Finite Group Databasebeta

← #31998228

Subgroups — #31998228

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×#17986479C211yesmaximal, characteristic
2C2×#17986041C211yesmaximal, characteristic, complemented
3C2×#17986035C211yesmaximal, characteristic, complemented
4C2×#17985967C211yesmaximal, characteristic
5#22590933C211yesmaximal, characteristic
6#22588542C212yesmaximal, complemented
7#22588540C212yesmaximal, complemented
8#22588534C212yesmaximal, complemented
9#22588529C212yesmaximal, complemented
10C2×#17985761C211yesmaximal, characteristic, complemented
11C71×#1736C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14#17986479—24no—
15#17986479—24no—
16C2×C4×Dic142—21no—
17C2×C4×D284—21no—
18C2×(C22⋊D71)⋊C4—22no—
19C2×#14556087C2211yescharacteristic
20C2×#14556086C2212yes—
21C22×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊C4×D71C2212yes—
23#17986041—24no—
24#17986035—24no—
25C2×#14556062C2212yes—
26C2×#14556061C2211yescharacteristic
27C2×#14556061C2211yescharacteristic
28C2×#14556060C2212yes—
29C2×#14556059C2211yescharacteristic
30C2×#14556059C2211yescharacteristic
31#17985967C418yescomplemented
32C2×(C2×D71)⋊C8C2212yes—
33C2×(C2×D71)⋊C8C2212yes—
34#17985904—21no—
35#17985900—21no—
36#17985878—22no—
37#17985856C2212yes—
38#17985856C2212yes—
39#17985761C2212yes—
40#17985761C2212yes—
41C22×M16×C71C2211yescharacteristic
42C2×#117×C71C2211yescharacteristic
43C22×C4⋊C4×C71C2211yescharacteristic
44#340×C71C2212yes—
45#340×C71C2212yes—
46C2×#11765865—22no—
47C2×#11765865—22no—
48C2×#11765865—22no—
49C2×#11765865—22no—
50C2×#11765865C2×C414yes—
51C2×#11765865Q814yes—
52C22×Dic142D411yescharacteristic
53C22×D284D411yescharacteristic
54C22×C22⋊D71D412yes—
55C23×Dic71—21no—
56C22×C4×D71—22no—
57C22×C4×D71C2312yes—
58C22×C71⋊C8—22no—
59C22×C71⋊C8—22no—
60C22×C71⋊C8C2×C414yescomplemented
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C4×Dic142—28no—
64C4×D284—28no—
65(C22⋊D71)⋊C4—216no—
66#14556087—22no—
67#14556087—22no—
68#14556086—24no—
69#14556086—24no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71D414yes—
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2312yes—
76C2×Dic71⋊C4—22no—
77C4⋊C4×D71—24no—
78C4⋊C4×D71—24no—
79C2×C22⋊Dic71—21no—
80C2×(C2×D71)⋊C4—22no—
81C2×(C2×D71)⋊C4—22no—
82C2×(C2×D71)⋊C4—22no—
83C2×C4×Dic71—22no—
84C2×C4×Dic71—21no—
85#14556062—24no—
86#14556061—22no—
87#14556061—22no—
88#14556060—24no—
89#14556059—22no—
90#14556059—22no—
91(C2×D71)⋊C8C2×C418yescomplemented
92(C2×D71)⋊C8C2×C418yescomplemented
93C71⋊#95—22no—
94C71⋊#95—22no—
95C23×C4×C71—21no—
96C23×C4×C71C2311yescharacteristic
97C2×M16×C71—22no—
98C2×M16×C71—22no—
99C2×M16×C71C2×C414yescomplemented
100C22×C8×C71C2312yes—
101#117×C71—22no—
102#117×C71D414yes—
103C2×C4⋊C4×C71—22no—
104C2×C4⋊C4×C71—22no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71C2312yes—
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C22⋊C4×C71—21no—
110C2×C42×C71—21no—
111C23×D71C2×D412yes—
112#11765865—28no—
113#11765865—28no—
114#11765865—28no—
115#11765865—28no—
116C2×Dic142—22no—
117C2×Dic142—22no—
118C2×Dic142C4⋊C414yes—
119C2×D284—22no—
120C2×D284—22no—
121C2×D284C4⋊C414yes—
122C2×C22⋊D71—24no—
