Finite Group Databasebeta

← #32007981

Subgroups — #32007981

Order 18176.

834 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986051C211yesmaximal, characteristic
2#22590736C211yesmaximal, characteristic
3#382×D71C211yesmaximal, characteristic
4#22589110C211yesmaximal, characteristic, complemented
5#22589108C211yesmaximal, characteristic, complemented
6#22589106C211yesmaximal, characteristic, complemented
7#22588534C211yesmaximal, characteristic
8#22588531C211yesmaximal, characteristic
9#22588529C211yesmaximal, characteristic
10#22588526C211yesmaximal, characteristic
11#22588503C211yesmaximal, characteristic, complemented
12#22588501C211yesmaximal, characteristic, complemented
13#22588469C211yesmaximal, characteristic, complemented
14#22588469C211yesmaximal, characteristic, complemented
15C71×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D71C2211yescharacteristic
17C2×C4⋊C4×D71C2211yescharacteristic
18C2×C42×D71C2211yescharacteristic
19#17986051C418yescomplemented
20C2×#14556062C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556060C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×C4⋊(C71⋊C8)C2211yescharacteristic
27C2×(C2×D71)⋊C8C2211yescharacteristic
28C2×(C2×D71)⋊C8C2211yescharacteristic
29C2×C8×Dic71C2211yescharacteristic
30C2×C8×Dic71C2211yescharacteristic
31#17985866—21no—
32#17985865C2211yescharacteristic
33#17985864C2211yescharacteristic
34#17985864C2211yescharacteristic
35#17985864C2211yescharacteristic
36#17985864C2211yescharacteristic
37#17985863—21no—
38#17985863—21no—
39#17985861—21no—
40#17985856C2211yescharacteristic
41#17985856C2211yescharacteristic
42C4×C4⋊Dic71C2211yescharacteristic
43C4×(C2×D71)⋊C4—21no—
44#17985845—21no—
45#95×D71—21no—
46#95×D71—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×D71—21no—
61C22×C4×D71—21no—
62C22×C4×D71D411yescharacteristic
63C22×C4×D71D411yescharacteristic
64C22×C4×D71C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×Dic71⋊C4—21no—
72C2×Dic71⋊C4—21no—
73C4⋊C4×D71—41no—
74C4⋊C4×D71—41no—
75C4⋊C4×D71—41no—
76C4⋊C4×D71—41no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71D411yescharacteristic
81C2×C4×Dic71D411yescharacteristic
82C2×C4×Dic71C2311yescharacteristic
83C42×D71C2×C414yes—
84C42×D71Q814yes—
85#14556062—41no—
86#14556061—41no—
87#14556061—41no—
88#14556060—41no—
89#14556059—41no—
90#14556059—41no—
91C4⋊(C71⋊C8)C2×C414yescomplemented
92(C2×D71)⋊C8C2×C414yes—
93(C2×D71)⋊C8C2×C414yes—
94C8×Dic71C2×C414yescomplemented
95C8×Dic71C2×C414yescomplemented
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×C414yescomplemented
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×D71—42no—
140C2×C4×D71—42no—
141C2×C4×D71—42no—
142C2×C4×D71—42no—
143C2×C4×D71—42no—
144C2×C4×D71—42no—
145C2×C4×D71C4⋊C414yes—
146C2×C4×D71C4⋊C414yes—
147C2×C4×D71C22×C412yes—
148C2×C4×D71C2×Q812yes—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8C22×C412yes—
154C2×C71⋊C8C22×C412yes—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—21no—
163Dic71⋊C4—42no—
164Dic71⋊C4—42no—
165(C2×D71)⋊C4—42no—
166(C2×D71)⋊C4—42no—
167C4×Dic71—42no—
168C4×Dic71C4⋊C414yes—
169C4×Dic71C4⋊C414yes—
170C4×Dic71C22×C412yes—
171C4×Dic71C2×Q812yes—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2411yescharacteristic
179C2×C8×C71—21no—
180C2×C8×C71—21no—
181C2×C8×C71—21no—
182C2×C8×C71—21no—
183C2×C8×C71C22×C412yes—
184C2×C8×C71C22×C412yes—
185C4⋊C4×C71—21no—
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—
193C42×C71—42no—
194C42×C71C22×C412yes—
195C42×C71C2×Q812yes—
196C23×C71C22×D411yescharacteristic
197C22×D71—21no—
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—21no—
201C22×D71C2×C4⋊C412yes—
202C22×D71#13512yes—
203C22×D71#13512yes—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
