Finite Group Databasebeta

← #32097441

Subgroups — #32097441

Order 18176.

784 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
1#22596650C211yesmaximal, characteristic
2#22589908C211yesmaximal, characteristic, complemented
3#22589888C211yesmaximal, characteristic, complemented
4#22589678C211yesmaximal, characteristic, complemented
5#22589491C211yesmaximal, characteristic, complemented
6#22588925C211yesmaximal, characteristic, complemented
7#22588911C211yesmaximal, characteristic, complemented
8#22588764C211yesmaximal, characteristic
9#22588764C211yesmaximal, characteristic
10#22588608C211yesmaximal, characteristic, complemented
11#22588594C211yesmaximal, characteristic, complemented
12#22588367C211yesmaximal, characteristic
13#22588363C211yesmaximal, characteristic
14C2×#17985754C211yesmaximal, characteristic
15C4×C22⋊Q8×C71C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C23×C71⋊C8C2211yescharacteristic
18C2×#14556120C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556052C2211yescharacteristic
24C2×#14556052C2211yescharacteristic
25C2×#14556052C2211yescharacteristic
26C2×#14556052C2211yescharacteristic
27#17985897C2211yescharacteristic
28#17985897C2211yescharacteristic
29#17985883—21no—
30#17985883—21no—
31#17985872—21no—
32#17985872—21no—
33#17985865C2211yescharacteristic
34#17985859C2211yescharacteristic
35#17985859C2211yescharacteristic
36#17985761C2211yescharacteristic
37#17985761C2211yescharacteristic
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985754C2211yescharacteristic
45#17985754C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C2×C22⋊Q8×C71C2211yescharacteristic
49C2×C4×Q8×C71C2211yescharacteristic
50C22×C42×C71C2211yescharacteristic
51#391×C71C2211yescharacteristic
52#389×C71C2211yescharacteristic
53#387×C71—21no—
54#385×C71—21no—
55#380×C71C2211yescharacteristic
56C4×C4⋊C4×C71—21no—
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C22×C71⋊C8—22no—
60C22×C71⋊C8—22no—
61C22×C71⋊C8—22no—
62C22×C71⋊C8—22no—
63C22×C71⋊C8—41no—
64C22×C71⋊C8—41no—
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69#14556120—41no—
70#14556120—41no—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71—21no—
81#14556061—22no—
82#14556061—41no—
83#14556059—22no—
84#14556059—41no—
85#14556052—22no—
86#14556052—22no—
87#14556052—41no—
88#14556052—41no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C22×Q8×C71C2311yescharacteristic
96C23×C4×C71D411yescharacteristic
97C23×C4×C71D411yescharacteristic
98C23×C4×C71C2311yescharacteristic
99C22⋊Q8×C71—22no—
100C22⋊Q8×C71Q814yes—
101C4×Q8×C71—41no—
102C4×Q8×C71—41no—
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×C22⋊C4×C71—21no—
110C2×C22⋊C4×C71—21no—
111C2×C42×C71—21no—
112C2×C42×C71—21no—
113C2×C42×C71—41no—
114C2×C42×C71—41no—
115C2×C42×C71D411yescharacteristic
116C2×C42×C71D411yescharacteristic
117C2×C42×C71C2311yescharacteristic
118C2×C42×C71C2311yescharacteristic
119#95×C71—21no—
120#95×C71—21no—
121#95×C71—21no—
122#95×C71—21no—
123#95×C71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—44no—
139C2×C71⋊C8—44no—
140C2×C71⋊C8#4112yes—
141C2×C71⋊C8#4112yes—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150Dic71⋊C4—42no—
151Dic71⋊C4—42no—
152Dic71⋊C4—42no—
153Dic71⋊C4—42no—
154C22⋊Dic71—42no—
155C22⋊Dic71—42no—
156C24×C71C2×D411yescharacteristic
157C2×Q8×C71—21no—
158C2×Q8×C71—21no—
159C2×Q8×C71—21no—
160C2×Q8×C71—21no—
161C2×Q8×C71—21no—
162C2×Q8×C71C2×Q812yes—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—22no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—41no—
175C22×C4×C71—41no—
176C22×C4×C71—41no—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×Q812yes—
