Finite Group Databasebeta

← #31920211

Subgroups — #31920211

Order 18176.

1862 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985856C211yesmaximal, characteristic
2C2×#17985846C211yesmaximal, characteristic
3C2×#95×D71C211yesmaximal, characteristic
4#22586599C211yesmaximal, characteristic, complemented
5#22586597C211yesmaximal, characteristic, complemented
6#22586596C211yesmaximal, characteristic, complemented
7#22586565C212yesmaximal, complemented
8#22586564C212yesmaximal, complemented
9(C2×D71)⋊#95C212yesmaximal
10(C2×D71)⋊#95C212yesmaximal
11C71×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C23×C4×D71C2211yescharacteristic
14C23×C4×D71C2211yescharacteristic
15C22×C4⋊Dic71C2211yescharacteristic
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2212yes—
18C22×C4×Dic71C2212yes—
19#17985856—24no—
20#17985846C418yescomplemented
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2211yescharacteristic
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2211yescharacteristic
27C2×C71⋊#95C2211yescharacteristic
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2211yescharacteristic
32#95×D71—24no—
33C22×C4⋊C4×C71C2211yescharacteristic
34C2×#95×C71C2212yes—
35C2×#95×C71C2211yescharacteristic
36C2×#95×C71C2212yes—
37C2×#95×C71C2211yescharacteristic
38C24×D71C2311yescharacteristic
39C23×Dic71C2312yes—
40C23×Dic71C2312yes—
41C23×Dic71C2311yescharacteristic
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C22×C4×D71—22no—
45C22×C4×D71—22no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—24no—
49C22×C4×D71—24no—
50C22×C4×D71—22no—
51C22×C4×D71—22no—
52C22×C4×D71—42no—
53C22×C4×D71—42no—
54C22×C4×D71—42no—
55C22×C4×D71—42no—
56C22×C4×D71C2×C414yes—
57C22×C4×D71D414yes—
58C22×C4×D71Q814yes—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71—42no—
61C2×(C2×D71)⋊C4—24no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4—24no—
65C2×(C2×D71)⋊C4—24no—
66C2×(C2×D71)⋊C4C2×C418yes—
67C2×C4×Dic71—42no—
68C2×C4×Dic71—42no—
69C2×C4×Dic71C2×C418yescomplemented
70C71⋊#95—22no—
71C71⋊#95—22no—
72C71⋊#95—24no—
73C71⋊#95—24no—
74C71⋊#95—24no—
75C71⋊#95—22no—
76C71⋊#95—24no—
77C71⋊#95—42no—
78C71⋊#95—42no—
79C71⋊#95—42no—
80C71⋊#95C2×C414yescomplemented
81C23×C4×C71C2312yes—
82C23×C4×C71C2312yes—
83C23×C4×C71C2311yescharacteristic
84C23×C4×C71C2311yescharacteristic
85C23×C4×C71C2311yescharacteristic
86C2×C4⋊C4×C71—22no—
87C2×C4⋊C4×C71—42no—
88#95×C71—22no—
89#95×C71—24no—
90#95×C71—42no—
91#95×C71C2×C414yescomplemented
92C23×D71—21no—
93C23×D71—21no—
94C23×D71—21no—
95C23×D71—21no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71C22×C412yes—
99C23×D71C2×D412yes—
100C23×D71C2×Q812yes—
101C23×D71#4114yes—
102C23×D71#4114yes—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—21no—
106C22×Dic71—21no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—21no—
110C22×Dic71—21no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71C22×C414yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C2×D412yes—
129C22×Dic71C2×Q812yes—
130C22×Dic71#4112yes—
131C22×Dic71#4112yes—
132C2×C4×D71—28no—
133C2×C4×D71—28no—
134C2×C4×D71—48no—
