Finite Group Databasebeta

← #32007825

Subgroups — #32007825

Order 18176.

818 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C8×D71C211yesmaximal, characteristic
2#22590736C211yesmaximal, characteristic
3#22590733C211yesmaximal, characteristic
4#22589111C211yesmaximal, characteristic, complemented
5#22589110C211yesmaximal, characteristic, complemented
6#22589109C211yesmaximal, characteristic, complemented
7#22589108C211yesmaximal, characteristic, complemented
8#22589107C211yesmaximal, characteristic, complemented
9#22589106C211yesmaximal, characteristic, complemented
10#22589105C211yesmaximal, characteristic, complemented
11#22588368C211yesmaximal, characteristic
12#22588366C211yesmaximal, characteristic
13#22588364C211yesmaximal, characteristic
14#22588362C211yesmaximal, characteristic
15C71×#1949C211yesmaximal, characteristic, complemented
16C22×C8×D71C2211yescharacteristic
17C22×C8×D71C2211yescharacteristic
18C2×C42×D71C2211yescharacteristic
19C2×#14556062C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556060C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×C4⋊(C71⋊C8)C2211yescharacteristic
26C2×Dic71⋊C8C2211yescharacteristic
27C2×Dic71⋊C8C2211yescharacteristic
28C4⋊C8×D71C418yescomplemented
29#17985866—21no—
30#17985866—21no—
31#17985865C2211yescharacteristic
32#17985864C2211yescharacteristic
33#17985864C2211yescharacteristic
34#17985861—21no—
35#17985861—21no—
36#17985856C2211yescharacteristic
37#17985856C2211yescharacteristic
38#17985856C2211yescharacteristic
39#17985856C2211yescharacteristic
40C4⋊C4×Dic71C2211yescharacteristic
41C4⋊C4×Dic71C2211yescharacteristic
42C4×C4⋊Dic71C2211yescharacteristic
43C4×(C2×D71)⋊C4—21no—
44#17985846—21no—
45#17985845—21no—
46#17985844—21no—
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C2×#107×C71C2211yescharacteristic
54C2×#106×C71C2211yescharacteristic
55C2×C4⋊C8×C71C2211yescharacteristic
56#382×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C4×D71D411yescharacteristic
61C22×C4×D71D411yescharacteristic
62C22×C4×D71C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C2×C8×D71—41no—
66C2×C8×D71—41no—
67C2×C8×D71—41no—
68C2×C8×D71—41no—
69C2×C8×D71C2×C414yes—
70C2×C8×D71C2×C414yes—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×(C2×D71)⋊C4—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71D411yescharacteristic
83C2×C4×Dic71D411yescharacteristic
84C2×C4×Dic71C2311yescharacteristic
85C42×D71C2×C414yes—
86C42×D71Q814yes—
87#14556062—41no—
88#14556061—41no—
89#14556061—41no—
90#14556060—41no—
91#14556059—41no—
92#14556059—41no—
93C4⋊(C71⋊C8)C2×C414yes—
94Dic71⋊C8C2×C414yes—
95Dic71⋊C8C2×C414yes—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C71⋊#95—21no—
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C22×C8×C71C2311yescharacteristic
110C22×C8×C71C2311yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C42×C71—21no—
116C2×C42×C71C2311yescharacteristic
117#107×C71—41no—
118#106×C71—41no—
119C4⋊C8×C71C2×C414yes—
120#95×C71—21no—
121#95×C71—21no—
122#95×C71—21no—
123C23×D71C2×D411yescharacteristic
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71C2×D411yescharacteristic
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C2×C4×D71—22no—
132C2×C4×D71—22no—
133C2×C4×D71—21no—
134C2×C4×D71—21no—
135C2×C4×D71—21no—
136C2×C4×D71—21no—
137C2×C4×D71—41no—
138C2×C4×D71—41no—
139C2×C4×D71C4⋊C414yes—
140C2×C4×D71C4⋊C414yes—
141C2×C4×D71C22×C412yes—
142C2×C4×D71C2×Q812yes—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8C22×C412yes—
148C2×C71⋊C8C22×C412yes—
149C8×D71—44no—
150C8×D71—44no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159(C2×D71)⋊C4—42no—
160(C2×D71)⋊C4—42no—
161(C2×D71)⋊C4—42no—
162(C2×D71)⋊C4—42no—
163C4×Dic71—42no—
164C4×Dic71—42no—
165C4×Dic71—42no—
166C4×Dic71C4⋊C414yes—
167C4×Dic71C4⋊C414yes—
168C4×Dic71C22×C412yes—
169C4×Dic71C2×Q812yes—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2411yescharacteristic
177C2×C8×C71—21no—
178C2×C8×C71—21no—