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×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71C2×C4⋊C412yes—
221C2×C4×C71C2×C4⋊C412yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C22×Q811yescharacteristic
224C2×C4×C71H(5,𝔽2)11yescharacteristic
225C2×C4×C71H(5,𝔽2)11yescharacteristic
226C2×C4×C712−1+411yescharacteristic
227C2×C4×C712−1+411yescharacteristic
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
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×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71C2×C4⋊C412yes—
247C2×Dic71C2×C4⋊C412yes—
248C2×Dic71C2×C4⋊C412yes—
249C4×D71—28no—
250C4×D71—22no—
251C4×D71—22no—
252C4×D71—22no—
253C4×D71—22no—
254C4×D71—44no—
255C4×D71—84no—
256C4×D71—84no—
257C8×C71—22no—
258C8×C71—22no—
259C71⋊C8—22no—
260C71⋊C8—22no—
261C22×C71C22×C4⋊C411yescharacteristic
262C22×C71#53511yescharacteristic
263C22×C71#53611yescharacteristic
264C22×C71C2×#13511yescharacteristic
265C22×C71C2×#13511yescharacteristic
266C22×C71#57511yescharacteristic
267C22×C71#57511yescharacteristic
268C2×D71—22no—
269C2×D71—22no—
270C2×D71—22no—
271C2×D71—22no—
272C2×D71—22no—
273C2×D71—22no—
274C2×D71#42714yes—
275C4×C71—22no—
276C4×C71—22no—
277C4×C71—81no—
278C4×C71—81no—
279C4×C71—81no—
280C4×C71—81no—
281C4×C71#52011yescharacteristic
282C4×C71#52011yescharacteristic
283C4×C71#54411yescharacteristic
284C4×C71#54411yescharacteristic
285Dic71—22no—
286Dic71—22no—
287Dic71—22no—
288Dic71—81no—
289Dic71—81no—
290Dic71—81no—
291Dic71—81no—
292#21983—711nomaximal, Sylow
293C2×C71#231711yescharacteristic
294C2×C71C2×#42711yescharacteristic
295C2×C71#291111yescharacteristic
296C2×C71#321111yescharacteristic
297C2×C71#321111yescharacteristic
298C2×C71#323211yescharacteristic
299C2×C71#323211yescharacteristic
300D71—24no—
301C2×#431—711no—
302#2283—711no—
303C2×#382—711no—
304#1949—711no—
305#1949—711no—
306#1949—711no—
307#1949D7111yescharacteristic, complemented
308#1853—711no—
309#1853—711no—
310#1852—711no—
311#1852—711no—
312#1846—711no—
313#1846—711no—
314#1835—711no—
315#1835—711no—
316C71#2198311yescharacteristic, complemented, Sylow
317C22×C4⋊C4—711no—
318C22×C4⋊C4—711no—
319C22×C42—711no—
320#431—718no—
321C2×#107—711no—
322C2×#107—711no—
323C2×#107—711no—
324C2×#107C2×D7111yescharacteristic
325C2×#106—711no—
326C2×#106—711no—
327C2×#106—711no—
328C2×#106C2×D7111yescharacteristic
329C2×C4⋊C8—711no—
330C2×C4⋊C8C2×D7111yescharacteristic
331C2×C22⋊C8—711no—
332C2×C22⋊C8—711no—
333C2×C4×C8—711no—
334C2×C4×C8—711no—
335#383—1421no—
336#382—711no—
337#382—711no—
338#382—711no—
339#382—711no—
340#382—711no—
341#382—1421no—
342#382—1421no—
343#382C2×D7111yescharacteristic
344#381—1421no—
345#380—711no—
346#380—711no—
347C4×C4⋊C4—711no—
348C4×C4⋊C4C2×D7111yescharacteristic
349C4×C22⋊C4—1421no—
350#376—1421no—
351C2×#95—1421no—
352C2×#95—1421no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340C2×D7111yescharacteristic
360#340C2×D7111yescharacteristic
361C23×C4—711no—
362C23×C4—711no—
363C23×C4—711no—
364C23×C4—1421no—
365C23×C4—1421no—
366C22×C8—711no—
367C22×C8—711no—
368C22×C8C22×D7111yescharacteristic
369C22×C8C22×D7111yescharacteristic
370C2×C4⋊C4—2841no—
371C2×C4⋊C4—2841no—
372C2×C4⋊C4—2841no—
373C2×C4⋊C4—2841no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C42—21no—
387C2×C42—711no—
388C2×C42—711no—
389C2×C42—711no—
390C2×C42—714no—
391C2×C42—714no—
392C2×C42—1421no—
393C2×C42C22×D7111yescharacteristic
394#107—41no—
395#107—2841no—
396#107—2841no—
397#107—2841no—
398#106—41no—
399#106—2841no—
400#106—2841no—
401#106—2841no—
402C4⋊C8—714no—
403C4⋊C8C4×D7114yescomplemented
404C22⋊C8—714no—
405C22⋊C8—714no—
406C4×C8—714no—
407C4×C8—714no—
408#95—21no—
