Finite Group Databasebeta

← #31964755

Subgroups — #31964755

Order 18176.

1058 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22601636C211yesmaximal, characteristic
2#22596973C211yesmaximal, characteristic, complemented
3#22586881C214yesmaximal, complemented
4#22586876C214yesmaximal, complemented
5#22586838C212yesmaximal, complemented
6#22586834C212yesmaximal, complemented
7C71×#2285C211yesmaximal, characteristic, complemented
8#17986556C411yescharacteristic, complemented
9#17986556C411yescharacteristic, complemented
10#17986556C2211yescharacteristic
11#17986549C411yescharacteristic, complemented
12#17986549C411yescharacteristic, complemented
13#17986549C2211yescharacteristic
14#17986533—21no—
15#17986533—21no—
16#17986533—21no—
17#17986533—21no—
18#17986533C412yescomplemented
19#17986533C2212yes—
20#17986510—21no—
21#17986510—21no—
22#17986510C2212yes—
23C4×#11765865—21no—
24C4×#11765865—21no—
25C4×#11765865C412yescomplemented
26#17986056—21no—
27#17986056C2212yes—
28#17986052—21no—
29#17986052C2212yes—
30#17986023—21no—
31#17985963C2212yes—
32C4×#11765849—21no—
33#17985735C2218yes—
34#17985732C2218yes—
35C2×#117×C71—21no—
36C22×C42×C71C2211yescharacteristic
37#384×C71—21no—
38#382×C71C2212yes—
39#380×C71—21no—
40#380×C71C2212yes—
41C4×C4⋊C4×C71—21no—
42C4×C4⋊C4×C71C2212yes—
43C2×#11765865—21no—
44C2×#11765865—21no—
45C2×#11765865C2×C412yes—
46C2×#11765849—22no—
47#14556128C2×C411yescharacteristic
48#14556128C2×C411yescharacteristic
49#14556128C2×C411yescharacteristic
50#14556125C2×C411yescharacteristic
51#14556122—22no—
52#14556122—22no—
53#14556120—21no—
54#14556120—21no—
55#14556120C2×C412yes—
56#14556116—21no—
57#14556116—21no—
58#14556116C2×C412yes—
59#14556111—22no—
60#14556111—22no—
61#14556111—22no—
62#14556107—21no—
63#14556107—21no—
64#14556107—21no—
65#14556107—21no—
66#14556107C2×C412yes—
67#14556107C2312yes—
68#14556102—21no—
69#14556102—21no—
70#14556102—21no—
71#14556102—21no—
72#14556102C2×C412yes—
73#14556102C2312yes—
74#14556097—22no—
75C4×Dic142—21no—
76C4×Dic142—21no—
77C4×Dic142—21no—
78C4×Dic142—21no—
79C4×Dic142C2×C412yes—
80C4×Dic142C2×C412yes—
81C4×D284—21no—
82C4×D284—21no—
83C4×D284—21no—
84C4×D284—21no—
85C4×D284C2×C412yes—
86C4×D284C2×C412yes—
87C4×C22⋊D71—22no—
88C4×C22⋊D71—22no—
89C4×C22⋊D71—22no—
90C4×C22⋊D71—22no—
91C4×C22⋊D71—22no—
92C4×C22⋊D71—22no—
93#14556087—21no—
94#14556087—21no—
95#14556087C2×C412yes—
96#14556085—22no—
97#14556085—22no—
98#14556085—22no—
99C42×D71—22no—
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
109#117×C71—44no—
110C2×C4⋊C4×C71—22no—
111C2×C4⋊C4×C71—22no—
112C2×C4⋊C4×C71—22no—
113C2×C22⋊C4×C71—22no—
114C2×C42×C71—22no—
115C2×C42×C71C2×C411yescharacteristic
116C2×C42×C71C2×C411yescharacteristic
117C2×C42×C71C2×C411yescharacteristic
118C2×C42×C71C2×C411yescharacteristic
119C2×C42×C71D411yescharacteristic
120C2×C42×C71D411yescharacteristic
121C2×C42×C71D411yescharacteristic
122C2×C42×C71D411yescharacteristic
123C2×C42×C71D411yescharacteristic
124C2×C42×C71D411yescharacteristic
125C2×C42×C71C2311yescharacteristic
126C2×C42×C71C2311yescharacteristic
127#95×C71—22no—
128#95×C71C2314yes—
129#11765865—44no—
130#11765865—42no—
131#11765865—42no—
132#11765865—44no—
133C2×Dic142—21no—
134C2×Dic142—21no—
135C2×Dic142C22×C412yes—
136C2×D284—21no—
