Finite Group Databasebeta

← #31965016

Subgroups — #31965016

Order 18176.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597375C211yesmaximal, characteristic, complemented
2#22590983C211yesmaximal, characteristic, complemented
3#22586912C212yesmaximal, complemented
4#22586896C212yesmaximal, complemented
5#22586883C212yesmaximal, complemented
6#22586873C212yesmaximal, complemented
7#22586865C211yesmaximal, characteristic, complemented
8#22586852C211yesmaximal, characteristic, complemented
9#22586839C211yesmaximal, characteristic, complemented
10#22586826C211yesmaximal, characteristic, complemented
11C71×#3520C211yesmaximal, characteristic
12#17986077—21no—
13#17986071—22no—
14D4×C71⋊C8—21no—
15#17986056C2211yescharacteristic
16#17986052C2211yescharacteristic
17#17985963C2212yes—
18C2×C22⋊(C71⋊C8)C2212yes—
19#17985902—21no—
20#17985898—21no—
21#17985892—22no—
22#17985862C2212yes—
23#17985859C2212yes—
24C4×C4⋊Dic71C2211yescharacteristic
25#17985845C2211yescharacteristic
26#17985736C2212yes—
27#17985736C2212yes—
28#17985736C2212yes—
29#17985736C2212yes—
30#17985736C2211yescharacteristic
31#17985736C2212yes—
32#17985736C2211yescharacteristic
33#17985733C2211yescharacteristic
34#17985733C2212yes—
35#17985733C2211yescharacteristic
36#556×C71C411yescharacteristic, complemented
37#554×C71—21no—
38#552×C71C412yescomplemented
39#551×C71—22no—
40D4×Q8×C71C411yescharacteristic, complemented
41#548×C71C412yescomplemented
42#547×C71C411yescharacteristic, complemented
43#547×C71C411yescharacteristic, complemented
44#546×C71—21no—
45#542×C71—21no—
46#539×C71—22no—
47#533×C71C2211yescharacteristic
48#529×C71C2211yescharacteristic
49#525×C71C2212yes—
50C2×C22⋊Q8×C71C2212yes—
51#520×C71—21no—
52C2×C4×D4×C71C2211yescharacteristic
53C2×#11765849—22no—
54C22×C71⋊C8—22no—
55C2×C4⋊Dic71—21no—
56C2×C4⋊Dic71—22no—
57C2×C22⋊Dic71—22no—
58C2×C22⋊Dic71—22no—
59C2×C4×Dic71—21no—
60C2×C4×Dic71—21no—
61C4⋊(C71⋊C8)—21no—
62C4⋊(C71⋊C8)—21no—
63C22⋊(C71⋊C8)—22no—
64C22⋊(C71⋊C8)—22no—
65C22⋊(C71⋊C8)C2311yescharacteristic
66C22⋊(C71⋊C8)C2311yescharacteristic
67C22⋊(C71⋊C8)C2312yes—
68#14556039—21no—
69C4×C71⋊C8—21no—
70C71⋊#95—21no—
71C71⋊#95—22no—
72C71⋊#95C2311yescharacteristic
73C71⋊#95C2311yescharacteristic
74C71⋊#95C2312yes—
75C2×#41×C71—21no—
76C2×#41×C71C2×C412yes—
77C22×Q8×C71C2×C412yes—
78C22×D4×C71D411yescharacteristic
79C23×C4×C71C2312yes—
80C4⋊Q8×C71—21no—
81C4⋊Q8×C71C2×C411yescharacteristic
82C4⋊Q8×C71C2×C411yescharacteristic
83#126×C71—22no—
84#126×C71—22no—
85#126×C71—21no—
86#125×C71—22no—
87#125×C71—21no—
88#125×C71C2×C411yescharacteristic
89#124×C71—21no—
90#124×C71—22no—
91#124×C71C2×C411yescharacteristic
92#123×C71—22no—
93#123×C71—22no—
94#123×C71—21no—
95#123×C71—22no—
96#123×C71—22no—
97C22⋊Q8×C71—22no—
98C22⋊Q8×C71—22no—
99C22⋊Q8×C71—21no—
100C22⋊Q8×C71—22no—
101C22⋊Q8×C71—21no—
102C22⋊Q8×C71—22no—
103C22⋊Q8×C71C2×C412yes—
