Finite Group Databasebeta

← #32016814

Subgroups — #32016814

Order 18176.

866 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986097C211yesmaximal, characteristic
2#22591251C211yesmaximal, characteristic
3#22589033C211yesmaximal, characteristic, complemented
4#22589019C211yesmaximal, characteristic, complemented
5#22588932C211yesmaximal, characteristic, complemented
6#22588911C211yesmaximal, characteristic, complemented
7#22588677C211yesmaximal, characteristic
8#22588677C211yesmaximal, characteristic
9#22588475C211yesmaximal, characteristic, complemented
10#22588355C211yesmaximal, characteristic, complemented
11#22588337C211yesmaximal, characteristic
12#22588215C211yesmaximal, characteristic, complemented
13#22588168C211yesmaximal, characteristic
14C4×#14556048C211yesmaximal, characteristic
15C71×#2355C211yesmaximal, characteristic, complemented
16C22×#11765852C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18C2×C4×D284C2211yescharacteristic
19C2×C4×D284C2211yescharacteristic
20#17986097C418yescomplemented
21C2×#14556061C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×#14556052C2211yescharacteristic
24C2×#14556052C2211yescharacteristic
25C2×#14556052C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556048C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×C4×C71⋊C8C2211yescharacteristic
32#17985897C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985870—21no—
35#17985870—21no—
36#17985865C2211yescharacteristic
37#17985858—21no—
38#95×D71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×Q8×C71C2211yescharacteristic
50#391×C71C2211yescharacteristic
51#391×C71C2211yescharacteristic
52#387×C71—21no—
53#387×C71—21no—
54#384×C71C2211yescharacteristic
55#382×C71—21no—
56C4×C4⋊C4×C71—21no—
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C2×#11765852—22no—
65C2×#11765852—22no—
66C2×#11765852C2×C414yes—
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C22×C71⋊C8C2311yescharacteristic
71C4×D284—22no—
72C4×D284—22no—
73C4×D284—22no—
74C4×D284—41no—
75C4×D284—41no—
76C4×D284C2×C414yes—
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×Dic71⋊C4—21no—
81C2×Dic71⋊C4—21no—
82C2×(C2×D71)⋊C4—21no—
83C2×(C2×D71)⋊C4—21no—
84C2×(C2×D71)⋊C4—21no—
85#14556061C2×C414yescomplemented
86C4⋊(C71⋊C8)—22no—
87#14556052—41no—
88#14556052—41no—
89#14556052C2×C414yescomplemented
90#14556048—22no—
91#14556048—22no—
92#14556048C2×C414yes—
93#14556039—41no—
94#14556039C2×C414yescomplemented
95C4×C71⋊C8—41no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C22×Q8×C71C2311yescharacteristic
101C4×Q8×C71—22no—
102C4×Q8×C71C2×C414yescomplemented
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71D411yescharacteristic
107C2×C4⋊C4×C71D411yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C42×C71—21no—
112C2×C42×C71—21no—
113C2×C42×C71—21no—
114C2×C42×C71C2311yescharacteristic
115C2×C42×C71C2311yescharacteristic
116C2×C42×C71C2311yescharacteristic
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120#95×C71—21no—
121#95×C71—21no—
122#95×C71—21no—
123#95×C71D411yescharacteristic
124#95×C71D411yescharacteristic
125C23×D71—21no—
126C2×D284—21no—
127C2×D284—21no—
128C2×D284—21no—
129C2×D284C22×C412yes—
130C2×D284#4112yes—
131C2×D284#4112yes—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C2×C4×D71—24no—
136C2×C4×D71—42no—
137C2×C4×D71—42no—
138C2×C4×D71—42no—
139C2×C4×D71—42no—
140C2×C4×D71—81no—
141C2×C4×D71—81no—
142C2×C4×D71—81no—
143C2×C4×D71—81no—
144#11765852—28no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8C22×C412yes—
156C2×C71⋊C8C22×C412yes—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71C22×C412yes—
163Dic71⋊C4—81no—
164Dic71⋊C4—81no—
165(C2×D71)⋊C4—24no—
166(C2×D71)⋊C4—42no—
167(C2×D71)⋊C4—42no—
168C2×Q8×C71—21no—
169C2×Q8×C71—21no—
170C2×Q8×C71—21no—
171C2×Q8×C71—21no—
172C2×Q8×C71—21no—
173C2×Q8×C71C22×C412yes—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71—21no—
177C22×C4×C71—21no—
