4×C71C2×#11711yescharacteristic
272C4×C71C2×#11711yescharacteristic
273C4×C71C2×#12311yescharacteristic
274C4×C71C2×#12311yescharacteristic
275Dic71—22no—
276Dic71—22no—
277Dic71—44no—
278Dic71—82no—
279#20700—711nomaximal, Sylow
280C2×C71—24no—
281C2×C71C2×#38111yescharacteristic
282C2×C71C2×#41811yescharacteristic
283C2×C71C2×#41911yescharacteristic
284C2×C71#309212yes—
285C2×C71#309212yes—
286D71—24no—
287D71—24no—
288C23×C4⋊C4—711no—
289C2×#407—711no—
290#2280—711no—
291#1824—714no—
292#1823—714no—
293#1736—711no—
294#1736—711no—
295#1736—711no—
296#1736D7111yescharacteristic, complemented
297C71#2070011yescharacteristic, complemented, Sylow
298C24×C4—711no—
299C24×C4—1421no—
300C22×M16—711no—
301C22×M16C2×D7111yescharacteristic
302C2×#117—711no—
303C2×#117C2×D7111yescharacteristic
304C22×C4⋊C4—711no—
305C22×C4⋊C4—718no—
306C22×C4⋊C4—712no—
307C22×C4⋊C4—712no—
308C22×C4⋊C4—1421no—
309C22×C4⋊C4—1421no—
310C22×C4⋊C4—1424no—
311C22×C4⋊C4—1421no—
312C22×C4⋊C4C2×D7111yescharacteristic
313#407—1424no—
314C2×C22⋊C8—712no—
315C2×C22⋊C8—712no—
316#381—1421no—
317#380—712no—
318#380—712no—
319#379—1421no—
320C4×C22⋊C4—1421no—
321C4×C22⋊C4—1421no—
322#340—714no—
323#340—714no—
324#340—714no—
325#340C2×D7114yescomplemented
326C25—711no—
327C23×C4—21no—
328C23×C4—2848no—
329C23×C4—711no—
330C23×C4—714no—
331C23×C4—711no—
332C23×C4—711no—
333C23×C4—714no—
334C23×C4—711no—
335C23×C4—711no—
336C23×C4—1422no—
337C23×C4—1422no—
338C23×C4—1428no—
339C23×C4—1421no—
340C23×C4—1422no—
341C23×C4—1422no—
342C23×C4—1422no—
343C23×C4—1422no—
344C23×C4C22×D7111yescharacteristic
345C2×M16—22no—
346C2×M16—22no—
347C2×M16—22no—
348C2×M16—1422no—
349C2×M16—1422no—
350C2×M16—1422no—
351C22×C8—712no—
352C22×C8C22×D7112yes—
353#117—22no—
354#117—22no—
355#117—1422no—
356#117—1422no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—714no—
360C2×C4⋊C4—712no—
361C2×C4⋊C4—718no—
362C2×C4⋊C4—718no—
363C2×C4⋊C4—712no—
364C2×C4⋊C4—1424no—
365C2×C4⋊C4—1424no—
366C2×C4⋊C4—1428no—
367C2×C4⋊C4—1428no—
368C2×C4⋊C4—1422no—
369C2×C4⋊C4—1422no—
370C2×C4⋊C4—1428no—
371C2×C4⋊C4—1422no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4C4×D7114yes—
374C2×C4⋊C4C22×D7112yes—
375C2×C4⋊C4C22×D7112yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1422no—
378C2×C22⋊C4—1422no—
379C2×C22⋊C4—1421no—
380C2×C42—21no—
381C2×C42—1421no—
382C22⋊C8—1424no—
383C22⋊C8—1424no—
384#95—1422no—
385#95—1422no—
386C24—711no—
387C24—711no—
388C24—714no—
389C24—1424no—
390C24—1424no—
391C24—1422no—
392C24—1422no—
393C24D4×D7111yescharacteristic
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—2848no—
399C22×C4—28432no—
400C22×C4—2848no—
401C22×C4—2844no—
402C22×C4—711no—
403C22×C4—711no—
404C22×C4—714no—
405C22×C4—712no—
406C22×C4—712no—
407C22×C4—712no—
408C22×C4—712no—
409C22×C4—714no—
410C22×C4—714no—
411C22×C4—1428no—
412C22×C4—1428no—
413C22×C4—1422no—
414C22×C4—1422no—
415C22×C4—1424no—
416C22×C4—1422no—
417C22×C4—1422no—
418C22×C4—1421no—
419C22×C4—1421no—
420C22×C4—1421no—
421C22×C4—1421no—
422C22×C4—1421no—
423C22×C4—1421no—
424C22×C4—1421no—
425C22×C4—1421no—
426C22×C4—1428no—
427C22×C4—1428no—
428C22×C4—1428no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1428no—
432C22×C4—1422no—
433C22×C4—1422no—
434C22×C4C2×C4×D7112yes—
435C22×C4D4×D7112yes—
436C22×C4D4×D7111yescharacteristic
