Finite Group Databasebeta

← #31918914

Subgroups — #31918914

Order 18176.

1790 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985861C211yesmaximal, characteristic
2C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
3C2×#17985846C211yesmaximal, characteristic
4#22586576C212yesmaximal
5#22586576C212yesmaximal
6#22586565C211yesmaximal, characteristic, complemented
7#22586564C212yesmaximal, complemented
8C4×C71⋊#95C211yesmaximal, characteristic, complemented
9#95×Dic71C211yesmaximal, characteristic, complemented
10#95×Dic71C212yesmaximal, complemented
11C71×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×(C2×D71)⋊C4C2212yes—
14C22×(C2×D71)⋊C4C2211yescharacteristic
15C22×(C2×D71)⋊C4C2211yescharacteristic
16C22×(C2×D71)⋊C4C2212yes—
17C22×C4×Dic71C2212yes—
18C22×C4×Dic71C2212yes—
19C22×C4×Dic71C2212yes—
20C22×C4×Dic71C2211yescharacteristic
21#17985861C418yescomplemented
22C4×(C2×D71)⋊C4C418yescomplemented
23#17985846C418yescomplemented
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2211yescharacteristic
29C2×C71⋊#95C2211yescharacteristic
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2211yescharacteristic
32C2×C71⋊#95C2211yescharacteristic
33C22×C42×C71C2211yescharacteristic
34C2×#95×C71C2212yes—
35C2×#95×C71C2212yes—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C24×D71C2311yescharacteristic
39C23×Dic71C2312yes—
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2311yescharacteristic
42C23×Dic71C2312yes—
43C23×Dic71C2311yescharacteristic
44C22×C4×D71—42no—
45C22×C4×D71—42no—
46C22×C4×D71C2×C414yes—
47C22×C4×D71C2×C414yes—
48C22×C4×D71C2×C414yes—
49C2×(C2×D71)⋊C4—24no—
50C2×(C2×D71)⋊C4—22no—
51C2×(C2×D71)⋊C4—24no—
52C2×(C2×D71)⋊C4—22no—
53C2×(C2×D71)⋊C4—22no—
54C2×(C2×D71)⋊C4—22no—
55C2×(C2×D71)⋊C4—42no—
56C2×(C2×D71)⋊C4—42no—
57C2×(C2×D71)⋊C4C2×C418yes—
58C2×(C2×D71)⋊C4C2×C414yes—
59C2×(C2×D71)⋊C4C2×C414yes—
60C2×(C2×D71)⋊C4C2×C418yes—
61C2×C4×Dic71—24no—
62C2×C4×Dic71—24no—
63C2×C4×Dic71—24no—
64C2×C4×Dic71—24no—
65C2×C4×Dic71—42no—
66C2×C4×Dic71—42no—
67C2×C4×Dic71—42no—
68C2×C4×Dic71—42no—
69C2×C4×Dic71—42no—
70C2×C4×Dic71C2×C418yes—
71C2×C4×Dic71C2×C414yes—
72C71⋊#95—24no—
73C71⋊#95—24no—
74C71⋊#95—24no—
75C71⋊#95—42no—
76C71⋊#95C2×C414yes—
77C71⋊#95C2×C414yes—
78C71⋊#95C2×C414yes—
79C71⋊#95C2×C418yes—
80C71⋊#95C2×C414yes—
81C23×C4×C71C2311yescharacteristic
82C23×C4×C71C2312yes—
83C23×C4×C71C2312yes—
84C23×C4×C71C2311yescharacteristic
85C23×C4×C71C2311yescharacteristic
86C2×C42×C71—42no—
87C2×C42×C71C2×C414yes—
88#95×C71—24no—
89#95×C71—42no—
90#95×C71C2×C414yes—
91#95×C71C2×C414yes—
92C23×D71—21no—
93C23×D71—22no—
94C23×D71—22no—
95C23×D71—21no—
96C23×D71—21no—
97C23×D71—21no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71C22×C412yes—
101C23×D71C22×C412yes—
102C23×D71C22×C412yes—
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—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—21no—
117C22×Dic71—21no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71C22×C414yes—
122C22×Dic71C22×C412yes—
123C22×Dic71C22×C412yes—
124C22×Dic71C22×C412yes—
125C22×Dic71C22×C412yes—
126C22×Dic71C22×C414yes—
127C22×Dic71C22×C412yes—
128C22×Dic71#4114yes—
129C22×Dic71#4112yes—
130C22×Dic71#4112yes—
131C22×Dic71#4114yes—
132C2×C4×D71—48no—
