Finite Group Databasebeta

← #31920073

Subgroups — #31920073

Order 18176.

1630 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985889C212yesmaximal
2C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
3#22586599C212yesmaximal, complemented
4#22586586C212yesmaximal
5#22586574C212yesmaximal, complemented
6#22586570C211yesmaximal, characteristic, complemented
7#22586565C211yesmaximal, characteristic, complemented
8(C2×D71)⋊#95C211yesmaximal, characteristic
9(C2×D71)⋊#95C211yesmaximal, characteristic
10#95×Dic71C211yesmaximal, characteristic, complemented
11C71×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×C4⋊Dic71C2212yes—
14C22×(C2×D71)⋊C4C2212yes—
15C22×(C2×D71)⋊C4C2212yes—
16C22×(C2×D71)⋊C4C2211yescharacteristic
17C22×(C2×D71)⋊C4C2211yescharacteristic
18C22×C4×Dic71C2212yes—
19C22×C4×Dic71C2211yescharacteristic
20#17985889—28no—
21C4×(C2×D71)⋊C4C418yescomplemented
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2211yescharacteristic
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2211yescharacteristic
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2211yescharacteristic
31C2×C71⋊#95C2211yescharacteristic
32C22×C4⋊C4×C71C2212yes—
33C22×C42×C71C2211yescharacteristic
34C2×#95×C71C2211yescharacteristic
35C2×#95×C71C2211yescharacteristic
36C2×#95×C71C2212yes—
37C24×D71C2311yescharacteristic
38C23×Dic71C2312yes—
39C23×Dic71C2311yescharacteristic
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—42no—
46C22×C4×D71—42no—
47C22×C4×D71C2×C414yes—
48C2×C4⋊Dic71—24no—
49C2×C4⋊Dic71—24no—
50C2×C4⋊Dic71—42no—
51C2×(C2×D71)⋊C4—24no—
52C2×(C2×D71)⋊C4—24no—
53C2×(C2×D71)⋊C4—24no—
54C2×(C2×D71)⋊C4—22no—
55C2×(C2×D71)⋊C4—22no—
56C2×(C2×D71)⋊C4—24no—
57C2×(C2×D71)⋊C4—42no—
58C2×(C2×D71)⋊C4—42no—
59C2×(C2×D71)⋊C4C2×C414yes—
60C2×(C2×D71)⋊C4C2×C414yes—
61C2×(C2×D71)⋊C4D414yes—
62C2×(C2×D71)⋊C4Q814yes—
63C2×C4×Dic71—24no—
64C2×C4×Dic71—24no—
65C2×C4×Dic71—42no—
66C2×C4×Dic71—42no—
67C2×C4×Dic71C2×C414yes—
68C71⋊#95—24no—
69C71⋊#95—24no—
70C71⋊#95—24no—
71C71⋊#95—22no—
72C71⋊#95—22no—
73C71⋊#95—42no—
74C71⋊#95—42no—
75C71⋊#95—42no—
76C71⋊#95C2×C414yes—
77C71⋊#95C2×C414yes—
78C23×C4×C71C2312yes—
79C23×C4×C71C2312yes—
80C23×C4×C71C2311yescharacteristic
81C23×C4×C71C2311yescharacteristic
82C23×C4×C71C2311yescharacteristic
83C2×C4⋊C4×C71—24no—
84C2×C4⋊C4×C71—24no—
85C2×C4⋊C4×C71—42no—
86C2×C42×C71—42no—
87C2×C42×C71C2×C414yes—
88#95×C71—22no—
89#95×C71—24no—
90#95×C71—22no—
91C23×D71—21no—
92C23×D71—21no—
93C23×D71—22no—
94C23×D71—21no—
95C23×D71—22no—
96C23×D71—21no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71C22×C412yes—
100C23×D71C2×D412yes—
101C23×D71C2×Q812yes—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—21no—
106C22×Dic71—21no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—21no—
111C22×Dic71—21no—
112C22×Dic71—22no—
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—22no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71C22×C412yes—
126C22×Dic71C22×C412yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C2×D412yes—
129C22×Dic71C2×Q812yes—
130C22×Dic71#4114yes—
131C2×C4×D71—28no—
