Finite Group Databasebeta

← #33054168

Subgroups — #33054168

Order 18688.

876 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618951C211yesmaximal, characteristic, complemented
2#22618582C211yesmaximal, characteristic, complemented
3#22608562C212yesmaximal, complemented
4#22608558C212yesmaximal, complemented
5#22608546C211yesmaximal, characteristic, complemented
6#22608538C211yesmaximal, characteristic, complemented
7#22608538C212yesmaximal, complemented
8#22608533C212yesmaximal, complemented
9#22608526C211yesmaximal, characteristic, complemented
10#22608519C211yesmaximal, characteristic, complemented
11C73×#3515C211yesmaximal, characteristic
12Q8×#10596831—21no—
13#17988665—21no—
14#17988665—22no—
15#17988648—22no—
16#17988644C2211yescharacteristic
17#17988644C2212yes—
18#17988644C2212yes—
19#17988644C2212yes—
20#17988614—21no—
21#17988554—21no—
22C4×#11924012C2211yescharacteristic
23C4×#11924012C2212yes—
24C4×#11924012C2211yescharacteristic
25C2×#14556502C2211yescharacteristic
26#17988329C2211yescharacteristic
27#17988329C2211yescharacteristic
28#17988329C2212yes—
29#17988329C2212yes—
30#17988329C2212yes—
31#17988329C2212yes—
32#17988329C2212yes—
33#17988327C2211yescharacteristic
34#17988327C2211yescharacteristic
35#17988327C2212yes—
36#557×C73C411yescharacteristic
37#554×C73C411yescharacteristic
38#554×C73C411yescharacteristic
39#553×C73C412yes—
40#552×C73—22no—
41#552×C73—21no—
42#552×C73C412yes—
43#547×C73—21no—
44#547×C73C411yescharacteristic
45#541×C73—21no—
46#539×C73—22no—
47#533×C73C2211yescharacteristic
48#529×C73C2211yescharacteristic
49#529×C73C2212yes—
50C2×#125×C73C2212yes—
51#520×C73—21no—
52C4×#41×C73C2211yescharacteristic
53C2×#11924012—21no—
54C2×#11924012—22no—
55C2×#11924012—21no—
56C22×#10596831—21no—
57C4⋊#10596831—22no—
58C4⋊#10596831—22no—
59C4⋊#10596831—21no—
60C4⋊#10596831—22no—
61C22⋊#10596831—22no—
62C22⋊#10596831—22no—
63C22⋊#10596831—22no—
64C22⋊#10596831—21no—
65#14556502—21no—
66#14556502C2311yescharacteristic
67#14556502C2311yescharacteristic
68#14556502C2312yes—
69#14556502C2311yescharacteristic
70#14556502C2312yes—
71#14556502C2311yescharacteristic
72C4×#10596831—21no—
73C4×#10596831—21no—
74C4×#10596831—22no—
75C2×#41×C73—21no—
76C2×#41×C73D411yescharacteristic
77C4⋊Q8×C73—21no—
78C4⋊Q8×C73C2×C411yescharacteristic
79C4⋊Q8×C73C2×C411yescharacteristic
80#126×C73—22no—
81#126×C73—22no—
82#126×C73—22no—
83#126×C73—21no—
84#125×C73—22no—
85#125×C73—22no—
86#125×C73—21no—
87#125×C73—22no—
88#125×C73—22no—
89#125×C73C2×C412yes—
90#125×C73C2×C412yes—
91#125×C73C2311yescharacteristic
92#125×C73C2312yes—
93#124×C73C2311yescharacteristic
94#123×C73—22no—
95#123×C73—21no—
96#123×C73—22no—
97#123×C73—22no—
98#123×C73—22no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—21no—
101C22⋊Q8×C73—21no—
102C22⋊Q8×C73—22no—
103C22⋊Q8×C73—21no—
104C22⋊Q8×C73—22no—
105C22⋊Q8×C73C2×C411yescharacteristic
106C22⋊Q8×C73C2×C412yes—
107#121×C73—22no—
108#121×C73C2×C412yes—
109#121×C73C2×C411yescharacteristic
110C4×Q8×C73—21no—
111C4×Q8×C73—21no—
112C4×Q8×C73—21no—
113C4×Q8×C73C2×C412yes—
114C4×Q8×C73C2×C411yescharacteristic
115C4×Q8×C73C2×C411yescharacteristic
116C4×Q8×C73D411yescharacteristic
