Finite Group Databasebeta

← #31964699

Subgroups — #31964699

Order 18176.

1024 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×#17986533C211yesmaximal, characteristic
2#22596973C211yesmaximal, characteristic, complemented
3#22586876C214yesmaximal, complemented
4#22586872C212yesmaximal, complemented
5#22586868C212yesmaximal
6#22586824C214yesmaximal, complemented
7C71×#2283C211yesmaximal, characteristic, complemented
8#17986533—21no—
9#17986533—21no—
10#17986533—21no—
11#17986533—21no—
12#17986533C412yescomplemented
13#17986533C412yescomplemented
14#17986533C2212yes—
15#17986533C2212yes—
16C2×#14556122—21no—
17C2×#14556120C411yescharacteristic, complemented
18C2×#14556116C411yescharacteristic, complemented
19C2×#14556111—21no—
20C2×#14556107C2211yescharacteristic
21C2×#14556102C2211yescharacteristic
22C2×C4×Dic142C411yescharacteristic, complemented
23C2×C4×D284C411yescharacteristic, complemented
24C2×C4×C22⋊D71—21no—
25C2×#14556085—21no—
26#17986056—21no—
27#17986056C2212yescomplemented
28#17986052—21no—
29#17986052C2212yescomplemented
30#17986023—21no—
31#17985963C2212yes—
32C4×#11765849—21no—
33#17985735C2218yescomplemented
34#17985732C2218yes—
35C22×C42×C71C2211yescharacteristic
36#383×C71—21no—
37#382×C71C2212yescomplemented
38#381×C71C2212yes—
39#380×C71—21no—
40C4×C4⋊C4×C71—21no—
41C4×C22⋊C4×C71—21no—
42#376×C71C2212yescomplemented
43C22×Dic142C2×C411yescharacteristic
44C22×D284C2×C411yescharacteristic
45C22×C22⋊D71—21no—
46C22×C4×D71—21no—
47C2×#11765849—22no—
48#14556122—22no—
49#14556122—22no—
50#14556122—22no—
51#14556122—22no—
52#14556120—21no—
53#14556120—21no—
54#14556120C2×C412yes—
55#14556120C2×C412yes—
56#14556116—21no—
57#14556116—21no—
58#14556116C2×C412yes—
59#14556116C2×C412yes—
60#14556111—22no—
61#14556111—22no—
62#14556111—22no—
63#14556111—22no—
64#14556107—21no—
65#14556107—21no—
66#14556107C2×C412yes—
67#14556107C2312yes—
68#14556102—21no—
69#14556102—21no—
70#14556102C2×C412yes—
71#14556102C2312yes—
72C4×Dic142—21no—
73C4×Dic142—21no—
74C4×Dic142C2×C412yes—
75C4×Dic142C2×C412yes—
76C4×D284—21no—
77C4×D284—21no—
78C4×D284C2×C412yes—
79C4×D284C2×C412yes—
80C4×C22⋊D71—22no—
81C4×C22⋊D71—22no—
82C4×C22⋊D71—22no—
83C4×C22⋊D71—22no—
84#14556085—22no—
85#14556085—22no—
86#14556085—22no—
87#14556085—22no—
88C2×C4⋊Dic71C2×C411yescharacteristic
89C2×C4⋊Dic71C2×C411yescharacteristic
90C2×Dic71⋊C4—21no—
91C2×Dic71⋊C4—21no—
92C2×Dic71⋊C4—21no—
93C2×Dic71⋊C4—21no—
94C2×C22⋊Dic71—21no—
95C2×(C2×D71)⋊C4—21no—
96C2×(C2×D71)⋊C4—21no—
97C2×(C2×D71)⋊C4—21no—
98C2×(C2×D71)⋊C4—21no—
99C2×C4×Dic71—21no—
100C4⋊(C71⋊C8)—22no—
101C4⋊(C71⋊C8)—22no—
102C22⋊(C71⋊C8)—22no—
103C22⋊(C71⋊C8)C2314yes—
104#14556039—22no—
105C4×C71⋊C8—22no—
106C23×C4×C71D411yescharacteristic
107C23×C4×C71D411yescharacteristic
108C23×C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71—22no—
