Finite Group Databasebeta

← #31920213

Subgroups — #31920213

Order 18176.

1862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985858C211yesmaximal, characteristic
2C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
3C2×#95×D71C211yesmaximal, characteristic
4#22586598C212yesmaximal, complemented
5#22586595C212yesmaximal, complemented
6#22586567C211yesmaximal, characteristic, complemented
7#22586566C211yesmaximal, characteristic, complemented
8#22586566C211yesmaximal, characteristic, complemented
9(C2×D71)⋊#95C212yesmaximal
10(C2×D71)⋊#95C212yesmaximal
11C71×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D71C2212yes—
13C23×C4×D71C2211yescharacteristic
14C22×Dic71⋊C4C2212yes—
15C22×(C2×D71)⋊C4C2212yes—
16C22×(C2×D71)⋊C4C2211yescharacteristic
17C22×(C2×D71)⋊C4C2211yescharacteristic
18C22×C4×Dic71C2211yescharacteristic
19#17985858—24no—
20C4×(C2×D71)⋊C4C418yescomplemented
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2211yescharacteristic
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2211yescharacteristic
32#95×D71—24no—
33C22×C42×C71C2211yescharacteristic
34C2×#95×C71C2212yes—
35C2×#95×C71C2211yescharacteristic
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2212yes—
38C24×D71C2311yescharacteristic
39C23×Dic71C2312yes—
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2311yescharacteristic
42C23×Dic71C2312yes—
43C23×Dic71C2311yescharacteristic
44C22×C4×D71—24no—
45C22×C4×D71—24no—
46C22×C4×D71—24no—
47C22×C4×D71—24no—
48C22×C4×D71—24no—
49C22×C4×D71—42no—
50C22×C4×D71—42no—
51C22×C4×D71—42no—
52C22×C4×D71—42no—
53C22×C4×D71C2×C414yes—
54C22×C4×D71D414yes—
55C22×C4×D71Q814yes—
56C2×Dic71⋊C4—24no—
57C2×Dic71⋊C4—42no—
58C2×Dic71⋊C4—42no—
59C2×(C2×D71)⋊C4—22no—
60C2×(C2×D71)⋊C4—22no—
61C2×(C2×D71)⋊C4—22no—
62C2×(C2×D71)⋊C4—22no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4—24no—
65C2×(C2×D71)⋊C4—24no—
66C2×(C2×D71)⋊C4C2×C414yes—
67C2×(C2×D71)⋊C4C2×C414yes—
68C2×C4×Dic71—42no—
69C2×C4×Dic71C2×C414yescomplemented
70C71⋊#95—24no—
71C71⋊#95—24no—
72C71⋊#95—24no—
73C71⋊#95—22no—
74C71⋊#95—22no—
75C71⋊#95—24no—
76C71⋊#95—42no—
77C71⋊#95—42no—
78C71⋊#95—42no—
79C71⋊#95C2×C414yescomplemented
80C71⋊#95C2×C414yescomplemented
81C23×C4×C71C2312yes—
82C23×C4×C71C2311yescharacteristic
83C23×C4×C71C2311yescharacteristic
84C23×C4×C71C2312yes—
85C23×C4×C71C2311yescharacteristic
86C2×C42×C71—42no—
87C2×C42×C71C2×C414yescomplemented
88#95×C71—22no—
89#95×C71—22no—
90#95×C71—24no—
91#95×C71—42no—
92C23×D71—22no—
93C23×D71—22no—
94C23×D71—22no—
95C23×D71—22no—
96C23×D71C22×C412yes—
97C23×D71C2×D412yes—
98C23×D71C2×Q812yes—
99C23×D71#4112yes—
100C23×D71#4112yes—
101C23×D71#4112yes—
102C23×D71#4112yes—
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—21no—
112C22×Dic71—21no—
113C22×Dic71—21no—
114C22×Dic71—21no—
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×C412yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C22×C412yes—
129C22×Dic71C2×D412yes—
130C22×Dic71C2×Q812yes—
131C22×Dic71#4114yes—
132C2×C4×D71—216no—
133C2×C4×D71—44no—
134C2×C4×D71—44no—
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×D71—48no—
