Finite Group Databasebeta

← #31918843

Subgroups — #31918843

Order 18176.

1958 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
2C2×#17985846C211yesmaximal, characteristic
3C2×#95×D71C211yesmaximal, characteristic
4#22586567C211yesmaximal, characteristic, complemented
5#22586566C212yesmaximal, complemented
6#22586565C211yesmaximal, characteristic, complemented
7#22586565C211yesmaximal, characteristic, complemented
8#22586564C212yesmaximal, complemented
9(C2×D71)⋊#95C212yesmaximal
10(C2×D71)⋊#95C212yesmaximal
11C71×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D71C2212yes—
13C23×C4×D71C2211yescharacteristic
14C22×(C2×D71)⋊C4C2212yes—
15C22×(C2×D71)⋊C4C2211yescharacteristic
16C22×(C2×D71)⋊C4C2211yescharacteristic
17C22×C4×Dic71C2212yes—
18C22×C4×Dic71C2211yescharacteristic
19C4×(C2×D71)⋊C4C418yescomplemented
20#17985846C418yescomplemented
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2211yescharacteristic
29C2×C71⋊#95C2211yescharacteristic
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2211yescharacteristic
32#95×D71C418yescomplemented
33C22×C42×C71C2211yescharacteristic
34C2×#95×C71C2212yes—
35C2×#95×C71C2212yes—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C24×D71C2311yescharacteristic
39C23×Dic71C2312yes—
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C22×C4×D71—24no—
45C22×C4×D71—24no—
46C22×C4×D71—42no—
47C22×C4×D71—42no—
48C22×C4×D71—42no—
49C22×C4×D71—42no—
50C22×C4×D71—42no—
51C22×C4×D71—42no—
52C22×C4×D71C2×C418yes—
53C22×C4×D71C2×C414yes—
54C22×C4×D71C2×C414yes—
55C22×C4×D71C2×C414yes—
56C2×(C2×D71)⋊C4—24no—
57C2×(C2×D71)⋊C4—24no—
58C2×(C2×D71)⋊C4—24no—
59C2×(C2×D71)⋊C4—24no—
60C2×(C2×D71)⋊C4C2×C418yes—
61C2×(C2×D71)⋊C4C2×C414yes—
62C2×(C2×D71)⋊C4C2×C414yes—
63C2×C4×Dic71—42no—
64C2×C4×Dic71—42no—
65C2×C4×Dic71—42no—
66C2×C4×Dic71C2×C418yescomplemented
67C2×C4×Dic71C2×C414yescomplemented
68C71⋊#95—42no—
69C71⋊#95—42no—
70C71⋊#95—42no—
71C71⋊#95—42no—
72C71⋊#95—42no—
73C71⋊#95—42no—
74C71⋊#95C2×C418yes—
75C71⋊#95C2×C414yes—
76C71⋊#95C2×C414yescomplemented
77C71⋊#95C2×C414yescomplemented
78C71⋊#95C2×C414yescomplemented
79C23×C4×C71C2312yes—
80C23×C4×C71C2311yescharacteristic
81C23×C4×C71C2311yescharacteristic
82C23×C4×C71C2312yes—
83C23×C4×C71C2311yescharacteristic
84C2×C42×C71—42no—
85C2×C42×C71C2×C414yescomplemented
86#95×C71—42no—
87#95×C71—42no—
88#95×C71C2×C414yes—
89#95×C71C2×C414yescomplemented
90C23×D71—22no—
91C23×D71—21no—
92C23×D71—22no—
93C23×D71—21no—
94C23×D71—21no—
95C23×D71—21no—
96C23×D71C22×C412yes—
97C23×D71C22×C412yes—
98C23×D71C22×C412yes—
99C23×D71#4114yes—
100C23×D71#4114yes—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—21no—
114C22×Dic71—22no—
115C22×Dic71—21no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—21no—
121C22×Dic71—22no—
122C22×Dic71—21no—
123C22×Dic71C22×C414yes—
124C22×Dic71C22×C414yes—
125C22×Dic71C22×C412yes—
126C22×Dic71C22×C412yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C22×C412yes—
129C22×Dic71C22×C412yes—
130C2×C4×D71—216no—
131C2×C4×D71—48no—
132C2×C4×D71—44no—
133C2×C4×D71—44no—
134C2×C4×D71—48no—
135C2×C4×D71—48no—
136C2×C4×D71—48no—
137C2×C4×D71—48no—
138C2×C4×D71—48no—
