Finite Group Databasebeta

← #32000522

Subgroups — #32000522

Order 18176.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986097C211yesmaximal, characteristic
2#22590702C211yesmaximal, characteristic
3#22588978C211yesmaximal, characteristic, complemented
4#22588971C211yesmaximal, characteristic, complemented
5#22588929C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588677C211yesmaximal, characteristic
8#22588677C211yesmaximal, characteristic
9#22588600C211yesmaximal, characteristic
10#22588586C211yesmaximal, characteristic
11#22588238C211yesmaximal, characteristic, complemented
12C4×#14556052C211yesmaximal, characteristic, complemented
13C4×#14556048C211yesmaximal, characteristic
14#22588092C211yesmaximal, characteristic, complemented
15C71×#2320C211yesmaximal, characteristic, complemented
16C22×#11765852C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18C2×C4×D284C2211yescharacteristic
19C2×C4×D284C2211yescharacteristic
20#17986097C418yescomplemented
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556052C2211yescharacteristic
25C2×#14556052C2211yescharacteristic
26C2×#14556052C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556048C2211yescharacteristic
29C2×#14556048C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×C4×C71⋊C8C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C4×C4⋊Dic71C2211yescharacteristic
34C4×C4⋊Dic71C2211yescharacteristic
35#17985846—21no—
36#17985846—21no—
37#17985846—21no—
38#95×D71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985754C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×Q8×C71C2211yescharacteristic
50#391×C71C2211yescharacteristic
51#391×C71C2211yescharacteristic
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#387×C71—21no—
55#387×C71—21no—
56#383×C71—21no—
57#382×C71—21no—
58C4×C4⋊C4×C71C2211yescharacteristic
59#376×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C22×C4×D71—21no—
63C22×C4×D71—21no—
64C2×#11765852—22no—
65C2×#11765852—22no—
66C2×#11765852C2×C414yes—
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C22×C71⋊C8C2311yescharacteristic
71C4×D284—22no—
72C4×D284—22no—
73C4×D284—22no—
74C4×D284—22no—
75C4×D284—22no—
76C4×D284C2×C414yes—
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82C2×(C2×D71)⋊C4—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86#14556061—22no—
87#14556061C2×C414yescomplemented
88#14556059—22no—
89#14556052—22no—
90#14556052—22no—
91#14556052C2×C414yescomplemented
92#14556048—22no—
93#14556048—22no—
94#14556048C2×C414yes—
95#14556039C2×C414yescomplemented
96C4×C71⋊C8—22no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C22×Q8×C71C2311yescharacteristic
101C4×Q8×C71—22no—
102C4×Q8×C71C2×C414yescomplemented
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C42×C71—21no—
110C2×C42×C71—21no—
111C2×C42×C71C2311yescharacteristic
112C2×C42×C71C2311yescharacteristic
113C2×C42×C71C2311yescharacteristic
114#95×C71—21no—
115#95×C71—21no—
116#95×C71—21no—
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120#95×C71—21no—
121#95×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123#95×C71D411yescharacteristic
124#95×C71D411yescharacteristic
125C23×D71—21no—
126C2×D284—21no—
127C2×D284—21no—
128C2×D284—21no—
129C2×D284C22×C412yes—
130C2×D284#4112yes—
131C2×D284#4112yes—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C2×C4×D71—24no—
136C2×C4×D71—24no—
137C2×C4×D71—24no—
138C2×C4×D71—81no—
139C2×C4×D71—81no—
140C2×C4×D71—81no—
141C2×C4×D71—81no—
142C2×C4×D71—81no—
143C2×C4×D71—81no—
144#11765852—28no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8C22×C412yes—
156C2×C71⋊C8C22×C412yes—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71C22×C412yes—
161C4⋊Dic71#4112yes—
162C4⋊Dic71#4112yes—
163(C2×D71)⋊C4—24no—
164(C2×D71)⋊C4—24no—
165(C2×D71)⋊C4—24no—
166C4×Dic71—81no—
167C4×Dic71—81no—
168C4×Dic71—81no—
169C2×Q8×C71—21no—
170C2×Q8×C71—21no—
171C2×Q8×C71—21no—
172C2×Q8×C71—21no—
173C2×Q8×C71—21no—