179C2×C8×C71—21no—
180C2×C8×C71—21no—
181C2×C8×C71C22×C412yes—
182C2×C8×C71C22×C412yes—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C42×C71—42no—
192C42×C71C22×C412yes—
193C42×C71C2×Q812yes—
194C23×C71C22×D411yescharacteristic
195C22×D71—21no—
196C22×D71—21no—
197C22×D71—21no—
198C22×D71—21no—
199C22×D71C2×C4⋊C412yes—
200C22×D71#13512yes—
201C22×D71#13512yes—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—41no—
207C2×C4×C71—41no—
208C2×C4×C71—41no—
209C2×C4×C71—41no—
210C2×C4×C71—41no—
211C2×C4×C71—41no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71C2×C4⋊C412yes—
219C2×C4×C71C2×C4⋊C412yes—
220C2×C4×C71C23×C411yescharacteristic
221C2×C4×C71C22×Q811yescharacteristic
222C2×C4×C71H(5,𝔽2)11yescharacteristic
223C2×C4×C71H(5,𝔽2)11yescharacteristic
224C2×C4×C712−1+411yescharacteristic
225C2×C4×C712−1+411yescharacteristic
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—41no—
233C2×Dic71—41no—
234C2×Dic71—41no—
235C2×Dic71—41no—
236C2×Dic71—41no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71C2×C4⋊C412yes—
245C2×Dic71C2×C4⋊C412yes—
246C2×Dic71C2×C4⋊C412yes—
247C4×D71—28no—
248C4×D71—22no—
249C4×D71—22no—
250C4×D71—22no—
251C4×D71—22no—
252C4×D71—44no—
253C8×C71—22no—
254C8×C71—22no—
255C71⋊C8—22no—
256C71⋊C8—22no—
257C22×C71C22×C4⋊C411yescharacteristic
258C22×C71#53711yescharacteristic
259C22×C71#53711yescharacteristic
260C22×C71C2×#13511yescharacteristic
261C22×C71C2×#13511yescharacteristic
262C22×C71#57511yescharacteristic
263C22×C71#57511yescharacteristic
264C2×D71—22no—
265C2×D71—22no—
266C2×D71—22no—
267C2×D71—22no—
268C2×D71—22no—
269C2×D71—22no—
270C2×D71#42714yes—
271C4×C71—22no—
272C4×C71—22no—
273C4×C71—81no—
274C4×C71—81no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71#51811yescharacteristic
278C4×C71#51911yescharacteristic
279C4×C71#54411yescharacteristic
280C4×C71#54411yescharacteristic
281Dic71—22no—
282Dic71—22no—
283Dic71—22no—
284Dic71—81no—
285Dic71—81no—
286Dic71—81no—
287Dic71—81no—
288#21960—711nomaximal, Sylow
289C2×C71#231611yescharacteristic
290C2×C71C2×#42711yescharacteristic
291C2×C71#291111yescharacteristic
292C2×C71#325011yescharacteristic
293C2×C71#325011yescharacteristic
294C2×C71#325011yescharacteristic
295C2×C71#325011yescharacteristic
296D71—24no—
297C22×C4⋊C8—711no—
298#2283—711no—
299#2283—711no—
300#1949—711no—
301#1949—711no—
302#1949—711no—
303#1949—711no—
304#1949—711no—
305#1949—711no—
306#1949—711no—
307#1949D7111yescharacteristic, complemented
308#1808—711no—
309#1808—711no—
310#1807—711no—
311#1807—711no—
312C71#2196011yescharacteristic, complemented, Sylow
313C23×C8—711no—
314C23×C8—711no—
315C22×C42—711no—
316C2×#107—711no—
317C2×#107—711no—
318C2×#107—711no—
319C2×#107C2×D7111yescharacteristic
320C2×#106—711no—
321C2×#106—711no—
322C2×#106—711no—
323C2×#106C2×D7111yescharacteristic
324C2×C4⋊C8—711no—
325C2×C4⋊C8—711no—
326C2×C4⋊C8—711no—
327C2×C4⋊C8—718no—
328C2×C4⋊C8C2×D7111yescharacteristic
329#383—1421no—
330#383—1421no—
331#382—711no—
332#382—711no—
333#382—711no—
334#382C2×D7111yescharacteristic
335#381—1421no—
336#381—1421no—
337#380—711no—
338#380—711no—
339#380—711no—
340#380—711no—
341C4×C4⋊C4—711no—
342C4×C4⋊C4—711no—
343C4×C4⋊C4—711no—
344C4×C4⋊C4C2×D7111yescharacteristic
345C4×C22⋊C4—1421no—
346C4×C22⋊C4—1421no—
347#376—1421no—
348#376—1421no—
349#340—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#340—711no—
355#340C2×D7111yescharacteristic
356#340C2×D7111yescharacteristic
357C23×C4—711no—
358C23×C4—711no—
359C23×C4—711no—
360C22×C8—2841no—
361C22×C8—2841no—
362C22×C8—2841no—
363C22×C8—2841no—
364C22×C8—714no—
365C22×C8—714no—
366C22×C8—711no—
367C22×C8—711no—
368C22×C8C22×D7111yescharacteristic
369C22×C8C22×D7111yescharacteristic
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4C22×D7111yescharacteristic
375C2×C4⋊C4C22×D7111yescharacteristic
376C2×C4⋊C4C22×D7111yescharacteristic
377C2×C4⋊C4C22×D7111yescharacteristic
378C2×C22⋊C4—1421no—
379C2×C22⋊C4—1421no—
380C2×C22⋊C4—1421no—
