Finite Group Databasebeta

← #31965297

Subgroups — #31965297

Order 18176.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597369C211yesmaximal, characteristic, complemented
2#22591280C211yesmaximal, characteristic, complemented
3#22586896C212yesmaximal, complemented
4#22586891C211yesmaximal, characteristic, complemented
5#22586887C211yesmaximal, characteristic, complemented
6#22586883C212yesmaximal, complemented
7#22586852C211yesmaximal, characteristic, complemented
8#22586848C211yesmaximal, characteristic, complemented
9#22586844C212yesmaximal, complemented
10#22586831C212yesmaximal, complemented
11C71×#3506C211yesmaximal, characteristic
12#17986071—22no—
13#17986071—22no—
14#17986056C2211yescharacteristic
15#17986056C2211yescharacteristic
16#17985963C2212yes—
17C2×C22⋊(C71⋊C8)C2212yes—
18#17985892—21no—
19#17985885—21no—
20#17985879—21no—
21#17985867C2211yescharacteristic
22#17985862C2211yescharacteristic
23#17985859C2211yescharacteristic
24C22⋊C4×Dic71—21no—
25C4×C22⋊Dic71C2211yescharacteristic
26#17985845C2211yescharacteristic
27C2×C71⋊#95C2211yescharacteristic
28#17985736C2212yes—
29#17985736C2212yes—
30#17985736C2212yes—
31#17985736C2212yes—
32#17985733C2212yes—
33#17985733C2212yes—
34#17985733C2212yes—
35#17985733C2212yes—
36#559×C71—22no—
37#556×C71C412yescomplemented
38#555×C71C412yescomplemented
39#553×C71—22no—
40#550×C71C412yescomplemented
41#547×C71C412yescomplemented
42#538×C71—21no—
43#538×C71—21no—
44#535×C71—21no—
45#533×C71C2211yescharacteristic
46#532×C71C2211yescharacteristic
47#525×C71C2212yes—
48#525×C71C2212yes—
49#520×C71—21no—
50C2×C4×D4×C71C2211yescharacteristic
51C23×Dic71—21no—
52C2×#11765849—22no—
53C22×C71⋊C8—22no—
54C2×C4⋊Dic71—21no—
55C2×C22⋊Dic71—21no—
56C2×C22⋊Dic71—21no—
57C2×C22⋊Dic71—21no—
58C2×C4×Dic71—21no—
59C2×C4×Dic71—21no—
60C4⋊(C71⋊C8)—21no—
61C4⋊(C71⋊C8)—21no—
62C22⋊(C71⋊C8)—22no—
63C22⋊(C71⋊C8)—22no—
64C22⋊(C71⋊C8)C2312yes—
65C22⋊(C71⋊C8)C2312yes—
66#14556039—21no—
67#14556039—21no—
68C71⋊#95—21no—
69C71⋊#95—21no—
70C71⋊#95—21no—
71C71⋊#95—21no—
72C71⋊#95—21no—
73C71⋊#95C2311yescharacteristic
74C71⋊#95C2311yescharacteristic
75C71⋊#95C2311yescharacteristic
76C71⋊#95C2311yescharacteristic
77C2×#41×C71—21no—
78C2×#41×C71C2×C412yes—
79C2×#41×C71C2×C412yescomplemented
80C22×D4×C71D411yescharacteristic
81C23×C4×C71C2311yescharacteristic
82C23×C4×C71C2311yescharacteristic
83C4⋊Q8×C71C2×C411yescharacteristic, complemented
84C4⋊Q8×C71C2×C411yescharacteristic, complemented
85C4⋊Q8×C71C2×C411yescharacteristic, complemented
86#127×C71—22no—
87#127×C71C2×C411yescharacteristic, complemented
88#126×C71—22no—
89#126×C71—22no—
90#125×C71—21no—
91#124×C71—22no—
92#124×C71—22no—
93#124×C71—22no—
94#124×C71—21no—
95#123×C71—22no—
96#123×C71—22no—
97#123×C71—22no—
98#123×C71—22no—
99C22⋊Q8×C71—21no—
100C22⋊Q8×C71C2×C412yes—
101C22⋊Q8×C71C2×C412yescomplemented