104C22⋊Q8×C71C2×C412yes—
105C22⋊Q8×C71C2×C412yes—
106C22⋊Q8×C71C2311yescharacteristic
107C22⋊Q8×C71C2312yes—
108#121×C71—22no—
109#121×C71—21no—
110#121×C71C2×C412yes—
111#121×C71C2311yescharacteristic
112#120×C71—22no—
113C4×Q8×C71—21no—
114C4×Q8×C71—21no—
115C4×Q8×C71C2×C411yescharacteristic
116C4×Q8×C71C2×C412yes—
117C4×D4×C71—21no—
118C4×D4×C71—22no—
119C4×D4×C71—22no—
120C4×D4×C71—21no—
121C4×D4×C71—22no—
122C4×D4×C71C2×C411yescharacteristic
123C4×D4×C71C2×C411yescharacteristic
124C4×D4×C71C2×C411yescharacteristic
125C4×D4×C71C2×C412yes—
126C4×D4×C71D411yescharacteristic
127C4×D4×C71D412yes—
128C4×D4×C71D411yescharacteristic
129#117×C71—21no—
130#117×C71—22no—
131#117×C71—21no—
132#117×C71C2×C412yes—
133C2×C4⋊C4×C71—22no—
134C2×C4⋊C4×C71—21no—
135C2×C4⋊C4×C71C2×C412yes—
136C2×C4⋊C4×C71D411yescharacteristic
137C2×C22⋊C4×C71—22no—
138C2×C22⋊C4×C71D412yes—
139C2×C42×C71C2311yescharacteristic
140C22×Dic71—21no—
141C22×Dic71—21no—
142C22×Dic71—22no—
143#11765849—82no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—22no—
147C2×C71⋊C8—82no—
148C4⋊Dic71—42no—
149C4⋊Dic71—82no—
150C22⋊Dic71—82no—
151C22⋊Dic71—82no—
152C4×Dic71—42no—
153C4×Dic71—81no—
154C24×C71C2×D412yes—
155#41×C71—44no—
156#41×C71—82no—
157C2×Q8×C71—21no—
158C2×Q8×C71—22no—
159C2×Q8×C71—82no—
160C2×Q8×C71C22×C411yescharacteristic
161C2×Q8×C71C22×C412yes—
162C2×D4×C71—22no—
163C2×D4×C71—21no—
164C2×D4×C71—22no—
165C2×D4×C71—42no—
166C2×D4×C71C22⋊C411yescharacteristic
167C2×D4×C71C22⋊C411yescharacteristic
168C2×D4×C71C22⋊C412yes—
169C2×D4×C71C22×C411yescharacteristic
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—21no—
173C22×C4×C71—22no—
174C22×C4×C71—82no—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D412yes—
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2411yescharacteristic
183C4⋊C4×C71—22no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71—22no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—22no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—22no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—22no—
196C4⋊C4×C71C22⋊C411yescharacteristic
197C4⋊C4×C71C22⋊C412yes—
198C4⋊C4×C71C22⋊C411yescharacteristic
199C4⋊C4×C71C22×C411yescharacteristic
200C4⋊C4×C71C22×C412yes—
201C4⋊C4×C71C22×C411yescharacteristic
202C22⋊C4×C71—22no—
203C22⋊C4×C71—22no—
204C22⋊C4×C71—22no—
205C22⋊C4×C71—22no—
206C22⋊C4×C71—21no—
207C22⋊C4×C71—22no—
208C22⋊C4×C71—21no—
209C22⋊C4×C71—21no—
210C22⋊C4×C71—22no—
211C22⋊C4×C71—21no—
212C22⋊C4×C71—21no—
213C22⋊C4×C71C22⋊C412yes—
214C22⋊C4×C71C22⋊C412yes—
215C22⋊C4×C71C22⋊C412yes—
216C22⋊C4×C71C22⋊C412yes—
217C42×C71—22no—
218C42×C71—22no—
219C42×C71—21no—
220C42×C71C22×C411yescharacteristic
221C42×C71C22×C411yescharacteristic
222C42×C71C2×D411yescharacteristic
223C42×C71C2×D411yescharacteristic
224C23×C71—22no—
225C23×C71—22no—
226C23×C71—21no—
227C23×C71—42no—
228C23×C71—42no—
