Finite Group Databasebeta

← #33159099

Subgroups — #33159099

Order 18688.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614333C211yesmaximal, characteristic
2#22612962C211yesmaximal, characteristic
3#22611352C211yesmaximal, characteristic, complemented
4#22611308C211yesmaximal, characteristic, complemented
5#22611131C211yesmaximal, characteristic
6#22610700C211yesmaximal, characteristic, complemented
7#22610668C211yesmaximal, characteristic, complemented
8#22610541C211yesmaximal, characteristic
9#22610526C211yesmaximal, characteristic, complemented
10#22610518C211yesmaximal, characteristic, complemented
11#22610183C211yesmaximal, characteristic, complemented
12#22610128C211yesmaximal, characteristic
13#22610027C211yesmaximal, characteristic
14#22609763C211yesmaximal, characteristic, complemented
15C73×#2035C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26C73⋊#401—21no—
27#17988512—21no—
28#17988494C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988480—21no—
33#17988480—21no—
34#17988475—21no—
35#17988474—21no—
36#17988473—21no—
37#17988452C2211yescharacteristic
38#17988447C2211yescharacteristic
39C4⋊C4×Dic73—21no—
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
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
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61Q8×Dic73—22no—
62Q8×Dic73—41no—
63#14556550—22no—
64#14556550—41no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×C22⋊Dic73—21no—
74C2×C22⋊Dic73—21no—
75C2×C22⋊Dic73D411yescharacteristic
76C2×C22⋊Dic73D411yescharacteristic
77C2×C4×Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73C2311yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83#14556524—22no—
84#14556524—22no—
85#14556522—41no—
86#14556520—22no—
87#14556515—22no—
88#14556515—22no—
89C22⋊#10596831—22no—
90#14556501—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103C22×Q8×C73C2311yescharacteristic
104C23×C4×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22⋊Q8×C73—22no—
108C22⋊Q8×C73—22no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115#106×C73—22no—
116#103×C73—22no—
117#103×C73—41no—
118C22⋊C8×C73—41no—
119#95×C73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831#4112yes—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—42no—
142C4⋊Dic73—42no—
143C4⋊Dic73—81no—
144C4⋊Dic73#4112yes—
145#11924007—42no—
146C22⋊Dic73—41no—
147C22⋊Dic73—41no—
148C22⋊Dic73—42no—
149C22⋊Dic73—42no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—42no—
155C4×Dic73—42no—
156C4×Dic73—42no—
157C4×Dic73—81no—
158C24×C73C2×D411yescharacteristic
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73#4112yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
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×C73C2×D411yescharacteristic
176C22×C4×C73C2411yescharacteristic
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C4⋊C4×C73—24no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73#4112yes—
190C22⋊C4×C73—24no—
191C22⋊C4×C73—81no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73C22×D411yescharacteristic
200Q8×C73—22no—
201Q8×C73—22no—
202Q8×C73—41no—
203Q8×C73—41no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
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×C73C2×#4111yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C73H(5,𝔽2)11yescharacteristic
228C2×C4×C732−1+411yescharacteristic
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—22no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C8×C73—41no—
254C8×C73—41no—
255#10596831—22no—
256#10596831—41no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73#52511yescharacteristic
265C22×C73#54611yescharacteristic
266C22×C73#54711yescharacteristic
267C22×C73#57511yescharacteristic
268C22×C73#57511yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57811yescharacteristic
271C4×C73—42no—
272C4×C73—41no—
273C4×C73—42no—
274C4×C73—81no—
275C4×C73—81no—
276C4×C73#52911yescharacteristic
