Finite Group Databasebeta

← #33158694

Subgroups — #33158694

Order 18688.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614355C211yesmaximal, characteristic
2#22612940C211yesmaximal, characteristic
3#22611342C211yesmaximal, characteristic, complemented
4#22611308C211yesmaximal, characteristic, complemented
5#22611124C211yesmaximal, characteristic
6#22610691C211yesmaximal, characteristic, complemented
7#22610658C211yesmaximal, characteristic, complemented
8#22610534C211yesmaximal, characteristic
9#22610518C211yesmaximal, characteristic, complemented
10#22610512C211yesmaximal, characteristic, complemented
11#22610197C211yesmaximal, characteristic, complemented
12#22610123C211yesmaximal, characteristic
13#22610022C211yesmaximal, characteristic
14#103×Dic73C211yesmaximal, characteristic, complemented
15C73×#2035C211yesmaximal, characteristic
16C2×#14556584—21no—
17C2×Q8×Dic73C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988529—21no—
28#17988515—21no—
29#17988494C2211yescharacteristic
30#17988486—21no—
31#17988486C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988473—21no—
35#17988469—21no—
36#17988462—21no—
37#17988456C2211yescharacteristic
38#17988447C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C22⋊Q8×C73C2211yescharacteristic
53C2×#106×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C23×Dic73—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556584—44no—
63Q8×Dic73—22no—
64Q8×Dic73—41no—
65C2×C4⋊Dic73—22no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—41no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73D411yescharacteristic
80C2×C22⋊Dic73D411yescharacteristic
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86#14556522—22no—
87#14556522—22no—
88#14556522—41no—
89#14556520—22no—
90#14556515—22no—
91#14556515—22no—
92C22⋊#10596831—22no—
93C8×Dic73—41no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104C22×Q8×C73C2311yescharacteristic
105C23×C4×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22⋊Q8×C73—22no—
109C22⋊Q8×C73—22no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116#106×C73—22no—
117#103×C73—22no—
118#103×C73—41no—
119C22⋊C8×C73—41no—
120#95×C73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—42no—
126C22×Dic73—42no—
127C22×Dic73—81no—
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831#4112yes—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—42no—
146C4⋊Dic73—42no—
147C4⋊Dic73—42no—
148C4⋊Dic73—81no—
149C4⋊Dic73#4112yes—
150#11924007—42no—
151#11924007—42no—
152#11924007—42no—
153C22⋊Dic73—41no—
154C22⋊Dic73—42no—
155C22⋊Dic73SD1614yes—
156C4×Dic73—21no—
157C4×Dic73—21no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—42no—
161C4×Dic73—42no—
162C4×Dic73—81no—
163C24×C73C2×D411yescharacteristic
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73#4112yes—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2411yescharacteristic
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C4⋊C4×C73—24no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73#4112yes—
195C22⋊C4×C73—24no—
196C22⋊C4×C73—42no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73C2×SD1612yes—
204C23×C73C22×D411yescharacteristic
205Q8×C73—22no—
206Q8×C73—22no—
207Q8×C73—41no—
208Q8×C73—41no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73C2×#4111yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73H(5,𝔽2)11yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—22no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—84no—
257C2×Dic73C2×SD1612yes—
258C2×Dic73C2×SD1612yes—
259C8×C73—41no—
260C8×C73—41no—
261#10596831—22no—
262#10596831—41no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73#52511yescharacteristic
271C22×C73#54611yescharacteristic
272C22×C73#54711yescharacteristic
273C22×C73C22×SD1611yescharacteristic
274C22×C73#57511yescharacteristic
275C22×C73#57711yescharacteristic
276C22×C73#57811yescharacteristic
277C4×C73—42no—
278C4×C73—41no—
279C4×C73—42no—
280C4×C73—81no—
281C4×C73—81no—
282C4×C73#52911yescharacteristic
283C4×C73#53911yescharacteristic
284C4×C73#55311yescharacteristic
285C4×C73#55611yescharacteristic
286Dic73—22no—
287Dic73—22no—
288Dic73—41no—
289Dic73—41no—
290Dic73—42no—
291Dic73—81no—
292Dic73—81no—
293Dic73—81no—
294#27034—731nomaximal, Sylow
295C2×C73—42no—
296C2×C73#246611yescharacteristic
297C2×C73#310311yescharacteristic
298C2×C73#312011yescharacteristic
299C2×C73#329811yescharacteristic
300C2×C73#331111yescharacteristic
301C2×C73#333511yescharacteristic
302C2×C73#335211yescharacteristic
303#2496—731no—
304#2360—731no—
305#2072—731no—
306#2066—731no—
307#2035—731no—
308#2035D7311yescharacteristic
309#1948—731no—
310#1941—731no—
311#1923—731no—