132C2×C4×D71—44no—
133C2×C4×D71—44no—
134C2×C4×D71—48no—
135C2×C4×D71—48no—
136C4⋊Dic71—48no—
137(C2×D71)⋊C4—28no—
138(C2×D71)⋊C4—48no—
139(C2×D71)⋊C4—48no—
140(C2×D71)⋊C4C4⋊C4116yescomplemented
141C4×Dic71—44no—
142C4×Dic71—48no—
143C24×C71C2411yescharacteristic
144C22×C4×C71—22no—
145C22×C4×C71—22no—
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—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71C22×C412yes—
164C22×C4×C71C22×C412yes—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C2×D412yes—
167C22×C4×C71C2×Q812yes—
168C22×C4×C71#4112yes—
169C22×C4×C71#4112yes—
170C22×C4×C71#4114yes—
171C22×C4×C71#4112yes—
172C22×C4×C71#4112yes—
173C4⋊C4×C71—48no—
174C42×C71—44no—
175C23×C71C23×C411yescharacteristic
176C23×C71C22×D411yescharacteristic
177C23×C71C22×Q811yescharacteristic
178C23×C71C2×#4112yes—
179C23×C71C2×#4111yescharacteristic
180C23×C71C2×#4111yescharacteristic
181C23×C71C2×#4112yes—
182C23×C71C2×#4111yescharacteristic
183C23×C71C2×#4111yescharacteristic
184C23×C71H(5,𝔽2)12yes—
185C23×C71H(5,𝔽2)12yes—
186C22×D71—22no—
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—22no—
190C22×D71—22no—
191C22×D71—22no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—22no—
196C22×D71—24no—
197C22×D71—24no—
198C22×D71—24no—
199C22×D71—24no—
200C22×D71—42no—
201C22×D71—42no—
202C22×D71—42no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—42no—
208C22×D71C2×C4⋊C414yes—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
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×C71C2×C4⋊C414yes—
236C2×C4×C71#11714yes—
237C2×C4×C71#11714yes—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
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×Dic71C2×C4⋊C414yes—
267C4×D71—416no—
268C22×C71C22×C4⋊C411yescharacteristic
269C22×C71C2×#11711yescharacteristic
270C22×C71C2×#11711yescharacteristic
271C22×C71C4×#4111yescharacteristic
272C22×C71C4×#4111yescharacteristic
273C22×C71#51811yescharacteristic
274C22×C71#51811yescharacteristic
275C22×C71#52912yes—
276C22×C71#52912yes—
277C22×C71#53211yescharacteristic
278C22×C71#53211yescharacteristic
279C22×C71#53311yescharacteristic
280C22×C71#53311yescharacteristic
281C22×C71#53812yes—
282C22×C71#53812yes—
283C22×C71#54612yes—
284C22×C71#54612yes—
285C22×C71#55112yes—
286C22×C71#55312yes—
287C22×C71#55312yes—
288C22×C71#55412yes—
289C22×C71#55412yes—
290C22×C71#55912yes—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71—24no—
294C2×D71—42no—
295C2×D71—44no—
296C2×D71—44no—
297C2×D71—42no—
298C2×D71—42no—
299C2×D71—42no—
300C2×D71—44no—
301C2×D71—44no—
302C4×C71—24no—
303C4×C71—24no—
304C4×C71—24no—
305C4×C71—44no—
306C4×C71—82no—
307Dic71—24no—
308Dic71—42no—
309Dic71—42no—
310Dic71—44no—
311Dic71—82no—
312#13948—711nomaximal, Sylow
313C2×C71#228511yescharacteristic
314C2×C71#229811yescharacteristic
315C2×C71#231211yescharacteristic
316C2×C71#234911yescharacteristic
317C2×C71#234911yescharacteristic
318C2×C71#235311yescharacteristic
319C2×C71#235711yescharacteristic
320C2×C71#255712yes—
321C2×C71#255912yes—
322C2×C71#258712yes—
323C2×C71#258912yes—
324D71—44no—
325C2×#388—712no—
