Finite Group Databasebeta

← #33053570

Subgroups — #33053570

Order 18688.

938 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619020C211yesmaximal, characteristic, complemented
2#22612582C211yesmaximal, characteristic, complemented
3#22608508C212yesmaximal, complemented
4#22608491C212yesmaximal, complemented
5#22608478C212yesmaximal, complemented
6#22608469C212yesmaximal, complemented
7#22608456C211yesmaximal, characteristic, complemented
8#22608439C211yesmaximal, characteristic, complemented
9#22608426C211yesmaximal, characteristic, complemented
10#22608417C211yesmaximal, characteristic, complemented
11C73×#3522C211yesmaximal, characteristic
12#17988718—21no—
13#17988718—22no—
14Q8×#10596831—21no—
15#17988644C2211yescharacteristic
16#17988644C2212yes—
17#17988640C2211yescharacteristic
18#17988640C2212yes—
19#17988494—21no—
20#17988486—21no—
21#17988473—22no—
22#17988455C2212yes—
23#17988453C2212yes—
24C4×C4⋊Dic73C2211yescharacteristic
25#17988433C2211yescharacteristic
26#17988324C2211yescharacteristic
27#17988324C2211yescharacteristic
28#17988324C2212yes—
29#17988321C2211yescharacteristic
30#17988321C2211yescharacteristic
31#17988321C2212yes—
32#17988321C2212yes—
33#17988321C2212yes—
34#17988321C2212yes—
35#17988321C2212yes—
36#559×C73C412yescomplemented
37#559×C73C411yescharacteristic, complemented
38Q82×C73C411yescharacteristic, complemented
39#554×C73C411yescharacteristic, complemented
40#554×C73C412yescomplemented
41#550×C73C411yescharacteristic, complemented
42#548×C73—21no—
43#547×C73—22no—
44#547×C73—21no—
45#542×C73—21no—
46#542×C73—22no—
47#532×C73C2211yescharacteristic
48#532×C73C2212yes—
49#529×C73C2211yescharacteristic
50#529×C73C2212yes—
51#520×C73—21no—
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4⋊Dic73—21no—
54C2×C4⋊Dic73—22no—
55C2×C4×Dic73—21no—
56C2×C4×Dic73—21no—
57C4⋊#10596831—22no—
58C4⋊#10596831—21no—
59C4⋊#10596831—21no—
60C4⋊#10596831—22no—
61C22⋊#10596831C2311yescharacteristic
62C22⋊#10596831C2311yescharacteristic
63C22⋊#10596831C2312yes—
64#14556502—22no—
65#14556502—21no—
66C4×#10596831—21no—
67C4×#10596831—22no—
68#14556498—21no—
69#14556498—22no—
70#14556498—22no—
71#14556498—22no—
72#14556498C2311yescharacteristic
73#14556498C2311yescharacteristic
74#14556498C2312yes—
75C2×#41×C73—21no—
76C2×#41×C73—21no—
77C2×#41×C73—22no—
78C22×Q8×C73D411yescharacteristic
79C4⋊Q8×C73—22no—
80C4⋊Q8×C73—21no—
81C4⋊Q8×C73C2×C411yescharacteristic
82C4⋊Q8×C73C2×C412yes—
83#127×C73C2×C411yescharacteristic
84#127×C73C2×C412yes—
85#126×C73—22no—
86#126×C73—21no—
87#125×C73C2×C411yescharacteristic
88#125×C73C2×C412yes—
89#124×C73—21no—
90#124×C73—22no—
91#124×C73—22no—
92#124×C73—21no—
93#124×C73C2×C411yescharacteristic
94#124×C73C2×C412yes—
95#123×C73—21no—
96#123×C73—22no—
97#123×C73—22no—
98#123×C73—22no—
99#123×C73—22no—
100C22⋊Q8×C73—21no—
101C22⋊Q8×C73—22no—
102C22⋊Q8×C73—22no—
103C22⋊Q8×C73—22no—
104C22⋊Q8×C73—22no—
105C22⋊Q8×C73C2311yescharacteristic
106#121×C73—21no—
107#121×C73—22no—
108#121×C73—22no—
109#121×C73—21no—
110#121×C73C2311yescharacteristic
111#121×C73C2312yes—
112C4×Q8×C73—21no—
113C4×Q8×C73C2×C411yescharacteristic
114C4×Q8×C73C2×C412yes—
115C4×Q8×C73C2×C411yescharacteristic
116C4×Q8×C73C2×C411yescharacteristic
117C4×Q8×C73C2×C412yes—
118C4×Q8×C73D411yescharacteristic
119C4×Q8×C73D411yescharacteristic
120C4×Q8×C73D412yes—
121C4×D4×C73—21no—
122C4×D4×C73—22no—