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2411yescharacteristic
185C4⋊C4×C71—24no—
186C4⋊C4×C71—22no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—41no—
193C4⋊C4×C71—42no—
194C4⋊C4×C71—81no—
195C4⋊C4×C71C22×C412yes—
196C42×C71—24no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—81no—
202C42×C71—81no—
203C42×C71C22×C412yes—
204C42×C71#4112yes—
205C23×C71C22×D411yescharacteristic
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—41no—
210C22×D71—41no—
211C22×D71—41no—
212C22×D71—41no—
213Q8×C71—22no—
214Q8×C71—22no—
215Q8×C71—41no—
216Q8×C71—41no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—41no—
244C2×C4×C71C4×D412yes—
245C2×C4×C71C4×D412yes—
246C2×C4×C71#13712yes—
247C2×C4×C71C23×C411yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71H(5,𝔽2)11yescharacteristic
252C2×C4×C712−1+411yescharacteristic
253C2×Dic71—22no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262D284—22no—
263D284—22no—
264D284—22no—
265D284#11714yes—
266C4×D71—84no—
267C4×D71—84no—
268C4×D71—84no—
269C71⋊C8—22no—
270C71⋊C8—22no—
271C71⋊C8—41no—
272C71⋊C8—41no—
273C22×C71C2×C4×D411yescharacteristic
274C22×C71#54611yescharacteristic
275C22×C71#54811yescharacteristic
276C22×C71C2×#13711yescharacteristic
277C22×C71#57511yescharacteristic
278C22×C71#57611yescharacteristic
279C22×C71#57711yescharacteristic
280C2×D71—24no—
281C2×D71—42no—
282C2×D71—42no—
283C2×D71—42no—
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—42no—
287C4×C71—22no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—42no—
291C4×C71—42no—
292C4×C71—41no—
293C4×C71—41no—
294C4×C71—41no—
295C4×C71—81no—
296C4×C71—81no—
297C4×C71C2×#11711yescharacteristic
298C4×C71#52011yescharacteristic
299C4×C71#55211yescharacteristic
300C4×C71#55311yescharacteristic
301Dic71—42no—
302Dic71—81no—
303Dic71—81no—
304#22776—711nomaximal, Sylow
305C2×C71#235211yescharacteristic
306C2×C71C4×#13711yescharacteristic
307C2×C71#294611yescharacteristic
308C2×C71#331111yescharacteristic
309C2×C71#331411yescharacteristic
310C2×C71#337011yescharacteristic
311C2×C71#337211yescharacteristic
312D71—44no—
313C2×#440—711no—
314#2355D7111yescharacteristic, complemented
315#2349—711no—
316#1935—711no—
317#1933—711no—
318#1921—711no—
319#1918—711no—
320#1877—711no—
321#1877—711no—
322#1836—711no—
323#1805—711no—
324#1802—711no—
325#1771—711no—
326#1762—711no—
327C4×#102—711no—
328C71#2277611yescharacteristic, complemented, Sylow
329C22×SD16—711no—
330C2×C4×Q8C2×D7111yescharacteristic
331C2×C4×D4—711no—
332C2×C4×D4—711no—
333C2×C4×D4—711no—
334#440—718no—
335C2×#107—711no—
336C2×C4⋊C8—711no—
337C2×#103—711no—
338C2×#103—711no—
339C2×#103—711no—
340C2×#102—711no—
341C2×#102—711no—
342C2×#102—711no—
343C2×#97—711no—
344C2×#97—711no—
345C2×C4×C8—711no—
346#391C2×D7111yescharacteristic
347#391C2×D7111yescharacteristic
348#389—711no—
349#389—711no—
350#387—21no—
351#387—21no—
352#385—1421no—
353#385—1421no—
354#384C2×D7111yescharacteristic
355#382—21no—
356#382—711no—
357#380—1421no—
358C4×C4⋊C4—21no—
359C4×C4⋊C4C2×D7111yescharacteristic
360C4×C4⋊C4C2×D7111yescharacteristic
361C4×C4⋊C4C2×D7111yescharacteristic
362C2×#95—1421no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#340—711no—
371#336—711no—
372#336—711no—
373C22×Q8C22×D7111yescharacteristic
374C22×D4—711no—
375C23×C4—1421no—
376C23×C4—1421no—
377C23×C4—1421no—
378C2×SD16—714no—
379C2×SD16—1422no—
380C2×SD16—1422no—
381C22×C8—711no—
382C22×C8—711no—
383C22×C8—711no—
384C22×C8—711no—
385C4×Q8—22no—
386C4×Q8C4×D7114yescomplemented
387C4×D4—2841no—
388C4×D4—2841no—
389C4×D4—714no—
390C4×D4—1422no—
391C4×D4—1422no—
392C4×D4—1422no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4—1421no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4C22⋊D7111yescharacteristic
402C2×C4⋊C4C22⋊D7111yescharacteristic
403C2×C4⋊C4C22×D7111yescharacteristic
404C2×C4⋊C4C22×D7111yescharacteristic
405C2×C4⋊C4C22×D7111yescharacteristic
406C2×C22⋊C4—1421no—
407C2×C22⋊C4—1421no—
