Finite Group Databasebeta

← #31919772

Subgroups — #31919772

Order 18176.

1730 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×#17985890C212yesmaximal
2C2×#17985863C211yesmaximal, characteristic, complemented
3C2×#17985858C211yesmaximal, characteristic
4C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
5#22586604C212yesmaximal, complemented
6#22586586C212yesmaximal
7#22586573C212yesmaximal, complemented
8#22586570C211yesmaximal, characteristic, complemented
9#22586564C211yesmaximal, characteristic, complemented
10(C2×D71)⋊#95C211yesmaximal, characteristic
11C4×#95×C71C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×C4⋊Dic71C2212yes—
14C22×Dic71⋊C4C2212yes—
15C22×Dic71⋊C4C2212yes—
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2212yes—
18C22×(C2×D71)⋊C4C2211yescharacteristic
19C22×(C2×D71)⋊C4C2211yescharacteristic
20C22×C4×Dic71C2211yescharacteristic
21#17985890—28no—
22#17985863—24no—
23#17985858—24no—
24C4×(C2×D71)⋊C4C418yescomplemented
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33C2×C71⋊#95C2211yescharacteristic
34C22×C42×C71C2212yes—
35C22×C42×C71C2211yescharacteristic
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2212yes—
38C2×#95×C71C2211yescharacteristic
39C24×D71C2311yescharacteristic
40C23×Dic71C2312yes—
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C23×Dic71C2311yescharacteristic
45C22×C4×D71—24no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—42no—
49C22×C4×D71C2×C414yes—
50C2×C4⋊Dic71—24no—
51C2×C4⋊Dic71—42no—
52C2×C4⋊Dic71—42no—
53C2×Dic71⋊C4—24no—
54C2×Dic71⋊C4—24no—
55C2×Dic71⋊C4—24no—
56C2×Dic71⋊C4—24no—
57C2×Dic71⋊C4—42no—
58C2×Dic71⋊C4—42no—
59C2×(C2×D71)⋊C4—24no—
60C2×(C2×D71)⋊C4—24no—
61C2×(C2×D71)⋊C4—24no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4—42no—
65C2×(C2×D71)⋊C4—42no—
66C2×(C2×D71)⋊C4C2×C414yes—
67C2×(C2×D71)⋊C4C2×C414yes—
68C2×(C2×D71)⋊C4D414yes—
69C2×(C2×D71)⋊C4Q814yes—
70C2×C4×Dic71—22no—
71C2×C4×Dic71—22no—
72C2×C4×Dic71C2×C414yes—
73C71⋊#95—24no—
74C71⋊#95—24no—
75C71⋊#95—22no—
76C71⋊#95—24no—
77C71⋊#95—22no—
78C71⋊#95—24no—
79C71⋊#95—24no—
80C71⋊#95C2×C414yes—
81C71⋊#95C2×C414yes—
82C23×C4×C71C2312yes—
83C23×C4×C71C2312yes—
84C23×C4×C71C2311yescharacteristic
85C23×C4×C71C2311yescharacteristic
86C23×C4×C71C2311yescharacteristic
87C2×C42×C71—24no—
88C2×C42×C71—24no—
89C2×C42×C71—42no—
90C2×C42×C71—42no—
91C2×C42×C71C2×C414yes—
92#95×C71—24no—
93#95×C71—22no—
94#95×C71D414yes—
95C23×D71—21no—
96C23×D71—22no—
97C23×D71—21no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—21no—
102C23×D71—21no—
103C23×D71C22×C412yes—
104C23×D71C2×D412yes—
105C23×D71C2×Q812yes—
106C22×Dic71—21no—
107C22×Dic71—21no—
108C22×Dic71—21no—
109C22×Dic71—21no—
110C22×Dic71—22no—
111C22×Dic71—21no—
112C22×Dic71—21no—
113C22×Dic71—21no—
114C22×Dic71—22no—
115C22×Dic71—21no—
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×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71C22×C412yes—
132C22×Dic71C22×C412yes—
133C22×Dic71C22×C412yes—
134C22×Dic71C2×D412yes—
135C22×Dic71C2×Q812yes—
136C22×Dic71#4114yes—
137C2×C4×D71—28no—
138C2×C4×D71—28no—
139C2×C4×D71—44no—
140C2×C4×D71—48no—
141C2×C4×D71—48no—