102C22⋊Q8×C71C2312yes—
103#121×C71—22no—
104#121×C71—22no—
105#121×C71—21no—
106#121×C71—22no—
107#121×C71—22no—
108#121×C71C2×C412yes—
109#121×C71C2×C412yescomplemented
110#121×C71C2312yes—
111#120×C71—22no—
112#120×C71—22no—
113C4×Q8×C71—22no—
114C4×Q8×C71—21no—
115C4×Q8×C71C2×C412yescomplemented
116C4×D4×C71—22no—
117C4×D4×C71—22no—
118C4×D4×C71—22no—
119C4×D4×C71—22no—
120C4×D4×C71—22no—
121C4×D4×C71—21no—
122C4×D4×C71C2×C412yes—
123C4×D4×C71C2×C412yes—
124C4×D4×C71C2×C412yescomplemented
125C4×D4×C71D412yes—
126C4×D4×C71D412yes—
127#117×C71—21no—
128#117×C71—22no—
129#117×C71C2×C412yes—
130#117×C71C2×C412yescomplemented
131C2×C4⋊C4×C71—22no—
132C2×C4⋊C4×C71D411yescharacteristic
133C2×C22⋊C4×C71D411yescharacteristic
134C2×C22⋊C4×C71D411yescharacteristic
135C2×C42×C71C2311yescharacteristic
136C22×Dic71—21no—
137C22×Dic71—21no—
138C22×Dic71—21no—
139C22×Dic71—21no—
140C22×Dic71—42no—
141C22×Dic71—42no—
142C22×Dic71—81no—
143#11765849—82no—
144C2×C71⋊C8—22no—
145C2×C71⋊C8—22no—
146C2×C71⋊C8—82no—
147C4⋊Dic71—81no—
148C22⋊Dic71—42no—
149C22⋊Dic71—42no—
150C22⋊Dic71—81no—
151C4×Dic71—42no—
152C4×Dic71—81no—
153C24×C71C2×D411yescharacteristic
154C24×C71C2×D411yescharacteristic
155#41×C71—42no—
156#41×C71—42no—
157#41×C71—82no—
158#41×C71—82no—
159C2×Q8×C71—21no—
160C2×Q8×C71—21no—
161C2×Q8×C71C22×C412yes—
162C2×D4×C71—22no—
163C2×D4×C71—22no—
164C2×D4×C71—22no—
165C2×D4×C71—22no—
166C2×D4×C71—22no—
167C2×D4×C71—21no—
168C2×D4×C71—21no—
169C2×D4×C71—41no—
170C2×D4×C71—41no—
171C2×D4×C71C22⋊C412yes—
172C2×D4×C71C22⋊C412yes—
173C2×D4×C71C22×C412yes—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—81no—
178C22×C4×C71—81no—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2×D412yes—
186C22×C4×C71C2×D412yes—
187C22×C4×C71C2411yescharacteristic
188C4⋊C4×C71—21no—
189C4⋊C4×C71—22no—
190C4⋊C4×C71—22no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—22no—
195C4⋊C4×C71—22no—
196C4⋊C4×C71C22⋊C412yes—
197C4⋊C4×C71C22⋊C412yes—
198C4⋊C4×C71C22×C412yes—
199C4⋊C4×C71C22×C412yes—
200C22⋊C4×C71—22no—
201C22⋊C4×C71—22no—
202C22⋊C4×C71—22no—
203C22⋊C4×C71—22no—
204C22⋊C4×C71—22no—
205C22⋊C4×C71—22no—
206C22⋊C4×C71—22no—
207C22⋊C4×C71—22no—
208C22⋊C4×C71—22no—
209C22⋊C4×C71C22⋊C412yes—
210C22⋊C4×C71C22⋊C412yes—
211C22⋊C4×C71C22⋊C412yes—
212C22⋊C4×C71C22⋊C412yes—
213C42×C71—22no—
214C42×C71—22no—
215C42×C71—22no—
216C42×C71C22×C411yescharacteristic
217C42×C71C22×C411yescharacteristic
218C42×C71C2×D411yescharacteristic
219C42×C71C2×D411yescharacteristic
220C23×C71—21no—
221C23×C71—21no—
222C23×C71—21no—
223C23×C71—21no—
224C23×C71—22no—
225C23×C71—41no—
226C23×C71—41no—
227C23×C71—41no—
228C23×C71—41no—
229C23×C71C2×C22⋊C412yes—
230C23×C71C2×C22⋊C412yes—
231C23×C71C22×D411yescharacteristic
232Q8×C71—42no—
233Q8×C71—42no—
234Q8×C71—81no—
235D4×C71—42no—
236D4×C71—42no—
237D4×C71—42no—
238D4×C71—42no—
239D4×C71—42no—