142C2×C4×D71—48no—
143C2×C4×D71C4⋊C4116yes—
144Dic71⋊C4—48no—
145(C2×D71)⋊C4—28no—
146(C2×D71)⋊C4—28no—
147(C2×D71)⋊C4—48no—
148C4×Dic71—44no—
149C24×C71C2411yescharacteristic
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—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
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×C71C22×C412yes—
174C22×C4×C71C22×C412yes—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C2×D412yes—
177C22×C4×C71C2×Q812yes—
178C22×C4×C71#4114yes—
179C42×C71—44no—
180C23×C71C23×C411yescharacteristic
181C23×C71C22×D411yescharacteristic
182C23×C71C22×Q811yescharacteristic
183C23×C71C2×#4111yescharacteristic
184C23×C71C2×#4111yescharacteristic
185C23×C71C2×#4111yescharacteristic
186C23×C71C2×#4111yescharacteristic
187C23×C71C2×#4112yes—
188C23×C71C2×#4112yes—
189C23×C71H(5,𝔽2)12yes—
190C23×C71H(5,𝔽2)12yes—
191C22×D71—22no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—24no—
196C22×D71—24no—
197C22×D71—24no—
198C22×D71—24no—
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×C4⋊C414yes—
208C22×D71#11714yes—
209C22×D71#11714yes—
210C22×D71#12414yes—
211C22×D71#12414yes—
212C22×D71#12514yes—
213C22×D71#12514yes—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—42no—
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×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71C2×C4⋊C414yes—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—24no—
246C2×Dic71—24no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—22no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—24no—
254C2×Dic71—24no—
255C2×Dic71—24no—
256C2×Dic71—42no—
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×Dic71—42no—
270C2×Dic71—42no—
271C2×Dic71C2×C4⋊C414yes—
272C4×D71—416no—
273C4×D71—432no—
274C22×C71C22×C4⋊C411yescharacteristic
275C22×C71C2×#11711yescharacteristic
276C22×C71C2×#11711yescharacteristic
277C22×C71C4×#4111yescharacteristic
278C22×C71C4×#4111yescharacteristic
279C22×C71#51811yescharacteristic
280C22×C71#51811yescharacteristic
281C22×C71#52512yes—
282C22×C71C2×#12411yescharacteristic
283C22×C71C2×#12411yescharacteristic
284C22×C71C2×#12511yescharacteristic
285C22×C71C2×#12511yescharacteristic
286C22×C71#53312yes—
287C22×C71#53412yes—
288C22×C71#54312yes—
289C22×C71#55112yes—
290C22×C71#55112yes—
291C22×C71#55112yes—
292C22×C71#55112yes—
293C22×C71#55312yes—
294C22×C71#55312yes—
295C22×C71#55312yes—
296C22×C71#55312yes—
297C2×D71—28no—
298C2×D71—28no—
299C2×D71—28no—
300C2×D71—24no—
301C2×D71—28no—
302C2×D71—24no—
303C2×D71—24no—
304C2×D71—24no—
305C2×D71—24no—
306C2×D71—24no—
307C2×D71#38218yes—
308C4×C71—24no—
309C4×C71—44no—
310C4×C71—42no—
311C4×C71—42no—
312C4×C71—82no—
313Dic71—24no—
314Dic71—44no—
315Dic71—42no—
316Dic71—42no—
317Dic71—82no—
318#13966—711nomaximal, Sylow
319C2×C71C2×#38211yescharacteristic
320C2×C71#229611yescharacteristic
321C2×C71#231511yescharacteristic
322C2×C71#235311yescharacteristic
323C2×C71#235311yescharacteristic
324C2×C71#235311yescharacteristic
325C2×C71#235311yescharacteristic
326C2×C71#256112yes—
327C2×C71#256912yes—
328C2×C71#256912yes—
329C2×C71#259112yes—
