Finite Group Databasebeta

← #31920186

Subgroups — #31920186

Order 18176.

2222 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556085C211yesmaximal, characteristic
2C2×#17985873C212yesmaximal
3C2×#17985861C211yesmaximal, characteristic
4C2×#95×D71C211yesmaximal, characteristic
5#22586594C212yesmaximal, complemented
6#22586591C212yesmaximal, complemented
7#22586570C211yesmaximal, characteristic, complemented
8Dic71⋊#95C211yesmaximal, characteristic, complemented
9#22586566C211yesmaximal, characteristic, complemented
10(C2×D71)⋊#95C212yesmaximal
11C71×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D71C2212yes—
13C23×C4×D71C2211yescharacteristic
14C2×#14556085—28no—
15C2×#14556085C418yescomplemented
16C22×C4⋊Dic71C2212yes—
17C22×Dic71⋊C4C2212yes—
18C22×Dic71⋊C4C2211yescharacteristic
19C22×Dic71⋊C4C2211yescharacteristic
20C22×(C2×D71)⋊C4C2212yes—
21C22×(C2×D71)⋊C4C2211yescharacteristic
22C22×(C2×D71)⋊C4C2211yescharacteristic
23C22×C4×Dic71C2211yescharacteristic
24#17985873—28no—
25#17985861—24no—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2211yescharacteristic
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2212yes—
33C2×C71⋊#95C2212yes—
34#95×D71—24no—
35C22×C42×C71C2211yescharacteristic
36C2×#95×C71C2212yes—
37C2×#95×C71C2212yes—
38C2×#95×C71C2211yescharacteristic
39C2×#95×C71C2211yescharacteristic
40C24×D71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2312yes—
44C23×Dic71C2311yescharacteristic
45C23×Dic71C2311yescharacteristic
46C22×C4×D71—24no—
47C22×C4×D71—24no—
48C22×C4×D71—24no—
49C22×C4×D71—24no—
50C22×C4×D71—24no—
51C22×C4×D71—24no—
52C22×C4×D71—24no—
53C22×C4×D71—42no—
54C22×C4×D71—42no—
55C22×C4×D71C2×C414yes—
56C22×C4×D71D414yes—
57C22×C4×D71Q814yes—
58#14556085—232no—
59C2×C4⋊Dic71—24no—
60C2×C4⋊Dic71—24no—
61C2×C4⋊Dic71—42no—
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—24no—
64C2×Dic71⋊C4—24no—
65C2×Dic71⋊C4—24no—
66C2×Dic71⋊C4—42no—
67C2×Dic71⋊C4C2×C414yescomplemented
68C2×Dic71⋊C4C2×C414yescomplemented
69C2×(C2×D71)⋊C4—24no—
70C2×(C2×D71)⋊C4—24no—
71C2×(C2×D71)⋊C4—24no—
72C2×(C2×D71)⋊C4—24no—
73C2×(C2×D71)⋊C4C2×C414yes—
74C2×(C2×D71)⋊C4C2×C414yes—
75C2×(C2×D71)⋊C4D418yes—
76C2×C4×Dic71—24no—
77C2×C4×Dic71C2×C414yescomplemented
78C71⋊#95—24no—
79C71⋊#95—24no—
80C71⋊#95—24no—
81C71⋊#95—24no—
82C71⋊#95—22no—
83C71⋊#95—22no—
84C71⋊#95—42no—
85C71⋊#95—42no—
86C23×C4×C71C2312yes—
87C23×C4×C71C2311yescharacteristic
88C23×C4×C71C2312yes—
89C23×C4×C71C2311yescharacteristic
90C23×C4×C71C2311yescharacteristic
91C2×C42×C71—24no—
92C2×C42×C71C2×C414yescomplemented
93#95×C71—22no—
94#95×C71—22no—
95#95×C71—24no—
96#95×C71—24no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—21no—
100C23×D71—21no—
101C23×D71—21no—
102C23×D71—21no—
103C23×D71C22×C412yes—
104C23×D71C2×D414yes—
105C23×D71C2×D412yes—
106C23×D71C2×Q812yes—
107C23×D71#4114yes—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—21no—
112C22×Dic71—21no—
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—21no—
122C22×Dic71—21no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71C22×C412yes—
131C22×Dic71C22×C412yes—
132C22×Dic71C22×C412yes—
133C22×Dic71C2×D414yes—
134C22×Dic71C2×D412yes—
135C22×Dic71C2×Q812yes—
136C22×Dic71#4114yes—
137C2×C4×D71—28no—
138C2×C4×D71—28no—
139C2×C4×D71—216no—
140C2×C4×D71—216no—