409#95—21no—
410#95—21no—
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—
420#95—1421no—
421#95—1421no—
422#95—1421no—
423#95—1421no—
424C24—711no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—2841no—
430C22×C4—2841no—
431C22×C4—2842no—
432C22×C4—2842no—
433C22×C4—2842no—
434C22×C4—2842no—
435C22×C4—2842no—
436C22×C4—2842no—
437C22×C4—714no—
438C22×C4—711no—
439C22×C4—711no—
440C22×C4—711no—
441C22×C4—712no—
442C22×C4—712no—
443C22×C4—714no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1421no—
452C22×C4—1421no—
453C22×C4—1421no—
454C22×C4D4×D7111yescharacteristic
455C22×C4D4×D7111yescharacteristic
456C22×C4C23×D7111yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—712no—
462C2×C8—712no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8C2×C4×D7112yes—
468C2×C8C2×C4×D7112yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—2842no—
478C4⋊C4—2842no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C4⋊C4—1421no—
486C4⋊C4—1421no—
487C22⋊C4—2842no—
488C22⋊C4—2842no—
489C42—42no—
490C42—2842no—
491C42—712no—
492C42—712no—
493C42—714no—
494C42—714no—
495C42C2×C4×D7112yes—
496C42Q8×D7112yes—
497C23—712no—
498C23—712no—
499C23—712no—
500C23—1421no—
501C23—1421no—
502C23—1421no—
503C23—1421no—
504C23C2×D4×D7111yescharacteristic
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
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—41no—
521C2×C4—2844no—
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—2841no—
534C2×C4—5684no—
535C2×C4—5684no—
536C2×C4—712no—
537C2×C4—712no—
538C2×C4—712no—
539C2×C4—1428no—
540C2×C4—1421no—
541C2×C4—1421no—
542C2×C4—1421no—
543C2×C4—1421no—
544C2×C4—1421no—
545C2×C4—1421no—
546C2×C4—1422no—
547C2×C4—1422no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4C4⋊C4×D7112yes—
551C2×C4C4⋊C4×D7112yes—
552C2×C4C22×C4×D7111yescharacteristic
553C2×C4C2×Q8×D7111yescharacteristic
554C2×C4#1455617311yescharacteristic
555C2×C4#1455617411yescharacteristic
556C2×C4#1455617511yescharacteristic
557C2×C4#1455617611yescharacteristic
558C8—22no—
559C8—22no—
560C8—1422no—
561C8—1422no—
562C22—714no—
563C22—1422no—
564C22—1422no—
565C22—1422no—
566C22—1422no—
567C22—1422no—
568C22—1422no—
569C22C2×C4⋊C4×D7111yescharacteristic
570C22#1798656511yescharacteristic
571C22#1798657211yescharacteristic
572C22#135×D7111yescharacteristic
573C22#135×D7111yescharacteristic
574C22#1798679511yescharacteristic
575C22#1798679811yescharacteristic
576C4—22no—
577C4—22no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—5681no—
583C4—5681no—
584C4—5681no—
585C4—5681no—
586C4—1422no—
587C4—1422no—
588C4—1422no—
589C4#1798647811yescharacteristic
590C4#1798648111yescharacteristic
591C4#1798662611yescharacteristic
592C4#1798662911yescharacteristic
593C2—1424no—
594C2#2259096211yescharacteristic
595C2#427×D7111yescharacteristic
596C2#2259692711yescharacteristic
597C2#2259930311yescharacteristic
598C2#2259930911yescharacteristic
599C2#2259950611yescharacteristic
600C2#2259951211yescharacteristic
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#21983
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. #22589110C2semidirect
5. #22589108C2semidirect
6. #22589106C2semidirect
11. #22588503C2semidirect
12. #22588501C2semidirect
13. #22588469C2semidirect
14. #22588469C2semidirect
15. C71×#1949C2semidirect
19. #17986051C4semidirect
91. C4⋊(C71⋊C8)C2×C4semidirect
94. C8×Dic71C2×C4semidirect
95. C8×Dic71C2×C4semidirect
119. C4⋊C8×C71C2×C4semidirect
307. #1949D71semidirect
316. C71#21983semidirect
403. C4⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860512
#225907362
#382×D712
#225891102
#225891082
#225891062
#225885342
#225885312
#225885292
#225885262
#225885032
#225885012
#225884692
C71×#19492
#2198371