326C2×C4×C22⋊C4—711no—
327#1442—712no—
328#1439—712no—
329#1436—712no—
330C4⋊#95—711no—
331C4⋊#95D7111yescharacteristic, complemented
332#1434—711no—
333C22⋊#95—711no—
334C22⋊#95—711no—
335C4×#95—711no—
336C71#1394811yescharacteristic, complemented, Sylow
337C24×C4—711no—
338C22×C4⋊C4—712no—
339C22×C4⋊C4C2×D7112yes—
340C22×C22⋊C4—712no—
341C22×C22⋊C4—712no—
342C22×C22⋊C4—711no—
343C22×C22⋊C4—711no—
344C22×C42—711no—
345C22×C42—712no—
346C22×C42C2×D7111yescharacteristic
347#388—1428no—
348C4×C22⋊C4—718no—
349C2×#95—712no—
350C2×#95—712no—
351C2×#95—712no—
352C2×#95—711no—
353C2×#95—711no—
354C2×#95—711no—
355C2×#95—712no—
356C2×#95—711no—
357C2×#95—712no—
358C2×#95—712no—
359C2×#95C2×D7111yescharacteristic
360C2×#95C2×D7111yescharacteristic
361C2×#95C2×D7112yes—
362C25—711no—
363C23×C4—2842no—
364C23×C4—2842no—
365C23×C4—714no—
366C23×C4—711no—
367C23×C4—712no—
368C23×C4—711no—
369C23×C4—711no—
370C23×C4—712no—
371C23×C4—1422no—
372C23×C4—1422no—
373C23×C4C22×D7112yes—
374C23×C4C22×D7112yes—
375C23×C4C22×D7111yescharacteristic
376C23×C4C22×D7111yescharacteristic
377C23×C4C22×D7111yescharacteristic
378C2×C4⋊C4—24no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—2842no—
382C2×C4⋊C4—1424no—
383C2×C4⋊C4—1424no—
384C2×C22⋊C4—2842no—
385C2×C22⋊C4—2842no—
386C2×C22⋊C4—714no—
387C2×C22⋊C4—714no—
388C2×C22⋊C4—714no—
389C2×C22⋊C4—714no—
390C2×C22⋊C4—1424no—
391C2×C22⋊C4—1424no—
392C2×C22⋊C4—1424no—
393C2×C22⋊C4—1424no—
394C2×C22⋊C4—1422no—
395C2×C22⋊C4—1422no—
396C2×C42—42no—
397C2×C42—2842no—
398C2×C42—2842no—
399C2×C42—714no—
400C2×C42—1424no—
401C2×C42—1424no—
402C2×C42C4×D7114yes—
403#95—22no—
404#95—24no—
405#95—22no—
406#95—2842no—
407#95—2842no—
408#95—2842no—
409#95—714no—
410#95—714no—
411#95—1424no—
412#95—1424no—
413#95—1424no—
414#95—1422no—
415#95—1422no—
416C24—712no—
417C24—712no—
418C24—712no—
419C24—1422no—
420C24—1421no—
421C24—1421no—
422C24—1422no—
423C24—1421no—
424C24—1421no—
425C24—1422no—
426C24—1422no—
427C24C23×D7111yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
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—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—2848no—
448C22×C4—2848no—
449C22×C4—2844no—
450C22×C4—2844no—
451C22×C4—712no—
452C22×C4—712no—
453C22×C4—712no—
454C22×C4—712no—
455C22×C4—712no—
456C22×C4—714no—
457C22×C4—1422no—
458C22×C4—1422no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1421no—
466C22×C4—1421no—
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—1428no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4C2×C4×D7112yes—
482C22×C4C2×C4×D7112yes—
483C22×C4C2×C4×D7112yes—
484C22×C4D4×D7112yes—
485C22×C4Q8×D7112yes—
486C22×C4#1176586412yes—
487C22×C4#1176586412yes—
488C22×C4#1176586514yes—
489C22×C4#1176586612yes—
490C22×C4#1176586612yes—
491C4⋊C4—48no—
492C4⋊C4—2848no—
493C22⋊C4—2848no—
494C22⋊C4—2848no—
495C22⋊C4—7116no—
496C22⋊C4—1428no—
497C42—44no—
498C42—2844no—
499C42—2848no—
500C23—2842no—
501C23—2842no—
502C23—2842no—
503C23—2842no—
504C23—2842no—
505C23—2842no—
506C23—2842no—
507C23—2842no—
508C23—714no—
509C23—1424no—
510C23—1424no—
511C23—1424no—