183C22×C4×C71#4112yes—
184C22×C4×C71C2411yescharacteristic
185C4⋊C4×C71—24no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—42no—
192C4⋊C4×C71—42no—
193C4⋊C4×C71C2×Q812yes—
194C22⋊C4×C71—24no—
195C22⋊C4×C71—42no—
196C42×C71—21no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71—41no—
201C42×C71—41no—
202C42×C71—42no—
203C42×C71—42no—
204C23×C71—21no—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71C22⋊Q812yes—
209C23×C71#13512yes—
210C23×C71#13712yes—
211C23×C71C22×D411yescharacteristic
212Q8×C71—22no—
213Q8×C71—22no—
214Q8×C71—41no—
215Q8×C71—41no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
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—21no—
234C2×C4×C71—21no—
235C2×C4×C71—44no—
236C2×C4×C71—44no—
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×C71—41no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—41no—
247C2×C4×C71C22⋊Q812yes—
248C2×C4×C71#13512yes—
249C2×C4×C71C22×Q811yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C2×C4×C71C2×#4111yescharacteristic
253C2×C4×C71H(5,𝔽2)11yescharacteristic
254C2×C4×C71H(5,𝔽2)11yescharacteristic
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C71⋊C8—41no—
270C71⋊C8—41no—
271C22×C71—22no—
272C22×C71—22no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—22no—
276C22×C71—22no—
277C22×C71#47914yes—
278C22×C71C2×C22⋊Q811yescharacteristic
279C22×C71#53211yescharacteristic
280C22×C71#53411yescharacteristic
281C22×C71C2×#13511yescharacteristic
282C22×C71C2×#13511yescharacteristic
283C22×C71C2×#13711yescharacteristic
284C22×C71#57511yescharacteristic
285C4×C71—22no—
286C4×C71—22no—
287C4×C71—42no—
288C4×C71—42no—
289C4×C71—41no—
290C4×C71—41no—
291C4×C71—41no—
292C4×C71—81no—
293C4×C71—81no—
294C4×C71#55111yescharacteristic
295C4×C71#55111yescharacteristic
296C4×C71#55411yescharacteristic
297C4×C71#55411yescharacteristic
298Dic71—81no—
299Dic71—81no—
300Dic71—81no—
301Dic71—81no—
302#29158—711nomaximal, Sylow
303C2×C71—24no—
304C2×C71#256411yescharacteristic
305C2×C71C2×#47911yescharacteristic
306C2×C71#307611yescharacteristic
307C2×C71#316811yescharacteristic
308C2×C71#318111yescharacteristic
309C2×C71#319011yescharacteristic
310C2×C71#319111yescharacteristic
311#2851—711no—
312C4×C22⋊Q8D7111yescharacteristic
313#2101—711no—
314#2097—711no—
315#2059—711no—
316#2026—711no—
317#1920—711no—
318#1918—711no—
319#1892—711no—
320#1892—711no—
321#1865—711no—
322#1863—711no—
323#1808—711no—
324#1807—711no—
325C2×#336—711no—
326C71#2915811yescharacteristic, complemented, Sylow
327C23×C8—711no—
328C23×C8—711no—
329C2×#125—711no—
330C2×C22⋊Q8C2×D7111yescharacteristic
331C2×C4×Q8C2×D7111yescharacteristic
332C22×C42C2×D7111yescharacteristic
333C2×#107—711no—
334C2×#107—711no—
335C2×#106—711no—
336C2×#106—711no—
337C2×#103—711no—
338C2×#103—711no—
339C2×#103—711no—
340C2×#103—711no—
341#391C2×D7111yescharacteristic
342#389—711no—
343#389—711no—
344#389C2×D7111yescharacteristic
345#387—21no—
346#387—1421no—
347#387—1421no—
348#385—21no—
349#385—1421no—
350#385—1421no—
351#382—711no—
352#380—711no—
353#380—711no—
354#380C2×D7111yescharacteristic
355C4×C4⋊C4—21no—
356C4×C4⋊C4C2×D7111yescharacteristic
357C4×C22⋊C4—21no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#336—711no—
367#336—711no—
368#336—711no—
369#336—711no—
370C22×Q8C22×D7111yescharacteristic
371C23×C4C22⋊D7111yescharacteristic
372C23×C4C22⋊D7111yescharacteristic
373C23×C4C22×D7111yescharacteristic
374C22×C8—2841no—
375C22×C8—2841no—
376C22×C8—711no—
377C22×C8—711no—
378C22×C8—711no—
379C22×C8—711no—
380C22×C8—1422no—
381C22×C8—1422no—
382C22×C8—1422no—
383C22×C8—1422no—
384#125—2841no—
385#125—2841no—
