Finite Group Databasebeta

← #31998223

Subgroups — #31998223

Order 18176.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986482C211yesmaximal, characteristic
2C2×#17986042C211yesmaximal, characteristic, complemented
3C2×#17986035C211yesmaximal, characteristic, complemented
4C2×#17985966C211yesmaximal, characteristic
5#22588537C212yesmaximal, complemented
6#22588537C212yesmaximal, complemented
7#22588532C212yesmaximal, complemented
8#22588527C212yesmaximal, complemented
9#22588059C211yesmaximal, characteristic, complemented
10C2×#17985761C211yesmaximal, characteristic, complemented
11C71×#2311C211yesmaximal, characteristic
12C22×#11765849C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14#17986482—24no—
15#17986482—24no—
16C2×Q8×Dic71—21no—
17C2×D4×Dic71—22no—
18C2×D4×Dic71—21no—
19C2×#14556087C2212yes—
20C2×#14556087C2211yescharacteristic
21C22×C4⋊Dic71C2212yes—
22C22×C4⋊Dic71C2211yescharacteristic
23#17986042—24no—
24#17986035—24no—
25C2×#14556061C2211yescharacteristic
26C2×#14556061C2211yescharacteristic
27C2×#14556061C2212yes—
28C2×#14556059C2211yescharacteristic
29C2×#14556059C2211yescharacteristic
30C2×#14556059C2212yes—
31#17985966C418yescomplemented
32C2×C22⋊(C71⋊C8)C2212yes—
33C2×C22⋊(C71⋊C8)C2212yes—
34#17985761C2212yes—
35#17985761C2212yes—
36#17985761C2212yes—
37#17985761C2212yes—
38C22×#41×C71C2211yescharacteristic
39C2×#117×C71C2211yescharacteristic
40C22×C4⋊C4×C71C2211yescharacteristic
41#391×C71—21no—
42#390×C71—21no—
43#386×C71—22no—
44#380×C71C2212yes—
45#380×C71C2212yes—
46C23×Dic71—22no—
47C23×Dic71—21no—
48C2×#11765849—22no—
49C2×#11765849—22no—
50C2×#11765849C2×C414yescomplemented
51C22×C71⋊C8—22no—
52C22×C71⋊C8—22no—
53C22×C71⋊C8C2×C414yescomplemented
54C22×C71⋊C8C2311yescharacteristic
55C22×C71⋊C8C2311yescharacteristic
56C22×C71⋊C8C2312yes—
57Q8×Dic71—28no—
58D4×Dic71—216no—
59D4×Dic71—28no—
60#14556087—24no—
61#14556087—24no—
62#14556087—22no—
63#14556087D414yes—
64C2×C4⋊Dic71—24no—
65C2×C4⋊Dic71—24no—
66C2×C4⋊Dic71—22no—
67C2×C4⋊Dic71—21no—
68C2×C4⋊Dic71—22no—
69C2×C4⋊Dic71D414yes—
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2312yes—
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C22⋊Dic71—22no—
74C2×C22⋊Dic71—21no—
75C2×C4×Dic71—22no—
76C2×C4×Dic71—21no—
77#14556061—22no—
78#14556061—22no—
79#14556061—24no—
80#14556059—22no—
81#14556059—22no—
82#14556059—24no—
83C22⋊(C71⋊C8)C2×C418yescomplemented
84C22⋊(C71⋊C8)C2×C418yescomplemented
85C2×#41×C71—22no—
86C2×#41×C71—22no—
87C2×#41×C71—22no—
88C2×#41×C71—22no—
89C2×#41×C71C2×C414yes—
90C2×#41×C71Q814yes—
91C22×Q8×C71D411yescharacteristic
92C22×D4×C71D412yes—
93C22×D4×C71D411yescharacteristic
94C23×C4×C71—21no—
95C23×C4×C71C2312yes—
96C23×C4×C71C2311yescharacteristic
97#117×C71—22no—
98#117×C71—22no—
99C2×C4⋊C4×C71—22no—
100C2×C4⋊C4×C71—22no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71C2312yes—
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71—22no—
107C2×C22⋊C4×C71—22no—
108C2×C42×C71—21no—
109#95×C71—22no—
110#95×C71—22no—
111C22×Dic71—22no—
112C22×Dic71—21no—
113C22×Dic71—21no—
114C22×Dic71—28no—
115C22×Dic71—24no—
116C22×Dic71—81no—
117C22×Dic71—82no—