137C2×D284—21no—
138C2×D284C22×C412yes—
139C2×C22⋊D71—22no—
140C2×C22⋊D71—22no—
141C2×C22⋊D71—22no—
142C2×C4×D71—22no—
143C2×C4×D71—22no—
144C2×C4×D71—22no—
145#11765849—44no—
146C2×C71⋊C8—24no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71C22×C412yes—
152C4⋊Dic71C22×C412yes—
153Dic71⋊C4—22no—
154Dic71⋊C4—22no—
155Dic71⋊C4—22no—
156Dic71⋊C4—22no—
157Dic71⋊C4—22no—
158Dic71⋊C4—22no—
159C22⋊Dic71—22no—
160C22⋊Dic71—22no—
161C22⋊Dic71—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—
170C4×Dic71—22no—
171C24×C71C2×D411yescharacteristic
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—24no—
175C22×C4×C71—42no—
176C22×C4×C71—42no—
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×C71C22×C411yescharacteristic
186C22×C4×C71C22×C411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71C2411yescharacteristic
195C4⋊C4×C71—44no—
196C4⋊C4×C71—44no—
197C4⋊C4×C71—44no—
198C22⋊C4×C71—44no—
199C42×C71—44no—
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×C71C22×C411yescharacteristic
209C42×C71C22×C411yescharacteristic
210C42×C71C22×C411yescharacteristic
211C42×C71C22×C411yescharacteristic
212C42×C71C2×D411yescharacteristic
213C42×C71C2×D411yescharacteristic
214C42×C71C2×D411yescharacteristic
215C42×C71C2×D411yescharacteristic
216C23×C71—21no—
217C23×C71—21no—
218C23×C71—41no—
219C23×C71—41no—
220C23×C71C2×C22⋊C411yescharacteristic
221C23×C71C2×C22⋊C411yescharacteristic
222C23×C71C22×D411yescharacteristic
223C22×D71—22no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
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—22no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—22no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—44no—
243C2×C4×C71—44no—
244C2×C4×C71—44no—
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×C71C2×C22⋊C411yescharacteristic
254C2×C4×C71C2×C22⋊C411yescharacteristic
255C2×C4×C71AGL(1,ℤ/8)12yes—
256C2×C4×C71#13712yes—
257C2×C4×C71C23×C411yescharacteristic
258C2×C4×C71C22×D411yescharacteristic
259C22⋊D71—44no—
260C22⋊D71—44no—
261C22⋊D71—42no—
262C22⋊D71—42no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—22no—
266D284—42no—
267D284—42no—
268D284—42no—
269C4×D71—44no—
270C4×D71—44no—
271C4×D71—44no—
272Dic142—42no—
273Dic142—42no—
274Dic142—42no—
275C71⋊C8—44no—
276C22×C71—21no—
277C22×C71—21no—
278C22×C71—21no—
279C22×C71—21no—
280C22×C71—21no—
281C22×C71—21no—
282C22×C71—21no—
283C22×C71—21no—
284C22×C71—42no—
285C22×C71—42no—
286C22×C71C22×C22⋊C411yescharacteristic
287C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C71C2×#13711yescharacteristic
289C2×D71—42no—
290C2×D71—42no—
291C2×D71—42no—
292C4×C71—21no—
293C4×C71—21no—
294C4×C71—21no—
295C4×C71—21no—
296C4×C71—21no—
297C4×C71—21no—
298C4×C71—21no—
299C4×C71—21no—
300C4×C71—84no—
301C4×C71#41012yes—
302C4×C71#41712yes—
303C4×C71#41712yes—
304C4×C71#42112yes—
305Dic71—42no—
306Dic71—42no—
307Dic71—42no—
308#17563—711nomaximal, Sylow
309C2×C71—21no—
310C2×C71—21no—
311C2×C71—41no—
312C2×C71—41no—
313C2×C71C2×#41011yescharacteristic
314C2×C71C2×#41711yescharacteristic
315C2×C71C2×#42111yescharacteristic
316D71—82no—
317#3444—711no—
318#2920—711no—
319#2285D7111yescharacteristic, complemented
320#1513—714no—
321#1512—714no—
322#1503—712no—
323#1502—712no—
324C71#1756311yescharacteristic, complemented, Sylow