229C23×C71C2×C22⋊C412yes—
230C23×C71C2×C22⋊C412yes—
231C23×C71C22×D411yescharacteristic
232Q8×C71—41no—
233Q8×C71—42no—
234Q8×C71—42no—
235Q8×C71—82no—
236D4×C71—22no—
237D4×C71—22no—
238D4×C71—42no—
239D4×C71—41no—
240D4×C71—44no—
241D4×C71—41no—
242D4×C71—41no—
243D4×C71—82no—
244C2×C4×C71—21no—
245C2×C4×C71—21no—
246C2×C4×C71—22no—
247C2×C4×C71—21no—
248C2×C4×C71—22no—
249C2×C4×C71—21no—
250C2×C4×C71—22no—
251C2×C4×C71—21no—
252C2×C4×C71—41no—
253C2×C4×C71—42no—
254C2×C4×C71—42no—
255C2×C4×C71—42no—
256C2×C4×C71—42no—
257C2×C4×C71—42no—
258C2×C4×C71—44no—
259C2×C4×C71—42no—
260C2×C4×C71—82no—
261C2×C4×C71—82no—
262C2×C4×C71C2×C22⋊C411yescharacteristic
263C2×C4×C71C2×C22⋊C412yes—
264C2×C4×C71C2×C22⋊C411yescharacteristic
265C2×C4×C71C2×C22⋊C411yescharacteristic
266C2×C4×C71C2×C22⋊C412yes—
267C2×C4×C71C2×C22⋊C411yescharacteristic
268C2×C4×C71C23×C411yescharacteristic
269C2×C4×C71C22×D411yescharacteristic
270C2×Dic71—41no—
271C2×Dic71—41no—
272C2×Dic71—41no—
273C2×Dic71—42no—
274C2×Dic71—42no—
275C2×Dic71—41no—
276C2×Dic71—42no—
277C2×Dic71—41no—
278C2×Dic71—41no—
279C71⋊C8—41no—
280C71⋊C8—42no—
281C71⋊C8—41no—
282C22×C71—21no—
283C22×C71—21no—
284C22×C71—22no—
285C22×C71—22no—
286C22×C71—22no—
287C22×C71—22no—
288C22×C71—22no—
289C22×C71—22no—
290C22×C71—41no—
291C22×C71—41no—
292C22×C71—41no—
293C22×C71—82no—
294C22×C71—82no—
295C22×C71C22×C22⋊C411yescharacteristic
296C22×C71#57711yescharacteristic
297C22×C71#57811yescharacteristic
298C4×C71—21no—
299C4×C71—21no—
300C4×C71—22no—
301C4×C71—22no—
302C4×C71—21no—
303C4×C71—21no—
304C4×C71—21no—
305C4×C71—42no—
306C4×C71—41no—
307C4×C71—41no—
308C4×C71—41no—
309C4×C71—42no—
310C4×C71#42112yes—
311Dic71—81no—
312Dic71—82no—
313Dic71—81no—
314#17600—711nomaximal, Sylow
315C2×C71—21no—
316C2×C71—21no—
317C2×C71—42no—
318C2×C71—42no—
319C2×C71—81no—
320C2×C71#288311yescharacteristic
321C2×C71#289811yescharacteristic
322C2×C71C2×#42111yescharacteristic
323#3520D7111yescharacteristic
324#2974—711no—
325#2319—711no—
326#1520—712no—
327#1516—712no—
328#1513—712no—
329#1511—712no—
330#1509—711no—
331#1506—711no—
332#1503—711no—
333#1500—711no—
334C71#1760011yescharacteristic, complemented, Sylow
335#556Dic7111yescharacteristic, complemented
336#554—21no—
337#552Dic7112yescomplemented
338#551—22no—
339D4×Q8Dic7111yescharacteristic, complemented
340#548Dic7112yescomplemented
341#547Dic7111yescharacteristic, complemented
342#547Dic7111yescharacteristic, complemented
343#546—21no—
344#542—21no—
345#539—22no—
346#533C2×D7111yescharacteristic
347#529C2×D7111yescharacteristic
348#525C2×D7112yes—
349C2×C22⋊Q8C2×D7112yes—
350#520—21no—
351C2×C4×D4C2×D7111yescharacteristic
352#436—1421no—
353#435—1422no—
354C8×D4—1421no—
355#432—711no—
356#431—711no—
357#407—712no—
358C2×C22⋊C8—712no—
359#390—1421no—
360#389—1421no—
361#388—1422no—
362#381—712no—
363#380—712no—
364C4×C4⋊C4—711no—