117C4×D4×C73—21no—
118C4×D4×C73—22no—
119C4×D4×C73—21no—
120C4×D4×C73—22no—
121C4×D4×C73C2×C411yescharacteristic
122C4×D4×C73C2×C412yes—
123C4×D4×C73C2×C411yescharacteristic
124C4×D4×C73C2×C412yes—
125C4×D4×C73D411yescharacteristic
126C4×D4×C73D412yes—
127#117×C73—22no—
128#117×C73—21no—
129#117×C73—22no—
130#117×C73C2×C412yes—
131#117×C73D411yescharacteristic
132#117×C73D412yes—
133C2×C4⋊C4×C73—22no—
134C2×C4⋊C4×C73—22no—
135C2×C4⋊C4×C73—21no—
136C2×C4⋊C4×C73C2×C412yes—
137C2×C4⋊C4×C73C2×C412yes—
138C2×C42×C73C2312yes—
139C2×C42×C73C2311yescharacteristic
140#11924012—44no—
141#11924012—42no—
142#11924012—42no—
143C2×#10596831—21no—
144C2×#10596831—22no—
145C2×#10596831—21no—
146C2×#10596831—22no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—42no—
150#41×C73—42no—
151#41×C73—44no—
152C2×Q8×C73—21no—
153C2×Q8×C73—21no—
154C2×Q8×C73C22⋊C411yescharacteristic
155C2×Q8×C73C22×C411yescharacteristic
156C2×D4×C73—22no—
157C2×D4×C73—21no—
158C2×D4×C73C22⋊C411yescharacteristic
159C2×D4×C73C22⋊C412yes—
160C2×D4×C73C22×C411yescharacteristic
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—21no—
164C22×C4×C73—22no—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×D412yes—
170C22×C4×C73C2×D411yescharacteristic
171C4⋊C4×C73—22no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—22no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—22no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73C22⋊C411yescharacteristic
188C4⋊C4×C73C22⋊C411yescharacteristic
189C4⋊C4×C73C22⋊C412yes—
190C4⋊C4×C73C22⋊C412yes—
191C4⋊C4×C73C22×C411yescharacteristic
192C4⋊C4×C73C22×C411yescharacteristic
193C4⋊C4×C73C22×C412yes—
194C4⋊C4×C73C22×C412yes—
195C22⋊C4×C73—22no—
196C22⋊C4×C73—22no—
197C22⋊C4×C73—21no—
198C22⋊C4×C73—21no—
199C22⋊C4×C73—22no—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—21no—
202C22⋊C4×C73—21no—
203C22⋊C4×C73—21no—
204C22⋊C4×C73C22⋊C412yes—
205C22⋊C4×C73C22⋊C412yes—
206C22⋊C4×C73C22⋊C412yes—
207C42×C73—22no—
208C42×C73—22no—
209C42×C73—21no—
210C42×C73C22×C412yes—
211C42×C73C22×C411yescharacteristic
212C42×C73C2×D412yes—
213C42×C73C2×D411yescharacteristic
214C42×C73C2×D411yescharacteristic
215C42×C73C2×D412yes—
216C42×C73C2411yescharacteristic
217C23×C73—21no—
218C23×C73C2×C22⋊C411yescharacteristic
219C23×C73C2×C22⋊C412yes—
220Q8×C73—22no—
221Q8×C73—41no—
222Q8×C73—42no—
223Q8×C73—81no—
224D4×C73—22no—
225D4×C73—42no—
226D4×C73—41no—
227D4×C73—44no—
228D4×C73—81no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—42no—
236C2×C4×C73—41no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—41no—
240C2×C4×C73—44no—
241C2×C4×C73—42no—
242C2×C4×C73—82no—
243C2×C4×C73—82no—
244C2×C4×C73C2×C22⋊C411yescharacteristic
245C2×C4×C73C2×C22⋊C412yes—
246C2×C4×C73C2×C22⋊C411yescharacteristic
247C2×C4×C73C2×C22⋊C412yes—
248C2×C4×C73C2×C22⋊C412yes—
249C2×C4×C73C2×C22⋊C411yescharacteristic
250C2×C4×C73C23×C411yescharacteristic
251C2×C4×C73C22×D411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253#10596831—22no—
254#10596831—41no—
255#10596831—41no—
256#10596831—42no—
257#10596831—41no—
258#10596831—42no—
259C22×C73—21no—
260C22×C73—22no—
