Finite Group Databasebeta

← #32008059

Subgroups — #32008059

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×#17986058C211yesmaximal, characteristic
2#22590965C211yesmaximal, characteristic
3#22590950C211yesmaximal, characteristic
4#22589109C211yesmaximal, characteristic, complemented
5#22589105C211yesmaximal, characteristic, complemented
6#22589100C211yesmaximal, characteristic, complemented
7#22589095C211yesmaximal, characteristic, complemented
8#22588538C211yesmaximal, characteristic
9#22588536C211yesmaximal, characteristic
10#22588531C211yesmaximal, characteristic
11#22588526C211yesmaximal, characteristic
12#22588509C211yesmaximal, characteristic, complemented
13#22588473C211yesmaximal, characteristic, complemented
14#22588468C211yesmaximal, characteristic, complemented
15C71×#1846C211yesmaximal, characteristic, complemented
16C2×#14556086C2211yescharacteristic
17C2×#14556085C2211yescharacteristic
18C2×C4⋊C4×D71C2211yescharacteristic
19#17986058C418yescomplemented
20C2×#14556061C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×Dic71⋊C8C2211yescharacteristic
25C2×Dic71⋊C8C2211yescharacteristic
26C2×(C2×D71)⋊C8C2211yescharacteristic
27C2×(C2×D71)⋊C8C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29#17985896—21no—
30#17985896—21no—
31#17985891—21no—
32#17985891—21no—
33#17985886—21no—
34#17985886—21no—
35#17985873—21no—
36#17985873—21no—
37#17985869C2211yescharacteristic
38#17985869C2211yescharacteristic
39#17985868C2211yescharacteristic
40#17985864C2211yescharacteristic
41#17985856C2211yescharacteristic
42#17985856C2211yescharacteristic
43C4⋊C4×Dic71C2211yescharacteristic
44C4⋊C4×Dic71C2211yescharacteristic
45#17985762C2211yescharacteristic
46#17985762C2211yescharacteristic
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C2×#107×C71C2211yescharacteristic
54C2×#107×C71C2211yescharacteristic
55C2×#106×C71C2211yescharacteristic
56C2×#106×C71C2211yescharacteristic
57C2×C4×C8×C71C2211yescharacteristic
58#382×C71C2211yescharacteristic
59#382×C71C2211yescharacteristic
60C22×C4×D71—21no—
61C22×C4×D71C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556086—41no—
65#14556086—41no—
66#14556085C2×C414yes—
67#14556085Q814yes—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4D411yescharacteristic
80C2×Dic71⋊C4D411yescharacteristic
81C4⋊C4×D71—41no—
82C4⋊C4×D71—41no—
83C2×(C2×D71)⋊C4—21no—
84C2×(C2×D71)⋊C4—21no—
85C2×(C2×D71)⋊C4—21no—
86C2×(C2×D71)⋊C4D411yescharacteristic
87C2×(C2×D71)⋊C4D411yescharacteristic
88C2×C4×Dic71—21no—
89C2×C4×Dic71—21no—
90C2×C4×Dic71—21no—
91C2×C4×Dic71C2311yescharacteristic
92#14556061—41no—
93#14556061—41no—
94#14556059—41no—
95#14556059—41no—
96Dic71⋊C8C2×C414yescomplemented
97Dic71⋊C8C2×C414yescomplemented
98(C2×D71)⋊C8C2×C414yes—
99(C2×D71)⋊C8C2×C414yes—
100#14556039C2×C414yescomplemented
101C71⋊#95—21no—
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C22×C8×C71C2311yescharacteristic
108C22×C8×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C42×C71C2311yescharacteristic
114#107×C71—41no—
115#107×C71—41no—
116#106×C71—41no—
117#106×C71—41no—
118C4×C8×C71C2×C414yescomplemented
119#95×C71—21no—
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—21no—
132C2×C4×D71—21no—
133C2×C4×D71—21no—
134C2×C4×D71—21no—
135C2×C4×D71—42no—
136C2×C4×D71—81no—
137C2×C4×D71—81no—
138C2×C4×D71C22×C412yes—
139C2×C4×D71C2×Q812yes—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8C22×C412yes—