142C4⋊Dic71—48no—
143Dic71⋊C4—48no—
144Dic71⋊C4—48no—
145(C2×D71)⋊C4—28no—
146(C2×D71)⋊C4—48no—
147(C2×D71)⋊C4—48no—
148(C2×D71)⋊C4C4⋊C4116yescomplemented
149C4×Dic71—28no—
150C24×C71C2411yescharacteristic
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
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×C71C22×C412yes—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C22×C412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D414yes—
175C22×C4×C71C2×D412yes—
176C22×C4×C71C2×Q812yes—
177C22×C4×C71#4114yes—
178C22×C4×C71#4114yes—
179C22×C4×C71#4112yes—
180C42×C71—44no—
181C42×C71—48no—
182C23×C71C23×C411yescharacteristic
183C23×C71C22×D411yescharacteristic
184C23×C71C22×D411yescharacteristic
185C23×C71C22×Q811yescharacteristic
186C23×C71C2×#4112yes—
187C23×C71C2×#4112yes—
188C23×C71C2×#4111yescharacteristic
189C23×C71C2×#4112yes—
190C23×C71H(5,𝔽2)12yes—
191C23×C71H(5,𝔽2)11yescharacteristic
192C23×C712−1+411yescharacteristic
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—22no—
202C22×D71—24no—
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—24no—
206C22×D71—22no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71C2×C4⋊C414yes—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
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—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
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×C4×C71#11714yes—
244C2×C4×C71C4×D414yes—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—22no—
250C2×Dic71—24no—
251C2×Dic71—22no—
252C2×Dic71—22no—
253C2×Dic71—24no—
254C2×Dic71—22no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—22no—
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×Dic71—42no—
272C2×Dic71—42no—
273C2×Dic71—42no—
274C2×Dic71C2×C4⋊C414yes—
275C2×Dic71#11714yes—
276C4×D71—416no—
277C22×C71C22×C4⋊C411yescharacteristic
278C22×C71C2×#11711yescharacteristic
279C22×C71C2×#11711yescharacteristic
280C22×C71C2×C4×D411yescharacteristic
281C22×C71C4×#4111yescharacteristic
282C22×C71#51811yescharacteristic
283C22×C71#52011yescharacteristic
284C22×C71#52912yes—
285C22×C71#53212yes—
286C22×C71#53212yes—
287C22×C71#53312yes—
288C22×C71#53812yes—
289C22×C71#53912yes—
290C22×C71D4211yescharacteristic
291C22×C71#54612yes—
292C22×C71#54612yes—
293C22×C71#54711yescharacteristic
294C22×C71D4×Q811yescharacteristic
295C22×C71#55112yes—
296C22×C71#55212yes—
297C22×C71#55412yes—
298C22×C71#55411yescharacteristic
299C22×C71#55512yes—
300C2×D71—24no—
301C2×D71—24no—
302C2×D71—24no—
303C2×D71—42no—
304C2×D71—44no—
305C2×D71—42no—
306C2×D71—42no—
307C2×D71—42no—
308C2×D71—44no—
309C2×D71—44no—
310C2×D71—44no—
311C4×C71—24no—
312C4×C71—24no—
313C4×C71—24no—
314C4×C71—44no—
315C4×C71—44no—
316Dic71—24no—
317Dic71—24no—
318Dic71—42no—
319Dic71—82no—
320Dic71—82no—
321#13915—711nomaximal, Sylow
322C2×C71#228511yescharacteristic
323C2×C71#229311yescharacteristic
324C2×C71#231811yescharacteristic
325C2×C71#234211yescharacteristic
326C2×C71#234611yescharacteristic
327C2×C71D4×C4⋊C411yescharacteristic
328C2×C71#234911yescharacteristic
329C2×C71#255312yes—
330C2×C71#255512yes—
331C2×C71#255612yes—
332C2×C71#257412yes—