261C22×C73—21no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—21no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73C22×C22⋊C411yescharacteristic
269C4×C73—21no—
270C4×C73—21no—
271C4×C73—22no—
272C4×C73—21no—
273C4×C73—22no—
274C4×C73—21no—
275C4×C73—22no—
276C4×C73—21no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—42no—
280C4×C73—41no—
281C4×C73—42no—
282C4×C73#42112yes—
283#17682—731nomaximal, Sylow
284C2×C73—41no—
285C2×C73—42no—
286C2×C73—81no—
287C2×C73#288911yescharacteristic
288C2×C73C2×#42111yescharacteristic
289C2×C73#290111yescharacteristic
290#3515D7311yescharacteristic
291#2971—731no—
292#2923—731no—
293#1535—732no—
294#1534—732no—
295#1531—731no—
296#1529—731no—
297#1529—732no—
298#1528—732no—
299#1526—731no—
300#1524—731no—
301C73#1768211yescharacteristic, complemented, Sylow
302#557Dic7311yescharacteristic
303#554Dic7311yescharacteristic
304#554Dic7311yescharacteristic
305#553Dic7312yes—
306#552—22no—
307#552—21no—
308#552Dic7312yes—
309#547—21no—
310#547Dic7311yescharacteristic
311#541—21no—
312#539—22no—
313#533C2×D7311yescharacteristic
314#529C2×D7311yescharacteristic
315#529C2×D7312yes—
316C2×#125C2×D7312yes—
317#520—21no—
318C4×#41C2×D7311yescharacteristic
319C8×Q8—1461no—
320#436—1461no—
321#436—1462no—
322#433—1462no—
323#432—732no—
324#432—732no—
325#432—731no—
326#432—732no—
327#424—1461no—
328#408—1461no—
329C4×M16—731no—
330C4×M16—731no—
331C4×M16—732no—
332C2×#97—731no—
333#330—731no—
334#330—731no—
335#330—732no—
336#330—732no—
337#330—732no—
338#330—732no—
339#330—732no—
340#329—731no—
341#329—731no—
342#329—732no—
343C2×#41—21no—
344C2×#41C22⋊D7311yescharacteristic
345C2×M16—1461no—
346C2×M16—1462no—
347C2×M16—1461no—
348C22×C8—1461no—
349C4⋊Q8—21no—
350C4⋊Q8C2×Dic7311yescharacteristic
351C4⋊Q8C2×Dic7311yescharacteristic
352#126—22no—
353#126—22no—
354#126—22no—
355#126—21no—
356#125—22no—
357#125—22no—
358#125—21no—
359#125—22no—
360#125—22no—
361#125C2×Dic7312yes—
362#125C2×Dic7312yes—
363#125C22×D7311yescharacteristic
364#125C22×D7312yes—
365#124C22×D7311yescharacteristic
366#123—22no—
367#123—21no—
368#123—22no—
369#123—22no—
370#123—22no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8C2×Dic7311yescharacteristic
378C22⋊Q8C2×Dic7312yes—
379#121—22no—
380#121C2×Dic7312yes—
381#121C2×Dic7311yescharacteristic
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8C2×Dic7312yes—
386C4×Q8C2×Dic7311yescharacteristic
387C4×Q8C2×Dic7311yescharacteristic
388C4×Q8C22⋊D7311yescharacteristic
389C4×D4—21no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—22no—
393C4×D4C2×Dic7311yescharacteristic
394C4×D4C2×Dic7312yes—
395C4×D4C2×Dic7311yescharacteristic
396C4×D4C2×Dic7312yes—
397C4×D4C22⋊D7311yescharacteristic
398C4×D4C22⋊D7312yes—
399#117—22no—
400#117—21no—
401#117—22no—
402#117C2×Dic7312yes—
403#117C22⋊D7311yescharacteristic
404#117C22⋊D7312yes—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4C2×Dic7312yes—
409C2×C4⋊C4C2×Dic7312yes—
410C2×C42C22×D7312yes—
411C2×C42C22×D7311yescharacteristic
412C4⋊C8—1462no—
413C4⋊C8—1462no—
414C4⋊C8—1461no—
415C4⋊C8—1462no—
416C22⋊C8—1462no—
417C22⋊C8—1462no—
418C22⋊C8—1462no—
419C22⋊C8—1461no—
420#97—731no—
421#97—731no—
422#97—731no—
423#97—731no—
424#97—732no—
425#97—732no—