123C2×C22⋊D71—24no—
124C2×C22⋊D71C4⋊C418yes—
125C22×Dic71—22no—
126C22×Dic71—24no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71—81no—
130C22×Dic71—81no—
131C22×Dic71C2×D412yes—
132C2×C4×D71—22no—
133C2×C4×D71—22no—
134C2×C4×D71—22no—
135C2×C4×D71—22no—
136C2×C4×D71—28no—
137C2×C4×D71—82no—
138C2×C4×D71—82no—
139C2×C4×D71C22×C414yes—
140C2×C4×D71C2×Q814yes—
141C2×C71⋊C8—28no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8C22×C412yes—
147C2×C71⋊C8C22×C412yes—
148C4⋊Dic71—24no—
149C4⋊Dic71—22no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71C2×D414yes—
155Dic71⋊C4—28no—
156C22⋊Dic71—42no—
157(C2×D71)⋊C4—28no—
158(C2×D71)⋊C4—82no—
159(C2×D71)⋊C4—82no—
160C4×Dic71—28no—
161C4×Dic71—42no—
162C24×C71C2×D411yescharacteristic
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—22no—
166C22×C4×C71—42no—
167C22×C4×C71—81no—
168C22×C4×C71—81no—
169C22×C4×C71C22×C412yes—
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×Q812yes—
174C22×C4×C71#4112yes—
175C22×C4×C71#4112yes—
176C22×C4×C71C2411yescharacteristic
177M16×C71—28no—
178C2×C8×C71—22no—
179C2×C8×C71—22no—
180C2×C8×C71C22×C414yes—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—42no—
186C4⋊C4×C71C2×D412yes—
187C4⋊C4×C71C2×D412yes—
188C22⋊C4×C71—24no—
189C42×C71—24no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71C2×C4⋊C412yes—
195C23×C71#12012yes—
196C23×C71#12312yes—
197C23×C71C22×D411yescharacteristic
198C22×D71—22no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71C2×C4⋊C414yes—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—41no—
208C2×C4×C71—41no—
209C2×C4×C71—41no—
210C2×C4×C71—41no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—84no—
215C2×C4×C71C2×C4⋊C412yes—
216C2×C4×C71C4×D414yes—
217C2×C4×C71C4×D414yes—
218C2×C4×C71#12012yes—
219C2×C4×C71C22⋊Q814yes—
220C2×C4×C71C22⋊Q814yes—
221C2×C4×C71#12312yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×D411yescharacteristic
225C2×C4×C71C22×Q811yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C22⋊D71—216no—
229C2×Dic71—22no—
230C2×Dic71—22no—
231C2×Dic71—24no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—41no—
235C2×Dic71—41no—
236C2×Dic71—41no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—84no—
241C2×Dic71C2×C4⋊C414yes—
242D284—28no—
243C4×D71—24no—
244C4×D71—24no—
245C4×D71—24no—
246C4×D71—24no—
247C4×D71—88no—
248Dic142—28no—
249C8×C71—24no—
250C71⋊C8—22no—
251C71⋊C8—22no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71#38514yes—
259C22×C71C22×C4⋊C411yescharacteristic
260C22×C71C2×#12011yescharacteristic
261C22×C71C2×#12311yescharacteristic
262C22×C71#57611yescharacteristic
263C22×C71#57711yescharacteristic
264C22×C71#57711yescharacteristic
265C22×C71#57811yescharacteristic
266C2×D71—24no—
267C2×D71—24no—
268C2×D71—24no—
269C2×D71—42no—
270C2×D71—42no—
271C2×D71—42no—
272C2×D71—42no—
273C4×C71—42no—
274C4×C71—42no—
275C4×C71—82no—
276C4×C71#38514yes—
277C4×C71C2×C4×D411yescharacteristic
278C4×C71C2×C4×D411yescharacteristic
279C4×C71C2×C22⋊Q811yescharacteristic
280C4×C71C2×C22⋊Q811yescharacteristic
281Dic71—24no—
282Dic71—44no—
283Dic71—81no—
284Dic71—81no—
285#20849—711nomaximal, Sylow
286C2×C71—24no—
