Finite Group Databasebeta

← #32056261

Subgroups — #32056261

Order 18176.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986398C211yesmaximal, characteristic, complemented
2C2×#17986039C211yesmaximal, characteristic
3#22592053C211yesmaximal, characteristic
4#22589822C211yesmaximal, characteristic, complemented
5#22589816C211yesmaximal, characteristic, complemented
6#22589678C211yesmaximal, characteristic, complemented
7#22589675C211yesmaximal, characteristic, complemented
8#22589075C211yesmaximal, characteristic, complemented
9#22589065C211yesmaximal, characteristic, complemented
10#22588537C211yesmaximal, characteristic
11#22588533C211yesmaximal, characteristic
12#22588509C211yesmaximal, characteristic, complemented
13#22588371C211yesmaximal, characteristic
14#22588124C211yesmaximal, characteristic
15C71×#2442C211yesmaximal, characteristic
16C22×#11765849C2211yescharacteristic
17C2×#14556120C2211yescharacteristic
18C2×#14556087C2211yescharacteristic
19#17986398—24no—
20#17986039—24no—
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556061C2211yescharacteristic
24C2×#14556061C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27C2×#14556059C2211yescharacteristic
28C2×#14556059C2211yescharacteristic
29C2×#14556059C2211yescharacteristic
30C2×#14556059C2211yescharacteristic
31C2×C22⋊(C71⋊C8)C2211yescharacteristic
32C2×#14556039C2211yescharacteristic
33C2×C4×C71⋊C8C2211yescharacteristic
34#17985935—21no—
35#17985922—21no—
36#17985897C2211yescharacteristic
37#17985897C2211yescharacteristic
38#17985896—21no—
39#17985882—21no—
40#17985869C2211yescharacteristic
41#17985859C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985754C2211yescharacteristic
49#17985754C2211yescharacteristic
50C2×#125×C71C2211yescharacteristic
51C22×C4⋊C4×C71C2211yescharacteristic
52#400×C71—21no—
53#398×C71—21no—
54#389×C71C2211yescharacteristic
55#389×C71C2211yescharacteristic
56#385×C71—21no—
57#382×C71C2211yescharacteristic
58#380×C71—21no—
59#380×C71C2211yescharacteristic
60C4×C22⋊C4×C71C2211yescharacteristic
61C2×#11765849—22no—
62C2×#11765849—22no—
63C2×#11765849Q814yes—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68#14556120—22no—
69#14556120—41no—
70#14556087—22no—
71#14556087—41no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×C4⋊Dic71C2311yescharacteristic
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4—21no—
80C2×Dic71⋊C4—21no—
81C2×Dic71⋊C4—21no—
82C2×C22⋊Dic71—21no—
83C2×C22⋊Dic71—21no—
84C2×C4×Dic71—21no—
85#14556061—22no—
86#14556061—22no—
87#14556061—41no—
88#14556061D414yes—
89#14556059—22no—
90#14556059—22no—
91#14556059—22no—
92#14556059—22no—
93#14556059—22no—
94#14556059—41no—
95C22⋊(C71⋊C8)Q814yes—
96#14556039—22no—
97C4×C71⋊C8—22no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C23×C4×C71—21no—
102C23×C4×C71C2311yescharacteristic
103#125×C71—22no—
104#125×C71—41no—
105C2×C4⋊C4×C71—22no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—21no—
111C2×C4⋊C4×C71—41no—
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C22⋊C4×C71—21no—
117C2×C22⋊C4×C71D411yescharacteristic
118C2×C22⋊C4×C71D411yescharacteristic
119C2×C42×C71C2311yescharacteristic
120C2×C42×C71C2311yescharacteristic
121#95×C71—21no—
122#95×C71—21no—
123#95×C71—21no—
124#95×C71—21no—
125#95×C71—21no—
126#95×C71D411yescharacteristic
127#95×C71D411yescharacteristic
128C22×Dic71—21no—
129C22×Dic71—21no—
130C22×Dic71—21no—
131C22×Dic71—21no—
132#11765849—28no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8C2×D412yes—
140C2×C71⋊C8C2×Q812yes—
141C2×C71⋊C8C2×Q812yes—
142C2×C71⋊C8C2×Q812yes—
143C2×C71⋊C8C2×Q812yes—
144C2×C71⋊C8#4112yes—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—42no—
153C4⋊Dic71—81no—