123C4×D4×C73—22no—
124C4×D4×C73—21no—
125C4×D4×C73—21no—
126C4×D4×C73—22no—
127C4×D4×C73C2×C411yescharacteristic
128C4×D4×C73C2×C412yes—
129C4×D4×C73C2×C412yes—
130C4×D4×C73C2×C412yes—
131C4×D4×C73C2×C412yes—
132#117×C73—21no—
133#117×C73—22no—
134C2×C4⋊C4×C73—22no—
135C2×C4⋊C4×C73—21no—
136C2×C4⋊C4×C73D412yes—
137C2×C4⋊C4×C73D411yescharacteristic
138C2×C42×C73C2311yescharacteristic
139C2×C42×C73C2312yes—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—22no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—22no—
146C4⋊Dic73—42no—
147C4⋊Dic73—82no—
148C4×Dic73—42no—
149C4×Dic73—81no—
150#41×C73—44no—
151#41×C73—81no—
152#41×C73—82no—
153#41×C73—81no—
154#41×C73—82no—
155C2×Q8×C73—22no—
156C2×Q8×C73—21no—
157C2×Q8×C73—42no—
158C2×Q8×C73C22⋊C411yescharacteristic
159C2×Q8×C73C22⋊C411yescharacteristic
160C2×Q8×C73C22⋊C412yes—
161C2×Q8×C73C22×C411yescharacteristic
162C2×D4×C73—21no—
163C2×D4×C73—22no—
164C2×D4×C73—22no—
165C2×D4×C73—21no—
166C2×D4×C73—22no—
167C2×D4×C73—21no—
168C2×D4×C73C22×C411yescharacteristic
169C2×D4×C73C22×C412yes—
170C22×C4×C73—22no—
171C22×C4×C73—21no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D412yes—
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D412yes—
179C22×C4×C73C2411yescharacteristic
180C4⋊C4×C73—21no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73C22⋊C412yes—
190C4⋊C4×C73C22⋊C411yescharacteristic
191C4⋊C4×C73C22⋊C412yes—
192C4⋊C4×C73C22⋊C411yescharacteristic
193C4⋊C4×C73C22⋊C412yes—
194C4⋊C4×C73C22⋊C412yes—
195C4⋊C4×C73C22⋊C412yes—
196C4⋊C4×C73C22×C411yescharacteristic
197C4⋊C4×C73C22×C412yes—
198C4⋊C4×C73C22×C411yescharacteristic
199C22⋊C4×C73—21no—
200C22⋊C4×C73—21no—
201C22⋊C4×C73—22no—
202C22⋊C4×C73—21no—
203C22⋊C4×C73—22no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—21no—
206C22⋊C4×C73—22no—
207C22⋊C4×C73—22no—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—21no—
210C22⋊C4×C73—22no—
211C42×C73—22no—
212C42×C73—21no—
213C42×C73C22×C411yescharacteristic
214C42×C73C22×C412yes—
215C42×C73C22×C412yes—
216C42×C73C22×C411yescharacteristic
217C42×C73C2×D411yescharacteristic
218C42×C73C2×D412yes—
219C42×C73C2×D412yes—
220C42×C73C2×D411yescharacteristic
221C23×C73—22no—
222C23×C73—21no—
223C23×C73C22×D411yescharacteristic
224Q8×C73—22no—
225Q8×C73—22no—
226Q8×C73—42no—
227Q8×C73—41no—
228Q8×C73—81no—
229Q8×C73—82no—
230D4×C73—41no—
231D4×C73—42no—
232D4×C73—44no—
233D4×C73—42no—
234D4×C73—41no—
235D4×C73—41no—
236D4×C73—42no—
237D4×C73—42no—
238D4×C73—81no—
239D4×C73—82no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—21no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—42no—
250C2×C4×C73—41no—
251C2×C4×C73—44no—
252C2×C4×C73—42no—
253C2×C4×C73—82no—
254C2×C4×C73—82no—
255C2×C4×C73—82no—
256C2×C4×C73C2×C22⋊C412yes—
257C2×C4×C73C2×C22⋊C411yescharacteristic
258C2×C4×C73C2×C22⋊C412yes—
259C2×C4×C73C2×C22⋊C411yescharacteristic
260C2×C4×C73C2×C22⋊C412yes—
261C2×C4×C73C2×C22⋊C412yes—
262C2×C4×C73C2×C22⋊C411yescharacteristic
263C2×C4×C73C2×C22⋊C411yescharacteristic
264C2×C4×C73C23×C411yescharacteristic
265C2×C4×C73C22×D411yescharacteristic
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—42no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275#10596831—41no—
276#10596831—41no—
277#10596831—42no—
278C22×C73—21no—
279C22×C73—21no—
280C22×C73—42no—
281C22×C73—41no—
282C22×C73—42no—
283C22×C73—41no—
284C22×C73—42no—