110C2×C22⋊C4×C71—22no—
111C2×C42×C71—22no—
112C2×C42×C71C2×C411yescharacteristic
113C2×C42×C71C2×C411yescharacteristic
114C2×C42×C71C2×C411yescharacteristic
115C2×C42×C71C2×C411yescharacteristic
116C2×C42×C71D411yescharacteristic
117C2×C42×C71D411yescharacteristic
118C2×C42×C71D411yescharacteristic
119C2×C42×C71D411yescharacteristic
120C2×C42×C71D411yescharacteristic
121C2×C42×C71D411yescharacteristic
122C2×C42×C71C2311yescharacteristic
123C2×C42×C71C2311yescharacteristic
124#95×C71—22no—
125#95×C71—22no—
126#95×C71—22no—
127#95×C71C2314yes—
128C23×D71—21no—
129C2×Dic142—21no—
130C2×Dic142—42no—
131C2×Dic142C22×C412yes—
132C2×D284—21no—
133C2×D284—42no—
134C2×D284C22×C412yes—
135C2×C22⋊D71—22no—
136C2×C22⋊D71—22no—
137C2×C22⋊D71—44no—
138C22×Dic71—21no—
139C22×Dic71—21no—
140C22×Dic71—21no—
141C2×C4×D71—22no—
142C2×C4×D71—22no—
143C2×C4×D71—44no—
144#11765849—44no—
145C2×C71⋊C8—24no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71C22×C412yes—
149C4⋊Dic71C22×C412yes—
150Dic71⋊C4—22no—
151Dic71⋊C4—22no—
152Dic71⋊C4—22no—
153Dic71⋊C4—22no—
154Dic71⋊C4—22no—
155Dic71⋊C4—22no—
156Dic71⋊C4—22no—
157Dic71⋊C4—22no—
158C22⋊Dic71—22no—
159C22⋊Dic71—22no—
160(C2×D71)⋊C4—22no—
161(C2×D71)⋊C4—22no—
162(C2×D71)⋊C4—22no—
163(C2×D71)⋊C4—22no—
164(C2×D71)⋊C4—22no—
165(C2×D71)⋊C4—22no—
166(C2×D71)⋊C4—22no—
167(C2×D71)⋊C4—22no—
168C4×Dic71—22no—
169C4×Dic71—22no—
170C24×C71C2×D411yescharacteristic
171C22×C4×C71—24no—
172C22×C4×C71—42no—
173C22×C4×C71—42no—
174C22×C4×C71—42no—
175C22×C4×C71C22⋊C411yescharacteristic
176C22×C4×C71C22⋊C411yescharacteristic
177C22×C4×C71C22⋊C411yescharacteristic
178C22×C4×C71C22⋊C411yescharacteristic
179C22×C4×C71C22⋊C411yescharacteristic
180C22×C4×C71C22⋊C411yescharacteristic
181C22×C4×C71C22⋊C411yescharacteristic
182C22×C4×C71C22⋊C411yescharacteristic
183C22×C4×C71C22×C411yescharacteristic
184C22×C4×C71C22×C411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2411yescharacteristic
193C4⋊C4×C71—44no—
194C22⋊C4×C71—44no—
195C42×C71—44no—
196C42×C71C22⋊C411yescharacteristic
197C42×C71C22⋊C411yescharacteristic
198C42×C71C22⋊C411yescharacteristic
199C42×C71C22⋊C411yescharacteristic
200C42×C71C22⋊C411yescharacteristic
201C42×C71C22⋊C411yescharacteristic
202C42×C71C22⋊C411yescharacteristic
203C42×C71C22⋊C411yescharacteristic
204C42×C71C22×C411yescharacteristic
205C42×C71C22×C411yescharacteristic
206C42×C71C22×C411yescharacteristic
207C42×C71C22×C411yescharacteristic
208C42×C71C2×D411yescharacteristic
209C42×C71C2×D411yescharacteristic
210C42×C71C2×D411yescharacteristic
211C42×C71C2×D411yescharacteristic
212C23×C71—22no—
213C23×C71—22no—
214C23×C71—41no—
215C23×C71C2×C22⋊C411yescharacteristic
216C23×C71C2×C22⋊C411yescharacteristic
217C23×C71C22×D411yescharacteristic
218C22×D71—22no—
219C22×D71—42no—
220C22×D71—42no—