133C2×C4×D71—44no—
134C2×C4×D71—44no—
135C2×C4×D71—48no—
136C2×C4×D71C42116yescomplemented
137(C2×D71)⋊C4—28no—
138(C2×D71)⋊C4—216no—
139(C2×D71)⋊C4—28no—
140(C2×D71)⋊C4—48no—
141C4×Dic71—48no—
142C4×Dic71—44no—
143C4×Dic71—48no—
144C4×Dic71—48no—
145C24×C71C2411yescharacteristic
146C22×C4×C71—22no—
147C22×C4×C71—22no—
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—21no—
158C22×C4×C71—22no—
159C22×C4×C71—21no—
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×C71C22×C414yes—
167C22×C4×C71C22×C414yes—
168C22×C4×C71C22×C412yes—
169C22×C4×C71C22×C412yes—
170C22×C4×C71C22×C412yes—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C22×C412yes—
173C22×C4×C71#4112yes—
174C22×C4×C71#4112yes—
175C42×C71—44no—
176C23×C71C23×C411yescharacteristic
177C23×C71C23×C411yescharacteristic
178C23×C71C23×C411yescharacteristic
179C23×C71C2×#4111yescharacteristic
180C23×C71C2×#4112yes—
181C23×C71C2×#4112yes—
182C23×C71C2×#4111yescharacteristic
183C23×C71C2×#4111yescharacteristic
184C23×C71C2×#4111yescharacteristic
185C23×C71H(5,𝔽2)12yes—
186C23×C71H(5,𝔽2)12yes—
187C22×D71—22no—
188C22×D71—24no—
189C22×D71—22no—
190C22×D71—22no—
191C22×D71—22no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—24no—
195C22×D71—22no—
196C22×D71—22no—
197C22×D71—24no—
198C22×D71—24no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—42no—
202C22×D71—42no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71C2×C4214yes—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—22no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—22no—
223C2×C4×C71—24no—
224C2×C4×C71—22no—
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×C71C2×C4214yes—
238C2×C4×C71#11714yes—
239C2×Dic71—22no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—22no—
244C2×Dic71—24no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—22no—
258C2×Dic71—24no—
259C2×Dic71—42no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—42no—
264C2×Dic71—42no—
265C2×Dic71C2×C4214yes—
266C2×Dic71#11714yes—
267C2×Dic71#11718yes—
268C4×D71—416no—
269C22×C71C22×C4211yescharacteristic
270C22×C71C2×#11711yescharacteristic
271C22×C71C2×#11712yes—
272C22×C71C2×#11711yescharacteristic
273C22×C71C4×#4111yescharacteristic
274C22×C71C4×#4111yescharacteristic
275C22×C71C4×#4111yescharacteristic
276C22×C71C4×#4111yescharacteristic
277C22×C71C4×#4112yes—
278C22×C71C4×#4111yescharacteristic
279C22×C71C4×#4111yescharacteristic
280C22×C71#51812yes—
281C22×C71#51812yes—
282C22×C71#51811yescharacteristic
283C22×C71#51811yescharacteristic
284C22×C71#55112yes—
285C22×C71#55112yes—
286C22×C71#55312yes—
287C22×C71#55312yes—
288C22×C71#55312yes—
289C22×C71#55312yes—
290C22×C71#55912yes—
291C22×C71#55912yes—
292C2×D71—24no—
293C2×D71—24no—
294C2×D71—24no—
295C2×D71—42no—
296C2×D71—44no—
297C2×D71—42no—
298C2×D71—42no—
299C2×D71—44no—
300C2×D71—42no—
301C2×D71—44no—
302C2×D71—44no—
303C4×C71—24no—
304C4×C71—24no—
305C4×C71—42no—
306C4×C71—44no—
307C4×C71—44no—
308Dic71—24no—
309Dic71—28no—
310Dic71—24no—
311Dic71—42no—
312Dic71—44no—
313#13807—711nomaximal, Sylow
314C2×C71C4×#11711yescharacteristic
315C2×C71C4×#11711yescharacteristic
316C2×C71#227311yescharacteristic
317C2×C71#235311yescharacteristic
318C2×C71#235312yes—
319C2×C71#235311yescharacteristic
