Finite Group Databasebeta

← #32008034

Subgroups — #32008034

Order 18176.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986057C211yesmaximal, characteristic
2#22590964C211yesmaximal, characteristic
3#22590947C211yesmaximal, characteristic
4#22589111C211yesmaximal, characteristic, complemented
5#22589109C211yesmaximal, characteristic, complemented
6#22589102C211yesmaximal, characteristic, complemented
7#22589100C211yesmaximal, characteristic, complemented
8#22588541C211yesmaximal, characteristic
9#22588537C211yesmaximal, characteristic
10#22588533C211yesmaximal, characteristic
11#22588505C211yesmaximal, characteristic, complemented
12#22588495C211yesmaximal, characteristic, complemented
13#22588472C211yesmaximal, characteristic, complemented
14#22588469C211yesmaximal, characteristic, complemented
15C71×#1853C211yesmaximal, characteristic
16C2×#14556087C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18#17986057C418yescomplemented
19C2×#14556062C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556060C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×Dic71⋊C8C2211yescharacteristic
26C2×Dic71⋊C8C2211yescharacteristic
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×#14556038C2211yescharacteristic
29C2×C8×Dic71C2211yescharacteristic
30#17985895—21no—
31#17985895—21no—
32#17985893—21no—
33#17985887—21no—
34#17985884—21no—
35#17985882—21no—
36#17985872—21no—
37#17985872—21no—
38#17985868C2211yescharacteristic
39#17985868C2211yescharacteristic
40#17985868C2211yescharacteristic
41#17985864C2211yescharacteristic
42#17985864C2211yescharacteristic
43#17985864C2211yescharacteristic
44#17985859C2211yescharacteristic
45C4⋊C4×Dic71C2211yescharacteristic
46C4⋊C4×Dic71C2211yescharacteristic
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C22×C4⋊C4×C71C2211yescharacteristic
54C2×#106×C71C2211yescharacteristic
55C2×#106×C71C2211yescharacteristic
56C2×C22⋊C8×C71C2211yescharacteristic
57#380×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62#14556087—41no—
63#14556087—41no—
64#14556087C2×C414yes—
65#14556087Q814yes—
66C2×C4⋊Dic71D411yescharacteristic
67C2×C4⋊Dic71D411yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×C22⋊Dic71—21no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71D411yescharacteristic
81C2×C22⋊Dic71D411yescharacteristic
82C2×C4×Dic71—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71C2311yescharacteristic
86C2×C4×Dic71C2311yescharacteristic
87#14556062—41no—
88#14556061—41no—
89#14556060—41no—
90#14556059—41no—
91#14556059—41no—
92#14556059—41no—
93Dic71⋊C8C2×C414yescomplemented
94Dic71⋊C8C2×C414yescomplemented
95C22⋊(C71⋊C8)C2×C414yes—
96#14556038C2×C414yescomplemented
97C8×Dic71C2×C414yescomplemented
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—
107C23×C4×C71—21no—
108C23×C4×C71C2311yescharacteristic
109C22×C8×C71C2311yescharacteristic
110C22×C8×C71C2311yescharacteristic
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—41no—
113C2×C4⋊C4×C71—41no—
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C22⋊C4×C71—21no—
119#106×C71—41no—
120#106×C71—41no—
121C22⋊C8×C71C2×C414yes—
122#95×C71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71C2×D411yescharacteristic
128C22×Dic71C2×D411yescharacteristic
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8C22×C412yes—
136C2×C71⋊C8C22×C412yes—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71C4⋊C414yes—
146C4⋊Dic71C4⋊C414yes—
147Dic71⋊C4—81no—
148Dic71⋊C4—81no—
149Dic71⋊C4—81no—
150Dic71⋊C4—81no—
151Dic71⋊C4—81no—
152Dic71⋊C4—81no—
153C22⋊Dic71—42no—
154C22⋊Dic71—81no—
155C22⋊Dic71C4⋊C414yes—
156C22⋊Dic71C4⋊C414yes—
157C4×Dic71—42no—
158C4×Dic71—81no—
159C4×Dic71—81no—
160C4×Dic71C22×C412yes—
161C4×Dic71C22×C412yes—
162C4×Dic71C2×Q812yes—