141C2×C4×D71—216no—
142C2×C4×D71—48no—
143C2×C4×D71—48no—
144C2×C4×D71—48no—
145C2×C4×D71—48no—
146C2×C4×D71—48no—
147C2×C4×D71—48no—
148C2×C4×D71C4⋊C4116yescomplemented
149C4⋊Dic71—48no—
150Dic71⋊C4—28no—
151Dic71⋊C4—28no—
152Dic71⋊C4—48no—
153(C2×D71)⋊C4—28no—
154(C2×D71)⋊C4—48no—
155(C2×D71)⋊C4C22⋊C4116yes—
156C4×Dic71—28no—
157C24×C71C2411yescharacteristic
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
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—21no—
172C22×C4×C71—21no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C22×C412yes—
182C22×C4×C71C22×C412yes—
183C22×C4×C71C2×D414yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71C2×Q812yes—
186C22×C4×C71#4114yes—
187C42×C71—28no—
188C23×C71C23×C411yescharacteristic
189C23×C71C22×D412yes—
190C23×C71C22×D411yescharacteristic
191C23×C71C22×Q811yescharacteristic
192C23×C71C2×#4112yes—
193C23×C71C2×#4112yes—
194C23×C71C2×#4112yes—
195C23×C71H(5,𝔽2)11yescharacteristic
196C23×C71H(5,𝔽2)11yescharacteristic
197C23×C712−1+411yescharacteristic
198C23×C712−1+411yescharacteristic
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—24no—
210C22×D71—24no—
211C22×D71—24no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—24no—
215C22×D71—24no—
216C22×D71—24no—
217C22×D71C2×C22⋊C414yes—
218C22×D71C2×C4⋊C414yes—
219C22×D71#11714yes—
220C22×D71#12118yes—
221C22×D71C22⋊Q818yes—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—24no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
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×C71—42no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71C2×C22⋊C414yes—
250C2×C4×C71C2×C4⋊C414yes—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—22no—
260C2×Dic71—22no—
261C2×Dic71—22no—
262C2×Dic71—24no—
263C2×Dic71—24no—
264C2×Dic71—24no—
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×Dic71—42no—
274C2×Dic71—42no—
275C2×Dic71—42no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71C2×C22⋊C414yes—
279C2×Dic71C2×C4⋊C414yes—
280C4×D71—232no—
281C4×D71—432no—
282C22×C71C22×C22⋊C411yescharacteristic
283C22×C71C22×C4⋊C411yescharacteristic
284C22×C71C2×#11711yescharacteristic
285C22×C71C4×#4111yescharacteristic
286C22×C71C4×#4111yescharacteristic
287C22×C71#52011yescharacteristic
288C22×C71#52011yescharacteristic
289C22×C71C2×#12112yes—
290C22×C71C2×C22⋊Q812yes—
291C22×C71#52512yes—
292C22×C71#53312yes—
293C22×C71#53511yescharacteristic
294C22×C71#53611yescharacteristic
295C22×C71#54411yescharacteristic
296C22×C71#54411yescharacteristic
297C22×C71#54612yes—
298C22×C71#54612yes—
299C22×C71#54712yes—
300C22×C71#54712yes—
301C22×C71#55112yes—
302C22×C71#55112yes—
303C22×C71#55212yes—
304C22×C71#55212yes—
305C2×D71—24no—
306C2×D71—24no—
307C2×D71—24no—
308C2×D71—24no—
309C2×D71—24no—
310C2×D71—28no—
311C2×D71—28no—
312C2×D71—28no—
313C2×D71—28no—
314C2×D71—24no—
315C2×D71#38018yes—
316C4×C71—24no—
317C4×C71—24no—
318C4×C71—42no—
319C4×C71—44no—
320C4×C71—82no—
321Dic71—24no—
322Dic71—24no—
323Dic71—42no—
324Dic71—44no—
325Dic71—82no—
326#13962—711nomaximal, Sylow
327C2×C71C2×#38011yescharacteristic
328C2×C71#229611yescharacteristic
329C2×C71#231711yescharacteristic
330C2×C71#233911yescharacteristic
331C2×C71#233911yescharacteristic
332C2×C71#234311yescharacteristic
333C2×C71#234311yescharacteristic
334C2×C71#241112yes—