386C22⋊Q8—22no—
387C22⋊Q8Dic14214yes—
388C4×Q8—41no—
389C4×Q8—41no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—711no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—1421no—
398C2×C4⋊C4—1421no—
399C2×C4⋊C4—1421no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4C22×D7111yescharacteristic
402C2×C4⋊C4C22×D7111yescharacteristic
403C2×C4⋊C4C22×D7111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1421no—
407C2×C22⋊C4—1421no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—41no—
411C2×C42—41no—
412C2×C42C22⋊D7111yescharacteristic
413C2×C42C22⋊D7111yescharacteristic
414C2×C42C22×D7111yescharacteristic
415C2×C42C22×D7111yescharacteristic
416#107—2841no—
417#107—1422no—
418#106—2841no—
419#106—1422no—
420#103—2841no—
421#103—2841no—
422#103—1422no—
423#103—1422no—
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#95—1421no—
433#95—1421no—
434#95—1421no—
435C24C2×C22⋊D7111yescharacteristic
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×Dic14212yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×Dic14212yes—
466C22×C4#1176586512yes—
467C22×C4C23×D7111yescharacteristic
468C2×C8—2844no—
469C2×C8—2844no—
470C2×C8—712no—
471C2×C8—712no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—2842no—
491C4⋊C4—2842no—
492C4⋊C4—2842no—
493C4⋊C4—2842no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C4⋊C4—1421no—
502C4⋊C4C2×Dic14212yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2842no—
506C22⋊C4—2842no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—42no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#1455610712yes—
520C23#1455614812yes—
521C23#1455615912yes—
522C23C22×C22⋊D7111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
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—44no—
547C2×C4—44no—
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—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—2841no—
570C2×C4#1455610712yes—
571C2×C4#1455614812yes—
572C2×C4C22×Dic14211yescharacteristic
573C2×C4#41×D7111yescharacteristic
574C2×C4#41×D7111yescharacteristic
575C2×C4C2×#1176586511yescharacteristic
576C2×C4#1455617311yescharacteristic
577C2×C4#1455617311yescharacteristic
578C8—2841no—
579C8—2841no—
580C8—1422no—
581C8—1422no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1798630614yes—
589C22C2×#1455610711yescharacteristic
590C22#1798654511yescharacteristic
591C22#1798655811yescharacteristic
592C22C2×#1455614811yescharacteristic
593C22C2×#1455614811yescharacteristic
594C22C2×#1455615911yescharacteristic
595C22#1798679711yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—5681no—
606C4—5681no—
607C4—5681no—
608C4—5681no—
609C4#1798667811yescharacteristic
610C4#1798667811yescharacteristic
611C4#1798670811yescharacteristic
612C4#1798670811yescharacteristic
613C2—24no—
614C2#2259351611yescharacteristic
615C2C2×#1798630611yescharacteristic
616C2#2259812811yescharacteristic
617C2#2259890411yescharacteristic
618C2#2259904311yescharacteristic
619C2#2259912111yescharacteristic
620C2#2259912811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenterC4×C22⋊Q8D71
hypocenterC71#29158
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22589908C2semidirect
3. #22589888C2semidirect
4. #22589678C2semidirect
5. #22589491C2semidirect
6. #22588925C2semidirect
7. #22588911C2semidirect
10. #22588608C2semidirect
11. #22588594C2semidirect
326. C71#29158semidirect

Maximal subgroups

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

subgroupindex
#225966502
#225899082
#225898882
#225896782
#225894912
#225889252
#225889112
#225887642
#225886082
#225885942
#225883672
#225883632
C2×#179857542
C4×C22⋊Q8×C712
#2915871