320C2×C71#235711yescharacteristic
321C2×C71#235712yes—
322C2×C71#235711yescharacteristic
323C2×C71#237412yes—
324C2×C71#237412yes—
325D71—44no—
326C2×#381—711no—
327C2×C4×C22⋊C4—711no—
328C2×C4×C22⋊C4—711no—
329#1437—712no—
330#1437—712no—
331#1434—712no—
332#1434—711no—
333#1434D7111yescharacteristic, complemented
334C4×#95—711no—
335C4×#95—712no—
336C4×#95—711no—
337C71#1380711yescharacteristic, complemented, Sylow
338C24×C4—711no—
339C22×C22⋊C4—712no—
340C22×C22⋊C4—711no—
341C22×C22⋊C4—712no—
342C22×C22⋊C4—711no—
343C22×C42—712no—
344C22×C42—711no—
345C22×C42—712no—
346C22×C42—712no—
347C22×C42C2×D7111yescharacteristic
348#381—718no—
349C4×C22⋊C4—718no—
350C4×C22⋊C4—718no—
351C2×#95—712no—
352C2×#95—711no—
353C2×#95—711no—
354C2×#95—712no—
355C2×#95—711no—
356C2×#95—711no—
357C2×#95—712no—
358C2×#95—712no—
359C2×#95—712no—
360C2×#95C2×D7112yes—
361C2×#95C2×D7112yes—
362C2×#95C2×D7111yescharacteristic
363C2×#95C2×D7111yescharacteristic
364C25—711no—
365C23×C4—2842no—
366C23×C4—2842no—
367C23×C4—714no—
368C23×C4—714no—
369C23×C4—714no—
370C23×C4—711no—
371C23×C4—712no—
372C23×C4—711no—
373C23×C4—712no—
374C23×C4—711no—
375C23×C4C22×D7111yescharacteristic
376C23×C4C22×D7112yes—
377C23×C4C22×D7112yes—
378C23×C4C22×D7111yescharacteristic
379C23×C4C22×D7111yescharacteristic
380C2×C22⋊C4—2842no—
381C2×C22⋊C4—2842no—
382C2×C22⋊C4—718no—
383C2×C22⋊C4—714no—
384C2×C22⋊C4—718no—
385C2×C22⋊C4—714no—
386C2×C22⋊C4—1424no—
387C2×C22⋊C4—1422no—
388C2×C22⋊C4—1424no—
389C2×C22⋊C4—1422no—
390C2×C22⋊C4—1422no—
391C2×C22⋊C4—1422no—
392C2×C42—42no—
393C2×C42—2842no—
394C2×C42—2842no—
395C2×C42—2842no—
396C2×C42—2842no—
397C2×C42—2842no—
398C2×C42—718no—
399C2×C42—714no—
400C2×C42—1424no—
401C2×C42—1424no—
402C2×C42—1424no—
403C2×C42—1424no—
404C2×C42C4×D7114yes—
405#95—24no—
406#95—42no—
407#95—2842no—
408#95—714no—
409#95—714no—
410#95—714no—
411#95—718no—
412#95—714no—
413#95—1424no—
414#95—1424no—
415#95—1424no—
416#95C4×D7114yes—
417#95C4×D7114yes—
418C24—712no—
419C24—712no—
420C24—712no—
421C24—1421no—
422C24—1422no—
423C24—1422no—
424C24—1421no—
425C24—1421no—
426C24—1421no—
427C24—1422no—
428C24—1422no—
429C24C23×D7111yescharacteristic
430C22×C4—22no—
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—22no—
440C22×C4—21no—
441C22×C4—22no—
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—2848no—
451C22×C4—2848no—
452C22×C4—2844no—
453C22×C4—2844no—
454C22×C4—7116no—
455C22×C4—712no—
456C22×C4—712no—
457C22×C4—712no—
458C22×C4—714no—
459C22×C4—712no—
460C22×C4—714no—
461C22×C4—712no—
462C22×C4—712no—
463C22×C4—714no—
464C22×C4—714no—
465C22×C4—712no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1421no—
471C22×C4—1422no—
472C22×C4—1421no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
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×C4C2×C4×D7114yes—
485C22×C4C2×C4×D7112yes—
486C22×C4C2×C4×D7114yes—
487C22×C4C2×C4×D7112yes—
488C22×C4C2×C4×D7112yes—
489C22×C4C2×C4×D7112yes—
490C22×C4C2×C4×D7112yes—
491C22×C4#1176586412yes—
492C22×C4#1176586612yes—
493C22⋊C4—2848no—
494C22⋊C4—1428no—
495C22⋊C4—14216no—
496C22⋊C4—1428no—
497C42—44no—
498C42—2848no—
499C42—2844no—
500C42—2848no—
501C42—2848no—
502C23—2842no—
503C23—2842no—
504C23—2842no—
505C23—2842no—
506C23—2842no—