135C2×C4×D71—44no—
136C2×C4×D71—44no—
137C2×C4×D71—48no—
138C2×C4×D71—48no—
139C2×C4×D71—48no—
140C2×C4×D71—48no—
141C2×C4×D71—48no—
142C2×C4×D71—44no—
143C2×C4×D71—44no—
144C2×C4×D71—44no—
145C2×C4×D71—44no—
146C2×C4×D71C4⋊C4116yes—
147C4⋊Dic71—44no—
148(C2×D71)⋊C4—216no—
149(C2×D71)⋊C4—48no—
150C4×Dic71—48no—
151C24×C71C2411yescharacteristic
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71C22×C414yes—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C2×D412yes—
178C22×C4×C71C2×Q812yes—
179C22×C4×C71#4112yes—
180C22×C4×C71#4112yes—
181C4⋊C4×C71—44no—
182C23×C71C23×C411yescharacteristic
183C23×C71C22×D411yescharacteristic
184C23×C71C22×Q811yescharacteristic
185C23×C71C2×#4112yes—
186C23×C71C2×#4112yes—
187C23×C71C2×#4111yescharacteristic
188C23×C71C2×#4111yescharacteristic
189C23×C71C2×#4111yescharacteristic
190C23×C71C2×#4111yescharacteristic
191C23×C71H(5,𝔽2)12yes—
192C23×C71H(5,𝔽2)12yes—
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—24no—
196C22×D71—22no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—24no—
211C22×D71C2×C4⋊C414yes—
212C22×D71#11718yes—
213C22×D71#12414yes—
214C22×D71#12414yes—
215C22×D71#12518yes—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—42no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—42no—
238C2×C4×C71—42no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71C2×C4⋊C414yes—
245C2×Dic71—24no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—24no—
255C2×Dic71—24no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—42no—
264C2×Dic71—42no—
265C2×Dic71—42no—
266C2×Dic71—42no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—42no—
270C2×Dic71—42no—
271C2×Dic71—42no—
272C2×Dic71—42no—
273C2×Dic71C2×C4⋊C414yes—
274C4×D71—416no—
275C4×D71—416no—
276C4×D71—416no—
277C22×C71C22×C4⋊C411yescharacteristic
278C22×C71C2×#11712yes—
279C22×C71C4×#4111yescharacteristic
280C22×C71C4×#4111yescharacteristic
281C22×C71#51812yes—
282C22×C71#52511yescharacteristic
283C22×C71#52511yescharacteristic
284C22×C71C2×#12411yescharacteristic
285C22×C71C2×#12411yescharacteristic
286C22×C71C2×#12512yes—
287C22×C71#53311yescharacteristic
288C22×C71#53311yescharacteristic
289C22×C71#53411yescharacteristic
290C22×C71#53411yescharacteristic
291C22×C71#54312yes—
292C22×C71#55112yes—
293C22×C71#55112yes—
294C22×C71#55112yes—
295C22×C71#55112yes—
296C22×C71#55312yes—
297C22×C71#55312yes—
298C22×C71#55312yes—
299C22×C71#55312yes—
300C2×D71—24no—
301C2×D71—24no—
302C2×D71—28no—
303C2×D71—28no—
304C2×D71—24no—
305C2×D71—24no—
306C2×D71—24no—
307C2×D71—28no—
308C2×D71—28no—
309C2×D71—24no—
310C2×D71#38218yes—
311C4×C71—24no—
312C4×C71—42no—
313C4×C71—42no—
314C4×C71—44no—
315C4×C71—82no—
316Dic71—24no—
317Dic71—42no—
318Dic71—42no—
319Dic71—44no—
320Dic71—82no—
321#13966—711nomaximal, Sylow
322C2×C71C2×#38211yescharacteristic
323C2×C71#229611yescharacteristic
324C2×C71#231511yescharacteristic
325C2×C71#235312yes—
326C2×C71#235312yes—
327C2×C71#256111yescharacteristic
328C2×C71#256111yescharacteristic
329C2×C71#256912yes—
330C2×C71#256912yes—
331C2×C71#259111yescharacteristic
332C2×C71#259111yescharacteristic