221C22×D71—42no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—44no—
241C2×C4×C71—44no—
242C2×C4×C71—44no—
243C2×C4×C71C2×C22⋊C411yescharacteristic
244C2×C4×C71C2×C22⋊C411yescharacteristic
245C2×C4×C71C2×C22⋊C411yescharacteristic
246C2×C4×C71C2×C22⋊C411yescharacteristic
247C2×C4×C71C2×C22⋊C411yescharacteristic
248C2×C4×C71C2×C22⋊C411yescharacteristic
249C2×C4×C71C2×C22⋊C411yescharacteristic
250C2×C4×C71C2×C22⋊C411yescharacteristic
251C2×C4×C71C2×C22⋊C411yescharacteristic
252C2×C4×C71C2×C22⋊C411yescharacteristic
253C2×C4×C71C23×C411yescharacteristic
254C2×C4×C71C22×D411yescharacteristic
255C22⋊D71—44no—
256C22⋊D71—44no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—22no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263D284—42no—
264D284—42no—
265C4×D71—44no—
266C4×D71—44no—
267Dic142—42no—
268Dic142—42no—
269C71⋊C8—44no—
270C22×C71—21no—
271C22×C71—21no—
272C22×C71—21no—
273C22×C71—21no—
274C22×C71—21no—
275C22×C71—21no—
276C22×C71—21no—
277C22×C71—44no—
278C22×C71#41012yes—
279C22×C71C22×C22⋊C411yescharacteristic
280C22×C71#57512yes—
281C2×D71—42no—
282C2×D71—42no—
283C2×D71—42no—
284C2×D71—84no—
285C4×C71—21no—
286C4×C71—21no—
287C4×C71—21no—
288C4×C71—21no—
289C4×C71—21no—
290C4×C71—21no—
291C4×C71—21no—
292C4×C71—21no—
293C4×C71—21no—
294C4×C71—21no—
295C4×C71—21no—
296C4×C71—84no—
297C4×C71#42112yes—
298Dic71—42no—
299Dic71—42no—
300Dic71—42no—
301#17554—711nomaximal, Sylow
302C2×C71—22no—
303C2×C71—22no—
304C2×C71—41no—
305C2×C71C2×#41011yescharacteristic
306C2×C71#289611yescharacteristic
307C2×C71C2×#42111yescharacteristic
308D71—82no—
309C2×#529—711no—
310#2920—711no—
311#2283D7111yescharacteristic, complemented
312#1512—714no—
313#1511—712no—
314#1510—712no—
315#1500—714no—
316C71#1755411yescharacteristic, complemented, Sylow
317#529—712no—
318#529—712no—
319#529—712no—
320#529—712no—
321#529—1421no—
322#529—1421no—
323#529—1421no—
324#529—1421no—
325C2×#126—1421no—
326C2×#125—711no—
327C2×#124—711no—
328C2×#123—1421no—
329C2×C22⋊Q8—711no—
330C2×#121—711no—
331C2×C4×Q8—711no—
332C2×C4×D4—711no—
333C2×C4×D4—1421no—
334C2×#117—1421no—
335C22×C42C2×D7111yescharacteristic
336#432—712no—
337#432—1421no—
338#431—712no—
339#431—1421no—
340#423—1421no—
341#407—712no—
342C4×M16—1421no—
343#383—21no—
344#382C2×D7112yescomplemented
345#381C2×D7112yes—
346#380—21no—
347C4×C4⋊C4—21no—
348C4×C22⋊C4—21no—
349#376C2×D7112yescomplemented
350#328—718no—
351#327—718no—
352C22×Q8—711no—
353C22×D4—711no—
354C22×D4—1421no—
355C23×C4—1421no—
356C23×C4C22⋊D7111yescharacteristic
357C23×C4C22⋊D7111yescharacteristic
358C23×C4C22×D7111yescharacteristic
359C2×M16—1422no—
360#126—1422no—
361#126—1422no—
362#126—1422no—
363#126—1422no—
364#125—712no—
365#125—712no—
366#125—1421no—
367#125—1421no—
368#124—712no—
369#124—712no—
370#124—1421no—
371#124—1421no—