325#533—711no—
326#533—711no—
327#533—711no—
328#532—711no—
329#532—711no—
330#532—711no—
331#529—712no—
332#529—712no—
333#529—1421no—
334#529—1421no—
335#529—1421no—
336#529—1421no—
337#525—712no—
338#525—1421no—
339#525—1421no—
340C4×#41—712no—
341C4×#41—1421no—
342C4×#41—1421no—
343C2×#117—21no—
344C22×C42C2×D7111yescharacteristic
345#432—712no—
346#432—1421no—
347#431—712no—
348#431—1421no—
349#423—1421no—
350#407—712no—
351C4×M16—1421no—
352#384—21no—
353#382C2×D7112yes—
354#380—21no—
355#380C2×D7112yes—
356C4×C4⋊C4—21no—
357C4×C4⋊C4C2×D7112yes—
358#328—718no—
359#327—718no—
360C2×#41—712no—
361C2×#41—1421no—
362C2×#41—1421no—
363C23×C4C22⋊D7111yescharacteristic
364C23×C4C22⋊D7111yescharacteristic
365C23×C4C22×D7111yescharacteristic
366C2×M16—1422no—
367C4⋊Q8—711no—
368C4⋊Q8—711no—
369C4⋊Q8—711no—
370#127—711no—
371#126—1422no—
372#126—1422no—
373#125—712no—
374#125—1421no—
375#125—1421no—
376#124—712no—
377#124—1421no—
378#124—1421no—
379#123—1422no—
380#123—1422no—
381#123—1422no—
382C22⋊Q8—712no—
383C22⋊Q8—712no—
384C22⋊Q8—1421no—
385C22⋊Q8—1421no—
386C22⋊Q8—1421no—
387C22⋊Q8—1421no—
388#121—712no—
389#121—712no—
390#121—1421no—
391#121—1421no—
392#121—1421no—
393#121—1421no—
394#120—1422no—
395C4×Q8—712no—
396C4×Q8—712no—
397C4×Q8—1421no—
398C4×Q8—1421no—
399C4×Q8—1421no—
400C4×Q8—1421no—
401C4×D4—712no—
402C4×D4—712no—
403C4×D4—1422no—
404C4×D4—1422no—
405C4×D4—1422no—
406C4×D4—1422no—
407C4×D4—1422no—
408C4×D4—1422no—
409C4×D4—1421no—
410C4×D4—1421no—
411C4×D4—1421no—
412C4×D4—1421no—
413#117—44no—
414#117—712no—
415#117—1421no—
416#117—1421no—
417#117—1422no—
418#117—1422no—
419#117—1422no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C42—22no—
425C2×C42—1422no—
426C2×C42C4×D7111yescharacteristic
427C2×C42C4×D7111yescharacteristic
428C2×C42C4×D7111yescharacteristic
429C2×C42C4×D7111yescharacteristic
430C2×C42D28411yescharacteristic
431C2×C42D28411yescharacteristic
432C2×C42D28411yescharacteristic
433C2×C42D28411yescharacteristic
434C2×C42C22⋊D7111yescharacteristic
435C2×C42C22⋊D7111yescharacteristic
436C2×C42C22×D7111yescharacteristic
437C2×C42C22×D7111yescharacteristic
438C4⋊C8—1422no—
439C4⋊C8—1422no—
440C22⋊C8—714no—
441C22⋊C8—1422no—
442#97—1422no—
443C4×C8—1422no—
444#95—22no—
445#95C22×D7114yes—
446C24C2×C22⋊D7111yescharacteristic
447#41—2844no—
448#41—2844no—
449#41—2842no—
450#41—2842no—
451C2×Q8—712no—
452C2×Q8—1421no—
453C2×Q8—1421no—
454C2×D4—712no—
455C2×D4—1422no—
456C2×D4—1422no—
457C2×D4—1422no—
458C2×D4—1421no—
459C2×D4—1421no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4(C2×D71)⋊C411yescharacteristic
469C22×C4(C2×D71)⋊C411yescharacteristic
470C22×C4(C2×D71)⋊C411yescharacteristic
471C22×C4(C2×D71)⋊C411yescharacteristic
472C22×C4(C2×D71)⋊C411yescharacteristic
473C22×C4(C2×D71)⋊C411yescharacteristic
474C22×C4(C2×D71)⋊C411yescharacteristic
475C22×C4(C2×D71)⋊C411yescharacteristic
476C22×C4C2×C4×D7111yescharacteristic
477C22×C4C2×C4×D7111yescharacteristic
478C22×C4C2×C22⋊D7111yescharacteristic
479C22×C4C2×C22⋊D7111yescharacteristic
480C22×C4C2×C22⋊D7111yescharacteristic
481C22×C4C2×C22⋊D7111yescharacteristic
482C22×C4C2×C22⋊D7111yescharacteristic
483C22×C4C2×D28411yescharacteristic
484C22×C4C2×D28411yescharacteristic
485C22×C4C23×D7111yescharacteristic
486M16—2844no—