330D71—28no—
331C2×#380—711no—
332C2×C4×C22⋊C4—711no—
333C22×#95—711no—
334#1442—712no—
335#1442—712no—
336#1434—711no—
337#1434—711no—
338#1434—711no—
339#1434D7111yescharacteristic, complemented
340C22⋊#95—712no—
341C22⋊#95—712no—
342C71#1396611yescharacteristic, complemented, Sylow
343C24×C4—711no—
344C24×C4—712no—
345C22×C4⋊C4—712no—
346C22×C22⋊C4—711no—
347C22×C22⋊C4—711no—
348C22×C22⋊C4—712no—
349C22×C42—711no—
350C22×C42C2×D7111yescharacteristic
351#380—1424no—
352C4×C22⋊C4—718no—
353C2×#95—712no—
354C2×#95—712no—
355C2×#95—712no—
356C2×#95—711no—
357C2×#95—711no—
358C2×#95—712no—
359C2×#95—712no—
360C2×#95—712no—
361C2×#95—711no—
362C2×#95—711no—
363C2×#95—712no—
364C2×#95—1424no—
365C2×#95C2×D7112yes—
366C2×#95C2×D7111yescharacteristic
367C2×#95C2×D7111yescharacteristic
368C2×#95C2×D7112yes—
369C25—711no—
370C23×C4—2842no—
371C23×C4—2842no—
372C23×C4—2842no—
373C23×C4—2842no—
374C23×C4—714no—
375C23×C4—714no—
376C23×C4—714no—
377C23×C4—711no—
378C23×C4—711no—
379C23×C4—711no—
380C23×C4—712no—
381C23×C4—712no—
382C23×C4—1424no—
383C23×C4—1424no—
384C23×C4—1424no—
385C23×C4—1424no—
386C23×C4—1424no—
387C23×C4C22×D7112yes—
388C23×C4C22×D7111yescharacteristic
389C23×C4C22×D7111yescharacteristic
390C23×C4C22×D7112yes—
391C23×C4C22×D7111yescharacteristic
392C2×C4⋊C4—2842no—
393C2×C4⋊C4—2842no—
394C2×C4⋊C4—1424no—
395C2×C22⋊C4—714no—
396C2×C22⋊C4—714no—
397C2×C22⋊C4—1424no—
398C2×C22⋊C4—1424no—
399C2×C22⋊C4—1424no—
400C2×C22⋊C4—1422no—
401C2×C22⋊C4—1422no—
402C2×C22⋊C4—1422no—
403C2×C22⋊C4—1422no—
404C2×C42—42no—
405C2×C42—2842no—
406C2×C42—714no—
407C2×C42C4×D7114yescomplemented
408#95—22no—
409#95—22no—
410#95—24no—
411#95—42no—
412#95—2842no—
413#95—2842no—
414#95—2842no—
415#95—714no—
416#95—714no—
417#95—1424no—
418#95—1424no—
419#95—1424no—
420#95—1422no—
421#95—1422no—
422#95—1424no—
423C24—712no—
424C24—712no—
425C24—712no—
426C24—712no—
427C24—712no—
428C24—712no—
429C24—712no—
430C24—1422no—
431C24—1422no—
432C24—1422no—
433C24—1422no—
434C24C23×D7111yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
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—2848no—
459C22×C4—2848no—
460C22×C4—2848no—
461C22×C4—2844no—
462C22×C4—2844no—
463C22×C4—2848no—
464C22×C4—2848no—
465C22×C4—2848no—
466C22×C4—2848no—
467C22×C4—2848no—
468C22×C4—7116no—
469C22×C4—712no—
470C22×C4—712no—
471C22×C4—712no—
472C22×C4—712no—
473C22×C4—712no—
474C22×C4—714no—
475C22×C4—14216no—
476C22×C4—1422no—
477C22×C4—1422no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1421no—
486C22×C4—1421no—
487C22×C4—1421no—
488C22×C4—1421no—
489C22×C4—1422no—
490C22×C4—1422no—
491C22×C4—1422no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4—1422no—
498C22×C4—1422no—
499C22×C4C2×C4×D7112yes—
500C22×C4C2×C4×D7112yes—
501C22×C4C2×C4×D7112yes—
502C22×C4D4×D7112yes—
503C22×C4Q8×D7112yes—
504C22×C4#1176586514yes—
505C4⋊C4—2848no—
506C22⋊C4—2848no—
507C22⋊C4—1428no—
508C22⋊C4—1428no—
509C42—44no—
510C42—2844no—
511C23—714no—
512C23—714no—
513C23—714no—
514C23—714no—
515C23—714no—
516C23—714no—
517C23—714no—
518C23—1424no—
519C23—1424no—
520C23—1424no—
521C23—1424no—