285C22×C73—41no—
286C22×C73C22×C22⋊C411yescharacteristic
287C22×C73#57611yescharacteristic
288C22×C73#57711yescharacteristic
289C4×C73—21no—
290C4×C73—21no—
291C4×C73—21no—
292C4×C73—22no—
293C4×C73—21no—
294C4×C73—22no—
295C4×C73—21no—
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—41no—
299C4×C73—42no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73#42112yes—
303Dic73—81no—
304Dic73—82no—
305Dic73—81no—
306#17602—731nomaximal, Sylow
307C2×C73—21no—
308C2×C73—21no—
309C2×C73—82no—
310C2×C73—81no—
311C2×C73#288411yescharacteristic
312C2×C73#289711yescharacteristic
313C2×C73C2×#42111yescharacteristic
314#3522D7311yescharacteristic
315#2984—731no—
316#2320—731no—
317#1521—732no—
318#1517—732no—
319#1514—732no—
320#1512—732no—
321#1509—731no—
322#1505—731no—
323#1502—731no—
324#1500—731no—
325C73#1760211yescharacteristic, complemented, Sylow
326#559Dic7312yescomplemented
327#559Dic7311yescharacteristic, complemented
328Q82Dic7311yescharacteristic, complemented
329#554Dic7311yescharacteristic, complemented
330#554Dic7312yescomplemented
331#550Dic7311yescharacteristic, complemented
332#548—21no—
333#547—22no—
334#547—21no—
335#542—21no—
336#542—22no—
337#532C2×D7311yescharacteristic
338#532C2×D7312yes—
339#529C2×D7311yescharacteristic
340#529C2×D7312yes—
341#520—21no—
342C2×C4×Q8C2×D7311yescharacteristic
343#446—1461no—
344#446—1462no—
345C8×Q8—1461no—
346#432—731no—
347#432—732no—
348#431—731no—
349#431—732no—
350#391—1461no—
351#389—1461no—
352#387—1462no—
353#383—732no—
354#382—732no—
355C4×C4⋊C4—731no—
356#376—731no—
357#328—731no—
358#328—731no—
359#328—732no—
360#327—731no—
361#327—731no—
362#327—732no—
363#327—732no—
364#327—732no—
365#327—732no—
366#327—732no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—22no—
370C22×Q8C22⋊D7311yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8C2×Dic7311yescharacteristic
374C4⋊Q8C2×Dic7312yes—
375#127C2×Dic7311yescharacteristic
376#127C2×Dic7312yes—
377#126—22no—
378#126—21no—
379#125C2×Dic7311yescharacteristic
380#125C2×Dic7312yes—
381#124—21no—
382#124—22no—
383#124—22no—
384#124—21no—
385#124C2×Dic7311yescharacteristic
386#124C2×Dic7312yes—
387#123—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8C22×D7311yescharacteristic
398#121—21no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121C22×D7311yescharacteristic
403#121C22×D7312yes—
404C4×Q8—21no—
405C4×Q8C2×Dic7311yescharacteristic
406C4×Q8C2×Dic7312yes—
407C4×Q8C2×Dic7311yescharacteristic
408C4×Q8C2×Dic7311yescharacteristic
409C4×Q8C2×Dic7312yes—
410C4×Q8C22⋊D7311yescharacteristic
411C4×Q8C22⋊D7311yescharacteristic
412C4×Q8C22⋊D7312yes—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4C2×Dic7311yescharacteristic
420C4×D4C2×Dic7312yes—
421C4×D4C2×Dic7312yes—
422C4×D4C2×Dic7312yes—
423C4×D4C2×Dic7312yes—
424#117—21no—
425#117—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4C22⋊D7312yes—
431C2×C4⋊C4C22⋊D7311yescharacteristic
432C2×C42—1461no—
433C2×C42—1461no—
434C2×C42C22×D7311yescharacteristic
435C2×C42C22×D7312yes—
436C4⋊C8—1462no—
437C4⋊C8—1461no—
438C4⋊C8—1461no—
439C4⋊C8—1462no—
440C22⋊C8—731no—
441C22⋊C8—731no—
442C22⋊C8—732no—
443#97—1462no—
444#97—1461no—
445C4×C8—1461no—
446C4×C8—1462no—
447#95—731no—
448#95—731no—
449#95—732no—
450#95—1461no—
451#95—1462no—
452#95—1462no—
453#95—1462no—
454#41—44no—
455#41—81no—
456#41—82no—
457#41—81no—
