Finite Group Databasebeta

← #31998175

Subgroups — #31998175

Order 18176.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D71C211yesmaximal, characteristic
2C2×#17986035C211yesmaximal, characteristic, complemented
3C22×(C2×D71)⋊C8C211yesmaximal, characteristic
4#22590719C211yesmaximal, characteristic
5#22588538C212yesmaximal, complemented
6#22588536C212yesmaximal, complemented
7#22588531C212yesmaximal, complemented
8#22588526C212yesmaximal, complemented
9#22588057C211yesmaximal, characteristic, complemented
10#22588056C211yesmaximal, characteristic, complemented
11C2×#427×C71C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14C2×#14556087C2211yescharacteristic
15C2×#14556087C2211yescharacteristic
16C2×#14556086C2212yes—
17C2×#14556085—21no—
18C2×#14556084—21no—
19#117×D71—24no—
20#117×D71—24no—
21C2×C4⋊C4×D71C2212yes—
22C2×C22⋊C4×D71—21no—
23C2×C42×D71—21no—
24#17986035—24no—
25C2×#14556061C2212yescomplemented
26C2×#14556059C2212yescomplemented
27C2×(C2×D71)⋊C8C418yescomplemented
28C2×(C2×D71)⋊C8C2211yescharacteristic
29C2×(C2×D71)⋊C8C2211yescharacteristic
30C2×(C2×D71)⋊C8C2211yescharacteristic
31C2×(C2×D71)⋊C8C2211yescharacteristic
32#17985860—21no—
33#17985856C2212yes—
34#17985856C2212yes—
35#17985854—21no—
36C4×(C2×D71)⋊C4—21no—
37C4×(C2×D71)⋊C4—21no—
38#17985762C2212yescomplemented
39#17985762C2212yescomplemented
40#17985761C2212yescomplemented
41#17985761C2212yescomplemented
42C23×C8×C71C2211yescharacteristic
43C2×#117×C71C2211yescharacteristic
44C2×#117×C71C2211yescharacteristic
45#427×C71—24no—
46C2×#107×C71C2212yescomplemented
47C2×#106×C71C2212yescomplemented
48C24×D71D411yescharacteristic
49C23×Dic71D411yescharacteristic
50C22×C4×D71—22no—
51C22×C4×D71—22no—
52C22×C4×D71—22no—
53C22×C4×D71—22no—
54C22×C4×D71—22no—
55C22×C4×D71C2×C414yes—
56C22×C4×D71D411yescharacteristic
57C22×C4×D71D411yescharacteristic
58C22×C4×D71Q814yes—
59C22×C4×D71C2311yescharacteristic
60C22×C4×D71C2311yescharacteristic
61C22×C71⋊C8—22no—
62C22×C71⋊C8—22no—
63C22×C71⋊C8C2×C414yescomplemented
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66#14556087—22no—
67#14556087—22no—
68#14556087—22no—
69#14556087D414yes—
70#14556086—24no—
71#14556086—24no—
72#14556085—28no—
73#14556084—28no—
74C2×C4⋊Dic71C2312yes—
75C2×C4⋊Dic71C2312yes—
76C2×Dic71⋊C4—22no—
77C4⋊C4×D71—24no—
78C4⋊C4×D71—24no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71—21no—
81C2×(C2×D71)⋊C4—22no—
82C2×(C2×D71)⋊C4—22no—
83C2×(C2×D71)⋊C4—22no—
84C22⋊C4×D71—28no—
85C2×C4×Dic71—22no—
86C2×C4×Dic71—21no—
87C2×C4×Dic71—21no—
88C42×D71—28no—
89#14556061—24no—
90#14556059—24no—
91(C2×D71)⋊C8C2×C414yes—
92(C2×D71)⋊C8C2×C414yes—
93(C2×D71)⋊C8C2×C414yescomplemented
94(C2×D71)⋊C8C2×C414yescomplemented
95C71⋊#95—22no—
96C71⋊#95—22no—
97C23×C4×C71C2311yescharacteristic
98C22×C8×C71—22no—
99C22×C8×C71—22no—
100C22×C8×C71C2×C414yescomplemented