145C2×C71⋊C8C22×C412yes—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—81no—
155C4⋊Dic71—81no—
156C4⋊Dic71—81no—
157C4⋊Dic71—81no—
158Dic71⋊C4—42no—
159Dic71⋊C4—81no—
160Dic71⋊C4—81no—
161Dic71⋊C4C4⋊C414yes—
162Dic71⋊C4C4⋊C414yes—
163(C2×D71)⋊C4—42no—
164(C2×D71)⋊C4—81no—
165(C2×D71)⋊C4—81no—
166(C2×D71)⋊C4C4⋊C414yes—
167(C2×D71)⋊C4C4⋊C414yes—
168C4×Dic71—42no—
169C4×Dic71—81no—
170C4×Dic71—81no—
171C4×Dic71C22×C412yes—
172C4×Dic71C2×Q812yes—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71—21no—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
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—
194C42×C71C22×C412yes—
195C42×C71C2×Q812yes—
196C23×C71C22×D411yescharacteristic
197C22×D71—21no—
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—21no—
201C22×D71—21no—
202C22×D71—21no—
203C22×D71C2×C4⋊C412yes—
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—22no—
250C4×D71—22no—
251C4×D71—22no—
252C4×D71—22no—
253C4×D71—84no—
254C8×C71—22no—
255C8×C71—22no—
256C71⋊C8—22no—
257C71⋊C8—22no—
258C22×C71C22×C4⋊C411yescharacteristic
259C22×C71#53511yescharacteristic
260C22×C71#53611yescharacteristic
261C22×C71#57611yescharacteristic
262C22×C71#57711yescharacteristic
263C22×C71#57711yescharacteristic
264C22×C71#57811yescharacteristic
265C2×D71—22no—
266C2×D71—22no—
267C2×D71—22no—
268C2×D71—41no—
269C2×D71—41no—
270C2×D71—41no—
271C2×D71—41no—
272C4×C71—22no—
273C4×C71—22no—
274C4×C71—81no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71—81no—
278C4×C71#52011yescharacteristic
279C4×C71#52011yescharacteristic
280C4×C71#54411yescharacteristic
281C4×C71#54411yescharacteristic
282Dic71—22no—
283Dic71—22no—
284Dic71—22no—
285Dic71—81no—
286Dic71—81no—
287Dic71—81no—
288Dic71—81no—
289#21992—711nomaximal, Sylow
290C2×C71#231711yescharacteristic
291C2×C71#291211yescharacteristic
292C2×C71#291311yescharacteristic
293C2×C71#322411yescharacteristic
294C2×C71#322511yescharacteristic
295C2×C71#324111yescharacteristic
296C2×C71#324211yescharacteristic
297D71—42no—
298C2×#433—711no—
299#2317—711no—
300#2314—711no—
301#1949—711no—
302#1949—711no—
303#1948—711no—
304#1947—711no—
305#1854—711no—
306#1854—711no—
307#1853—711no—
308#1852—711no—
309#1847—711no—
310#1846D7111yescharacteristic, complemented
311#1836—711no—
312#1834—711no—
313C71#2199211yescharacteristic, complemented, Sylow
314C2×#117—711no—
315C2×#117—711no—
316C22×C4⋊C4—711no—
317#433—718no—
318C2×#107—711no—
319C2×#107—711no—
320C2×#107C2×D7111yescharacteristic
321C2×#107C2×D7111yescharacteristic
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⋊C8—711no—
330C2×#97—711no—
331C2×C4×C8C2×D7111yescharacteristic
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#382C2×D7111yescharacteristic
345#382C2×D7111yescharacteristic
346#380—711no—
347#380—711no—
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#340—711no—
357#340—711no—
358C23×C4—711no—
359C23×C4—1421no—
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—2841no—
369C2×C4⋊C4—2841no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—1421no—
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—711no—
388C2×C22⋊C4—711no—
389C2×C22⋊C4—1421no—
390C2×C22⋊C4—1421no—
391C2×C22⋊C4—1421no—
392C2×C42—711no—
393C2×C42—1421no—
394C2×C42—1421no—
395C2×C42—1421no—
396C2×C42C22×D7111yescharacteristic
397#107—41no—
398#107—41no—
399#107—2841no—
400#107—2841no—
401#106—41no—
402#106—41no—
403#106—2841no—
404#106—2841no—
405C4⋊C8—714no—
