Finite Group Databasebeta

← #33154084

Subgroups — #33154084

Order 18688.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614272C211yesmaximal, characteristic
2#22611253C211yesmaximal, characteristic, complemented
3#22611124C211yesmaximal, characteristic
4#22611082C211yesmaximal, characteristic, complemented
5#22610939C211yesmaximal, characteristic
6#22610888C211yesmaximal, characteristic, complemented
7#22610606C211yesmaximal, characteristic, complemented
8#22610565C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic
10#22610513C211yesmaximal, characteristic, complemented
11#22610272C211yesmaximal, characteristic, complemented
12#22610238C211yesmaximal, characteristic
13#22610017C211yesmaximal, characteristic
14C4×#14556515C211yesmaximal, characteristic, complemented
15C73×#2360C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988528—21no—
31#17988515—21no—
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988475—21no—
35#17988451—21no—
36#17988447C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C22⋊Q8×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C22×C4⋊C4×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#389×C73—21no—
53#389×C73C2211yescharacteristic
54#387×C73—21no—
55#385×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C22⋊C4×C73—21no—
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C2×#11924018—22no—
61C2×#11924018—41no—
62C2×#11924018—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556583—22no—
68#14556583—41no—
69#14556570—22no—
70#14556570—41no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80#14556522—22no—
81C4⋊#10596831—41no—
82#14556515—22no—
83#14556515—22no—
84#14556515—41no—
85#14556514—22no—
86#14556514—22no—
87#14556514—41no—
88C22⋊#10596831—41no—
89C22⋊#10596831—41no—
90C4×#10596831—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73C2311yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73C2311yescharacteristic
100C22⋊Q8×C73—22no—
101C22⋊Q8×C73—22no—
102C4×Q8×C73—22no—
103C4×Q8×C73—41no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—41no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73—21no—
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136#11924018—44no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73#4112yes—
155#11924007—42no—
156#11924007—42no—
157#11924007—42no—
158C22⋊Dic73—42no—
159C22⋊Dic73—42no—
160C4×Dic73—42no—
161C24×C73C2×D411yescharacteristic
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73#4112yes—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
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—42no—
179C22×C4×C73—42no—
180C22×C4×C73—81no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—24no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—42no—
193C4⋊C4×C73—42no—
194C4⋊C4×C73—81no—
195C4⋊C4×C73#4112yes—
196C4⋊C4×C73#4112yes—
197C22⋊C4×C73—24no—
198C22⋊C4×C73—22no—
199C22⋊C4×C73—22no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—42no—
205C42×C73—81no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73#13512yes—
213C23×C73C22×D411yescharacteristic
214Q8×C73—22no—
215Q8×C73—22no—
216Q8×C73—41no—
217Q8×C73—41no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—84no—
249C2×C4×C73#13712yes—
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73H(5,𝔽2)11yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×Dic73—22no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268Dic146—41no—
269Dic146—41no—
270Dic146—41no—
271Dic146—41no—
272#10596831—41no—
273#10596831—41no—
274#10596831—41no—
275#10596831—41no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73#52511yescharacteristic
284C22×C73#54611yescharacteristic
285C22×C73#54611yescharacteristic
286C22×C73C2×#13511yescharacteristic
287C22×C73C2×#13711yescharacteristic
288C22×C73#57711yescharacteristic
289C22×C73#57811yescharacteristic
290C4×C73—22no—
291C4×C73—41no—
292C4×C73—42no—
293C4×C73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73—42no—
297C4×C73—41no—
298C4×C73—81no—
299C4×C73#52911yescharacteristic
300C4×C73#53811yescharacteristic
301C4×C73#55511yescharacteristic
302C4×C73#55911yescharacteristic
303Dic73—42no—
304Dic73—81no—
305Dic73—81no—
306Dic73—81no—
307#26725—731nomaximal, Sylow
308C2×C73—42no—
309C2×C73#245911yescharacteristic
310C2×C73#310811yescharacteristic
311C2×C73#312011yescharacteristic
312C2×C73#329611yescharacteristic
313C2×C73#330011yescharacteristic
314C2×C73#331211yescharacteristic
315C2×C73#333111yescharacteristic
316#2489—731no—
317#2360D7311yescharacteristic