426#97—1461no—
427C4×C8—1461no—
428C4×C8—1461no—
429C4×C8—1462no—
430#41—42no—
431#41—44no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C22⋊Dic7311yescharacteristic
435C2×Q8C22×Dic7311yescharacteristic
436C2×D4—22no—
437C2×D4—21no—
438C2×D4C22⋊Dic7311yescharacteristic
439C2×D4C22⋊Dic7312yes—
440C2×D4C22×Dic7311yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4C22×Dic7312yes—
446C22×C4C22×Dic7311yescharacteristic
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7312yes—
449C22×C4C2×C22⋊D7312yes—
450C22×C4C2×C22⋊D7311yescharacteristic
451M16—2922no—
452M16—2924no—
453M16—2922no—
454C2×C8—2922no—
455C2×C8—1462no—
456C2×C8—1461no—
457C2×C8—1462no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4C22⋊Dic7311yescharacteristic
478C4⋊C4C22⋊Dic7311yescharacteristic
479C4⋊C4C22⋊Dic7312yes—
480C4⋊C4C22⋊Dic7312yes—
481C4⋊C4C22×Dic7311yescharacteristic
482C4⋊C4C22×Dic7311yescharacteristic
483C4⋊C4C22×Dic7312yes—
484C4⋊C4C22×Dic7312yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4C22⋊Dic7312yes—
495C22⋊C4C22⋊Dic7312yes—
496C22⋊C4C22⋊Dic7312yes—
497C42—22no—
498C42—22no—
499C42—21no—
500C42C22×Dic7312yes—
501C42C22×Dic7311yescharacteristic
502C42C2×C22⋊D7311yescharacteristic
503C42C2×C22⋊D7311yescharacteristic
504C42C2×C22⋊D7312yes—
505C42C2×C22⋊D7312yes—
506C42C23×D7311yescharacteristic
507C23—21no—
508C23C2×C22⋊Dic7311yescharacteristic
509C23C2×C22⋊Dic7312yes—
510Q8—22no—
511Q8—41no—
512Q8—42no—
513Q8—81no—
514D4—22no—
515D4—42no—
516D4—41no—
517D4—44no—
518D4—81no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4C2×C22⋊Dic7311yescharacteristic
535C2×C4C2×C22⋊Dic7312yes—
536C2×C4C2×C22⋊Dic7311yescharacteristic
537C2×C4C2×C22⋊Dic7312yes—
538C2×C4C2×C22⋊Dic7312yes—
539C2×C4C2×C22⋊Dic7311yescharacteristic
540C2×C4C23×Dic7311yescharacteristic
541C2×C4C22×C22⋊D7311yescharacteristic
542C2×C4C22×C22⋊D7311yescharacteristic
543C8—2921no—
544C8—2921no—
545C8—2922no—
546C8—2921no—
547C8—2922no—
548C8—1462no—
549C22—21no—
550C22—22no—
551C22—21no—
552C22—22no—
553C22—22no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22C22×C22⋊Dic7311yescharacteristic
559C4—21no—
560C4—21no—
561C4—22no—
562C4—21no—
563C4—22no—
564C4—21no—
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4#1798860412yes—
573C2—41no—
574C2—42no—
575C2—81no—
576C2#2261841411yescharacteristic
577C2C2×#1798860411yescharacteristic
578C2#2261847811yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
Fitting, prime radicalC73×#3515C2
socleC22×C73C22×C22⋊C4
cosocleC42×C73C24
hypercenter#3515D73
hypocenterC73#17682
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3515

Complements

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

NH ≅ G/Ntype
1. #22618951C2semidirect
2. #22618582C2semidirect
3. #22608562C2semidirect
4. #22608558C2semidirect
5. #22608546C2semidirect
6. #22608538C2semidirect
7. #22608538C2semidirect
8. #22608533C2semidirect
9. #22608526C2semidirect
10. #22608519C2semidirect
301. C73#17682semidirect

Maximal subgroups

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

subgroupindex
#226189512
#226185822
#226085622
#226085582
#226085462
#226085382
#226085332
#226085262
#226085192
C73×#35152
#1768273