487C2×C8—1424no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—712no—
492C4⋊C4—712no—
493C4⋊C4—1422no—
494C4⋊C4—1422no—
495C4⋊C4—1422no—
496C4⋊C4—1422no—
497C4⋊C4—1422no—
498C4⋊C4—1422no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C4⋊C4—1421no—
502C4⋊C4—1421no—
503C22⋊C4—44no—
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—
513C42—44no—
514C42—1422no—
515C42—1422no—
516C42—1422no—
517C42(C2×D71)⋊C411yescharacteristic
518C42(C2×D71)⋊C411yescharacteristic
519C42(C2×D71)⋊C411yescharacteristic
520C42(C2×D71)⋊C411yescharacteristic
521C42(C2×D71)⋊C411yescharacteristic
522C42(C2×D71)⋊C411yescharacteristic
523C42(C2×D71)⋊C411yescharacteristic
524C42(C2×D71)⋊C411yescharacteristic
525C42C2×C4×D7111yescharacteristic
526C42C2×C4×D7111yescharacteristic
527C42C2×C4×D7111yescharacteristic
528C42C2×C4×D7111yescharacteristic
529C42C2×D28411yescharacteristic
530C42C2×D28411yescharacteristic
531C42C2×D28411yescharacteristic
532C42C2×D28411yescharacteristic
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—1422no—
538C23C2×(C2×D71)⋊C411yescharacteristic
539C23C2×(C2×D71)⋊C411yescharacteristic
540C23C22×C22⋊D7111yescharacteristic
541Q8—2842no—
542Q8—2842no—
543Q8—2842no—
544D4—2842no—
545D4—2844no—
546D4—2844no—
547D4—2842no—
548D4—2842no—
549D4—2842no—
550D4—2842no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—2844no—
573C2×C4—2844no—
574C2×C4—2844no—
575C2×C4—1422no—
576C2×C4—1422no—
577C2×C4—1422no—
578C2×C4C2×(C2×D71)⋊C411yescharacteristic
579C2×C4C2×(C2×D71)⋊C411yescharacteristic
580C2×C4C2×(C2×D71)⋊C411yescharacteristic
581C2×C4C2×(C2×D71)⋊C411yescharacteristic
582C2×C4C2×(C2×D71)⋊C411yescharacteristic
583C2×C4C2×(C2×D71)⋊C411yescharacteristic
584C2×C4C2×(C2×D71)⋊C411yescharacteristic
585C2×C4C2×(C2×D71)⋊C411yescharacteristic
586C2×C4C2×(C2×D71)⋊C411yescharacteristic
587C2×C4C2×(C2×D71)⋊C411yescharacteristic
588C2×C4#1455615412yes—
589C2×C4#1455615912yes—
590C2×C4C22×C4×D7111yescharacteristic
591C2×C4C22×D28411yescharacteristic
592C8—2844no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—42no—
602C22—42no—
603C22—2842no—
604C22—2842no—
605C22—2842no—
606C22C22×(C2×D71)⋊C411yescharacteristic
607C22C2×#1455615411yescharacteristic
608C22C2×#1455615911yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—84no—
618C4—2842no—
619C4—2842no—
620C4—2842no—
621C4#1798597212yes—
622C4#1798599812yes—
623C4#1798599812yes—
624C4#1798601712yes—
625C2—21no—
626C2—21no—
627C2—41no—
628C2—41no—
629C2—5682no—
630C2C2×#1798597211yescharacteristic
631C2C2×#1798599811yescharacteristic
632C2C2×#1798601711yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2285C2
socleC22×C71C22×C22⋊C4
prime radical#22601636C2
cosocleC22×C4×C71C24
hypercenter#2285D71
hypocenterC71#17563
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #22596973C2semidirect
3. #22586881C2semidirect
4. #22586876C2semidirect
5. #22586838C2semidirect
6. #22586834C2semidirect
7. C71×#2285C2semidirect
8. #17986556C4semidirect
9. #17986556C4semidirect
11. #17986549C4semidirect
12. #17986549C4semidirect
18. #17986533C4semidirect
25. C4×#11765865C4semidirect
319. #2285D71semidirect
324. C71#17563semidirect

Maximal subgroups

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

subgroupindex
#226016362
#225969732
#225868812
#225868762
#225868382
#225868342
C71×#22852
#1756371