240D4×C71—42no—
241D4×C71—81no—
242D4×C71—82no—
243D4×C71—82no—
244D4×C71—82no—
245D4×C71—82no—
246C2×C4×C71—21no—
247C2×C4×C71—21no—
248C2×C4×C71—22no—
249C2×C4×C71—22no—
250C2×C4×C71—22no—
251C2×C4×C71—41no—
252C2×C4×C71—42no—
253C2×C4×C71—42no—
254C2×C4×C71—41no—
255C2×C4×C71—41no—
256C2×C4×C71—42no—
257C2×C4×C71—42no—
258C2×C4×C71—41no—
259C2×C4×C71—44no—
260C2×C4×C71—82no—
261C2×C4×C71—82no—
262C2×C4×C71C2×C22⋊C411yescharacteristic
263C2×C4×C71C2×C22⋊C412yes—
264C2×C4×C71C2×C22⋊C412yes—
265C2×C4×C71C2×C22⋊C411yescharacteristic
266C2×C4×C71C2×C22⋊C411yescharacteristic
267C2×C4×C71C2×C22⋊C411yescharacteristic
268C2×C4×C71C23×C411yescharacteristic
269C2×C4×C71C22×D411yescharacteristic
270C2×Dic71—22no—
271C2×Dic71—41no—
272C2×Dic71—41no—
273C2×Dic71—41no—
274C2×Dic71—41no—
275C2×Dic71—41no—
276C2×Dic71—41no—
277C2×Dic71—41no—
278C2×Dic71—41no—
279C2×Dic71—41no—
280C2×Dic71—41no—
281C2×Dic71—41no—
282C2×Dic71—84no—
283C71⋊C8—42no—
284C71⋊C8—42no—
285C22×C71—21no—
286C22×C71—21no—
287C22×C71—22no—
288C22×C71—22no—
289C22×C71—22no—
290C22×C71—22no—
291C22×C71—22no—
292C22×C71—22no—
293C22×C71—42no—
294C22×C71—42no—
295C22×C71—42no—
296C22×C71—42no—
297C22×C71—42no—
298C22×C71—81no—
299C22×C71—81no—
300C22×C71C22×C22⋊C411yescharacteristic
301C22×C71#57511yescharacteristic
302C22×C71#57511yescharacteristic
303C4×C71—21no—
304C4×C71—21no—
305C4×C71—22no—
306C4×C71—22no—
307C4×C71—21no—
308C4×C71—21no—
309C4×C71—21no—
310C4×C71—21no—
311C4×C71—42no—
312C4×C71—42no—
313C4×C71—42no—
314Dic71—42no—
315Dic71—81no—
316Dic71—81no—
317Dic71—81no—
318#17629—711nomaximal, Sylow
319C2×C71—21no—
320C2×C71—21no—
321C2×C71—42no—
322C2×C71—42no—
323C2×C71—82no—
324C2×C71#288311yescharacteristic
325C2×C71#289611yescharacteristic
326C2×C71#290111yescharacteristic
327#3506D7111yescharacteristic
328#2973—711no—
329#2353—711no—
330#1516—712no—
331#1515—711no—
332#1514—711no—
333#1513—712no—
334#1506—711no—
335#1505—711no—
336#1504—712no—
337#1501—712no—
338C71#1762911yescharacteristic, complemented, Sylow
339#559—22no—
340#556Dic7112yescomplemented
341#555Dic7112yescomplemented
342#553—22no—
343#550Dic7112yescomplemented
344#547Dic7112yescomplemented
345#538—21no—
346#538—21no—
347#535—21no—
348#533C2×D7111yescharacteristic
349#532C2×D7111yescharacteristic
350#525C2×D7112yes—
351#525C2×D7112yes—
352#520—21no—
353C2×C4×D4C2×D7111yescharacteristic
354#435—1422no—
355#435—1422no—
356#432—711no—
357#432—711no—
358#407—712no—
359C2×C22⋊C8—712no—
360#388—1421no—
361#387—1421no—
362#386—1421no—
363#383—711no—
364#381—711no—
365#380—711no—
366C4×C22⋊C4—711no—
367C4×C22⋊C4—1421no—
368#376—711no—
369C2×#95—711no—
370#328—712no—
371#328—712no—
372#328—712no—
373#328—712no—
374#327—712no—
375#327—712no—
376#327—712no—
377#327—712no—
378C2×#41—21no—
379C2×#41C2×Dic7112yes—
380C2×#41C2×Dic7112yescomplemented
381C22×D4C22⋊D7111yescharacteristic