408C2×C22⋊C4—1421no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C22×D7111yescharacteristic
413C2×C42C22×D7111yescharacteristic
414C2×C42C22×D7111yescharacteristic
415#107—714no—
416C4⋊C8—1422no—
417#103—2841no—
418#103—2841no—
419#103—714no—
420#102—714no—
421#102—1422no—
422#102—1422no—
423#97—2841no—
424#97—714no—
425C4×C8—2841no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436#95C22⋊D7111yescharacteristic
437#95C22⋊D7111yescharacteristic
438C24—1421no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D7112yes—
445C2×D4—712no—
446C2×D4—712no—
447C2×D4—712no—
448C2×D4—1421no—
449C2×D4—1421no—
450C2×D4—1421no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—2842no—
458C22×C4—2842no—
459C22×C4—5681no—
460C22×C4—5681no—
461C22×C4—5681no—
462C22×C4—5681no—
463C22×C4—1421no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4—1424no—
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×C22⋊D7111yescharacteristic
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4C23×D7111yescharacteristic
474SD16—1428no—
475C2×C8—712no—
476C2×C8—712no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8—1421no—
483C2×C8—1421no—
484C2×C8—1421no—
485C2×C8—1421no—
486C2×C8—1421no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—5681no—
498C4⋊C4—5681no—
499C4⋊C4—712no—
500C4⋊C4—1421no—
501C4⋊C4—1421no—
502C4⋊C4—1421no—
503C4⋊C4—1421no—
504C4⋊C4—1421no—
505C4⋊C4C2×C4×D7112yes—
506C22⋊C4—2842no—
507C22⋊C4—2842no—
508C22⋊C4—1424no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—24no—
514C42—81no—
515C42—81no—
516C42C2×C4×D7112yes—
517C42#1176586512yes—
518C23—2841no—
519C23—2841no—
520C23—2841no—
521C23—2841no—
522C23—1422no—
523C23—1422no—
524C23—1422no—
525C23C22×C22⋊D7111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530D4—714no—
531D4—1422no—
532D4—1422no—
533D4—1422no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—5684no—
570C2×C4—5684no—
571C2×C4—5684no—
572C2×C4—1422no—
573C2×C4C4×C22⋊D7112yes—
574C2×C4C4×C22⋊D7112yes—
575C2×C4#1455615912yes—
576C2×C4C22×C4×D7111yescharacteristic
577C2×C4#41×D7111yescharacteristic
578C2×C4#41×D7111yescharacteristic
579C2×C4C2×#1176586511yescharacteristic
580C2×C4#1455617411yescharacteristic
581C2×C4#1455617511yescharacteristic
582C8—2841no—
583C8—2841no—
584C8—1422no—
585C8—1422no—
586C22—2842no—
587C22—2842no—
588C22—2842no—
589C22—2842no—
590C22—2842no—
591C22—2842no—
592C22—1424no—
593C22C2×C4×C22⋊D7111yescharacteristic
594C22#1798663711yescharacteristic
595C22#1798665611yescharacteristic
596C22C2×#1455615911yescharacteristic
597C22#1798679711yescharacteristic
598C22#1798680211yescharacteristic
599C22#1798680711yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—2842no—
611C4—5681no—
612C4—5681no—
613C4#117×D7111yescharacteristic
614C4#1798647911yescharacteristic
615C4#1798668811yescharacteristic
616C4#1798669311yescharacteristic
617C2—2844no—
618C2#2259127011yescharacteristic
619C2C4×#1455615911yescharacteristic
620C2#2259716811yescharacteristic
621C2#2260037011yescharacteristic
622C2#2260041511yescharacteristic
623C2#2260108711yescharacteristic
624C2#2260111711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2355C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2355D71
hypocenterC71#22776
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2355

Complements

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

NH ≅ G/Ntype
3. #22589033C2semidirect
4. #22589019C2semidirect
5. #22588932C2semidirect
6. #22588911C2semidirect
9. #22588475C2semidirect
10. #22588355C2semidirect
12. #22588215C2semidirect
15. C71×#2355C2semidirect
20. #17986097C4semidirect
85. #14556061C2×C4semidirect
89. #14556052C2×C4semidirect
94. #14556039C2×C4semidirect
102. C4×Q8×C71C2×C4semidirect
314. #2355D71semidirect
328. C71#22776semidirect
386. C4×Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860972
#225912512
#225890332
#225890192
#225889322
#225889112
#225886772
#225884752
#225883552
#225883372
#225882152
#225881682
C4×#145560482
C71×#23552
#2277671