Finite Group Databasebeta

← #31923886

Subgroups — #31923886

Order 18176.

1782 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985871C211yesmaximal, characteristic
2C22×C71⋊#95C211yesmaximal, characteristic
3#22586606C212yesmaximal, complemented
4#22586600C212yesmaximal, complemented
5#22586599C212yesmaximal, complemented
6#22586590C212yesmaximal, complemented
7#22586563C211yesmaximal, characteristic
8#22586562C211yesmaximal, characteristic
9#22586562C211yesmaximal, characteristic
10#22586562C211yesmaximal, characteristic
11C2×#385×C71C211yesmaximal, characteristic
12C24×Dic71C2211yescharacteristic
13C24×Dic71C2211yescharacteristic
14C22×Dic71⋊C4C2212yes—
15C22×C22⋊Dic71C2211yescharacteristic
16C22×C22⋊Dic71C2211yescharacteristic
17#17985871C418yescomplemented
18C2×C71⋊#95—24no—
19C2×C71⋊#95C2212yes—
20C2×C71⋊#95C2212yes—
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C24×C4×C71C2211yescharacteristic
31C22×C4⋊C4×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C22⋊C4×C71C2211yescharacteristic
34C22×C22⋊C4×C71C2211yescharacteristic
35#385×C71—24no—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C23×Dic71—22no—
39C23×Dic71—22no—
40C23×Dic71—22no—
41C23×Dic71—22no—
42C23×Dic71—42no—
43C23×Dic71—42no—
44C23×Dic71—42no—
45C23×Dic71—42no—
46C23×Dic71C2×C414yes—
47C23×Dic71Q814yes—
48C23×Dic71C2312yes—
49C23×Dic71C2312yes—
50C23×Dic71C2312yes—
51C23×Dic71C2312yes—
52C2×Dic71⋊C4—42no—
53C2×Dic71⋊C4—42no—
54C2×Dic71⋊C4C2×C418yescomplemented
55C2×C22⋊Dic71—22no—
56C2×C22⋊Dic71—22no—
57C2×C22⋊Dic71—22no—
58C2×C22⋊Dic71—22no—
59C2×C22⋊Dic71—22no—
60C2×C22⋊Dic71C2×C414yes—
61C71⋊#95—24no—
62C71⋊#95—24no—
63C71⋊#95—42no—
64C71⋊#95—42no—
65C71⋊#95—42no—
66C71⋊#95—42no—
67C71⋊#95—42no—
68C71⋊#95—42no—
69C71⋊#95—42no—
70C71⋊#95—42no—
71C71⋊#95C2×C418yescomplemented
72C25×C71C2311yescharacteristic
73C23×C4×C71—22no—
74C23×C4×C71—22no—
75C23×C4×C71—42no—
76C23×C4×C71—42no—
77C23×C4×C71D414yes—
78C23×C4×C71C2311yescharacteristic
79C23×C4×C71C2311yescharacteristic
80C23×C4×C71C2311yescharacteristic
81C23×C4×C71C2311yescharacteristic
82C23×C4×C71C2311yescharacteristic
83C23×C4×C71C2311yescharacteristic
84C2×C4⋊C4×C71—22no—
85C2×C4⋊C4×C71—22no—
86C2×C4⋊C4×C71—42no—
87C2×C4⋊C4×C71—42no—
88C2×C22⋊C4×C71—22no—
89C2×C22⋊C4×C71—22no—
90C2×C22⋊C4×C71—22no—
91C2×C22⋊C4×C71—22no—
92C2×C22⋊C4×C71—22no—
93C2×C22⋊C4×C71C2×C414yes—
94#95×C71—22no—
95#95×C71—22no—
96C22×Dic71—28no—
97C22×Dic71—28no—
98C22×Dic71—22no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—22no—
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—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—48no—
124C22×Dic71—48no—
125C22×Dic71—44no—
126C22×Dic71—44no—
127C22×Dic71—44no—
128C22×Dic71—44no—
129C22×Dic71—48no—
130C22×Dic71—48no—
131C22×Dic71C22×C414yes—
132C22×Dic71C22×C414yes—
133C22×Dic71C2×Q814yes—
134Dic71⋊C4—48no—
135C22⋊Dic71—28no—
136C22⋊Dic71—44no—
137C24×C71—22no—
138C24×C71—22no—
139C24×C71—22no—
140C24×C71—22no—
141C24×C71C22×C412yes—
142C24×C71C2×D412yes—
143C24×C71C2×Q812yes—
144C24×C71#4112yes—
145C24×C71#4112yes—
146C24×C71#4112yes—
147C24×C71#4112yes—