154C4⋊Dic71C2×D412yes—
155Dic71⋊C4—42no—
156Dic71⋊C4—42no—
157Dic71⋊C4—42no—
158Dic71⋊C4—81no—
159C22⋊Dic71—42no—
160C22⋊Dic71—81no—
161C4×Dic71—42no—
162C24×C71C2×D411yescharacteristic
163C22×C4×C71—21no—
164C22×C4×C71—21no—
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—42no—
172C22×C4×C71—42no—
173C22×C4×C71—81no—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×Q812yes—
180C22×C4×C71C2×Q812yes—
181C22×C4×C71C2411yescharacteristic
182C4⋊C4×C71—21no—
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—42no—
190C4⋊C4×C71—42no—
191C4⋊C4×C71—42no—
192C4⋊C4×C71—81no—
193C4⋊C4×C71—81no—
194C4⋊C4×C71C2×D412yes—
195C22⋊C4×C71—41no—
196C22⋊C4×C71—41no—
197C22⋊C4×C71—42no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—21no—
202C23×C71—21no—
203C23×C71—21no—
204C23×C71—21no—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71C4⋊Q812yes—
209C23×C71C22×D411yescharacteristic
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—22no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
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—84no—
241C2×C4×C71C4⋊Q812yes—
242C2×C4×C71C22×D411yescharacteristic
243C2×C4×C71C22×Q811yescharacteristic
244C2×C4×C71C22×Q811yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71H(5,𝔽2)11yescharacteristic
247C2×C4×C712−1+411yescharacteristic
248C2×Dic71—22no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C71⋊C8—22no—
261C71⋊C8—22no—
262C71⋊C8—22no—
263C71⋊C8C22⋊Q814yes—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—41no—
271C22×C71C2×C4⋊Q811yescharacteristic
272C22×C71D4211yescharacteristic
273C22×C71#54711yescharacteristic
274C22×C71#57511yescharacteristic
275C22×C71#57511yescharacteristic
276C22×C71#57711yescharacteristic
277C22×C71#57711yescharacteristic
278C4×C71—22no—
279C4×C71—41no—
280C4×C71—41no—
281C4×C71—41no—
282C4×C71—42no—
283C4×C71—41no—
284C4×C71—81no—
285C4×C71—81no—
286C4×C71—81no—
287C4×C71C2×C22⋊Q811yescharacteristic
288C4×C71#54411yescharacteristic
289C4×C71D4×Q811yescharacteristic
290C4×C71#55411yescharacteristic
291Dic71—42no—
292Dic71—81no—
293Dic71—81no—
294Dic71—81no—
295#26075—711nomaximal, Sylow
296C2×C71—42no—
297C2×C71#244011yescharacteristic
298C2×C71#316311yescharacteristic
299C2×C71#316611yescharacteristic
300C2×C71#326711yescharacteristic
301C2×C71#327111yescharacteristic
302C2×C71#335111yescharacteristic
303C2×C71#335111yescharacteristic
304C2×#499—711no—
305C2×#428—711no—
306#2442D7111yescharacteristic
307#2441—711no—
308#2085—711no—
309#2084—711no—
310#2059—711no—
311#2059—711no—
312#1943—711no—
313#1941—711no—
314#1854—711no—
315#1853—711no—
316#1847—711no—
317#1809—711no—
318#1753—711no—
319C71#2607511yescharacteristic, complemented, Sylow
320C22×M16—711no—
321C2×#125—711no—
322C2×#125C2×D7111yescharacteristic
323C2×#117—711no—
324C22×C4⋊C4C2×D7111yescharacteristic
325#499—1424no—
326#428—1424no—
327C2×#107—711no—
328C2×#107—711no—
329C2×#107—711no—
330C2×#107—711no—
331C2×#106—711no—
332C2×#106—711no—
333C2×#106—711no—
334C2×#106—711no—
335C2×#106—711no—
336C2×#106—711no—
337C2×C22⋊C8—711no—
338C2×#97—711no—
339C2×C4×C8—711no—
340#400—21no—
341#399—1421no—
342#398—21no—
343#396—1421no—
344#389—711no—
345#389—711no—
346#389—1421no—
347#389C2×D7111yescharacteristic
348#389C2×D7111yescharacteristic
349#387—1421no—
350#385—21no—
351#384—711no—
352#382C2×D7111yescharacteristic
353#380—21no—
354#380—711no—
355#380C2×D7111yescharacteristic
356C4×C22⋊C4C2×D7111yescharacteristic
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#336—711no—
364#336—711no—
365C23×C4—21no—
366C23×C4C22×D7111yescharacteristic
367C2×M16—714no—
368C2×M16—1422no—
369C2×M16—1422no—
370C22×C8—711no—
371C22×C8—711no—
372C22×C8—711no—