507C23—2842no—
508C23—2842no—
509C23—2842no—
510C23—714no—
511C23—1422no—
512C23—1424no—
513C23—1422no—
514C23—1422no—
515C23—1422no—
516C23—1422no—
517C23—1422no—
518C23—1424no—
519C23—1422no—
520C23—1422no—
521C23—1424no—
522C23—1424no—
523C23—1422no—
524C23—1422no—
525C23C22×C4×D7111yescharacteristic
526C23C22×C4×D7111yescharacteristic
527C23C22×C4×D7111yescharacteristic
528C23#41×D7111yescharacteristic
529C23#41×D7112yes—
530C23#41×D7112yes—
531C23#41×D7111yescharacteristic
532C23C2×#1176586411yescharacteristic
533C23C2×#1176586611yescharacteristic
534C23#1455617312yes—
535C23#1455617412yes—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—28416no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2842no—
567C2×C4—2842no—
568C2×C4—2842no—
569C2×C4—2842no—
570C2×C4—714no—
571C2×C4—714no—
572C2×C4—718no—
573C2×C4—1422no—
574C2×C4—1422no—
575C2×C4—1424no—
576C2×C4—1424no—
577C2×C4—1424no—
578C2×C4—1422no—
579C2×C4—1424no—
580C2×C4—1424no—
581C2×C4—1424no—
582C2×C4—1424no—
583C2×C4—1424no—
584C2×C4—1424no—
585C2×C4—1422no—
586C2×C4—1424no—
587C2×C4—1424no—
588C2×C4—1424no—
589C2×C4—1424no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4—1422no—
593C2×C4C42×D7114yes—
594C2×C4#1455608614yes—
595C22—2842no—
596C22—2844no—
597C22—2842no—
598C22—2842no—
599C22—2844no—
600C22—2842no—
601C22—2844no—
602C22—2844no—
603C22—1424no—
604C22—1424no—
605C22—1424no—
606C22C2×C42×D7111yescharacteristic
607C22#117×D7111yescharacteristic
608C22#117×D7112yes—
609C22C2×#1455608611yescharacteristic
610C22C4×#1176586411yescharacteristic
611C22C4×#1176586411yescharacteristic
612C22#1798646711yescharacteristic
613C22#1798646711yescharacteristic
614C22#1798646712yes—
615C22C4×#1176586611yescharacteristic
616C22C4×#1176586611yescharacteristic
617C22#1798647112yes—
618C22#1798647112yes—
619C22#1798647211yescharacteristic
620C22#1798647311yescharacteristic
621C22#1798667412yes—
622C22#1798667612yes—
623C22#1798669212yes—
624C22#1798669412yes—
625C22#1798670012yes—
626C22#1798670212yes—
627C22#1798673112yes—
628C22#1798673312yes—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—2842no—
635C4—2844no—
636C4—1424no—
637C4—1428no—
638C4—1424no—
639C2—2844no—
640C2#2259067911yescharacteristic
641C2C4×#1455608611yescharacteristic
642C2#2259069211yescharacteristic
643C2#2259127311yescharacteristic
644C2#2259127912yes—
645C2#2259128211yescharacteristic
646C2#2259130811yescharacteristic
647C2#2259131012yes—
648C2#2259131511yescharacteristic
649C2#2259146812yes—
650C2#2259147212yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC71×#1434C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1434D71
hypocenterC71#13807
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
6. #22586565C2semidirect
7. #22586564C2semidirect
8. C4×C71⋊#95C2semidirect
9. #95×Dic71C2semidirect
10. #95×Dic71C2semidirect
11. C71×#1434C2semidirect
21. #17985861C4semidirect
22. C4×(C2×D71)⋊C4C4semidirect
23. #17985846C4semidirect
136. C2×C4×D71C42semidirect
333. #1434D71semidirect
337. C71#13807semidirect

Maximal subgroups

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

subgroupindex
C2×#179858612
C2×C4×(C2×D71)⋊C42
C2×#179858462
#225865762
#225865652
#225865642
C4×C71⋊#952
#95×Dic712
C71×#14342
#1380771