372#123—1422no—
373#123—1422no—
374#123—1422no—
375#123—1422no—
376C22⋊Q8—712no—
377C22⋊Q8—712no—
378C22⋊Q8—1421no—
379C22⋊Q8—1421no—
380#121—712no—
381#121—712no—
382#121—1421no—
383#121—1421no—
384C4×Q8—712no—
385C4×Q8—712no—
386C4×Q8—1421no—
387C4×Q8—1421no—
388C4×D4—712no—
389C4×D4—712no—
390C4×D4—1422no—
391C4×D4—1422no—
392C4×D4—1422no—
393C4×D4—1422no—
394C4×D4—1421no—
395C4×D4—1421no—
396#117—1422no—
397#117—1422no—
398#117—1422no—
399#117—1422no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—711no—
403C2×C4⋊C4—1421no—
404C2×C4⋊C4—1421no—
405C2×C4⋊C4—1421no—
406C2×C4⋊C4—1421no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—1421no—
409C2×C22⋊C4—1421no—
410C2×C22⋊C4—1421no—
411C2×C22⋊C4—1421no—
412C2×C22⋊C4—1421no—
413C2×C42—22no—
414C2×C42—1421no—
415C2×C42C4×D7111yescharacteristic
416C2×C42C4×D7111yescharacteristic
417C2×C42C4×D7111yescharacteristic
418C2×C42C4×D7111yescharacteristic
419C2×C42D28411yescharacteristic
420C2×C42D28411yescharacteristic
421C2×C42D28411yescharacteristic
422C2×C42D28411yescharacteristic
423C2×C42C22⋊D7111yescharacteristic
424C2×C42C22⋊D7111yescharacteristic
425C2×C42C22×D7111yescharacteristic
426C2×C42C22×D7111yescharacteristic
427C4⋊C8—1422no—
428C4⋊C8—1422no—
429C22⋊C8—714no—
430C22⋊C8—1422no—
431#97—1422no—
432C4×C8—1422no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95C22×D7114yes—
437C24—1421no—
438C24C2×C22⋊D7111yescharacteristic
439C2×Q8—2842no—
440C2×Q8—712no—
441C2×Q8—1421no—
442C2×D4—2842no—
443C2×D4—2844no—
444C2×D4—712no—
445C2×D4—1422no—
446C2×D4—1422no—
447C2×D4—1421no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—2844no—
453C22×C4—1421no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1422no—
457C22×C4—1422no—
458C22×C4(C2×D71)⋊C411yescharacteristic
459C22×C4(C2×D71)⋊C411yescharacteristic
460C22×C4(C2×D71)⋊C411yescharacteristic
461C22×C4(C2×D71)⋊C411yescharacteristic
462C22×C4(C2×D71)⋊C411yescharacteristic
463C22×C4(C2×D71)⋊C411yescharacteristic
464C22×C4(C2×D71)⋊C411yescharacteristic
465C22×C4(C2×D71)⋊C411yescharacteristic
466C22×C4C2×C4×D7111yescharacteristic
467C22×C4C2×C4×D7111yescharacteristic
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×C22⋊D7111yescharacteristic
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4C2×D28411yescharacteristic
474C22×C4C2×D28411yescharacteristic
475C22×C4C23×D7111yescharacteristic
476M16—2844no—
477C2×C8—1424no—
478C4⋊C4—44no—
479C4⋊C4—712no—
480C4⋊C4—712no—
481C4⋊C4—1422no—
482C4⋊C4—1422no—
483C4⋊C4—1422no—
484C4⋊C4—1422no—
485C4⋊C4—1422no—
486C4⋊C4—1422no—
487C4⋊C4—1422no—
488C4⋊C4—1422no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C22⋊C4—44no—
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—
502C42—44no—
503C42—1422no—
504C42—1422no—
505C42(C2×D71)⋊C411yescharacteristic
506C42(C2×D71)⋊C411yescharacteristic