139C2×C4×D71—48no—
140C2×C4×D71—48no—
141C2×C4×D71C42116yes—
142(C2×D71)⋊C4—216no—
143(C2×D71)⋊C4—28no—
144(C2×D71)⋊C4—28no—
145C4×Dic71—48no—
146C4×Dic71—44no—
147C24×C71C2411yescharacteristic
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—21no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—21no—
166C22×C4×C71—22no—
167C22×C4×C71—21no—
168C22×C4×C71—22no—
169C22×C4×C71—21no—
170C22×C4×C71C22×C414yes—
171C22×C4×C71C22×C414yes—
172C22×C4×C71C22×C412yes—
173C22×C4×C71C22×C412yes—
174C22×C4×C71C22×C412yes—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C22×C412yes—
177C42×C71—44no—
178C23×C71C23×C411yescharacteristic
179C23×C71C23×C411yescharacteristic
180C23×C71C23×C411yescharacteristic
181C23×C71C2×#4112yes—
182C23×C71C2×#4112yes—
183C23×C71H(5,𝔽2)12yes—
184C23×C71H(5,𝔽2)11yescharacteristic
185C23×C71H(5,𝔽2)12yes—
186C23×C71H(5,𝔽2)11yescharacteristic
187C23×C712−1+411yescharacteristic
188C23×C712−1+411yescharacteristic
189C22×D71—22no—
190C22×D71—24no—
191C22×D71—24no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—22no—
196C22×D71—22no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—24no—
200C22×D71—24no—
201C22×D71—24no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71C2×C4214yes—
208C22×D71#11714yes—
209C22×D71#11718yes—
210C22×D71#11714yes—
211C22×D71#11718yes—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—24no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
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×C71C2×C4214yes—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—24no—
253C2×Dic71—24no—
254C2×Dic71—22no—
255C2×Dic71—22no—
256C2×Dic71—24no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—42no—
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×Dic71C2×C4214yes—
270C4×D71—416no—
271C4×D71—432no—
272C22×C71C22×C4211yescharacteristic
273C22×C71C2×#11712yes—
274C22×C71C2×#11711yescharacteristic
275C22×C71C2×#11712yes—
276C22×C71C2×#11711yescharacteristic
277C22×C71#51811yescharacteristic
278C22×C71#51812yes—
279C22×C71#51812yes—
280C22×C71#51811yescharacteristic
281C22×C71#51811yescharacteristic
282C22×C71#51911yescharacteristic
283C22×C71#52011yescharacteristic
284C22×C71#52011yescharacteristic
285C22×C71#52011yescharacteristic
286C22×C71#52011yescharacteristic
287C22×C71#53812yes—
288C22×C71#53812yes—
289C22×C71#53812yes—
290C22×C71#53812yes—
291C22×C71#53912yes—
292C22×C71#53912yes—
293C22×C71#53912yes—
294C22×C71#53912yes—
295C2×D71—28no—
296C2×D71—28no—
297C2×D71—24no—
298C2×D71—24no—
299C2×D71—28no—
300C2×D71—24no—
301C2×D71—28no—
302C2×D71—24no—
303C2×D71—24no—
304C2×D71—24no—
305C2×D71#37618yes—
306C4×C71—24no—
307C4×C71—44no—
308C4×C71—42no—
309C4×C71—44no—
310C4×C71—42no—
311Dic71—24no—
312Dic71—44no—
313Dic71—42no—
314Dic71—42no—
315Dic71—44no—
316#13796—711nomaximal, Sylow
317C2×C71C2×#37611yescharacteristic
318C2×C71#227311yescharacteristic
319C2×C71#227411yescharacteristic
320C2×C71#231212yes—
321C2×C71#231312yes—
322C2×C71#231811yescharacteristic
323C2×C71#231811yescharacteristic
324C2×C71#231812yes—
325C2×C71#231811yescharacteristic
326C2×C71#231812yes—
327C2×C71#231811yescharacteristic
328D71—28no—
329C2×C4×C22⋊C4—711no—