101C22×C8×C71C2311yescharacteristic
102C22×C8×C71C2311yescharacteristic
103#117×C71—22no—
104#117×C71—22no—
105#117×C71—22no—
106#117×C71D414yes—
107C2×C4⋊C4×C71C2312yes—
108C2×C4⋊C4×C71C2312yes—
109C2×C22⋊C4×C71—21no—
110C2×C22⋊C4×C71—21no—
111C2×C42×C71—21no—
112C2×C42×C71—21no—
113#107×C71—24no—
114#106×C71—24no—
115C23×D71—22no—
116C23×D71—22no—
117C23×D71—24no—
118C23×D71—42no—
119C23×D71C4⋊C414yes—
120C23×D71C2×D411yescharacteristic
121C23×D71C2×D411yescharacteristic
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71C4⋊C414yes—
127C22×Dic71C2×D411yescharacteristic
128C22×Dic71C2×D411yescharacteristic
129C2×C4×D71—28no—
130C2×C4×D71—28no—
131C2×C4×D71—22no—
132C2×C4×D71—22no—
133C2×C4×D71—28no—
134C2×C4×D71—28no—
135C2×C4×D71—22no—
136C2×C4×D71—22no—
137C2×C4×D71—21no—
138C2×C4×D71—21no—
139C2×C4×D71—21no—
140C2×C4×D71—21no—
141C2×C4×D71—21no—
142C2×C4×D71—21no—
143C2×C4×D71—21no—
144C2×C4×D71—21no—
145C2×C4×D71—28no—
146C2×C4×D71—44no—
147C2×C4×D71—44no—
148C2×C4×D71C4⋊C414yes—
149C2×C4×D71C4⋊C414yes—
150C2×C4×D71C22×C412yes—
151C2×C4×D71C22×C412yes—
152C2×C4×D71C2×Q812yes—
153C2×C4×D71C2×Q812yes—
154C2×C71⋊C8—28no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8C22×C412yes—
160C2×C71⋊C8C22×C412yes—
161C4⋊Dic71—22no—
162C4⋊Dic71—22no—
163C4⋊Dic71—22no—
164C4⋊Dic71C2×D414yes—
165Dic71⋊C4—28no—
166C22⋊Dic71—24no—
167C22⋊Dic71—42no—
168(C2×D71)⋊C4—28no—
169(C2×D71)⋊C4—44no—
170(C2×D71)⋊C4—44no—
171C4×Dic71—28no—
172C4×Dic71—24no—
173C4×Dic71—42no—
174C24×C71C2×D411yescharacteristic
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71C22×C412yes—
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×Q812yes—
182C22×C4×C71#4112yes—
183C22×C4×C71#4112yes—
184C22×C4×C71C2411yescharacteristic
185C2×C8×C71—28no—
186C2×C8×C71—21no—
187C2×C8×C71—21no—
188C2×C8×C71—21no—
189C2×C8×C71—21no—
190C2×C8×C71C22×C412yes—
191C2×C8×C71C22×C412yes—
192C4⋊C4×C71—22no—
193C4⋊C4×C71—22no—
194C4⋊C4×C71—22no—
195C4⋊C4×C71C2×D414yes—
196C22⋊C4×C71—24no—
197C22⋊C4×C71—42no—
198C42×C71—24no—
199C42×C71—42no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71C2×C4⋊C412yes—
203C23×C71#12012yes—
204C23×C71#12312yes—
205C23×C71C22×D411yescharacteristic
206C22×D71—21no—
207C22×D71—21no—
208C22×D71—22no—
209C22×D71—21no—
210C22×D71—21no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—28no—
214C22×D71—28no—
215C22×D71—44no—
216C22×D71—48no—
217C22×D71—48no—
218C22×D71C2×C4⋊C412yes—
219C22×D71C2×C4⋊C412yes—
220C22×D71#13514yes—
221C2×C4×C71—24no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71C2×C4⋊C412yes—
228C2×C4×C71C4×D414yes—
229C2×C4×C71C4×D414yes—
230C2×C4×C71#12012yes—
231C2×C4×C71C22⋊Q814yes—
232C2×C4×C71C22⋊Q814yes—
233C2×C4×C71#12312yes—
234C2×C4×C71C23×C411yescharacteristic
235C2×C4×C71C22×D411yescharacteristic
236C2×C4×C71C22×D411yescharacteristic