287C2×C71C2×#38511yescharacteristic
288C2×C71#291211yescharacteristic
289C2×C71#291311yescharacteristic
290C2×C71#301211yescharacteristic
291C2×C71#301311yescharacteristic
292C2×C71#309611yescharacteristic
293C2×C71#309711yescharacteristic
294D71—44no—
295C2×#520—711no—
296C2×#428—711no—
297C2×#427—711no—
298C2×#408—711no—
299#2311—711no—
300#1854—712no—
301#1854—712no—
302#1853—712no—
303#1852—712no—
304#1736D7111yescharacteristic, complemented
305C2×#340—711no—
306C71#2084911yescharacteristic, complemented, Sylow
307C22×#41—711no—
308C22×M16C2×D7111yescharacteristic
309C23×C8—711no—
310#520—1424no—
311#520—1424no—
312C2×C4×Q8—1421no—
313C2×C4×D4—1421no—
314C2×C4×D4—1422no—
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—712no—
324C2×#107—711no—
325C2×#107—711no—
326C2×#106—712no—
327C2×#106—711no—
328C2×#106—711no—
329#408—718no—
330C2×C22⋊C8—712no—
331C2×C22⋊C8—712no—
332#391—1421no—
333#390—1421no—
334#386—1422no—
335#380—712no—
336#380—712no—
337#340—712no—
338#340—712no—
339#340C2×D7112yes—
340#340C2×D7112yes—
341C2×#41—714no—
342C2×#41—714no—
343C2×#41—1422no—
344C2×#41—1422no—
345C2×#41—1422no—
346C2×#41—1422no—
347C22×Q8—711no—
348C22×D4—712no—
349C22×D4—711no—
350C23×C4—21no—
351C23×C4—712no—
352C23×C4—1422no—
353C23×C4—1421no—
354C23×C4C22×D7111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D7114yescomplemented
358C22×C8—714no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—1422no—
362C22×C8—1422no—
363C22×C8C22×D7112yes—
364C4×Q8—1428no—
365C4×D4—1428no—
366C4×D4—14216no—
367#117—22no—
368#117—1422no—
369#117—1422no—
370#117—1424no—
371#117—1424no—
372#117D28414yes—
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—1422no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1422no—
384C2×C4⋊C4—1424no—
385C2×C4⋊C4C22×D7112yes—
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C4⋊C4C22×D7111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1421no—
390C2×C22⋊C4—1422no—
391C2×C22⋊C4—1422no—
392C2×C22⋊C4—1422no—
393C2×C42—21no—
394C2×C42—1421no—
395C2×C42—1422no—
396#107—1424no—
397#107—1422no—
398#107—1422no—
399#106—1424no—
400#106—1422no—
401#106—1422no—
402C22⋊C8—718no—
403C22⋊C8—718no—
404#95—1422no—
405#95—1422no—
406C24—712no—
407C24D4×D7111yescharacteristic
408#41—1428no—
409#41—1428no—
410#41—1428no—
411#41—1428no—
412C2×Q8—714no—
413C2×Q8—1422no—
414C2×Q8—1422no—
415C2×D4—718no—
416C2×D4—714no—
417C2×D4—1424no—
418C2×D4—1424no—
419C2×D4—1422no—
420C2×D4—1422no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—5682no—
428C22×C4—5682no—
429C22×C4—5681no—
430C22×C4—5681no—
431C22×C4—712no—
432C22×C4—714no—
433C22×C4—714no—
434C22×C4—1428no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1422no—
438C22×C4—1424no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4C2×C4×D7112yes—
444C22×C4D4×D7112yes—
445C22×C4D4×D7111yescharacteristic
446C22×C4C2×D28412yes—
447C22×C4Q8×D7112yes—
448C22×C4#1176586412yes—
449C22×C4#1176586512yes—
450C22×C4C23×D7111yescharacteristic
451M16—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—712no—
455C2×C8—712no—
456C2×C8—1421no—
457C2×C8—1421no—
458C2×C8—1421no—
459C2×C8—1421no—
460C2×C8—1428no—