148C24×C71C2411yescharacteristic
149C22×C4×C71—28no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
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—21no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—44no—
177C22×C4×C71—44no—
178C22×C4×C71—48no—
179C22×C4×C71—48no—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C22×C412yes—
182C22×C4×C71C2×D412yes—
183C22×C4×C71C2×D412yes—
184C4⋊C4×C71—44no—
185C4⋊C4×C71—44no—
186C22⋊C4×C71—28no—
187C22⋊C4×C71—44no—
188C23×C71—24no—
189C23×C71—24no—
190C23×C71—22no—
191C23×C71—22no—
192C23×C71—22no—
193C23×C71—22no—
194C23×C71—24no—
195C23×C71—24no—
196C23×C71—24no—
197C23×C71—24no—
198C23×C71—24no—
199C23×C71—24no—
200C23×C71—24no—
201C23×C71—24no—
202C23×C71—24no—
203C23×C71—24no—
204C23×C71#11714yes—
205C23×C71C4×D414yes—
206C23×C71C4×Q814yes—
207C23×C71C22⋊Q814yes—
208C23×C71#12314yes—
209C23×C71#12514yes—
210C23×C71#12614yes—
211C23×C71C23×C411yescharacteristic
212C23×C71C22×D411yescharacteristic
213C23×C71C22×Q811yescharacteristic
214C23×C71C2×#4111yescharacteristic
215C23×C71C2×#4111yescharacteristic
216C23×C71C2×#4111yescharacteristic
217C23×C71C2×#4111yescharacteristic
218C23×C71H(5,𝔽2)12yes—
219C23×C71H(5,𝔽2)12yes—
220C23×C71H(5,𝔽2)12yes—
221C23×C712−1+412yes—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—416no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
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—41no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—41no—
247C2×C4×C71—41no—
248C2×C4×C71—41no—
249C2×C4×C71—42no—
250C2×C4×C71—42no—
251C2×C4×C71—42no—
252C2×C4×C71—42no—
253C2×Dic71—24no—
254C2×Dic71—24no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—416no—
263C2×Dic71—416no—
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×Dic71—42no—
275C2×Dic71—42no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71—42no—
279C2×Dic71—42no—
280C2×Dic71—42no—
281C2×Dic71—42no—
282C2×Dic71—42no—
283C22×C71—28no—
284C22×C71—24no—
285C22×C71—24no—
286C22×C71—28no—
287C22×C71—24no—
288C22×C71—24no—
289C22×C71—24no—
290C22×C71—24no—
291C22×C71—28no—
292C22×C71—28no—
293C22×C71#38718yes—
294C22×C71C2×#11711yescharacteristic
295C22×C71C2×C4×D411yescharacteristic
296C22×C71C2×C4×Q811yescharacteristic
297C22×C71#51812yes—
298C22×C71#52012yes—
299C22×C71C2×C22⋊Q811yescharacteristic
300C22×C71C2×#12311yescharacteristic
301C22×C71C2×#12511yescharacteristic
302C22×C71C2×#12611yescharacteristic
303C22×C71#53411yescharacteristic
304C22×C71#53411yescharacteristic
305C22×C71#53511yescharacteristic
306C22×C71#53611yescharacteristic
307C22×C71#53812yes—
308C22×C71#53812yes—
309C22×C71#53812yes—
310C22×C71#53812yes—
311C22×C71#53912yes—
312C22×C71#53912yes—
313C22×C71#53912yes—
314C22×C71#53912yes—
315C22×C71#54312yes—
316C22×C71#54412yes—
317C4×C71—42no—
318C4×C71—42no—
319C4×C71—42no—
320C4×C71—42no—
321C4×C71—81no—
322C4×C71—81no—
323Dic71—44no—
324Dic71—44no—
325Dic71—44no—
326Dic71—82no—
327#14265—711nomaximal, Sylow
328C2×C71—28no—
329C2×C71C2×#38711yescharacteristic
330C2×C71#231812yes—
331C2×C71#232111yescharacteristic
332C2×C71#232211yescharacteristic