365#376—711no—
366#328—711no—
367#328—711no—
368#328—712no—
369#328—712no—
370#328—712no—
371#328—712no—
372#328—712no—
373#327—711no—
374#327—711no—
375#327—712no—
376C2×#41—21no—
377C2×#41C2×Dic7112yes—
378C22×Q8C2×Dic7112yes—
379C22×D4C22⋊D7111yescharacteristic
380C23×C4C22×D7112yes—
381C2×M16—1422no—
382C22×C8—1422no—
383C4⋊Q8—21no—
384C4⋊Q8C2×Dic7111yescharacteristic
385C4⋊Q8C2×Dic7111yescharacteristic
386#126—22no—
387#126—22no—
388#126—21no—
389#125—22no—
390#125—21no—
391#125C2×Dic7111yescharacteristic
392#124—21no—
393#124—22no—
394#124C2×Dic7111yescharacteristic
395#123—22no—
396#123—22no—
397#123—21no—
398#123—22no—
399#123—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C2×Dic7112yes—
407C22⋊Q8C2×Dic7112yes—
408C22⋊Q8C2×Dic7112yes—
409C22⋊Q8C22×D7111yescharacteristic
410C22⋊Q8C22×D7112yes—
411#121—22no—
412#121—21no—
413#121C2×Dic7112yes—
414#121C22×D7111yescharacteristic
415#120—22no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8C2×Dic7111yescharacteristic
419C4×Q8C2×Dic7112yes—
420C4×D4—21no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—21no—
424C4×D4—22no—
425C4×D4C2×Dic7111yescharacteristic
426C4×D4C2×Dic7111yescharacteristic
427C4×D4C2×Dic7111yescharacteristic
428C4×D4C2×Dic7112yes—
429C4×D4C22⋊D7111yescharacteristic
430C4×D4C22⋊D7112yes—
431C4×D4C22⋊D7111yescharacteristic
432#117—21no—
433#117—22no—
434#117—21no—
435#117C2×Dic7112yes—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—1421no—
439C2×C4⋊C4—1422no—
440C2×C4⋊C4C2×Dic7112yes—
441C2×C4⋊C4C22⋊D7111yescharacteristic
442C2×C22⋊C4—22no—
443C2×C22⋊C4—1422no—
444C2×C22⋊C4—1422no—
445C2×C22⋊C4C22⋊D7112yes—
446C2×C42—1421no—
447C2×C42—1421no—
448C2×C42C22×D7111yescharacteristic
449C4⋊C8—1421no—
450C4⋊C8—1421no—
451C22⋊C8—711no—
452C22⋊C8—712no—
453C22⋊C8—711no—
454C22⋊C8—1422no—
455C22⋊C8—1422no—
456#97—1421no—
457C4×C8—1421no—
458#95—711no—
459#95—711no—
460#95—712no—
461#95—1421no—
462#95—1422no—
463C24C2×C22⋊D7112yes—
464#41—44no—
465#41—82no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—82no—
469C2×Q8C22×Dic7111yescharacteristic
470C2×Q8C22×Dic7112yes—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—42no—
475C2×D4C22⋊Dic7111yescharacteristic
476C2×D4C22⋊Dic7111yescharacteristic
477C2×D4C22⋊Dic7112yes—
478C2×D4C22×Dic7111yescharacteristic
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—82no—
484C22×C4—1421no—
485C22×C4—1421no—
486C22×C4—1422no—
487C22×C4C22×Dic7112yes—
488C22×C4C22×Dic7112yes—
489C22×C4C2×C22⋊D7111yescharacteristic
490C22×C4C2×C22⋊D7111yescharacteristic
491C22×C4C2×C22⋊D7112yes—
492C22×C4C2×C22⋊D7112yes—
493C22×C4C2×C22⋊D7112yes—
494C22×C4C23×D7111yescharacteristic
495M16—5682no—
496C2×C8—5682no—
497C2×C8—1421no—
498C2×C8—1421no—
499C2×C8—1422no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—2842no—
514C4⋊C4—5682no—
515C4⋊C4C22⋊Dic7111yescharacteristic
516C4⋊C4C22⋊Dic7112yes—