461C2×C8C2×C4×D7114yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—714no—
468C4⋊C4—1428no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1422no—
474C4⋊C4—1424no—
475C4⋊C4C2×D28412yes—
476C4⋊C4C2×D28412yes—
477C22⋊C4—24no—
478C22⋊C4—2842no—
479C22⋊C4—5682no—
480C22⋊C4—5682no—
481C22⋊C4—1428no—
482C42—24no—
483C42—2842no—
484C42—1428no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—714no—
490C23—1422no—
491C23—1422no—
492C23—1422no—
493C23—1422no—
494C23—1422no—
495C23—1422no—
496C23C4⋊C4×D7112yes—
497C23#1455609812yes—
498C23#1455611012yes—
499C23C2×D4×D7111yescharacteristic
500Q8—1428no—
501D4—14216no—
502D4—1428no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4—2842no—
514C2×C4—2842no—
515C2×C4—2841no—
516C2×C4—2841no—
517C2×C4—2841no—
518C2×C4—2841no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—5688no—
522C2×C4—5684no—
523C2×C4—714no—
524C2×C4—1424no—
525C2×C4—1422no—
526C2×C4—1422no—
527C2×C4—1424no—
528C2×C4—1424no—
529C2×C4—1424no—
530C2×C4—1424no—
531C2×C4C4⋊C4×D7112yes—
532C2×C4(C22⋊D71)⋊C414yes—
533C2×C4C4×D28414yes—
534C2×C4#1455609812yes—
535C2×C4#1455610614yes—
536C2×C4#1455610814yes—
537C2×C4#1455611012yes—
538C2×C4C22×C4×D7111yescharacteristic
539C2×C4C2×D4×D7111yescharacteristic
540C2×C4C22×D28411yescharacteristic
541C2×C4C2×Q8×D7111yescharacteristic
542C2×C4C2×#1176586411yescharacteristic
543C2×C4C2×#1176586511yescharacteristic
544C8—24no—
545C8—1422no—
546C8—1422no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—2842no—
554C22—2842no—
555C22—2842no—
556C22—2842no—
557C22—1424no—
558C22—1424no—
559C22—1424no—
560C22#1798587314yes—
561C22C2×C4⋊C4×D7111yescharacteristic
562C22C2×#1455609811yescharacteristic
563C22C2×#1455611011yescharacteristic
564C22#1798680311yescharacteristic
565C22#1798680811yescharacteristic
566C22#1798680911yescharacteristic
567C22#1798681511yescharacteristic
568C4—42no—
569C4—42no—
570C4—82no—
571C4—2844no—
572C4—5681no—
573C4—5681no—
574C4—1424no—
575C4#1798587314yes—
576C4C2×(C22⋊D71)⋊C411yescharacteristic
577C4C2×C4×D28411yescharacteristic
578C4C2×#1455610611yescharacteristic
579C4C2×#1455610811yescharacteristic
580C2—24no—
581C2—2844no—
582C2C2×#1798587311yescharacteristic
583C2#2259693411yescharacteristic
584C2#2259693911yescharacteristic
585C2#2259764611yescharacteristic
586C2#2259765711yescharacteristic
587C2#2259829611yescharacteristic
588C2#2259830711yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1736C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenter#1736D71
hypocenterC71#20849
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
2. C2×#17986041C2semidirect
3. C2×#17986035C2semidirect
6. #22588542C2semidirect
7. #22588540C2semidirect
8. #22588534C2semidirect
9. #22588529C2semidirect
10. C2×#17985761C2semidirect
11. C71×#1736C2semidirect
31. #17985967C4semidirect
60. C22×C71⋊C8C2×C4semidirect
91. (C2×D71)⋊C8C2×C4semidirect
92. (C2×D71)⋊C8C2×C4semidirect
99. C2×M16×C71C2×C4semidirect
304. #1736D71semidirect
306. C71#20849semidirect
357. C2×M16C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864792
C2×#179860412
C2×#179860352
C2×#179859672
#225909332
#225885422
#225885402
#225885342
#225885292
C2×#179857612
C71×#17362
#2084971