333C2×C71#232912yes—
334C2×C71#262211yescharacteristic
335C2×C71#262311yescharacteristic
336C2×C71#262611yescharacteristic
337C2×C71#262711yescharacteristic
338C2×C71#264312yes—
339C2×C71#264612yes—
340C2×#385—711no—
341C2×#385D7111yescharacteristic
342C22×#95—711no—
343#1445—712no—
344#1443—712no—
345#1442—712no—
346#1441—712no—
347C22⋊#95—711no—
348C22⋊#95—711no—
349C22⋊#95—711no—
350C22⋊#95—711no—
351C71#1426511yescharacteristic, complemented, Sylow
352C24×C4—711no—
353C24×C4—711no—
354C24×C4C2×D7111yescharacteristic
355C22×C4⋊C4—712no—
356C22×C4⋊C4C2×D7111yescharacteristic
357C22×C4⋊C4C2×D7111yescharacteristic
358C22×C22⋊C4—711no—
359C22×C22⋊C4—711no—
360C22×C22⋊C4C2×D7111yescharacteristic
361C22×C22⋊C4C2×D7111yescharacteristic
362#385—24no—
363#385—718no—
364C2×#95—712no—
365C2×#95—712no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95—712no—
369C2×#95—712no—
370C2×#95—712no—
371C2×#95—712no—
372C2×#95—712no—
373C2×#95—712no—
374C2×#95—712no—
375C2×#95—1424no—
376C2×#95C2×D7111yescharacteristic
377C2×#95C2×D7111yescharacteristic
378C25C22×D7111yescharacteristic
379C23×C4—22no—
380C23×C4—22no—
381C23×C4—42no—
382C23×C4—42no—
383C23×C4—2842no—
384C23×C4—2842no—
385C23×C4—2842no—
386C23×C4—2842no—
387C23×C4—714no—
388C23×C4—714no—
389C23×C4—712no—
390C23×C4—712no—
391C23×C4—712no—
392C23×C4—712no—
393C23×C4—1422no—
394C23×C4—1422no—
395C23×C4—1422no—
396C23×C4—1422no—
397C23×C4C22⋊D7114yes—
398C23×C4C22×D7111yescharacteristic
399C23×C4C22×D7111yescharacteristic
400C23×C4C22×D7111yescharacteristic
401C23×C4C22×D7111yescharacteristic
402C23×C4C22×D7111yescharacteristic
403C23×C4C22×D7111yescharacteristic
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—42no—
408C2×C4⋊C4—2842no—
409C2×C4⋊C4—2842no—
410C2×C4⋊C4—718no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—714no—
417C2×C22⋊C4—1422no—
418C2×C22⋊C4—1422no—
419C2×C22⋊C4—1422no—
420C2×C22⋊C4—1422no—
421C2×C22⋊C4—1422no—
422C2×C22⋊C4C4×D7114yes—
423#95—22no—
424#95—22no—
425#95—2842no—
426#95—2842no—
427#95—2842no—
428#95—2842no—
429#95—2842no—
430#95—2842no—
431#95—2842no—
432#95—2842no—
433#95—718no—
434#95—1424no—
435#95—1424no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24C2×C4×D7112yes—
441C24C2×C22⋊D7112yes—
442C24Q8×D7112yes—
443C24#1176586412yes—
444C24#1176586412yes—
445C24#1176586512yes—
446C24#1176586612yes—
447C24C23×D7111yescharacteristic
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—2844no—
480C22×C4—2844no—
481C22×C4—2848no—
482C22×C4—2848no—
483C22×C4—2844no—
484C22×C4—2844no—
485C22×C4—2848no—
486C22×C4—2848no—
487C22×C4—714no—
488C22×C4—714no—
489C22×C4—714no—
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—1422no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1422no—
506C22×C4—1422no—
507C22×C4—1422no—
508C22×C4—1422no—
509C22×C4—1422no—
510C22×C4—1422no—
511C22×C4—1422no—
512C22×C4—1422no—
513C22×C4—1422no—
514C22×C4—1422no—
515C22×C4—1428no—
516C22×C4—1428no—
517C22×C4C2×C4×D7112yes—
518C22×C4C2×C4×D7112yes—
519C22×C4C2×C22⋊D7112yes—
520C22×C4C2×C22⋊D7112yes—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—2848no—