333D71—28no—
334C2×#380—711no—
335C2×C4×C22⋊C4—711no—
336C22×#95—711no—
337#1442—711no—
338#1442—711no—
339#1442—711no—
340#1442D7111yescharacteristic, complemented
341#1434—712no—
342#1434—712no—
343C22⋊#95—712no—
344C22⋊#95—712no—
345C71#1396611yescharacteristic, complemented, Sylow
346C24×C4—711no—
347C24×C4—711no—
348C24×C4—711no—
349C22×C4⋊C4—711no—
350C22×C4⋊C4C2×D7111yescharacteristic
351C22×C22⋊C4—712no—
352C22×C22⋊C4—712no—
353C22×C42—712no—
354#380—1424no—
355C4×C22⋊C4—718no—
356C2×#95—712no—
357C2×#95—712no—
358C2×#95—712no—
359C2×#95—711no—
360C2×#95—712no—
361C2×#95—712no—
362C2×#95—712no—
363C2×#95—711no—
364C2×#95—711no—
365C2×#95—712no—
366C2×#95—711no—
367C2×#95—1424no—
368C2×#95C2×D7112yes—
369C2×#95C2×D7111yescharacteristic
370C2×#95C2×D7112yes—
371C2×#95C2×D7111yescharacteristic
372C25—711no—
373C23×C4—2842no—
374C23×C4—2842no—
375C23×C4—2842no—
376C23×C4—2842no—
377C23×C4—714no—
378C23×C4—714no—
379C23×C4—714no—
380C23×C4—711no—
381C23×C4—712no—
382C23×C4—711no—
383C23×C4—711no—
384C23×C4—712no—
385C23×C4—1424no—
386C23×C4—1424no—
387C23×C4—1422no—
388C23×C4—1422no—
389C23×C4—1422no—
390C23×C4—1422no—
391C23×C4—1422no—
392C23×C4—1422no—
393C23×C4C22×D7112yes—
394C23×C4C22×D7112yes—
395C23×C4C22×D7111yescharacteristic
396C23×C4C22×D7111yescharacteristic
397C23×C4C22×D7111yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4—2842no—
401C2×C4⋊C4—1422no—
402C2×C22⋊C4—718no—
403C2×C22⋊C4—1424no—
404C2×C22⋊C4—1424no—
405C2×C22⋊C4—1424no—
406C2×C22⋊C4—1424no—
407C2×C22⋊C4—1424no—
408C2×C42—2842no—
409C2×C42—2842no—
410C2×C42—718no—
411#95—22no—
412#95—24no—
413#95—42no—
414#95—2842no—
415#95—2842no—
416#95—2842no—
417#95—714no—
418#95—1422no—
419#95—1422no—
420#95—1424no—
421#95—1424no—
422#95—1424no—
423#95—1422no—
424#95—1424no—
425#95C4×D7114yescomplemented
426C24—712no—
427C24—712no—
428C24—712no—
429C24—714no—
430C24—714no—
431C24—1421no—
432C24—1421no—
433C24—1421no—
434C24—1421no—
435C24—1422no—
436C24—1422no—
437C24C23×D7111yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—2848no—
462C22×C4—2848no—
463C22×C4—2844no—
464C22×C4—2848no—
465C22×C4—2848no—
466C22×C4—2844no—
467C22×C4—2844no—
468C22×C4—2844no—
469C22×C4—2844no—
470C22×C4—2844no—
471C22×C4—2848no—
472C22×C4—2848no—
473C22×C4—7116no—
474C22×C4—712no—
475C22×C4—712no—
476C22×C4—712no—
477C22×C4—714no—
478C22×C4—712no—
479C22×C4—712no—
480C22×C4—1428no—
481C22×C4—1428no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1421no—
490C22×C4—1421no—
491C22×C4—1422no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1421no—
497C22×C4—1422no—
498C22×C4—1421no—
499C22×C4—1422no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4—1422no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4C2×C4×D7114yes—
506C22×C4C2×C4×D7112yes—
507C22×C4D4×D7112yes—
508C22×C4Q8×D7112yes—
509C22×C4#1176586412yes—
510C22×C4#1176586612yes—
511C4⋊C4—44no—
512C4⋊C4—2844no—
513C22⋊C4—2848no—
514C22⋊C4—14216no—
515C42—2848no—
516C23—714no—
517C23—718no—
518C23—714no—
519C23—714no—
520C23—718no—
521C23—1422no—
522C23—1424no—
523C23—1424no—
524C23—1424no—
525C23—1424no—