381C2×C22⋊C4—1421no—
382C2×C42—21no—
383C2×C42—711no—
384C2×C42—711no—
385C2×C42—711no—
386C2×C42—714no—
387C2×C42—714no—
388C2×C42—1421no—
389C2×C42—1421no—
390C2×C42—1421no—
391C2×C42C22×D7111yescharacteristic
392#107—41no—
393#107—2841no—
394#107—2841no—
395#107—2841no—
396#106—41no—
397#106—2841no—
398#106—2841no—
399#106—2841no—
400C4⋊C8—714no—
401C4⋊C8—714no—
402C4⋊C8—714no—
403C4⋊C8C4×D7114yes—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95—1421no—
411#95—1421no—
412#95—1421no—
413#95—1421no—
414#95—1421no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420C24—711no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—2841no—
426C22×C4—2841no—
427C22×C4—714no—
428C22×C4—711no—
429C22×C4—711no—
430C22×C4—711no—
431C22×C4—712no—
432C22×C4—712no—
433C22×C4—714no—
434C22×C4—1422no—
435C22×C4—1422no—
436C22×C4—1421no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4—1421no—
442C22×C4—1421no—
443C22×C4—1421no—
444C22×C4D4×D7111yescharacteristic
445C22×C4D4×D7111yescharacteristic
446C22×C4C23×D7111yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—2844no—
452C2×C8—2844no—
453C2×C8—712no—
454C2×C8—712no—
455C2×C8—1421no—
456C2×C8—1421no—
457C2×C8—1421no—
458C2×C8—1421no—
459C2×C8C2×C4×D7112yes—
460C2×C8C2×C4×D7112yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C22⋊C4—2842no—
478C22⋊C4—2842no—
479C22⋊C4—2842no—
480C22⋊C4—2842no—
481C42—42no—
482C42—2842no—
483C42—2842no—
484C42—2842no—
485C42—712no—
486C42—712no—
487C42—714no—
488C42—714no—
489C42C2×C4×D7112yes—
490C42Q8×D7112yes—
491C23—712no—
492C23—712no—
493C23—712no—
494C23—1421no—
495C23—1421no—
496C23—1421no—
497C23—1421no—
498C23C2×D4×D7111yescharacteristic
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—2844no—
516C2×C4—2841no—
517C2×C4—2841no—
518C2×C4—2841no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—712no—
529C2×C4—712no—
530C2×C4—712no—
531C2×C4—1428no—
532C2×C4—1421no—
533C2×C4—1421no—
534C2×C4—1421no—
535C2×C4—1421no—
536C2×C4—1421no—
537C2×C4—1421no—
538C2×C4—1422no—
539C2×C4—1422no—
540C2×C4—1422no—
541C2×C4—1422no—
542C2×C4C4⋊C4×D7112yes—
543C2×C4C4⋊C4×D7112yes—
544C2×C4C22×C4×D7111yescharacteristic
545C2×C4C2×Q8×D7111yescharacteristic
546C2×C4#1455617311yescharacteristic
547C2×C4#1455617411yescharacteristic
548C2×C4#1455617511yescharacteristic
549C2×C4#1455617611yescharacteristic
550C8—22no—
551C8—22no—
552C8—1422no—
553C8—1422no—
554C22—714no—
555C22—1422no—
556C22—1422no—
557C22—1422no—
558C22—1422no—
559C22—1422no—
560C22—1422no—
561C22C2×C4⋊C4×D7111yescharacteristic
562C22#1798657811yescharacteristic
563C22#1798657911yescharacteristic
564C22#135×D7111yescharacteristic
565C22#135×D7111yescharacteristic
566C22#1798679511yescharacteristic
567C22#1798679811yescharacteristic
568C4—22no—
569C4—22no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—5681no—
575C4—5681no—
576C4—5681no—
577C4—5681no—
578C4—1422no—
579C4—1422no—
580C4—1422no—
581C4#1798647111yescharacteristic
582C4#1798647511yescharacteristic
583C4#1798662611yescharacteristic
584C4#1798662911yescharacteristic
585C2—1424no—
586C2#2259095511yescharacteristic
587C2#427×D7111yescharacteristic
588C2#2259692711yescharacteristic
589C2#2259968911yescharacteristic
590C2#2259969011yescharacteristic
591C2#2259969111yescharacteristic
592C2#2259969411yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1949C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1949D71
hypocenterC71#21960
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #22589111C2semidirect
5. #22589110C2semidirect
6. #22589109C2semidirect
7. #22589108C2semidirect
8. #22589107C2semidirect
9. #22589106C2semidirect
10. #22589105C2semidirect
15. C71×#1949C2semidirect
28. C4⋊C8×D71C4semidirect
307. #1949D71semidirect
312. C71#21960semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C8×D712
#225907362
#225907332
#225891112
#225891102
#225891092
#225891082
#225891072
#225891062
#225891052
#225883682
#225883662
#225883642
#225883622
C71×#19492
#2196071