382C23×C4—1421no—
383C23×C4C22×D7111yescharacteristic
384C23×C4C22×D7111yescharacteristic
385C2×M16—1422no—
386C22×C8—1422no—
387C4⋊Q8C2×Dic7111yescharacteristic, complemented
388C4⋊Q8C2×Dic7111yescharacteristic, complemented
389C4⋊Q8C2×Dic7111yescharacteristic, complemented
390#127—22no—
391#127C2×Dic7111yescharacteristic, complemented
392#126—22no—
393#126—22no—
394#125—21no—
395#124—22no—
396#124—22no—
397#124—22no—
398#124—21no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403C22⋊Q8—21no—
404C22⋊Q8C2×Dic7112yes—
405C22⋊Q8C2×Dic7112yescomplemented
406C22⋊Q8C22×D7112yes—
407#121—22no—
408#121—22no—
409#121—21no—
410#121—22no—
411#121—22no—
412#121C2×Dic7112yes—
413#121C2×Dic7112yescomplemented
414#121C22×D7112yes—
415#120—22no—
416#120—22no—
417C4×Q8—22no—
418C4×Q8—21no—
419C4×Q8C2×Dic7112yescomplemented
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—22no—
425C4×D4—21no—
426C4×D4C2×Dic7112yes—
427C4×D4C2×Dic7112yes—
428C4×D4C2×Dic7112yescomplemented
429C4×D4C22⋊D7112yes—
430C4×D4C22⋊D7112yes—
431#117—21no—
432#117—22no—
433#117C2×Dic7112yes—
434#117C2×Dic7112yescomplemented
435C2×C4⋊C4—22no—
436C2×C4⋊C4—1421no—
437C2×C4⋊C4C22⋊D7111yescharacteristic
438C2×C22⋊C4—1421no—
439C2×C22⋊C4—1421no—
440C2×C22⋊C4—1421no—
441C2×C22⋊C4C22⋊D7111yescharacteristic
442C2×C22⋊C4C22⋊D7111yescharacteristic
443C2×C42—1421no—
444C2×C42—1421no—
445C2×C42C22×D7111yescharacteristic
446C4⋊C8—1421no—
447C4⋊C8—1421no—
448C22⋊C8—712no—
449C22⋊C8—712no—
450C22⋊C8—1422no—
451C22⋊C8—1422no—
452#97—1421no—
453#97—1421no—
454#95—711no—
455#95—711no—
456#95—711no—
457#95—711no—
458#95—1421no—
459#95—1421no—
460#95—1421no—
461#95—1421no—
462#95—1421no—
463C24C2×C22⋊D7111yescharacteristic
464C24C2×C22⋊D7111yescharacteristic
465#41—42no—
466#41—42no—
467#41—82no—
468#41—82no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8C22×Dic7112yes—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4C22⋊Dic7112yes—
482C2×D4C22⋊Dic7112yes—
483C2×D4C22×Dic7112yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—2842no—
490C22×C4—2842no—
491C22×C4—5681no—
492C22×C4—1421no—
493C22×C4—1421no—
494C22×C4—1421no—
495C22×C4—1421no—
496C22×C4C22×Dic7112yes—
497C22×C4C22×Dic7112yes—
498C22×C4C2×C22⋊D7111yescharacteristic
499C22×C4C2×C22⋊D7112yes—
500C22×C4C2×C22⋊D7112yes—
501C22×C4C2×C22⋊D7111yescharacteristic
502C22×C4C2×C22⋊D7111yescharacteristic
503C22×C4C2×C22⋊D7111yescharacteristic
504C22×C4C23×D7111yescharacteristic
505M16—5682no—
506C2×C8—5682no—
507C2×C8—1422no—
508C2×C8—1422no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—5681no—
518C4⋊C4C22⋊Dic7112yes—
519C4⋊C4C22⋊Dic7112yes—
520C4⋊C4C22×Dic7112yes—
521C4⋊C4C22×Dic7112yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—2842no—
532C22⋊C4—2842no—
533C22⋊C4—5681no—
534C22⋊C4C22⋊Dic7112yes—
535C22⋊C4C22⋊Dic7112yes—
536C22⋊C4C22⋊Dic7112yes—