526C23—1424no—
527C23—1424no—
528C23—1424no—
529C23—1424no—
530C23—1424no—
531C23—1422no—
532C23—1422no—
533C23—1424no—
534C23—1422no—
535C23—1422no—
536C23—1422no—
537C23—1422no—
538C23—1422no—
539C23C22×C4×D7111yescharacteristic
540C23C2×D4×D7111yescharacteristic
541C23C2×Q8×D7111yescharacteristic
542C23#41×D7112yes—
543C23#41×D7111yescharacteristic
544C23#41×D7111yescharacteristic
545C23#41×D7112yes—
546C23C2×#1176586411yescharacteristic
547C23C2×#1176586611yescharacteristic
548C23#1455617312yes—
549C23#1455617412yes—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—28416no—
579C2×C4—28416no—
580C2×C4—28416no—
581C2×C4—2841no—
582C2×C4—2842no—
583C2×C4—2842no—
584C2×C4—2842no—
585C2×C4—2842no—
586C2×C4—2842no—
587C2×C4—2842no—
588C2×C4—2841no—
589C2×C4—2841no—
590C2×C4—2841no—
591C2×C4—2842no—
592C2×C4—2842no—
593C2×C4—2842no—
594C2×C4—2842no—
595C2×C4—2842no—
596C2×C4—2842no—
597C2×C4—2842no—
598C2×C4—714no—
599C2×C4—1424no—
600C2×C4—1424no—
601C2×C4—1424no—
602C2×C4—1424no—
603C2×C4—1424no—
604C2×C4—1422no—
605C2×C4—1422no—
606C2×C4—1422no—
607C2×C4—1424no—
608C2×C4—1424no—
609C2×C4—1422no—
610C2×C4C4⋊C4×D7114yes—
611C22—718no—
612C22—1428no—
613C22—1424no—
614C22—1428no—
615C22—1424no—
616C22—1428no—
617C22—1428no—
618C22—1424no—
619C22—1424no—
620C22—1424no—
621C22—1424no—
622C22C2×C4⋊C4×D7111yescharacteristic
623C22#117×D7112yes—
624C22C4×#1176586411yescharacteristic
625C22C4×#1176586611yescharacteristic
626C22#1798647112yes—
627C22#1798650911yescharacteristic
628C22#1798651211yescharacteristic
629C22#124×D7111yescharacteristic
630C22#124×D7111yescharacteristic
631C22#125×D7112yes—
632C22#1798655411yescharacteristic
633C22#1798655511yescharacteristic
634C22#1798655911yescharacteristic
635C22#1798656011yescharacteristic
636C22#1798662412yes—
637C22#1798667212yes—
638C22#1798667412yes—
639C22#1798667612yes—
640C22#1798667912yes—
641C22#1798669412yes—
642C22#1798669612yes—
643C22#1798670012yes—
644C22#1798670212yes—
645C4—24no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—2844no—
651C4—2842no—
652C4—2842no—
653C4—5682no—
654C4—1424no—
655C2—1428no—
656C2#382×D7111yescharacteristic
657C2#2259082111yescharacteristic
658C2#2259095111yescharacteristic
659C2#2259127512yes—
660C2#2259127912yes—
661C2#2259349211yescharacteristic
662C2#2259349411yescharacteristic
663C2#2259355812yes—
664C2#2259356012yes—
665C2#2259381411yescharacteristic
666C2#2259381611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×#1442C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1442D71
hypocenterC71#13966
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #22586599C2semidirect
5. #22586597C2semidirect
6. #22586596C2semidirect
7. #22586565C2semidirect
8. #22586564C2semidirect
11. C71×#1442C2semidirect
20. #17985846C4semidirect
69. C2×C4×Dic71C2×C4semidirect
80. C71⋊#95C2×C4semidirect
91. #95×C71C2×C4semidirect
340. #1442D71semidirect
345. C71#13966semidirect
425. #95C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858562
C2×#179858462
C2×#95×D712
#225865992
#225865972
#225865962
#225865652
#225865642
(C2×D71)⋊#952
C71×#14422
#1396671