174C2×Q8×C71C22×C412yes—
175C22×C4×C71—21no—
176C22×C4×C71—21no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2411yescharacteristic
186C4⋊C4×C71—24no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—81no—
193C4⋊C4×C71—81no—
194C4⋊C4×C71C22×C412yes—
195C42×C71—24no—
196C42×C71—21no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71—81no—
201C42×C71C22×C412yes—
202C42×C71#4112yes—
203C23×C71C22×D411yescharacteristic
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—41no—
208C22×D71—41no—
209C22×D71—41no—
210C22×D71—41no—
211Q8×C71—22no—
212Q8×C71—22no—
213Q8×C71—41no—
214Q8×C71—41no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
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—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
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—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71C4×D412yes—
244C2×C4×C71C4×D412yes—
245C2×C4×C71C23×C411yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260D284—22no—
261D284—22no—
262D284—22no—
263D284#11714yes—
264C4×D71—84no—
265C4×D71—84no—
266C4×D71—84no—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C71⋊C8—41no—
270C71⋊C8—41no—
271C22×C71C2×C4×D411yescharacteristic
272C22×C71#52511yescharacteristic
273C22×C71#52511yescharacteristic
274C22×C71#57611yescharacteristic
275C22×C71#57711yescharacteristic
276C22×C71#57711yescharacteristic
277C22×C71#57811yescharacteristic
278C2×D71—24no—
279C2×D71—42no—
280C2×D71—42no—
281C2×D71—42no—
282C2×D71—42no—
283C2×D71—42no—
284C2×D71—42no—
285C4×C71—22no—
286C4×C71—22no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—41no—
290C4×C71—41no—
291C4×C71—42no—
292C4×C71—42no—
293C4×C71—81no—
294C4×C71—81no—
295C4×C71C2×#11711yescharacteristic
296C4×C71C4×#4111yescharacteristic
297C4×C71#52911yescharacteristic
298C4×C71#52911yescharacteristic
299Dic71—42no—
300Dic71—42no—
301Dic71—42no—
302#21125—711nomaximal, Sylow
303C2×C71#229011yescharacteristic
304C2×C71#294611yescharacteristic
305C2×C71#294711yescharacteristic
306C2×C71#311811yescharacteristic
307C2×C71#312011yescharacteristic
308C2×C71#312311yescharacteristic
309C2×C71#312411yescharacteristic
310D71—44no—
311C2×#440—711no—
312#2320D7111yescharacteristic, complemented
313#2275—711no—
314#1928—711no—
315#1927—711no—
316#1921—711no—
317#1918—711no—
318#1877—711no—
319#1877—711no—
320#1864—711no—
321#1862—711no—
322#1776—711no—
323C4×#103—711no—
324C4×#102—711no—
325#1746—711no—
326C71#2112511yescharacteristic, complemented, Sylow
327C22×SD16—711no—
328C2×C4×Q8C2×D7111yescharacteristic
329C2×C4×D4—711no—
330C2×C4×D4—711no—
331C2×C4×D4—711no—
332#440—718no—
333C2×#107—711no—
334C2×#107—711no—
335C2×#106—711no—
336C2×#103—711no—
337C2×#103—711no—
338C2×#103—711no—
339C2×#102—711no—
340C2×#102—711no—
341C2×#102—711no—
342C2×#97—711no—
343C2×C4×C8—711no—
344#391C2×D7111yescharacteristic
345#391C2×D7111yescharacteristic
346#389C2×D7111yescharacteristic
347#389C2×D7111yescharacteristic
348#387—21no—
349#387—21no—
350#383—21no—
351#382—21no—
352C4×C4⋊C4—711no—
353C4×C4⋊C4—711no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4C2×D7111yescharacteristic
356C4×C22⋊C4—1421no—
357C4×C22⋊C4—1421no—
358C4×C22⋊C4—1421no—
359#376C2×D7111yescharacteristic
360C2×#95—1421no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#336—711no—
368#336—711no—
369#336—711no—
370#336—711no—
371C22×Q8C22×D7111yescharacteristic
372C22×D4—711no—
373C23×C4—1421no—
374C23×C4—1421no—
375C23×C4—1421no—
376C2×SD16—714no—
377C2×SD16—1422no—
378C2×SD16—1422no—
379C22×C8—711no—
380C22×C8—711no—
381C22×C8—711no—
382C22×C8—711no—
383C4×Q8—22no—
384C4×Q8C4×D7114yescomplemented
385C4×D4—714no—
386C4×D4—1422no—
387C4×D4—1422no—
388C4×D4—1422no—
389C4×D4—1422no—
390C4×D4—1422no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C4⋊C4C22×D7111yescharacteristic
399C2×C4⋊C4C22×D7111yescharacteristic
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C22⋊C4—1421no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—1421no—