437C22×C4#1176586412yes—
438C22×C4#1176586512yes—
439C22×C4#1176586512yes—
440C22×C4#1176586612yes—
441C22×C4C23×D7111yescharacteristic
442M16—44no—
443M16—2844no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1422no—
448C2×C8—1422no—
449C2×C8—1422no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—714no—
455C4⋊C4—1428no—
456C4⋊C4—1428no—
457C4⋊C4—1422no—
458C4⋊C4—1422no—
459C4⋊C4—1422no—
460C4⋊C4—1424no—
461C4⋊C4C2×C4×D7114yes—
462C22⋊C4—42no—
463C22⋊C4—2844no—
464C22⋊C4—2844no—
465C22⋊C4—2842no—
466C42—42no—
467C42—2842no—
468C23—22no—
469C23—22no—
470C23—712no—
471C23—712no—
472C23—714no—
473C23—714no—
474C23—1428no—
475C23—1428no—
476C23—14216no—
477C23—14216no—
478C23—1428no—
479C23—1421no—
480C23—1421no—
481C23—1421no—
482C23—1421no—
483C23—1422no—
484C23—1422no—
485C23C22⋊C4×D7112yes—
486C23#1455611012yes—
487C23#1455611312yes—
488C23C2×D4×D7111yescharacteristic
489C2×C4—24no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—28416no—
496C2×C4—28416no—
497C2×C4—2848no—
498C2×C4—2844no—
499C2×C4—2842no—
500C2×C4—2842no—
501C2×C4—2842no—
502C2×C4—712no—
503C2×C4—712no—
504C2×C4—1424no—
505C2×C4—1428no—
506C2×C4—1421no—
507C2×C4—1421no—
508C2×C4—1421no—
509C2×C4—1421no—
510C2×C4—1428no—
511C2×C4—1428no—
512C2×C4—1422no—
513C2×C4—1422no—
514C2×C4—1422no—
515C2×C4—1422no—
516C2×C4—1422no—
517C2×C4—1422no—
518C2×C4—1422no—
519C2×C4—1422no—
520C2×C4C22⋊C4×D7112yes—
521C2×C4#1455608514yes—
522C2×C4#1455608614yes—
523C2×C4#1455611012yes—
524C2×C4#1455611014yes—
525C2×C4#1455611312yes—
526C2×C4#1455611314yes—
527C2×C4C22×C4×D7111yescharacteristic
528C2×C4C2×D4×D7111yescharacteristic
529C2×C4C2×#1176586411yescharacteristic
530C2×C4C2×#1176586511yescharacteristic
531C2×C4C2×#1176586511yescharacteristic
532C2×C4C2×#1176586611yescharacteristic
533C8—42no—
534C8—2842no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—714no—
540C22—714no—
541C22—14232no—
542C22—1424no—
543C22—1424no—
544C22—1424no—
545C22—1424no—
546C22—1422no—
547C22—1422no—
548C22—1422no—
549C22—1422no—
550C22#1798586114yes—
551C22C2×C22⋊C4×D7111yescharacteristic
552C22C2×#1455611011yescharacteristic
553C22C2×#1455611311yescharacteristic
554C22AGL(1,ℤ/8)×D7112yes—
555C22#137×D7112yes—
556C4—44no—
557C4—82no—
558C4—2844no—
559C4—5682no—
560C4—1422no—
561C4—1422no—
562C4#1798586114yes—
563C4C2×#1455608511yescharacteristic
564C4C2×#1455608611yescharacteristic
565C4C2×#1455611011yescharacteristic
566C4C2×#1455611311yescharacteristic
567C2—24no—
568C2—1424no—
569C2—1424no—
570C2C2×#1798586111yescharacteristic
571C2#418×D7111yescharacteristic
572C2#419×D7111yescharacteristic
573C2#2259826112yes—
574C2#2259826612yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1736C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1736D71
hypocenterC71#20700
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #22588429C2semidirect
5. #22588425C2semidirect
6. #22588061C2semidirect
7. #22588060C2semidirect
8. #22588059C2semidirect
9. C71×#1736C2semidirect
19. C2×C4⋊C4×D71C4semidirect
31. #17985762C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
37. #340×C71C22semidirect
296. #1736D71semidirect
297. C71#20700semidirect
325. #340C2×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×#179859622
#225907192
#225884292
#225884252
#225880612
#225880602
#225880592
C71×#17362
#2070071