237C2×C4×C71C22×Q811yescharacteristic
238C2×C4×C71C2×#4111yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×Dic71—21no—
241C2×Dic71—21no—
242C2×Dic71—21no—
243C2×Dic71—21no—
244C2×Dic71—28no—
245C2×Dic71—24no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71C2×C4⋊C412yes—
252C2×Dic71C2×C4⋊C412yes—
253C4×D71—22no—
254C4×D71—22no—
255C4×D71—22no—
256C4×D71—22no—
257C4×D71—22no—
258C4×D71—22no—
259C4×D71—22no—
260C4×D71—22no—
261C4×D71—28no—
262C4×D71—28no—
263C4×D71—416no—
264C8×C71—22no—
265C8×C71—22no—
266C71⋊C8—22no—
267C71⋊C8—22no—
268C22×C71—24no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71—24no—
272C22×C71#38514yes—
273C22×C71C22×C4⋊C411yescharacteristic
274C22×C71C2×#12011yescharacteristic
275C22×C71C2×#12311yescharacteristic
276C22×C71C2×#13512yes—
277C22×C71#57512yes—
278C2×D71—24no—
279C2×D71—22no—
280C2×D71—24no—
281C2×D71—22no—
282C2×D71—22no—
283C2×D71—22no—
284C2×D71—22no—
285C2×D71—416no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71#42714yescomplemented
289C4×C71—44no—
290C4×C71—82no—
291C4×C71#38514yes—
292C4×C71C2×C4×D411yescharacteristic
293C4×C71C2×C4×D411yescharacteristic
294C4×C71C2×C22⋊Q811yescharacteristic
295C4×C71C2×C22⋊Q811yescharacteristic
296Dic71—22no—
297Dic71—22no—
298Dic71—44no—
299Dic71—82no—
300#20843—711nomaximal, Sylow
301C2×C71—24no—
302C2×C71C2×#38511yescharacteristic
303C2×C71C2×#42711yescharacteristic
304C2×C71#291111yescharacteristic
305C2×C71#300212yes—
306C2×C71#309112yes—
307D71—24no—
308D71—42no—
309C22×#117—711no—
310C2×#427—711no—
311C2×#427D7111yescharacteristic, complemented
312C22×C22⋊C8—711no—
313#2280—711no—
314#1854—712no—
315#1854—712no—
316#1853—712no—
317#1852—712no—
318#1735—711no—
319#1735—711no—
320C71#2084311yescharacteristic, complemented, Sylow
321C24×C4—711no—
322C23×C8—711no—
323C23×C8C2×D7111yescharacteristic
324C2×#117—711no—
325C2×#117—711no—
326C2×#117—712no—
327C2×#117—1421no—
328C2×#117—1421no—
329C2×#117—1424no—
330C2×#117—1424no—
331C2×#117C2×D7111yescharacteristic
332C2×#117C2×D7111yescharacteristic
333C22×C4⋊C4—712no—
334C22×C22⋊C4—1421no—
335C22×C42—1421no—
336#427—24no—
337#427—1424no—
338C2×#107—712no—
339C2×#107C2×D7112yescomplemented
340C2×#106—712no—
341C2×#106C2×D7112yescomplemented
342C2×C22⋊C8—718no—
343C2×C22⋊C8—711no—
344C2×C22⋊C8—711no—
345C2×C22⋊C8—711no—
346C2×C22⋊C8—711no—
347#381—1421no—
348#380—712no—
349#380—712no—
350#379—1421no—
351C4×C22⋊C4—1421no—
352C4×C22⋊C4—1421no—
353#340—712no—
354#340—712no—
355#340—712no—
356#340—712no—
357C25—711no—
358C23×C4—711no—
359C23×C4—714no—
360C23×C4—711no—
361C23×C4—711no—
362C23×C4—714no—
363C23×C4—711no—
364C23×C4—711no—
365C23×C4—1422no—
366C23×C4—1422no—
367C23×C4—1422no—
368C23×C4—1422no—
369C23×C4—1422no—
370C23×C4C22×D7111yescharacteristic
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—714no—
374C22×C8—711no—
375C22×C8—711no—