406C2×C42—1421no—
407C2×C42—1421no—
408C2×C42C22×D7111yescharacteristic
409C2×C42C22×D7111yescharacteristic
410C2×C42C22×D7111yescharacteristic
411#107—714no—
412#107—1422no—
413#106—1422no—
414#103—714no—
415#103—1422no—
416#103—1422no—
417#102—714no—
418#102—1422no—
419#102—1422no—
420#97—714no—
421C4×C8—1422no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1421no—
430#95—1421no—
431#95—1421no—
432#95C22⋊D7111yescharacteristic
433#95C22⋊D7111yescharacteristic
434#95C22⋊D7111yescharacteristic
435#95C22⋊D7111yescharacteristic
436C24—1421no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7112yes—
443C2×D4—712no—
444C2×D4—712no—
445C2×D4—712no—
446C2×D4—1421no—
447C2×D4—1421no—
448C2×D4—1421no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—5681no—
454C22×C4—5681no—
455C22×C4—5681no—
456C22×C4—5681no—
457C22×C4—5681no—
458C22×C4—5681no—
459C22×C4—1424no—
460C22×C4—1424no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1424no—
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C23×D7111yescharacteristic
472SD16—1428no—
473C2×C8—712no—
474C2×C8—712no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8—1421no—
483C2×C8—1421no—
484C2×C8—1421no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—712no—
494C4⋊C4—712no—
495C4⋊C4—712no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4C2×C4×D7112yes—
500C22⋊C4—1424no—
501C22⋊C4—1424no—
502C22⋊C4—1424no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—24no—
508C42—81no—
509C42—5681no—
510C42—5681no—
511C42—5681no—
512C42C2×C4×D7112yes—
513C42#1176586512yes—
514C23—2841no—
515C23—2841no—
516C23—2841no—
517C23—2841no—
518C23—1422no—
519C23—1422no—
520C23—1422no—
521C23C22×C22⋊D7111yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—714no—
527D4—1422no—
528D4—1422no—
529D4—1422no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—5684no—
565C2×C4—5684no—
566C2×C4—5684no—
567C2×C4—1422no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4C4×C22⋊D7112yes—
571C2×C4C4×C22⋊D7112yes—
572C2×C4C22×C4×D7111yescharacteristic
573C2×C4#41×D7111yescharacteristic
574C2×C4#41×D7111yescharacteristic
575C2×C4#41×D7111yescharacteristic
576C2×C4#41×D7111yescharacteristic
577C2×C4C2×#1176586511yescharacteristic
578C8—2841no—
579C8—2841no—
580C8—1422no—
581C8—1422no—
582C22—2842no—
583C22—2842no—
584C22—2842no—
585C22—2842no—
586C22—2842no—
587C22—2842no—
588C22—1424no—
589C22C2×C4×C22⋊D7111yescharacteristic
590C22#1798650611yescharacteristic
591C22#1798650611yescharacteristic
592C22#1798680211yescharacteristic
593C22#1798680711yescharacteristic
594C22#1798680711yescharacteristic
595C22#1798681411yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—2842no—
607C4—2842no—
608C4—2842no—
609C4#117×D7111yescharacteristic
610C4#1798646711yescharacteristic
611C4#1798652711yescharacteristic
612C4#1798652711yescharacteristic
613C2—2844no—
614C2#2259077711yescharacteristic
615C2#2259716811yescharacteristic
616C2#2259718311yescharacteristic
617C2#2259851811yescharacteristic
618C2#2259854811yescharacteristic
619C2#2259858711yescharacteristic
620C2#2259859611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2320C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2320D71
hypocenterC71#21125
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
3. #22588978C2semidirect
4. #22588971C2semidirect
5. #22588929C2semidirect
6. #22588908C2semidirect
11. #22588238C2semidirect
12. C4×#14556052C2semidirect
14. #22588092C2semidirect
15. C71×#2320C2semidirect
20. #17986097C4semidirect
87. #14556061C2×C4semidirect
91. #14556052C2×C4semidirect
95. #14556039C2×C4semidirect
102. C4×Q8×C71C2×C4semidirect
312. #2320D71semidirect
326. C71#21125semidirect
384. C4×Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860972
#225907022
#225889782
#225889712
#225889292
#225889082
#225886772
#225886002
#225885862
#225882382
C4×#145560522
C4×#145560482
#225880922
C71×#23202
#2112571