330C2×C4×C22⋊C4—711no—
331C22×#95—711no—
332#1434—711no—
333#1434—712no—
334#1434—711no—
335#1434—711no—
336#1434—712no—
337#1434D7111yescharacteristic, complemented
338C22⋊#95—712no—
339C22⋊#95—712no—
340C71#1379611yescharacteristic, complemented, Sylow
341C24×C4—711no—
342C24×C4—712no—
343C22×C22⋊C4—711no—
344C22×C22⋊C4—711no—
345C22×C22⋊C4—712no—
346C22×C42—712no—
347C22×C42—711no—
348C22×C42C2×D7111yescharacteristic
349C4×C22⋊C4—718no—
350C4×C22⋊C4—718no—
351C2×#95—712no—
352C2×#95—712no—
353C2×#95—712no—
354C2×#95—712no—
355C2×#95—712no—
356C2×#95—712no—
357C2×#95—712no—
358C2×#95—711no—
359C2×#95—711no—
360C2×#95—711no—
361C2×#95—711no—
362C2×#95—718no—
363C2×#95C2×D7112yes—
364C2×#95C2×D7112yes—
365C2×#95C2×D7111yescharacteristic
366C2×#95C2×D7111yescharacteristic
367C25—711no—
368C23×C4—2842no—
369C23×C4—2842no—
370C23×C4—2842no—
371C23×C4—2842no—
372C23×C4—2842no—
373C23×C4—2842no—
374C23×C4—714no—
375C23×C4—714no—
376C23×C4—714no—
377C23×C4—718no—
378C23×C4—711no—
379C23×C4—711no—
380C23×C4—712no—
381C23×C4—711no—
382C23×C4—712no—
383C23×C4—1424no—
384C23×C4—1424no—
385C23×C4C22×D7111yescharacteristic
386C23×C4C22×D7111yescharacteristic
387C23×C4C22×D7112yes—
388C23×C4C22×D7112yes—
389C23×C4C22×D7111yescharacteristic
390C2×C22⋊C4—714no—
391C2×C22⋊C4—714no—
392C2×C22⋊C4—718no—
393C2×C22⋊C4—1424no—
394C2×C22⋊C4—1424no—
395C2×C22⋊C4—1424no—
396C2×C22⋊C4—1424no—
397C2×C42—42no—
398C2×C42—2842no—
399C2×C42—2842no—
400C2×C42—2842no—
401C2×C42—718no—
402C2×C42—714no—
403C2×C42C4×D7114yescomplemented
404#95—42no—
405#95—42no—
406#95—2842no—
407#95—2842no—
408#95—2842no—
409#95—2842no—
410#95—2842no—
411#95—2842no—
412#95—718no—
413#95—714no—
414#95—714no—
415#95—714no—
416#95—714no—
417#95C4×D7114yes—
418#95C4×D7114yescomplemented
419C24—712no—
420C24—712no—
421C24—712no—
422C24—714no—
423C24—714no—
424C24—1421no—
425C24—1422no—
426C24—1422no—
427C24—1421no—
428C24—1421no—
429C24—1421no—
430C24C23×D7111yescharacteristic
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—2848no—
454C22×C4—2848no—
455C22×C4—2844no—
456C22×C4—2844no—
457C22×C4—2848no—
458C22×C4—2848no—
459C22×C4—2848no—
460C22×C4—2848no—
461C22×C4—2848no—
462C22×C4—2848no—
463C22×C4—7116no—
464C22×C4—712no—
465C22×C4—712no—
466C22×C4—712no—
467C22×C4—714no—
468C22×C4—714no—
469C22×C4—712no—
470C22×C4—712no—
471C22×C4—14216no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1422no—
478C22×C4—1421no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1421no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1421no—
490C22×C4—1422no—
491C22×C4—1421no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4C2×C4×D7114yes—
495C22×C4C2×C4×D7112yes—
496C22×C4C2×C4×D7112yes—
497C22×C4C2×C4×D7114yes—
498C22×C4C2×C4×D7112yes—
499C22×C4C2×C4×D7112yes—
500C22×C4C2×C4×D7112yes—
501C22⋊C4—1428no—
502C22⋊C4—14216no—
503C22⋊C4—1428no—
504C42—44no—
505C42—2844no—
506C42—2848no—
507C23—714no—
508C23—714no—
509C23—718no—
510C23—714no—
511C23—718no—
512C23—1424no—
513C23—1424no—
514C23—1424no—
515C23—1424no—
516C23—1424no—
517C23—1424no—
518C23—1424no—
519C23—1424no—
520C23—1422no—
521C23—1424no—
522C23—1424no—
523C23—1422no—
524C23—1422no—