512C23—1424no—
513C23—1422no—
514C23—1422no—
515C23—1422no—
516C23—1422no—
517C23—1422no—
518C23—1422no—
519C23—1422no—
520C23—1422no—
521C23—1422no—
522C23—1422no—
523C23C22×C4×D7111yescharacteristic
524C23C2×D4×D7111yescharacteristic
525C23C2×Q8×D7111yescharacteristic
526C23#41×D7112yes—
527C23C2×#1176586411yescharacteristic
528C23C2×#1176586411yescharacteristic
529C23C2×#1176586512yes—
530C23C2×#1176586611yescharacteristic
531C23C2×#1176586611yescharacteristic
532C23#1455617312yes—
533C23#1455617412yes—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—28416no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2842no—
567C2×C4—2842no—
568C2×C4—2842no—
569C2×C4—2842no—
570C2×C4—2842no—
571C2×C4—2842no—
572C2×C4—2842no—
573C2×C4—2842no—
574C2×C4—2842no—
575C2×C4—2842no—
576C2×C4—714no—
577C2×C4—1424no—
578C2×C4—1424no—
579C2×C4—1422no—
580C2×C4—1422no—
581C2×C4—1422no—
582C2×C4—1422no—
583C2×C4—1422no—
584C2×C4—1422no—
585C2×C4—1424no—
586C2×C4—1424no—
587C2×C4—1424no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4C4⋊C4×D7114yes—
591C2×C4#1455608614yes—
592C2×C4#1455608614yes—
593C22—2844no—
594C22—2842no—
595C22—2842no—
596C22—2842no—
597C22—2844no—
598C22—2844no—
599C22—2842no—
600C22—2844no—
601C22—1424no—
602C22—1424no—
603C22—1424no—
604C22C2×C4⋊C4×D7111yescharacteristic
605C22C2×#1455608611yescharacteristic
606C22C2×#1455608611yescharacteristic
607C22#1798646711yescharacteristic
608C22C4×#1176586511yescharacteristic
609C22#1798647211yescharacteristic
610C22#1798647311yescharacteristic
611C22#1798652912yes—
612C22#1798653612yes—
613C22#1798654711yescharacteristic
614C22#1798655011yescharacteristic
615C22#1798655411yescharacteristic
616C22#1798655411yescharacteristic
617C22#1798658612yes—
618C22#1798658912yes—
619C22#1798663912yes—
620C22#1798664212yes—
621C22#1798667612yes—
622C22#1798669212yes—
623C22#1798669512yes—
624C22#1798670412yes—
625C22#1798670512yes—
626C22#1798673412yes—
627C4—24no—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—82no—
632C4—2842no—
633C4—2844no—
634C4—2842no—
635C4—5682no—
636C4—1424no—
637C2—2844no—
638C2#2259074511yescharacteristic
639C2#2259083311yescharacteristic
640C2#2259093711yescharacteristic
641C2#2259124711yescharacteristic
642C2#2259124811yescharacteristic
643C2#2259128211yescharacteristic
644C2#2259131111yescharacteristic
645C2#2259344612yes—
646C2#2259347412yes—
647C2#2259376812yes—
648C2#2259379612yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C4⋊#95C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4⋊#95D71
hypocenterC71#13948
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. #22586599C2semidirect
5. #22586574C2semidirect
6. #22586570C2semidirect
7. #22586565C2semidirect
10. #95×Dic71C2semidirect
11. C71×C4⋊#95C2semidirect
21. C4×(C2×D71)⋊C4C4semidirect
140. (C2×D71)⋊C4C4⋊C4semidirect
331. C4⋊#95D71semidirect
336. C71#13948semidirect

Maximal subgroups

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

subgroupindex
C2×#179858892
C2×C4×(C2×D71)⋊C42
#225865992
#225865862
#225865742
#225865702
#225865652
(C2×D71)⋊#952
#95×Dic712
C71×C4⋊#952
#1394871