118C22×Dic71—81no—
119C22×Dic71—82no—
120#11765849—28no—
121C2×C71⋊C8—28no—
122C2×C71⋊C8—21no—
123C2×C71⋊C8—22no—
124C2×C71⋊C8—21no—
125C2×C71⋊C8—21no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8C22×C412yes—
129C2×C71⋊C8C22×C412yes—
130C2×C71⋊C8C22×C414yes—
131C4⋊Dic71—28no—
132C4⋊Dic71—24no—
133C4⋊Dic71—21no—
134C4⋊Dic71—22no—
135C4⋊Dic71—21no—
136C4⋊Dic71C2×D412yes—
137C4⋊Dic71C2×D414yes—
138C4⋊Dic71C2×D412yes—
139C22⋊Dic71—28no—
140C22⋊Dic71—24no—
141C4×Dic71—28no—
142C4×Dic71—24no—
143C24×C71C2×D412yes—
144C24×C71C2×D411yescharacteristic
145#41×C71—28no—
146#41×C71—28no—
147#41×C71—28no—
148#41×C71—28no—
149C2×Q8×C71—22no—
150C2×Q8×C71—22no—
151C2×Q8×C71C4⋊C414yes—
152C2×D4×C71—24no—
153C2×D4×C71—24no—
154C2×D4×C71—22no—
155C2×D4×C71—22no—
156C2×D4×C71C4⋊C418yes—
157C2×D4×C71C4⋊C414yes—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—22no—
165C22×C4×C71—42no—
166C22×C4×C71—81no—
167C22×C4×C71—81no—
168C22×C4×C71C22×C414yes—
169C22×C4×C71C22×C412yes—
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×D412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×Q814yes—
175C22×C4×C71C2×Q812yes—
176C22×C4×C71#4112yes—
177C22×C4×C71#4112yes—
178C22×C4×C71C2411yescharacteristic
179C4⋊C4×C71—21no—
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—42no—
186C22⋊C4×C71—42no—
187C22⋊C4×C71—82no—
188C22⋊C4×C71—82no—
189C42×C71—42no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—22no—
195C23×C71—22no—
196C23×C71—22no—
197C23×C71—22no—
198C23×C71—22no—
199C23×C71—22no—
200C23×C71C2×C4⋊C414yes—
201C23×C71C2×C4⋊C412yes—
202C23×C71#12012yes—
203C23×C71#12312yes—
204C23×C71C22×D411yescharacteristic
205Q8×C71—28no—
206D4×C71—216no—
207D4×C71—28no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—24no—
211C2×C4×C71—24no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—42no—
217C2×C4×C71—41no—
218C2×C4×C71—42no—
219C2×C4×C71—41no—
220C2×C4×C71—42no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—84no—
224C2×C4×C71C2×C4⋊C414yes—
225C2×C4×C71C2×C4⋊C412yes—
226C2×C4×C71C4×D414yes—
227C2×C4×C71C4×D414yes—
228C2×C4×C71#12012yes—
229C2×C4×C71C22⋊Q814yes—
230C2×C4×C71C22⋊Q814yes—
231C2×C4×C71#12312yes—
232C2×C4×C71C23×C411yescharacteristic
233C2×C4×C71C22×D411yescharacteristic
234C2×C4×C71C22×D411yescharacteristic
235C2×C4×C71C22×Q811yescharacteristic
236C2×C4×C71C2×#4111yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C2×Dic71—22no—
239C2×Dic71—24no—
240C2×Dic71—22no—
241C2×Dic71—42no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—42no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—84no—
248C2×Dic71—88no—
249C71⋊C8—22no—
250C71⋊C8—22no—
251C71⋊C8—24no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—24no—
259C22×C71—24no—
260C22×C71—24no—
261C22×C71—42no—
262C22×C71—42no—
263C22×C71—42no—
264C22×C71—42no—
265C22×C71#38514yes—
266C22×C71C22×C4⋊C411yescharacteristic
267C22×C71C2×#12011yescharacteristic
268C22×C71C2×#12311yescharacteristic
269C22×C71#57611yescharacteristic
270C22×C71#57711yescharacteristic
271C22×C71#57711yescharacteristic
272C22×C71#57811yescharacteristic
273C4×C71—24no—
274C4×C71—82no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71#38514yes—