507C42(C2×D71)⋊C411yescharacteristic
508C42(C2×D71)⋊C411yescharacteristic
509C42(C2×D71)⋊C411yescharacteristic
510C42(C2×D71)⋊C411yescharacteristic
511C42(C2×D71)⋊C411yescharacteristic
512C42(C2×D71)⋊C411yescharacteristic
513C42C2×C4×D7111yescharacteristic
514C42C2×C4×D7111yescharacteristic
515C42C2×C4×D7111yescharacteristic
516C42C2×C4×D7111yescharacteristic
517C42C2×D28411yescharacteristic
518C42C2×D28411yescharacteristic
519C42C2×D28411yescharacteristic
520C42C2×D28411yescharacteristic
521C23—22no—
522C23—22no—
523C23—41no—
524C23—2842no—
525C23—2842no—
526C23—2842no—
527C23—1422no—
528C23C2×(C2×D71)⋊C411yescharacteristic
529C23C2×(C2×D71)⋊C411yescharacteristic
530C23C22×C22⋊D7111yescharacteristic
531Q8—2842no—
532Q8—2842no—
533D4—2842no—
534D4—2842no—
535D4—2844no—
536D4—2844no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—2844no—
559C2×C4—2842no—
560C2×C4—2842no—
561C2×C4—2842no—
562C2×C4—2844no—
563C2×C4—1422no—
564C2×C4—1422no—
565C2×C4—1422no—
566C2×C4C2×(C2×D71)⋊C411yescharacteristic
567C2×C4C2×(C2×D71)⋊C411yescharacteristic
568C2×C4C2×(C2×D71)⋊C411yescharacteristic
569C2×C4C2×(C2×D71)⋊C411yescharacteristic
570C2×C4C2×(C2×D71)⋊C411yescharacteristic
571C2×C4C2×(C2×D71)⋊C411yescharacteristic
572C2×C4C2×(C2×D71)⋊C411yescharacteristic
573C2×C4C2×(C2×D71)⋊C411yescharacteristic
574C2×C4C2×(C2×D71)⋊C411yescharacteristic
575C2×C4C2×(C2×D71)⋊C411yescharacteristic
576C2×C4C22×C4×D7111yescharacteristic
577C2×C4C22×D28411yescharacteristic
578C8—2844no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—44no—
587C22—2842no—
588C22—2842no—
589C22—2842no—
590C22—5684no—
591C22#1798597212yes—
592C22C22×(C2×D71)⋊C411yescharacteristic
593C22#1798679712yes—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—84no—
606C4—2842no—
607C4—2842no—
608C4—2842no—
609C4#1798601712yes—
610C2—22no—
611C2—22no—
612C2—41no—
613C2—5682no—
614C2C2×#1798597211yescharacteristic
615C2#2259685711yescharacteristic
616C2C2×#1798601711yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2283C2
socleC22×C71C22×C22⋊C4
prime radicalC2×#14556102C22
cosocleC22×C4×C71C24
hypercenter#2283D71
hypocenterC71#17554
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. #22596973C2semidirect
3. #22586876C2semidirect
4. #22586872C2semidirect
6. #22586824C2semidirect
7. C71×#2283C2semidirect
12. #17986533C4semidirect
13. #17986533C4semidirect
17. C2×#14556120C4semidirect
18. C2×#14556116C4semidirect
22. C2×C4×Dic142C4semidirect
23. C2×C4×D284C4semidirect
27. #17986056C22semidirect
29. #17986052C22semidirect
33. #17985735C22semidirect
37. #382×C71C22semidirect
42. #376×C71C22semidirect
311. #2283D71semidirect
316. C71#17554semidirect
344. #382C2×D71semidirect
349. #376C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179865332
#225969732
#225868762
#225868722
#225868682
#225868242
C71×#22832
#1755471