458#41—82no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C22⋊Dic7311yescharacteristic
463C2×Q8C22⋊Dic7311yescharacteristic
464C2×Q8C22⋊Dic7312yes—
465C2×Q8C22×Dic7311yescharacteristic
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4C22×Dic7311yescharacteristic
473C2×D4C22×Dic7312yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1462no—
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C2×C22⋊D7312yes—
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C2×C22⋊D7312yes—
486C22×C4C23×D7311yescharacteristic
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1462no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—2922no—
500C4⋊C4—5842no—
501C4⋊C4C22⋊Dic7312yes—
502C4⋊C4C22⋊Dic7311yescharacteristic
503C4⋊C4C22⋊Dic7312yes—
504C4⋊C4C22⋊Dic7311yescharacteristic
505C4⋊C4C22⋊Dic7312yes—
506C4⋊C4C22⋊Dic7312yes—
507C4⋊C4C22⋊Dic7312yes—
508C4⋊C4C22×Dic7311yescharacteristic
509C4⋊C4C22×Dic7312yes—
510C4⋊C4C22×Dic7311yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C42—22no—
524C42—21no—
525C42—2922no—
526C42—5841no—
527C42C22×Dic7311yescharacteristic
528C42C22×Dic7312yes—
529C42C22×Dic7312yes—
530C42C22×Dic7311yescharacteristic
531C42C2×C22⋊D7311yescharacteristic
532C42C2×C22⋊D7312yes—
533C42C2×C22⋊D7312yes—
534C42C2×C22⋊D7311yescharacteristic
535C23—22no—
536C23—21no—
537C23C22×C22⋊D7311yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—41no—
542Q8—81no—
543Q8—82no—
544D4—41no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—81no—
553D4—82no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2922no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4C2×C22⋊Dic7312yes—
580C2×C4C2×C22⋊Dic7311yescharacteristic
581C2×C4C2×C22⋊Dic7312yes—
582C2×C4C2×C22⋊Dic7311yescharacteristic
583C2×C4C2×C22⋊Dic7312yes—
584C2×C4C2×C22⋊Dic7312yes—
585C2×C4C2×C22⋊Dic7311yescharacteristic
586C2×C4C2×C22⋊Dic7311yescharacteristic
587C2×C4C23×Dic7311yescharacteristic
588C2×C4C22×C22⋊D7311yescharacteristic
589C8—2921no—
590C8—2921no—
591C8—2922no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22C22×C22⋊Dic7311yescharacteristic
601C22#1798939011yescharacteristic
602C22#1798939811yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—5841no—
617C4—5842no—
618C4—5841no—
619C4#1798860412yes—
620C2—21no—
621C2—21no—
622C2—82no—
623C2—81no—
624C2#2261839511yescharacteristic
625C2#2261845711yescharacteristic
626C2C2×#1798860411yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
Fitting, prime radicalC73×#3522C2
socleC22×C73C22×C22⋊C4
cosocleC22×C4×C73C24
hypercenter#3522D73
hypocenterC73#17602
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #22619020C2semidirect
2. #22612582C2semidirect
3. #22608508C2semidirect
4. #22608491C2semidirect
5. #22608478C2semidirect
6. #22608469C2semidirect
7. #22608456C2semidirect
8. #22608439C2semidirect
9. #22608426C2semidirect
10. #22608417C2semidirect
36. #559×C73C4semidirect
37. #559×C73C4semidirect
38. Q82×C73C4semidirect
39. #554×C73C4semidirect
40. #554×C73C4semidirect
41. #550×C73C4semidirect
325. C73#17602semidirect
326. #559Dic73semidirect
327. #559Dic73semidirect
328. Q82Dic73semidirect
329. #554Dic73semidirect
330. #554Dic73semidirect
331. #550Dic73semidirect

Maximal subgroups

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

subgroupindex
#226190202
#226125822
#226085082
#226084912
#226084782
#226084692
#226084562
#226084392
#226084262
#226084172
C73×#35222
#1760273