524C22⋊C4—28no—
525C22⋊C4—44no—
526C22⋊C4—2844no—
527C22⋊C4—1428no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23#1455608614yes—
545C23C4×C22⋊D7114yes—
546C23#1455609314yes—
547C23#1455610414yes—
548C23#1455610914yes—
549C23#1455611814yes—
550C23#1455612314yes—
551C23C22×C4×D7111yescharacteristic
552C23C22×C22⋊D7111yescharacteristic
553C23C2×Q8×D7111yescharacteristic
554C23C2×#1176586411yescharacteristic
555C23C2×#1176586411yescharacteristic
556C23C2×#1176586511yescharacteristic
557C23C2×#1176586611yescharacteristic
558C23#1455617312yes—
559C23#1455617312yes—
560C23#1455617412yes—
561C23#1455617512yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—416no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—28416no—
594C2×C4—28416no—
595C2×C4—2842no—
596C2×C4—2842no—
597C2×C4—2842no—
598C2×C4—2842no—
599C2×C4—2842no—
600C2×C4—2842no—
601C2×C4—2842no—
602C2×C4—2842no—
603C2×C4—2842no—
604C2×C4—2842no—
605C2×C4—2842no—
606C2×C4—2842no—
607C2×C4—2842no—
608C2×C4—2842no—
609C2×C4—2842no—
610C2×C4—2842no—
611C2×C4—2842no—
612C2×C4—2842no—
613C2×C4—2842no—
614C2×C4—1424no—
615C2×C4—1424no—
616C2×C4—1424no—
617C2×C4—1424no—
618C2×C4—1424no—
619C2×C4—1424no—
620C2×C4—1424no—
621C2×C4—1424no—
622C2×C4—1424no—
623C22—28no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—28no—
629C22—24no—
630C22—24no—
631C22—28no—
632C22—28no—
633C22#1798588018yes—
634C22C2×#1455608611yescharacteristic
635C22C2×C4×C22⋊D7111yescharacteristic
636C22C2×#1455609311yescharacteristic
637C22#1798647112yes—
638C22#1798647812yes—
639C22C2×#1455610411yescharacteristic
640C22C2×#1455610911yescharacteristic
641C22C2×#1455611811yescharacteristic
642C22C2×#1455612311yescharacteristic
643C22#1798655811yescharacteristic
644C22#1798655811yescharacteristic
645C22#1798656311yescharacteristic
646C22#1798657011yescharacteristic
647C22#1798658212yes—
648C22#1798658412yes—
649C22#1798658412yes—
650C22#1798658912yes—
651C22#1798659112yes—
652C22#1798659312yes—
653C22#1798659312yes—
654C22#1798659812yes—
655C22#1798662412yes—
656C22#1798662812yes—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—2844no—
664C4—2844no—
665C4—2844no—
666C4—5682no—
667C2—28no—
668C2C2×#1798588011yescharacteristic
669C2#2259096812yes—
670C2#2259099411yescharacteristic
671C2#2259100511yescharacteristic
672C2#2259107112yes—
673C2#2259411511yescharacteristic
674C2#2259412211yescharacteristic
675C2#2259414711yescharacteristic
676C2#2259415511yescharacteristic
677C2#2259429612yes—
678C2#2259431212yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#385×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC2×#385D71
hypocenterC71#14265
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. #22586606C2semidirect
4. #22586600C2semidirect
5. #22586599C2semidirect
6. #22586590C2semidirect
17. #17985871C4semidirect
54. C2×Dic71⋊C4C2×C4semidirect
71. C71⋊#95C2×C4semidirect
351. C71#14265semidirect

Maximal subgroups

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

subgroupindex
C2×#179858712
C22×C71⋊#952
#225866062
#225866002
#225865992
#225865902
#225865632
#225865622
C2×#385×C712
#1426571