333D71—44no—
334C2×#388—712no—
335C2×#382—711no—
336C2×#380—711no—
337C2×C4×C22⋊C4—711no—
338#1444—712no—
339#1439—712no—
340#1436—712no—
341C4⋊#95—711no—
342#1434—711no—
343C22⋊#95—711no—
344C4×#95D7111yescharacteristic, complemented
345C71#1391511yescharacteristic, complemented, Sylow
346C24×C4—711no—
347C22×C4⋊C4—712no—
348C22×C4⋊C4—712no—
349C22×C4⋊C4—712no—
350C22×C22⋊C4—712no—
351C22×C22⋊C4—712no—
352C22×C22⋊C4—711no—
353C22×C22⋊C4—711no—
354C22×C42—711no—
355C22×C42C2×D7112yes—
356C22×C42C2×D7111yescharacteristic
357#388—1428no—
358#382—1424no—
359#380—1424no—
360C4×C22⋊C4—718no—
361C2×#95—711no—
362C2×#95—711no—
363C2×#95—711no—
364C2×#95—712no—
365C2×#95—711no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95—712no—
369C2×#95—712no—
370C2×#95C2×D7111yescharacteristic
371C2×#95C2×D7112yes—
372C2×#95C2×D7111yescharacteristic
373C25—711no—
374C23×C4—2842no—
375C23×C4—714no—
376C23×C4—711no—
377C23×C4—711no—
378C23×C4—711no—
379C23×C4—712no—
380C23×C4—712no—
381C23×C4—1424no—
382C23×C4—1422no—
383C23×C4—1422no—
384C23×C4C22×D7112yes—
385C23×C4C22×D7112yes—
386C23×C4C22×D7111yescharacteristic
387C23×C4C22×D7111yescharacteristic
388C23×C4C22×D7111yescharacteristic
389C2×C4⋊C4—2842no—
390C2×C4⋊C4—2842no—
391C2×C4⋊C4—2842no—
392C2×C4⋊C4—2842no—
393C2×C4⋊C4—1424no—
394C2×C4⋊C4—1424no—
395C2×C4⋊C4—1424no—
396C2×C4⋊C4—1424no—
397C2×C4⋊C4—1424no—
398C2×C22⋊C4—2842no—
399C2×C22⋊C4—2842no—
400C2×C22⋊C4—714no—
401C2×C22⋊C4—714no—
402C2×C22⋊C4—714no—
403C2×C22⋊C4—714no—
404C2×C22⋊C4—1424no—
405C2×C22⋊C4—1424no—
406C2×C22⋊C4—1424no—
407C2×C22⋊C4—1424no—
408C2×C22⋊C4—1424no—
409C2×C42—24no—
410C2×C42—24no—
411C2×C42—42no—
412C2×C42—42no—
413C2×C42—714no—
414C2×C42—1422no—
415C2×C42—1422no—
416C2×C42C4×D7114yes—
417#95—24no—
418#95—22no—
419#95—714no—
420#95—714no—
421#95—1424no—
422#95—1424no—
423#95—1424no—
424#95—1422no—
425#95—1422no—
426#95—1424no—
427#95—1424no—
428#95D28414yes—
429C24—712no—
430C24—712no—
431C24—712no—
432C24—1421no—
433C24—1422no—
434C24—1422no—
435C24—1422no—
436C24—1421no—
437C24—1422no—
438C24—1421no—
439C24—1421no—
440C24C23×D7111yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
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—22no—
459C22×C4—22no—
460C22×C4—2848no—
461C22×C4—2848no—
462C22×C4—2844no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—712no—
466C22×C4—712no—
467C22×C4—714no—
468C22×C4—712no—
469C22×C4—1428no—
470C22×C4—1422no—
471C22×C4—1422no—
472C22×C4—1422no—
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×C4—1421no—
485C22×C4—1422no—
486C22×C4—1421no—
487C22×C4—1422no—
488C22×C4—1428no—
489C22×C4—1421no—
490C22×C4—1421no—
491C22×C4—1421no—
492C22×C4—1421no—
493C22×C4—1422no—
494C22×C4—1421no—
495C22×C4—1421no—
496C22×C4C2×C4×D7112yes—
497C22×C4C2×C4×D7112yes—
498C22×C4C2×C4×D7112yes—
499C22×C4D4×D7112yes—
500C22×C4C2×D28414yes—
501C22×C4C2×D28412yes—
502C22×C4Q8×D7112yes—
503C22×C4#1176586514yes—
504C22×C4#1176586514yes—
505C22×C4#1176586512yes—
506C4⋊C4—2848no—
507C4⋊C4—2848no—
508C4⋊C4—2848no—
509C22⋊C4—2848no—
510C22⋊C4—2848no—
511C22⋊C4—7116no—
512C22⋊C4—1428no—
513C42—44no—
514C42—48no—
515C42—1428no—