373C22×C8—711no—
374#125—22no—
375#125—41no—
376#125—2841no—
377#125—1422no—
378#117—2841no—
379#117—1422no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—41no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4—1421no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C4⋊C4C22×D7111yescharacteristic
399C2×C4⋊C4C22×D7111yescharacteristic
400C2×C4⋊C4C22×D7111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1421no—
403C2×C22⋊C4—1421no—
404C2×C22⋊C4C22⋊D7111yescharacteristic
405C2×C22⋊C4C22⋊D7111yescharacteristic
406C2×C42—1421no—
407C2×C42C22×D7111yescharacteristic
408C2×C42C22×D7111yescharacteristic
409#107—2841no—
410#107—714no—
411#107—1422no—
412#107—1422no—
413#106—2841no—
414#106—1422no—
415#106—1422no—
416#106—1422no—
417#106—1422no—
418#106—1422no—
419C22⋊C8—714no—
420#97—1422no—
421C4×C8—1422no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1421no—
428#95—1421no—
429#95—1421no—
430#95C22⋊D7111yescharacteristic
431#95C22⋊D7111yescharacteristic
432C24C2×C22⋊D7111yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4C2×C22⋊D7111yescharacteristic
449C22×C4C2×C22⋊D7111yescharacteristic
450C22×C4C2×C22⋊D7111yescharacteristic
451C22×C4C2×C22⋊D7111yescharacteristic
452C22×C4C2×C22⋊D7111yescharacteristic
453C22×C4Q8×D7112yes—
454C22×C4Q8×D7112yes—
455C22×C4C23×D7111yescharacteristic
456M16—1428no—
457C2×C8—712no—
458C2×C8—712no—
459C2×C8—712no—
460C2×C8—712no—
461C2×C8—712no—
462C2×C8—712no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—2842no—
482C4⋊C4—2842no—
483C4⋊C4—2842no—
484C4⋊C4—2842no—
485C4⋊C4—5681no—
486C4⋊C4—5681no—
487C4⋊C4—712no—
488C4⋊C4—1421no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4D4×D7112yes—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—2842no—
500C22⋊C4—5681no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—2842no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1455612612yes—
513C23C22×C22⋊D7111yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—1422no—
557C2×C4#1455612612yes—
558C2×C4C2×D4×D7111yescharacteristic
559C2×C4C2×Q8×D7111yescharacteristic
560C2×C4C2×Q8×D7111yescharacteristic
561C2×C4#41×D7111yescharacteristic
562C2×C4#1455617311yescharacteristic
563C2×C4#1455617511yescharacteristic
564C8—714no—
565C8—1422no—
566C8—1422no—
567C8—1422no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1455612611yescharacteristic
576C22D4×C22⋊D7111yescharacteristic
577C22#1798664811yescharacteristic
578C22#1798679711yescharacteristic
579C22#1798679711yescharacteristic
580C22#1798680711yescharacteristic
581C22#1798681011yescharacteristic
582C4—22no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—2842no—
592C4—5681no—
593C4—5681no—
594C4—5681no—
595C4C22⋊Q8×D7111yescharacteristic
596C4#1798662811yescharacteristic
597C4#1798666211yescharacteristic
598C4#1798670311yescharacteristic
599C2—42no—
600C2#2259203911yescharacteristic
601C2#2259886611yescharacteristic
602C2#2259888911yescharacteristic
603C2#2259986111yescharacteristic
604C2#2259992611yescharacteristic
605C2#2260089411yescharacteristic
606C2#2260090011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2442C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2442D71
hypocenterC71#26075
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#17986398C2semidirect
4. #22589822C2semidirect
5. #22589816C2semidirect
6. #22589678C2semidirect
7. #22589675C2semidirect
8. #22589075C2semidirect
9. #22589065C2semidirect
12. #22588509C2semidirect
319. C71#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#179863982
C2×#179860392
#225920532
#225898222
#225898162
#225896782
#225896752
#225890752
#225890652
#225885372
#225885332
#225885092
#225883712
#225881242
C71×#24422
#2607571