318#2056—731no—
319#2035—731no—
320#2026—731no—
321#2001—731no—
322#1989—731no—
323#1933—731no—
324#1927—731no—
325#1923—731no—
326#1920—731no—
327#1878—731no—
328#1872—731no—
329#1823—731no—
330C4×#103—731no—
331C73#2672511yescharacteristic, complemented, Sylow
332C22×Q16—731no—
333C2×#125—731no—
334C2×C22⋊Q8—731no—
335C2×C22⋊Q8C2×D7311yescharacteristic
336C2×C4×Q8C2×D7311yescharacteristic
337C22×C4⋊C4C2×D7311yescharacteristic
338C2×#106—731no—
339C2×C4⋊C8—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#103—731no—
346C2×C22⋊C8—731no—
347C2×C22⋊C8—731no—
348C2×C4×C8—731no—
349#400—1461no—
350#397—1461no—
351#391—731no—
352#391—731no—
353#391C2×D7311yescharacteristic
354#389—21no—
355#389C2×D7311yescharacteristic
356#388—1461no—
357#387—21no—
358#385—21no—
359#382—1461no—
360#380—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4C2×D7311yescharacteristic
363C4×C22⋊C4—21no—
364C4×C22⋊C4C2×D7311yescharacteristic
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#336—731no—
374#336—731no—
375C22×Q8—731no—
376C22×Q8C22×D7311yescharacteristic
377C23×C4—21no—
378C23×C4C22×D7311yescharacteristic
379C2×Q16—2921no—
380C2×Q16—2921no—
381C2×Q16—1462no—
382C22×C8—731no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—731no—
386#125—2921no—
387#125—1462no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—2921no—
391C22⋊Q8—1462no—
392C4×Q8—22no—
393C4×Q8—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4C22⋊D7311yescharacteristic
413C2×C22⋊C4C22⋊D7311yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1461no—
417C2×C42C22×D7311yescharacteristic
418C2×C42C22×D7311yescharacteristic
419#106—1462no—
420C4⋊C8—2921no—
421#103—2921no—
422#103—2921no—
423#103—1462no—
424#103—1462no—
425#103—1462no—
426#103—1462no—
427C22⋊C8—2921no—
428C22⋊C8—2921no—
429C4×C8—2921no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95C22⋊D7311yescharacteristic
441#95C22⋊D7311yescharacteristic
442C24C2×C22⋊D7311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×Q8—1461no—
454C2×Q8#1192402812yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C23×D7311yescharacteristic
478Q16—2924no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—2922no—
500C4⋊C4—2922no—
501C4⋊C4—2922no—
502C4⋊C4—732no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4#1192402712yes—
509C4⋊C4#1192402812yes—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—2922no—
514C22⋊C4—2922no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—81no—
521C42—2922no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1455661112yes—
529C23C22×C22⋊D7311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—2921no—
535Q8—2921no—
536Q8—2921no—
537Q8—2921no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—1462no—
581C2×C4#1455662212yes—
582C2×C4#41×D7311yescharacteristic
583C2×C4C2×#1192402711yescharacteristic
584C2×C4C2×#1192402811yescharacteristic
585C2×C4C2×#1192402811yescharacteristic
586C2×C4#1455663611yescharacteristic
587C2×C4#1455663711yescharacteristic
588C8—2921no—
589C8—2921no—
590C8—2921no—
591C8—2921no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1798909811yescharacteristic
600C22#1798922511yescharacteristic
601C22#1798923111yescharacteristic
602C22C2×#1455661111yescharacteristic
603C22C2×#1455662211yescharacteristic
604C22#1798939811yescharacteristic
605C22#1798940211yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—2922no—
616C4—5841no—
617C4—5841no—
618C4—5841no—
619C4#1798911711yescharacteristic
620C4#1798917011yescharacteristic
621C4#1798930211yescharacteristic
622C4#1798932011yescharacteristic
623C2—42no—
624C2#2261386211yescharacteristic
625C2#2262001111yescharacteristic
626C2#2262014511yescharacteristic
627C2#2262176611yescharacteristic
628C2#2262181211yescharacteristic
629C2#2262198211yescharacteristic
630C2#2262224111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2360C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2360D73
hypocenterC73#26725
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #22611253C2semidirect
4. #22611082C2semidirect
6. #22610888C2semidirect
7. #22610606C2semidirect
8. #22610565C2semidirect
10. #22610513C2semidirect
11. #22610272C2semidirect
14. C4×#14556515C2semidirect
331. C73#26725semidirect

Maximal subgroups

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

subgroupindex
#226142722
#226112532
#226111242
#226110822
#226109392
#226108882
#226106062
#226105652
#226105392
#226105132
#226102722
#226102382
#226100172
C4×#145565152
C73×#23602
#2672573