516C23—2842no—
517C23—2842no—
518C23—2842no—
519C23—2842no—
520C23—2842no—
521C23—2842no—
522C23—2842no—
523C23—2842no—
524C23—714no—
525C23—1422no—
526C23—1424no—
527C23—1422no—
528C23—1422no—
529C23—1424no—
530C23—1422no—
531C23—1422no—
532C23—1422no—
533C23—1422no—
534C23—1422no—
535C23—1422no—
536C23—1422no—
537C23—1424no—
538C23—1424no—
539C23C22×C4×D7111yescharacteristic
540C23C2×D4×D7111yescharacteristic
541C23C22×D28411yescharacteristic
542C23C2×Q8×D7111yescharacteristic
543C23#41×D7112yes—
544C23C2×#1176586512yes—
545C23C2×#1176586512yes—
546C23C2×#1176586511yescharacteristic
547C23#1455617312yes—
548C23#1455617411yescharacteristic
549C23#1455617511yescharacteristic
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—28416no—
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—2842no—
591C2×C4—2842no—
592C2×C4—2842no—
593C2×C4—714no—
594C2×C4—714no—
595C2×C4—1422no—
596C2×C4—1422no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4—1424no—
600C2×C4—1422no—
601C2×C4—1422no—
602C2×C4—1424no—
603C2×C4—1422no—
604C2×C4—1422no—
605C2×C4—1422no—
606C2×C4—1422no—
607C2×C4—1422no—
608C2×C4C4⋊C4×D7114yes—
609C2×C4#1455608514yes—
610C2×C4C4×D28414yes—
611C22—2844no—
612C22—2842no—
613C22—2844no—
614C22—2844no—
615C22—2844no—
616C22—2842no—
617C22—2842no—
618C22—2842no—
619C22—1424no—
620C22—1424no—
621C22—1424no—
622C22C2×C4⋊C4×D7111yescharacteristic
623C22#117×D7111yescharacteristic
624C22C2×#1455608511yescharacteristic
625C22C2×C4×D28411yescharacteristic
626C22C4×#1176586511yescharacteristic
627C22#1798647211yescharacteristic
628C22#1798647911yescharacteristic
629C22#1798653312yes—
630C22#1798654612yes—
631C22#1798654912yes—
632C22#1798655312yes—
633C22#1798658212yes—
634C22#1798659112yes—
635C22D4×D28411yescharacteristic
636C22#1798663912yes—
637C22#1798664412yes—
638C22#1798665411yescharacteristic
639C22Q8×D28411yescharacteristic
640C22#1798667812yes—
641C22#1798668212yes—
642C22#1798670412yes—
643C22#1798670911yescharacteristic
644C22#1798671212yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—2842no—
651C4—5682no—
652C4—5682no—
653C4—1424no—
654C4—1424no—
655C2—2844no—
656C2#2259074411yescharacteristic
657C2#2259080411yescharacteristic
658C2#2259096711yescharacteristic
659C2#2259119211yescharacteristic
660C2#2259122611yescharacteristic
661C2C4⋊C4×D28411yescharacteristic
662C2#2259124711yescharacteristic
663C2#2259338512yes—
664C2#2259340912yes—
665C2#2259341812yes—
666C2#2259360812yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC4×#95×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4×#95D71
hypocenterC71#13915
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×#17985863C2semidirect
5. #22586604C2semidirect
7. #22586573C2semidirect
8. #22586570C2semidirect
9. #22586564C2semidirect
11. C4×#95×C71C2semidirect
24. C4×(C2×D71)⋊C4C4semidirect
148. (C2×D71)⋊C4C4⋊C4semidirect
344. C4×#95D71semidirect
345. C71#13915semidirect

Maximal subgroups

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

subgroupindex
C2×#179858902
C2×#179858632
C2×#179858582
C2×C4×(C2×D71)⋊C42
#225866042
#225865862
#225865732
#225865702
#225865642
(C2×D71)⋊#952
C4×#95×C712
#1391571