163C4×Dic71C2×Q812yes—
164C24×C71C2×D411yescharacteristic
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—42no—
174C22×C4×C71—81no—
175C22×C4×C71—81no—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×Q812yes—
179C22×C4×C71C2411yescharacteristic
180C2×C8×C71—21no—
181C2×C8×C71—21no—
182C2×C8×C71—21no—
183C2×C8×C71—21no—
184C2×C8×C71C22×C412yes—
185C2×C8×C71C22×C412yes—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—42no—
195C22⋊C4×C71—81no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71—21no—
202C23×C71C2×C4⋊C412yes—
203C23×C71C22×D411yescharacteristic
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
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×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—84no—
223C2×C4×C71C2×C4⋊C412yes—
224C2×C4×C71C23×C411yescharacteristic
225C2×C4×C71C22×Q811yescharacteristic
226C2×C4×C71H(5,𝔽2)11yescharacteristic
227C2×C4×C71H(5,𝔽2)11yescharacteristic
228C2×C4×C712−1+411yescharacteristic
229C2×C4×C712−1+411yescharacteristic
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237C2×Dic71—21no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71C2×C4⋊C412yes—
251C2×Dic71C2×C4⋊C412yes—
252C2×Dic71C2×C4⋊C412yes—
253C2×Dic71C2×C4⋊C412yes—
254C8×C71—22no—
255C8×C71—22no—
256C71⋊C8—22no—
257C71⋊C8—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71C22×C4⋊C411yescharacteristic
266C22×C71#53511yescharacteristic
267C22×C71#53611yescharacteristic
268C22×C71#57711yescharacteristic
269C22×C71#57711yescharacteristic
270C22×C71#57711yescharacteristic
271C22×C71#57711yescharacteristic
272C4×C71—22no—
273C4×C71—81no—
274C4×C71—81no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71#52011yescharacteristic
278C4×C71#52011yescharacteristic
279C4×C71#54411yescharacteristic
280C4×C71#54411yescharacteristic
281Dic71—22no—
282Dic71—22no—
283Dic71—22no—
284Dic71—22no—
285Dic71—81no—
286Dic71—81no—
287Dic71—81no—
288Dic71—81no—
289#21990—711nomaximal, Sylow
290C2×C71—42no—
291C2×C71#231711yescharacteristic
292C2×C71#291211yescharacteristic
293C2×C71#291211yescharacteristic
294C2×C71#322111yescharacteristic
295C2×C71#322211yescharacteristic
296C2×C71#324411yescharacteristic
297C2×C71#324411yescharacteristic
298C2×#433—711no—
299#2317—711no—
300#2314—711no—
301#1949—711no—
302#1949—711no—
303#1948—711no—
304#1948—711no—
305#1854—711no—
306#1854—711no—
307#1853—711no—
308#1853D7111yescharacteristic
309#1847—711no—
310#1844—711no—
311#1836—711no—
312#1835—711no—
313C71#2199011yescharacteristic, complemented, Sylow
314C2×#117—711no—
315C2×#117—711no—
316C22×C4⋊C4C2×D7111yescharacteristic
317#433—718no—
318C2×#107—711no—
319C2×#107—711no—
320C2×#106—711no—
321C2×#106—711no—
322C2×#106—711no—
323C2×#106—711no—
324C2×#106C2×D7111yescharacteristic
325C2×#106C2×D7111yescharacteristic
326C2×C4⋊C8—711no—
327C2×C4⋊C8—711no—
328C2×C22⋊C8—711no—
329C2×C22⋊C8C2×D7111yescharacteristic
330C2×#97—711no—
331C2×C4×C8—711no—
332#389—1421no—
333#389—1421no—
334#388—1421no—
335#388—1421no—
336#387—1421no—
337#387—1421no—
338#385—1421no—
339#385—1421no—
340#384—711no—
341#384—711no—
342#384—711no—
343#382—711no—
344#382—711no—
345#382—711no—
346#380—711no—
347#380C2×D7111yescharacteristic
348C4×C4⋊C4—711no—
349C4×C4⋊C4—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340C2×D7111yescharacteristic
357#340C2×D7111yescharacteristic
358C23×C4—21no—
359C23×C4C22×D7111yescharacteristic
360C22×C8—711no—
361C22×C8—711no—
362C22×C8C22×D7111yescharacteristic
363C22×C8C22×D7111yescharacteristic
364#117—2841no—
365#117—2841no—
366#117—714no—
367#117—714no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4—1421no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C4⋊C4C22×D7111yescharacteristic
385C2×C4⋊C4C22×D7111yescharacteristic