278C4×C71C2×C4×D411yescharacteristic
279C4×C71C2×C4×D411yescharacteristic
280C4×C71C2×C22⋊Q811yescharacteristic
281C4×C71C2×C22⋊Q811yescharacteristic
282Dic71—44no—
283Dic71—42no—
284Dic71—42no—
285#20849—711nomaximal, Sylow
286C2×C71—24no—
287C2×C71—44no—
288C2×C71C2×#38511yescharacteristic
289C2×C71#291211yescharacteristic
290C2×C71#291311yescharacteristic
291C2×C71#301211yescharacteristic
292C2×C71#301311yescharacteristic
293C2×C71#309611yescharacteristic
294C2×C71#309711yescharacteristic
295C2×#520—711no—
296C2×#428—711no—
297C2×#427—711no—
298C2×#408—711no—
299#2311D7111yescharacteristic
300#1854—712no—
301#1854—712no—
302#1853—712no—
303#1852—712no—
304#1736—711no—
305C2×#340—711no—
306C71#2084911yescharacteristic, complemented, Sylow
307C22×#41C2×D7111yescharacteristic
308C22×M16—711no—
309C23×C8—711no—
310#520—1424no—
311#520—1424no—
312C2×C4×Q8—1421no—
313C2×C4×D4—1422no—
314C2×C4×D4—1421no—
315C2×#117—711no—
316C2×#117—712no—
317C2×#117C2×D7111yescharacteristic
318C22×C4⋊C4—711no—
319C22×C4⋊C4—712no—
320C22×C4⋊C4C2×D7111yescharacteristic
321#428—1424no—
322#427—1424no—
323C2×#107—711no—
324C2×#107—712no—
325C2×#107—711no—
326C2×#106—711no—
327C2×#106—712no—
328C2×#106—711no—
329#408—718no—
330C2×C22⋊C8—712no—
331C2×C22⋊C8—712no—
332#391—21no—
333#390—21no—
334#386—22no—
335#380C2×D7112yes—
336#380C2×D7112yes—
337#340—712no—
338#340—712no—
339#340—712no—
340#340—712no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Dic14214yes—
346C2×#41C4×D7114yes—
347C22×Q8C22⋊D7111yescharacteristic
348C22×D4C22⋊D7112yes—
349C22×D4C22⋊D7111yescharacteristic
350C23×C4—21no—
351C23×C4—1422no—
352C23×C4—1421no—
353C23×C4C22×D7112yes—
354C23×C4C22×D7111yescharacteristic
355C2×M16—714no—
356C2×M16—1422no—
357C2×M16—1422no—
358C22×C8—714no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—712no—
362C22×C8—1422no—
363C22×C8—1422no—
364C4×Q8—1428no—
365C4×D4—14216no—
366C4×D4—1428no—
367#117—22no—
368#117—22no—
369#117—714no—
370#117—1424no—
371#117—1424no—
372#117—1422no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—714no—
377C2×C4⋊C4—712no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—711no—
380C2×C4⋊C4—1424no—
381C2×C4⋊C4—1424no—
382C2×C4⋊C4—1422no—
383C2×C4⋊C4—1422no—
384C2×C4⋊C4—1421no—
385C2×C4⋊C4C22×D7112yes—
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C4⋊C4C22×D7111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—1422no—
392C2×C22⋊C4—1421no—
393C2×C42—21no—
394C2×C42—1422no—
395C2×C42—1421no—
396#107—1422no—
397#107—1422no—
398#107—1424no—
399#106—1422no—
400#106—1422no—
401#106—1424no—
402C22⋊C8—718no—
403C22⋊C8—718no—
404#95—22no—
405#95—22no—
406C24C2×C22⋊D7112yes—
407C24C2×C22⋊D7111yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8Dic71⋊C414yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4Dic71⋊C418yes—
420C2×D4Dic71⋊C414yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—5681no—
432C22×C4—5682no—
433C22×C4—5681no—
434C22×C4—5682no—
435C22×C4—1421no—
436C22×C4—1422no—
437C22×C4—1421no—
438C22×C4—1428no—
439C22×C4—1424no—
440C22×C4C2×C4×D7114yes—
441C22×C4C2×C4×D7112yes—
442C22×C4D4×D7112yes—
443C22×C4D4×D7112yes—