335C2×C71#241312yes—
336C2×C71#243812yes—
337C2×C71#243812yes—
338D71—28no—
339C22×#117—711no—
340C2×#385—712no—
341C2×#381—711no—
342C22×#95—711no—
343#1441—712no—
344#1441—712no—
345C4⋊#95—711no—
346C4⋊#95—711no—
347#1434—711no—
348#1434D7111yescharacteristic, complemented
349C22⋊#95—712no—
350C71#1396211yescharacteristic, complemented, Sylow
351C24×C4—712no—
352C24×C4—711no—
353C2×#117—718no—
354C2×#117—1428no—
355C22×C4⋊C4—712no—
356C22×C4⋊C4—712no—
357C22×C4⋊C4—711no—
358C22×C4⋊C4—711no—
359C22×C22⋊C4—711no—
360C22×C22⋊C4—711no—
361C22×C22⋊C4—712no—
362C22×C42—711no—
363C22×C42C2×D7111yescharacteristic
364#385—1428no—
365#381—1424no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95—712no—
369C2×#95—711no—
370C2×#95—712no—
371C2×#95—711no—
372C2×#95—712no—
373C2×#95—712no—
374C2×#95—1424no—
375C2×#95C2×D7112yes—
376C2×#95C2×D7111yescharacteristic
377C2×#95C2×D7111yescharacteristic
378C2×#95C2×D7112yes—
379C25—711no—
380C23×C4—2842no—
381C23×C4—2842no—
382C23×C4—711no—
383C23×C4—711no—
384C23×C4—714no—
385C23×C4—714no—
386C23×C4—714no—
387C23×C4—711no—
388C23×C4—712no—
389C23×C4—712no—
390C23×C4—1424no—
391C23×C4—1424no—
392C23×C4—1424no—
393C23×C4—1424no—
394C23×C4—1424no—
395C23×C4—1424no—
396C23×C4—1424no—
397C23×C4C22×D7112yes—
398C23×C4C22×D7111yescharacteristic
399C23×C4C22×D7112yes—
400C23×C4C22×D7111yescharacteristic
401C23×C4C22×D7111yescharacteristic
402#117—14232no—
403C2×C4⋊C4—2842no—
404C2×C4⋊C4—2842no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—714no—
407C2×C4⋊C4—1424no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1424no—
411C2×C4⋊C4—1424no—
412C2×C4⋊C4—1424no—
413C2×C22⋊C4—714no—
414C2×C22⋊C4—718no—
415C2×C22⋊C4—714no—
416C2×C22⋊C4—1424no—
417C2×C22⋊C4—1424no—
418C2×C22⋊C4—1424no—
419C2×C22⋊C4—1424no—
420C2×C42—24no—
421C2×C42—714no—
422C2×C42—1424no—
423C2×C42C4×D7114yescomplemented
424#95—22no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95—2842no—
429#95—2842no—
430#95—1424no—
431#95—1424no—
432#95—1424no—
433#95—1424no—
434#95—1422no—
435#95—1422no—
436C24—712no—
437C24—714no—
438C24—712no—
439C24—712no—
440C24—714no—
441C24—1422no—
442C24—1422no—
443C24—1421no—
444C24—1421no—
445C24—1421no—
446C24—1421no—
447C24C23×D7111yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—2848no—
471C22×C4—2848no—
472C22×C4—2848no—
473C22×C4—2848no—
474C22×C4—2848no—
475C22×C4—2848no—
476C22×C4—712no—
477C22×C4—712no—
478C22×C4—712no—
479C22×C4—714no—
480C22×C4—712no—
481C22×C4—7116no—
482C22×C4—712no—
483C22×C4—714no—
484C22×C4—14216no—
485C22×C4—1428no—
486C22×C4—1428no—
487C22×C4—14216no—
488C22×C4—14216no—
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×C4—1422no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4—1421no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1421no—
506C22×C4—1422no—
507C22×C4—1422no—
508C22×C4—1422no—
509C22×C4—1421no—
510C22×C4—1421no—
511C22×C4C2×C4×D7112yes—
512C22×C4C2×C4×D7112yes—
513C22×C4C2×C4×D7112yes—
514C22×C4D4×D7114yes—
515C22×C4D4×D7112yes—
516C22×C4Q8×D7112yes—
517C22×C4#1176586514yes—
518C4⋊C4—2848no—
519C4⋊C4—2848no—
520C4⋊C4—1428no—
521C4⋊C4—1428no—
522C22⋊C4—2848no—
523C22⋊C4—7116no—
524C22⋊C4—1428no—
525C42—28no—
526C42—1428no—
527C23—714no—
528C23—714no—
529C23—714no—
530C23—718no—
531C23—718no—