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—711no—
389C2×C22⋊C4—711no—
390C2×C22⋊C4—1421no—
391C2×C22⋊C4—1421no—
392C2×C42—711no—
393C2×C42—711no—
394C2×C42—1421no—
395C2×C42—1421no—
396C2×C42—1421no—
397#107—2841no—
398#107—2841no—
399#106—41no—
400#106—41no—
401#106—2841no—
402#106—2841no—
403#106—2841no—
404#106—2841no—
405C4⋊C8—714no—
406C4⋊C8—714no—
407C22⋊C8—714no—
408C22⋊C8C4×D7114yes—
409#97—714no—
410C4×C8—714no—
411#95—21no—
412#95—1421no—
413#95—1421no—
414#95—1421no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420#95—1421no—
421C24D4×D7111yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—711no—
434C22×C4—711no—
435C22×C4—711no—
436C22×C4—711no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4C2×C4×D7112yes—
442C22×C4D4×D7111yescharacteristic
443C22×C4Q8×D7112yes—
444C22×C4C23×D7111yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—712no—
450C2×C8—712no—
451C2×C8—1421no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8C2×C4×D7112yes—
456C2×C8C2×C4×D7112yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—5681no—
467C4⋊C4—5681no—
468C4⋊C4—5681no—
469C4⋊C4—5681no—
470C4⋊C4—5681no—
471C4⋊C4—5681no—
472C4⋊C4—714no—
473C4⋊C4—714no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C22⋊C4—81no—
483C22⋊C4—2842no—
484C22⋊C4—5681no—
485C22⋊C4—714no—
486C22⋊C4—714no—
487C42—2842no—
488C42—5681no—
489C42—5681no—
490C42—712no—
491C42—712no—
492C42—712no—
493C42—712no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D7112yes—
501C23C2×D4×D7111yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—712no—
534C2×C4—712no—
535C2×C4—712no—
536C2×C4—712no—
537C2×C4—1421no—
538C2×C4—1421no—
539C2×C4—1421no—
540C2×C4—1421no—
541C2×C4—1421no—
542C2×C4—1421no—
543C2×C4—1421no—
544C2×C4—1421no—
545C2×C4C4⋊C4×D7112yes—
546C2×C4C22×C4×D7111yescharacteristic
547C2×C4C2×Q8×D7111yescharacteristic
548C2×C4#1455617311yescharacteristic
549C2×C4#1455617311yescharacteristic
550C2×C4#1455617511yescharacteristic
551C2×C4#1455617511yescharacteristic
552C8—22no—
553C8—22no—
554C8—1422no—
555C8—1422no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D7111yescharacteristic
564C22#1798656411yescharacteristic
565C22#1798657111yescharacteristic
566C22#1798680511yescharacteristic
567C22#1798680511yescharacteristic
568C22#1798680811yescharacteristic
569C22#1798680911yescharacteristic
570C4—22no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—5681no—
576C4—5681no—
577C4—5681no—
578C4—5681no—
579C4—1422no—
580C4—1422no—
581C4—1422no—
582C4—1422no—
583C4#1798647811yescharacteristic
584C4#1798647811yescharacteristic
585C4#1798662811yescharacteristic
586C4#1798662811yescharacteristic
587C2—42no—
588C2#2259095911yescharacteristic
589C2#2259693111yescharacteristic
590C2#2259693411yescharacteristic
591C2#2259939711yescharacteristic
592C2#2259940811yescharacteristic
593C2#2259963411yescharacteristic
594C2#2259964211yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1853C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#1853D71
hypocenterC71#21990
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #22589111C2semidirect
5. #22589109C2semidirect
6. #22589102C2semidirect
7. #22589100C2semidirect
11. #22588505C2semidirect
12. #22588495C2semidirect
13. #22588472C2semidirect
14. #22588469C2semidirect
18. #17986057C4semidirect
93. Dic71⋊C8C2×C4semidirect
94. Dic71⋊C8C2×C4semidirect
96. #14556038C2×C4semidirect
97. C8×Dic71C2×C4semidirect
313. C71#21990semidirect

Maximal subgroups

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

subgroupindex
C2×#179860572
#225909642
#225909472
#225891112
#225891092
#225891022
#225891002
#225885412
#225885372
#225885332
#225885052
#225884952
#225884722
#225884692
C71×#18532
#2199071