444C22×C4C2×C22⋊D7112yes—
445C22×C4C2×C22⋊D7111yescharacteristic
446C22×C4C2×Dic14214yes—
447C22×C4C2×Dic14212yes—
448C22×C4#1176586412yes—
449C22×C4#1176586412yes—
450C22×C4C23×D7111yescharacteristic
451M16—1428no—
452C2×C8—712no—
453C2×C8—712no—
454C2×C8—714no—
455C2×C8—1421no—
456C2×C8—1422no—
457C2×C8—1421no—
458C2×C8—1421no—
459C2×C8—1421no—
460C2×C8—1422no—
461C2×C8—1428no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—714no—
470C4⋊C4—712no—
471C4⋊C4—712no—
472C4⋊C4—1428no—
473C4⋊C4—1421no—
474C4⋊C4—1422no—
475C4⋊C4—1421no—
476C4⋊C4—1424no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—1428no—
481C22⋊C4—1424no—
482C42—42no—
483C42—1428no—
484C42—1424no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×Dic71⋊C414yes—
496C23C2×Dic71⋊C412yes—
497C23#1455609612yes—
498C23#1455610912yes—
499C23C22×C22⋊D7111yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4—2842no—
520C2×C4—2841no—
521C2×C4—2842no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—5684no—
526C2×C4—5688no—
527C2×C4—1422no—
528C2×C4—1424no—
529C2×C4—1422no—
530C2×C4C2×Dic71⋊C414yes—
531C2×C4C2×Dic71⋊C412yes—
532C2×C4(C22⋊D71)⋊C414yes—
533C2×C4(C22⋊D71)⋊C414yes—
534C2×C4#1455609612yes—
535C2×C4#1455610514yes—
536C2×C4#1455610514yes—
537C2×C4#1455610912yes—
538C2×C4C22×C4×D7111yescharacteristic
539C2×C4C2×D4×D7111yescharacteristic
540C2×C4C2×D4×D7111yescharacteristic
541C2×C4C22×Dic14211yescharacteristic
542C2×C4C2×#1176586411yescharacteristic
543C2×C4C2×#1176586411yescharacteristic
544C8—1422no—
545C8—1422no—
546C8—1424no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1798587114yes—
561C22C22×Dic71⋊C411yescharacteristic
562C22C2×#1455609611yescharacteristic
563C22C2×#1455610911yescharacteristic
564C22#1798680211yescharacteristic
565C22#1798680711yescharacteristic
566C22#1798681011yescharacteristic
567C22#1798681411yescharacteristic
568C4—24no—
569C4—81no—
570C4—81no—
571C4—82no—
572C4—2844no—
573C4—2842no—
574C4—2842no—
575C4#1798587114yes—
576C4C2×(C22⋊D71)⋊C411yescharacteristic
577C4C2×(C22⋊D71)⋊C411yescharacteristic
578C4C2×#1455610511yescharacteristic
579C4C2×#1455610511yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1798587111yescharacteristic
583C2#2259693311yescharacteristic
584C2#2259693811yescharacteristic
585C2#2259764111yescharacteristic
586C2#2259765211yescharacteristic
587C2#2259829211yescharacteristic
588C2#2259830311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2311C2
socleC23×C71C22×D4
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenter#2311D71
hypocenterC71#20849
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#17986042C2semidirect
3. C2×#17986035C2semidirect
5. #22588537C2semidirect
6. #22588537C2semidirect
7. #22588532C2semidirect
8. #22588527C2semidirect
9. #22588059C2semidirect
10. C2×#17985761C2semidirect
31. #17985966C4semidirect
50. C2×#11765849C2×C4semidirect
53. C22×C71⋊C8C2×C4semidirect
83. C22⋊(C71⋊C8)C2×C4semidirect
84. C22⋊(C71⋊C8)C2×C4semidirect
306. C71#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#179864822
C2×#179860422
C2×#179860352
C2×#179859662
#225885372
#225885322
#225885272
#225880592
C2×#179857612
C71×#23112
#2084971