525C23—1422no—
526C23—1422no—
527C23—1422no—
528C23—1422no—
529C23—1422no—
530C23C22×C4×D7111yescharacteristic
531C23C22×C4×D7111yescharacteristic
532C23C22×C4×D7111yescharacteristic
533C23#41×D7112yes—
534C23#41×D7112yes—
535C23#1455617311yescharacteristic
536C23#1455617312yes—
537C23#1455617412yes—
538C23#1455617411yescharacteristic
539C23#1455617511yescharacteristic
540C23#1455617611yescharacteristic
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—28416no—
570C2×C4—28432no—
571C2×C4—2842no—
572C2×C4—2842no—
573C2×C4—2842no—
574C2×C4—2842no—
575C2×C4—2842no—
576C2×C4—2842no—
577C2×C4—2842no—
578C2×C4—2842no—
579C2×C4—2842no—
580C2×C4—2842no—
581C2×C4—2842no—
582C2×C4—2842no—
583C2×C4—714no—
584C2×C4—1422no—
585C2×C4—1422no—
586C2×C4—1424no—
587C2×C4—1424no—
588C2×C4—1424no—
589C2×C4—1424no—
590C2×C4—1424no—
591C2×C4—1422no—
592C2×C4—1422no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1424no—
596C2×C4—1424no—
597C2×C4—1424no—
598C2×C4—1422no—
599C2×C4—1422no—
600C2×C4C42×D7114yes—
601C22—718no—
602C22—1424no—
603C22—1428no—
604C22—1428no—
605C22—1424no—
606C22—1424no—
607C22—1424no—
608C22—1428no—
609C22—1424no—
610C22—1428no—
611C22—1424no—
612C22C2×C42×D7111yescharacteristic
613C22#117×D7111yescharacteristic
614C22#117×D7112yes—
615C22#117×D7112yes—
616C22#117×D7111yescharacteristic
617C22#1798647111yescharacteristic
618C22#1798647112yes—
619C22#1798647112yes—
620C22#1798647211yescharacteristic
621C22#1798647311yescharacteristic
622C22#1798647511yescharacteristic
623C22#1798647811yescharacteristic
624C22#1798647911yescharacteristic
625C22#1798648011yescharacteristic
626C22#1798648111yescharacteristic
627C22#1798658112yes—
628C22#1798658312yes—
629C22#1798658512yes—
630C22#1798658812yes—
631C22#1798659012yes—
632C22#1798659212yes—
633C22#1798659512yes—
634C22#1798659712yes—
635C4—24no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—2842no—
641C4—2844no—
642C4—2842no—
643C4—2844no—
644C4—1424no—
645C2—1428no—
646C2#376×D7111yescharacteristic
647C2#2259069211yescharacteristic
648C2#2259069411yescharacteristic
649C2#2259093412yes—
650C2#2259093912yes—
651C2#2259096911yescharacteristic
652C2#2259097012yes—
653C2#2259097111yescharacteristic
654C2#2259097212yes—
655C2#2259097511yescharacteristic
656C2#2259097611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC71×#1434C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1434D71
hypocenterC71#13796
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #22586567C2semidirect
5. #22586566C2semidirect
6. #22586565C2semidirect
7. #22586565C2semidirect
8. #22586564C2semidirect
11. C71×#1434C2semidirect
19. C4×(C2×D71)⋊C4C4semidirect
20. #17985846C4semidirect
32. #95×D71C4semidirect
66. C2×C4×Dic71C2×C4semidirect
67. C2×C4×Dic71C2×C4semidirect
76. C71⋊#95C2×C4semidirect
77. C71⋊#95C2×C4semidirect
78. C71⋊#95C2×C4semidirect
85. C2×C42×C71C2×C4semidirect
89. #95×C71C2×C4semidirect
337. #1434D71semidirect
340. C71#13796semidirect
403. C2×C42C4×D71semidirect
418. #95C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C2×D71)⋊C42
C2×#179858462
C2×#95×D712
#225865672
#225865662
#225865652
#225865642
(C2×D71)⋊#952
C71×#14342
#1379671