522C23—1424no—
523C23—1424no—
524C23—1424no—
525C23—1424no—
526C23—1424no—
527C23—1422no—
528C23—1422no—
529C23—1422no—
530C23—1422no—
531C23—1424no—
532C23—1424no—
533C23—1424no—
534C23C22×C4×D7111yescharacteristic
535C23C2×D4×D7111yescharacteristic
536C23C2×Q8×D7111yescharacteristic
537C23#41×D7111yescharacteristic
538C23#41×D7111yescharacteristic
539C23#41×D7112yes—
540C23#41×D7111yescharacteristic
541C23#41×D7111yescharacteristic
542C23C2×#1176586512yes—
543C23#1455617312yes—
544C23#1455617412yes—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
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—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—28416no—
574C2×C4—28432no—
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—2842no—
584C2×C4—2842no—
585C2×C4—2842no—
586C2×C4—2842no—
587C2×C4—2842no—
588C2×C4—2842no—
589C2×C4—2842no—
590C2×C4—714no—
591C2×C4—1424no—
592C2×C4—1424no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—1422no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4—1424no—
600C2×C4—1424no—
601C2×C4—1422no—
602C2×C4—1422no—
603C2×C4—1424no—
604C2×C4C4⋊C4×D7114yes—
605C22—718no—
606C22—1428no—
607C22—1428no—
608C22—1428no—
609C22—1424no—
610C22—1424no—
611C22—1424no—
612C22—1424no—
613C22—1424no—
614C22—1428no—
615C22—1424no—
616C22C2×C4⋊C4×D7111yescharacteristic
617C22#117×D7111yescharacteristic
618C22#117×D7111yescharacteristic
619C22#1798646711yescharacteristic
620C22C4×#1176586511yescharacteristic
621C22#1798647211yescharacteristic
622C22#1798647311yescharacteristic
623C22#1798650812yes—
624C22#124×D7111yescharacteristic
625C22#124×D7111yescharacteristic
626C22#125×D7111yescharacteristic
627C22#125×D7111yescharacteristic
628C22#1798655312yes—
629C22#1798655712yes—
630C22#1798662412yes—
631C22#1798667312yes—
632C22#1798667412yes—
633C22#1798667712yes—
634C22#1798667812yes—
635C22#1798669312yes—
636C22#1798669412yes—
637C22#1798669912yes—
638C22#1798670012yes—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—2842no—
645C4—2842no—
646C4—2844no—
647C4—5682no—
648C4—1424no—
649C2—1428no—
650C2#382×D7111yescharacteristic
651C2#2259082011yescharacteristic
652C2#2259095311yescharacteristic
653C2#2259127311yescharacteristic
654C2#2259127711yescharacteristic
655C2#2259128111yescharacteristic
656C2#2259128311yescharacteristic
657C2#2259349612yes—
658C2#2259356212yes—
659C2#2259356312yes—
660C2#2259381812yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×#1434C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1434D71
hypocenterC71#13966
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #22586598C2semidirect
5. #22586595C2semidirect
6. #22586567C2semidirect
7. #22586566C2semidirect
8. #22586566C2semidirect
11. C71×#1434C2semidirect
20. C4×(C2×D71)⋊C4C4semidirect
69. C2×C4×Dic71C2×C4semidirect
79. C71⋊#95C2×C4semidirect
80. C71⋊#95C2×C4semidirect
87. C2×C42×C71C2×C4semidirect
339. #1434D71semidirect
342. C71#13966semidirect
407. C2×C42C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858582
C2×C4×(C2×D71)⋊C42
C2×#95×D712
#225865982
#225865952
#225865672
#225865662
(C2×D71)⋊#952
C71×#14342
#1396671