537C22⋊C4C22⋊Dic7112yes—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—2842no—
542C42—5681no—
543C42C22×Dic7111yescharacteristic
544C42C22×Dic7111yescharacteristic
545C42C2×C22⋊D7111yescharacteristic
546C42C2×C22⋊D7111yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×C22⋊Dic7112yes—
557C23C2×C22⋊Dic7112yes—
558C23C22×C22⋊D7111yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—81no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—82no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—2841no—
590C2×C4—2841no—
591C2×C4—2841no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4—2841no—
595C2×C4—2841no—
596C2×C4—2841no—
597C2×C4—2841no—
598C2×C4—2841no—
599C2×C4—2841no—
600C2×C4—5684no—
601C2×C4—1422no—
602C2×C4C2×C22⋊Dic7111yescharacteristic
603C2×C4C2×C22⋊Dic7112yes—
604C2×C4C2×C22⋊Dic7112yes—
605C2×C4C2×C22⋊Dic7111yescharacteristic
606C2×C4C2×C22⋊Dic7111yescharacteristic
607C2×C4C2×C22⋊Dic7111yescharacteristic
608C2×C4C23×Dic7111yescharacteristic
609C2×C4C22×C22⋊D7111yescharacteristic
610C8—2842no—
611C8—2842no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22C22×C22⋊Dic7111yescharacteristic
628C22#1798679711yescharacteristic
629C22#1798679711yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—2842no—
642C4—5681no—
643C4—5681no—
644C4—5681no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—82no—
650C2#2259680111yescharacteristic
651C2#2259685911yescharacteristic
652C2#2259688711yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
Fitting, prime radicalC71×#3506C2
socleC22×C71C22×C22⋊C4
cosocleC22×C4×C71C24
hypercenter#3506D71
hypocenterC71#17629
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #22597369C2semidirect
2. #22591280C2semidirect
3. #22586896C2semidirect
4. #22586891C2semidirect
5. #22586887C2semidirect
6. #22586883C2semidirect
7. #22586852C2semidirect
8. #22586848C2semidirect
9. #22586844C2semidirect
10. #22586831C2semidirect
37. #556×C71C4semidirect
38. #555×C71C4semidirect
40. #550×C71C4semidirect
41. #547×C71C4semidirect
79. C2×#41×C71C2×C4semidirect
83. C4⋊Q8×C71C2×C4semidirect
84. C4⋊Q8×C71C2×C4semidirect
85. C4⋊Q8×C71C2×C4semidirect
87. #127×C71C2×C4semidirect
101. C22⋊Q8×C71C2×C4semidirect
109. #121×C71C2×C4semidirect
115. C4×Q8×C71C2×C4semidirect
124. C4×D4×C71C2×C4semidirect
130. #117×C71C2×C4semidirect
338. C71#17629semidirect
340. #556Dic71semidirect
341. #555Dic71semidirect
343. #550Dic71semidirect
344. #547Dic71semidirect
380. C2×#41C2×Dic71semidirect
387. C4⋊Q8C2×Dic71semidirect
388. C4⋊Q8C2×Dic71semidirect
389. C4⋊Q8C2×Dic71semidirect
391. #127C2×Dic71semidirect
405. C22⋊Q8C2×Dic71semidirect
413. #121C2×Dic71semidirect
419. C4×Q8C2×Dic71semidirect
428. C4×D4C2×Dic71semidirect
434. #117C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225912802
#225868962
#225868912
#225868872
#225868832
#225868522
#225868482
#225868442
#225868312
C71×#35062
#1762971