376C22×C8—1422no—
377C22×C8—1422no—
378C22×C8C4×D7114yescomplemented
379C22×C8C22×D7111yescharacteristic
380C22×C8C22×D7111yescharacteristic
381#117—22no—
382#117—22no—
383#117—22no—
384#117—714no—
385#117—1428no—
386#117—1422no—
387#117—1422no—
388#117—1428no—
389#117—1424no—
390#117—1422no—
391#117—1424no—
392#117C22⋊D7114yes—
393C2×C4⋊C4—712no—
394C2×C4⋊C4—712no—
395C2×C4⋊C4—1424no—
396C2×C4⋊C4—1422no—
397C2×C4⋊C4—1424no—
398C2×C4⋊C4C22×D7112yes—
399C2×C4⋊C4C22×D7112yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1428no—
403C2×C22⋊C4—1422no—
404C2×C22⋊C4—1422no—
405C2×C22⋊C4—1421no—
406C2×C22⋊C4—1422no—
407C2×C22⋊C4—1421no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1428no—
411C2×C42—1421no—
412C2×C42—1422no—
413C2×C42—1421no—
414#107—24no—
415#107—1424no—
416#106—24no—
417#106—1424no—
418C22⋊C8—714no—
419C22⋊C8—714no—
420C22⋊C8—714no—
421C22⋊C8—714no—
422#95—1422no—
423#95—1422no—
424C24—2842no—
425C24—714no—
426C24—711no—
427C24—711no—
428C24—1424no—
429C24—1422no—
430C24—1422no—
431C24D4×D7111yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—2844no—
435C22×C4—2844no—
436C22×C4—711no—
437C22×C4—711no—
438C22×C4—714no—
439C22×C4—712no—
440C22×C4—712no—
441C22×C4—712no—
442C22×C4—712no—
443C22×C4—714no—
444C22×C4—714no—
445C22×C4—1428no—
446C22×C4—1422no—
447C22×C4—1422no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1421no—
451C22×C4—1421no—
452C22×C4—1421no—
453C22×C4—1421no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1428no—
459C22×C4—1428no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1428no—
463C22×C4—1428no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4C2×C4×D7112yes—
467C22×C4D4×D7112yes—
468C22×C4D4×D7111yescharacteristic
469C22×C4C2×C22⋊D7112yes—
470C22×C4Q8×D7112yes—
471C22×C4#1176586512yes—
472C22×C4#1176586612yes—
473C22×C4C23×D7111yescharacteristic
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—28no—
479C2×C8—712no—
480C2×C8—712no—
481C2×C8—1421no—
482C2×C8—1421no—
483C2×C8—1421no—
484C2×C8—1421no—
485C2×C8—1428no—
486C2×C8C2×C4×D7112yes—
487C2×C8C2×C4×D7112yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—714no—
492C4⋊C4—1428no—
493C4⋊C4—1422no—
494C4⋊C4—1422no—
495C4⋊C4—1422no—
496C4⋊C4C2×C22⋊D7114yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—2844no—
500C22⋊C4—2844no—
501C22⋊C4—2842no—
502C22⋊C4—1428no—
503C22⋊C4—1424no—
504C42—24no—
505C42—42no—
506C42—2842no—
507C42—1428no—
508C42—1424no—
509C23—22no—
510C23—22no—
511C23—2848no—
512C23—2848no—
513C23—2844no—
514C23—712no—
515C23—714no—
516C23—712no—
517C23—1428no—
518C23—1428no—
519C23—1421no—
520C23—1421no—
521C23—1422no—
522C23—1422no—
523C23—1421no—
524C23—1421no—
525C23—1422no—
526C23C4⋊C4×D7112yes—
527C23#1455609612yes—
528C23#1455611312yes—
529C23C2×D4×D7111yescharacteristic
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—28416no—