312#1920—731no—
313#1919—731no—
314#1865—731no—
315#1853—731no—
316#1824—731no—
317C4×#103—731no—
318C73#2703411yescharacteristic, complemented, Sylow
319C2×#126—1461no—
320C2×C22⋊Q8C2×D7311yescharacteristic
321C2×C4×Q8—731no—
322C22×C4⋊C4—731no—
323C2×#106—731no—
324C2×#106—731no—
325C2×#106—731no—
326C2×#106C2×D7311yescharacteristic
327C2×C4⋊C8—731no—
328C2×#103—731no—
329C2×#103—731no—
330C2×#103C2×D7311yescharacteristic
331C2×#103C2×D7311yescharacteristic
332C2×C22⋊C8—731no—
333C2×C22⋊C8C2×D7311yescharacteristic
334C2×C4×C8—731no—
335#400—1461no—
336#397—1461no—
337#391—731no—
338#389—731no—
339#389—731no—
340#389—731no—
341#389—1461no—
342#387—1461no—
343#387—1461no—
344#385—1461no—
345#384—731no—
346#380—731no—
347#380C2×D7311yescharacteristic
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350C4×C22⋊C4—731no—
351C4×C22⋊C4—1461no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340C2×D7311yescharacteristic
360#336—731no—
361#336—731no—
362C22×Q8C22×D7311yescharacteristic
363C23×C4—1461no—
364C23×C4C22×D7311yescharacteristic
365C22×C8—731no—
366C22×C8—731no—
367C22×C8C22×D7311yescharacteristic
368C22×C8C22×D7311yescharacteristic
369#126—2924no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—2921no—
373C4×Q8—1462no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—2921no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1462no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C4⋊C4C22×D7311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—731no—
394C2×C22⋊C4—731no—
395C2×C22⋊C4—1461no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42—1461no—
401#106—22no—
402#106—2921no—
403#106—1462no—
404#106—1462no—
405C4⋊C8—1462no—
406#103—22no—
407#103—41no—
408#103—1462no—
409#103—1462no—
410C22⋊C8—41no—
411C22⋊C8—1462no—
412C4×C8—2921no—
413#95—21no—
414#95—731no—
415#95—731no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424C24D4×D7311yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#1192402712yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2922no—
442C22×C4—2922no—
443C22×C4—5841no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4D4×D7311yescharacteristic
453C22×C4C23×D7311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—732no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—2922no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—2922no—
478C4⋊C4—5841no—
479C4⋊C4—732no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4#1192402712yes—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—2922no—
491C22⋊C4—2921no—
492C22⋊C4—734no—
493C42—2922no—
494C42—2922no—
495C42—5841no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23SD16×D7312yes—
507C23C2×D4×D7311yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—5844no—
543C2×C4—732no—
544C2×C4—732no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1462no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4#41×D7311yescharacteristic
557C2×C4#41×D7311yescharacteristic
558C2×C4C2×#1192402711yescharacteristic
559C2×C4C2×#1192402711yescharacteristic
560C2×C4#1455663611yescharacteristic
561C2×C4#1455663811yescharacteristic
562C8—41no—
563C8—41no—
564C8—2921no—
565C8—1462no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1798909311yescharacteristic
574C22#1798922611yescharacteristic
575C22#1798923511yescharacteristic
576C22C2×SD16×D7311yescharacteristic
577C22#1798938611yescharacteristic
578C22#1798939611yescharacteristic
579C22#1798940111yescharacteristic
580C4—41no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—2922no—
586C4—2921no—
587C4—2921no—
588C4—5841no—
589C4—5841no—
590C4—5841no—
591C4—1462no—
592C4—1462no—
593C4#1798911411yescharacteristic
594C4#1798918211yescharacteristic
595C4#1798928411yescharacteristic
596C4#1798930611yescharacteristic
597C2—42no—
598C2#2261396011yescharacteristic
599C2#2261995711yescharacteristic
600C2#2262014611yescharacteristic
601C2#2262179111yescharacteristic
602C2#2262196211yescharacteristic
603C2#2262228111yescharacteristic
604C2#2262249911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2035C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2035D73
hypocenterC73#27034
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #22611342C2semidirect
4. #22611308C2semidirect
6. #22610691C2semidirect
7. #22610658C2semidirect
9. #22610518C2semidirect
10. #22610512C2semidirect
11. #22610197C2semidirect
14. #103×Dic73C2semidirect
318. C73#27034semidirect

Maximal subgroups

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

subgroupindex
#226143552
#226129402
#226113422
#226113082
#226111242
#226106912
#226106582
#226105342
#226105182
#226105122
#226101972
#226101232
#226100222
#103×Dic732
C73×#20352
#2703473