517C4⋊C4C22⋊Dic7111yescharacteristic
518C4⋊C4C22×Dic7111yescharacteristic
519C4⋊C4C22×Dic7112yes—
520C4⋊C4C22×Dic7111yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—21no—
532C22⋊C4—5682no—
533C22⋊C4—5682no—
534C22⋊C4C22⋊Dic7112yes—
535C22⋊C4C22⋊Dic7112yes—
536C22⋊C4C22⋊Dic7112yes—
537C22⋊C4C22⋊Dic7112yes—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—2842no—
542C42—5681no—
543C42C22×Dic7111yescharacteristic
544C42C22×Dic7111yescharacteristic
545C42C2×C22⋊D7111yescharacteristic
546C42C2×C22⋊D7111yescharacteristic
547C23—22no—
548C23—22no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23C2×C22⋊Dic7112yes—
553C23C2×C22⋊Dic7112yes—
554C23C22×C22⋊D7111yescharacteristic
555Q8—41no—
556Q8—42no—
557Q8—42no—
558Q8—82no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—41no—
563D4—44no—
564D4—41no—
565D4—41no—
566D4—82no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—2841no—
586C2×C4—2841no—
587C2×C4—2841no—
588C2×C4—2842no—
589C2×C4—2842no—
590C2×C4—2841no—
591C2×C4—2842no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4C2×C22⋊Dic7111yescharacteristic
595C2×C4C2×C22⋊Dic7112yes—
596C2×C4C2×C22⋊Dic7111yescharacteristic
597C2×C4C2×C22⋊Dic7111yescharacteristic
598C2×C4C2×C22⋊Dic7112yes—
599C2×C4C2×C22⋊Dic7111yescharacteristic
600C2×C4C23×Dic7111yescharacteristic
601C2×C4C22×C22⋊D7111yescharacteristic
602C8—2841no—
603C8—2842no—
604C8—2841no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—82no—
617C22—82no—
618C22C22×C22⋊Dic7111yescharacteristic
619C22#1798680711yescharacteristic
620C22#1798681411yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—5681no—
634C4—5682no—
635C4—5681no—
636C4#1798601612yes—
637C2—21no—
638C2—21no—
639C2—42no—
640C2—42no—
641C2—81no—
642C2#2259680111yescharacteristic
643C2#2259687311yescharacteristic
644C2C2×#1798601611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
Fitting, prime radicalC71×#3520C2
socleC22×C71C22×C22⋊C4
cosocleC22×C4×C71C24
hypercenter#3520D71
hypocenterC71#17600
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3520

Complements

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

NH ≅ G/Ntype
1. #22597375C2semidirect
2. #22590983C2semidirect
3. #22586912C2semidirect
4. #22586896C2semidirect
5. #22586883C2semidirect
6. #22586873C2semidirect
7. #22586865C2semidirect
8. #22586852C2semidirect
9. #22586839C2semidirect
10. #22586826C2semidirect
36. #556×C71C4semidirect
38. #552×C71C4semidirect
40. D4×Q8×C71C4semidirect
41. #548×C71C4semidirect
42. #547×C71C4semidirect
43. #547×C71C4semidirect
334. C71#17600semidirect
335. #556Dic71semidirect
337. #552Dic71semidirect
339. D4×Q8Dic71semidirect
340. #548Dic71semidirect
341. #547Dic71semidirect
342. #547Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973752
#225909832
#225869122
#225868962
#225868832
#225868732
#225868652
#225868522
#225868392
#225868262
C71×#35202
#1760071