537C2×C4—2842no—
538C2×C4—2842no—
539C2×C4—2842no—
540C2×C4—2842no—
541C2×C4—2842no—
542C2×C4—712no—
543C2×C4—712no—
544C2×C4—1424no—
545C2×C4—1428no—
546C2×C4—1421no—
547C2×C4—1421no—
548C2×C4—1421no—
549C2×C4—1421no—
550C2×C4—1428no—
551C2×C4—1428no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4—1422no—
555C2×C4—1422no—
556C2×C4—1422no—
557C2×C4—1422no—
558C2×C4—1422no—
559C2×C4—1422no—
560C2×C4C4⋊C4×D7112yes—
561C2×C4C4×C22⋊D7114yes—
562C2×C4#1455609014yes—
563C2×C4#1455609612yes—
564C2×C4#1455610414yes—
565C2×C4#1455610614yes—
566C2×C4#1455611312yes—
567C2×C4C22×C4×D7111yescharacteristic
568C2×C4C2×D4×D7111yescharacteristic
569C2×C4C22×C22⋊D7111yescharacteristic
570C2×C4C2×Q8×D7111yescharacteristic
571C2×C4C2×#1176586511yescharacteristic
572C2×C4C2×#1176586611yescharacteristic
573C8—22no—
574C8—22no—
575C8—1422no—
576C8—1422no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—24no—
581C22—28416no—
582C22—2842no—
583C22—2842no—
584C22—714no—
585C22—1422no—
586C22—1422no—
587C22—1422no—
588C22—1422no—
589C22—1422no—
590C22—1424no—
591C22—1424no—
592C22#1798587014yes—
593C22C2×C4⋊C4×D7111yescharacteristic
594C22C2×#1455609611yescharacteristic
595C22C2×#1455611311yescharacteristic
596C22#135×D7112yes—
597C22#1798679612yes—
598C4—44no—
599C4—82no—
600C4—2844no—
601C4—5682no—
602C4—1422no—
603C4—1422no—
604C4#1798587014yes—
605C4C2×C4×C22⋊D7111yescharacteristic
606C4C2×#1455609011yescharacteristic
607C4C2×#1455610411yescharacteristic
608C4C2×#1455610611yescharacteristic
609C2—24no—
610C2—2842no—
611C2—1424no—
612C2C2×#1798587011yescharacteristic
613C2#427×D7111yescharacteristic
614C2#2259692811yescharacteristic
615C2#2259755412yes—
616C2#2259825712yes—
subgroupG/H
centerC22×C4D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#427×C71C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenterC2×#427D71
hypocenterC71#20843
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#17986035C2semidirect
5. #22588538C2semidirect
6. #22588536C2semidirect
7. #22588531C2semidirect
8. #22588526C2semidirect
9. #22588057C2semidirect
10. #22588056C2semidirect
11. C2×#427×C71C2semidirect
25. C2×#14556061C22semidirect
26. C2×#14556059C22semidirect
27. C2×(C2×D71)⋊C8C4semidirect
38. #17985762C22semidirect
39. #17985762C22semidirect
40. #17985761C22semidirect
41. #17985761C22semidirect
46. C2×#107×C71C22semidirect
47. C2×#106×C71C22semidirect
63. C22×C71⋊C8C2×C4semidirect
93. (C2×D71)⋊C8C2×C4semidirect
94. (C2×D71)⋊C8C2×C4semidirect
100. C22×C8×C71C2×C4semidirect
288. C2×D71#427semidirect
311. C2×#427D71semidirect
320. C71#20843semidirect
339. C2×#107C2×D71semidirect
341. C2×#106C2×D71semidirect
378. C22×C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D712
C2×#179860352
C22×(C2×D71)⋊C82
#225907192
#225885382
#225885362
#225885312
#225885262
#225880572
#225880562
C2×#427×C712
#2084371