406C4⋊C8—714no—
407C22⋊C8—714no—
408C22⋊C8—714no—
409#97—714no—
410C4×C8C4×D7114yescomplemented
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420#95—1421no—
421C24—711no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—2842no—
427C22×C4—5681no—
428C22×C4—5681no—
429C22×C4—711no—
430C22×C4—711no—
431C22×C4—711no—
432C22×C4—712no—
433C22×C4—712no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4—1421no—
442C22×C4D4×D7111yescharacteristic
443C22×C4D4×D7111yescharacteristic
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—2842no—
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—2842no—
483C22⋊C4—5681no—
484C22⋊C4—5681no—
485C22⋊C4—714no—
486C22⋊C4—714no—
487C42—2842no—
488C42—5681no—
489C42—5681no—
490C42—712no—
491C42—712no—
492C42C2×C4×D7112yes—
493C42Q8×D7112yes—
494C23—712no—
495C23—1421no—
496C23—1421no—
497C23—1421no—
498C23—1421no—
499C23—1421no—
500C23—1421no—
501C23C2×D4×D7111yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
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—41no—
516C2×C4—41no—
517C2×C4—41no—
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—2841no—
529C2×C4—2841no—
530C2×C4—5684no—
531C2×C4—712no—
532C2×C4—712no—
533C2×C4—712no—
534C2×C4—1421no—
535C2×C4—1421no—
536C2×C4—1421no—
537C2×C4—1421no—
538C2×C4—1421no—
539C2×C4—1421no—
540C2×C4—1422no—
541C2×C4—1422no—
542C2×C4—1422no—
543C2×C4—1422no—
544C2×C4C4⋊C4×D7112yes—
545C2×C4C4⋊C4×D7112yes—
546C2×C4C22×C4×D7111yescharacteristic
547C2×C4C2×Q8×D7111yescharacteristic
548C2×C4#1455617311yescharacteristic
549C2×C4#1455617311yescharacteristic
550C2×C4#1455617611yescharacteristic
551C2×C4#1455617611yescharacteristic
552C8—22no—
553C8—22no—
554C8—1422no—
555C8—1422no—
556C22—2841no—
557C22—2841no—
558C22—2841no—
559C22—2841no—
560C22—1422no—
561C22—1422no—
562C22—1422no—
563C22C2×C4⋊C4×D7111yescharacteristic
564C22#1798656211yescharacteristic
565C22#1798657411yescharacteristic
566C22#1798680111yescharacteristic
567C22#1798680611yescharacteristic
568C22#1798680611yescharacteristic
569C22#1798681311yescharacteristic
570C4—22no—
571C4—22no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—5681no—
577C4—5681no—
578C4—5681no—
579C4—5681no—
580C4—1422no—
581C4—1422no—
582C4—1422no—
583C4#1798648011yescharacteristic
584C4#1798648011yescharacteristic
585C4#1798662611yescharacteristic
586C4#1798662611yescharacteristic
587C2—2842no—
588C2#2259096111yescharacteristic
589C2#2259693211yescharacteristic
590C2#2259693711yescharacteristic
591C2#2259942711yescharacteristic
592C2#2259944211yescharacteristic
593C2#2259959611yescharacteristic
594C2#2259961111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1846C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1846D71
hypocenterC71#21992
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
4. #22589109C2semidirect
5. #22589105C2semidirect
6. #22589100C2semidirect
7. #22589095C2semidirect
12. #22588509C2semidirect
13. #22588473C2semidirect
14. #22588468C2semidirect
15. C71×#1846C2semidirect
19. #17986058C4semidirect
96. Dic71⋊C8C2×C4semidirect
97. Dic71⋊C8C2×C4semidirect
100. #14556039C2×C4semidirect
118. C4×C8×C71C2×C4semidirect
310. #1846D71semidirect
313. C71#21992semidirect
410. C4×C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860582
#225909652
#225909502
#225891092
#225891052
#225891002
#225890952
#225885382
#225885362
#225885312
#225885262
#225885092
#225884732
#225884682
C71×#18462
#2199271