532C23—1422no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1424no—
537C23—1424no—
538C23—1424no—
539C23—1424no—
540C23—1424no—
541C23—1424no—
542C23—1424no—
543C23—1424no—
544C23—1422no—
545C23—1422no—
546C23—1422no—
547C23—1422no—
548C23—1424no—
549C23—1424no—
550C23C22×C4×D7111yescharacteristic
551C23C2×D4×D7111yescharacteristic
552C23C2×D4×D7112yes—
553C23C2×Q8×D7111yescharacteristic
554C23#41×D7112yes—
555C23#41×D7112yes—
556C23C2×#1176586512yes—
557C23#1455617411yescharacteristic
558C23#1455617411yescharacteristic
559C23#1455617511yescharacteristic
560C23#1455617511yescharacteristic
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—28432no—
589C2×C4—2842no—
590C2×C4—2842no—
591C2×C4—2842no—
592C2×C4—2842no—
593C2×C4—2842no—
594C2×C4—2842no—
595C2×C4—2842no—
596C2×C4—2842no—
597C2×C4—2842no—
598C2×C4—2842no—
599C2×C4—2842no—
600C2×C4—2842no—
601C2×C4—2842no—
602C2×C4—714no—
603C2×C4—714no—
604C2×C4—14232no—
605C2×C4—1424no—
606C2×C4—1424no—
607C2×C4—1422no—
608C2×C4—1422no—
609C2×C4—1422no—
610C2×C4—1424no—
611C2×C4—1424no—
612C2×C4—1424no—
613C2×C4—1422no—
614C2×C4—1422no—
615C2×C4—1422no—
616C2×C4—1422no—
617C2×C4—1424no—
618C2×C4—1424no—
619C2×C4C22⋊C4×D7114yes—
620C2×C4C4⋊C4×D7114yes—
621C22—718no—
622C22—1428no—
623C22—1428no—
624C22—1428no—
625C22—1424no—
626C22—1424no—
627C22—1424no—
628C22—1424no—
629C22—1424no—
630C22—1428no—
631C22—1424no—
632C22C2×C22⋊C4×D7111yescharacteristic
633C22C2×C4⋊C4×D7111yescharacteristic
634C22#117×D7111yescharacteristic
635C22#1798646711yescharacteristic
636C22C4×#1176586511yescharacteristic
637C22#1798647911yescharacteristic
638C22#1798647911yescharacteristic
639C22#121×D7112yes—
640C22C22⋊Q8×D7112yes—
641C22#1798650812yes—
642C22#1798655312yes—
643C22#1798656711yescharacteristic
644C22#1798656911yescharacteristic
645C22#1798662911yescharacteristic
646C22#1798662911yescharacteristic
647C22#1798664112yes—
648C22#1798664512yes—
649C22#1798664712yes—
650C22#1798665412yes—
651C22#1798667312yes—
652C22#1798667412yes—
653C22#1798668512yes—
654C22#1798668812yes—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—44no—
659C4—82no—
660C4—2842no—
661C4—2844no—
662C4—5682no—
663C4—1424no—
664C4—1424no—
665C2—1428no—
666C2#380×D7111yescharacteristic
667C2#2259082011yescharacteristic
668C2#2259096511yescharacteristic
669C2#2259116411yescharacteristic
670C2#2259117111yescharacteristic
671C2#2259119711yescharacteristic
672C2#2259120111yescharacteristic
673C2#2259172412yes—
674C2#2259174612yes—
675C2#2259201512yes—
676C2#2259202112yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×#1434C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1434D71
hypocenterC71#13962
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
5. #22586594C2semidirect
6. #22586591C2semidirect
7. #22586570C2semidirect
8. Dic71⋊#95C2semidirect
9. #22586566C2semidirect
11. C71×#1434C2semidirect
15. C2×#14556085C4semidirect
67. C2×Dic71⋊C4C2×C4semidirect
68. C2×Dic71⋊C4C2×C4semidirect
77. C2×C4×Dic71C2×C4semidirect
92. C2×C42×C71C2×C4semidirect
148. C2×C4×D71C4⋊C4semidirect
348. #1434D71semidirect
350. C71#13962semidirect
423. C2×C42C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×#145560852
C2×#179858732
C2×#179858612
C2×#95×D712
#225865942
#225865912
#225865702
Dic71⋊#952
#225865662
(C2×D71)⋊#952
C71×#14342
#1396271