277C4×C73#53911yescharacteristic
278C4×C73#55311yescharacteristic
279C4×C73#55611yescharacteristic
280Dic73—41no—
281Dic73—41no—
282Dic73—41no—
283Dic73—42no—
284Dic73—41no—
285Dic73—81no—
286Dic73—81no—
287Dic73—81no—
288#27061—731nomaximal, Sylow
289C2×C73—81no—
290C2×C73#246611yescharacteristic
291C2×C73#311111yescharacteristic
292C2×C73#312111yescharacteristic
293C2×C73#330711yescharacteristic
294C2×C73#332711yescharacteristic
295C2×C73#334911yescharacteristic
296C2×C73#335011yescharacteristic
297#2494—731no—
298#2362—731no—
299#2074—731no—
300#2066—731no—
301#2036—731no—
302#2035D7311yescharacteristic
303#1949—731no—
304#1943—731no—
305#1924—731no—
306#1921—731no—
307#1920—731no—
308#1863—731no—
309#1854—731no—
310#1825—731no—
311#1763—731no—
312C73#2706111yescharacteristic, complemented, Sylow
313C2×C22⋊Q8C2×D7311yescharacteristic
314C2×C4×Q8—731no—
315C2×#117—731no—
316C2×#107—731no—
317C2×#107—731no—
318C2×#106—731no—
319C2×#106C2×D7311yescharacteristic
320C2×C4⋊C8—731no—
321C2×#103—731no—
322C2×#103—731no—
323C2×#103C2×D7311yescharacteristic
324C2×#103C2×D7311yescharacteristic
325C2×C22⋊C8—731no—
326C2×C22⋊C8C2×D7311yescharacteristic
327C2×#97—731no—
328#401—1461no—
329#397—1461no—
330#391—731no—
331#389—731no—
332#389—731no—
333#389—731no—
334#388—1461no—
335#388—1461no—
336#388—1461no—
337#387—1461no—
338#387—1461no—
339#382—731no—
340#380—731no—
341#380C2×D7311yescharacteristic
342C4×C4⋊C4—731no—
343C4×C4⋊C4—731no—
344C4×C4⋊C4—1461no—
345C4×C22⋊C4—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340C2×D7311yescharacteristic
354#336—731no—
355#336—731no—
356C22×Q8C22×D7311yescharacteristic
357C23×C4C22×D7311yescharacteristic
358C22×C8—731no—
359C22×C8—731no—
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C4×Q8—2921no—
365C4×Q8—1462no—
366#117—2921no—
367#117—1462no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—731no—
387C2×C42—731no—
388C2×C42—1461no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42—1461no—
392#107—1462no—
393#107—1462no—
394#106—22no—
395#106—2921no—
396C4⋊C8—1462no—
397#103—22no—
398#103—41no—
399#103—1462no—
400#103—1462no—
401C22⋊C8—41no—
402C22⋊C8—1462no—
403#97—2921no—
404#95—21no—
405#95—731no—
406#95—731no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416#95—1461no—
417C24D4×D7311yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8#1192402712yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—731no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4D4×D7311yescharacteristic
443C22×C4C23×D7311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—732no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—2922no—
463C4⋊C4—2922no—
464C4⋊C4—2922no—
465C4⋊C4—5841no—
466C4⋊C4—732no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4#1192402712yes—
475C22⋊C4—24no—
476C22⋊C4—81no—
477C22⋊C4—2922no—
478C22⋊C4—2922no—
479C22⋊C4—2921no—
480C22⋊C4—2921no—
481C42—2922no—
482C42—2922no—
483C42—2922no—
484C42—5841no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42—1461no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C2×D4×D7311yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1462no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4#41×D7311yescharacteristic
545C2×C4#41×D7311yescharacteristic
546C2×C4C2×#1192402711yescharacteristic
547C2×C4C2×#1192402711yescharacteristic
548C2×C4#1455663611yescharacteristic
549C2×C4#1455663811yescharacteristic
550C8—41no—
551C8—41no—
552C8—2921no—
553C8—1462no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1798909311yescharacteristic
562C22#1798922611yescharacteristic
563C22#1798923511yescharacteristic
564C22#1798938411yescharacteristic
565C22#1798938611yescharacteristic
566C22#1798939311yescharacteristic
567C22#1798940311yescharacteristic
568C4—42no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—2921no—
574C4—2922no—
575C4—2921no—
576C4—2921no—
577C4—2921no—
578C4—5841no—
579C4—5841no—
580C4—5841no—
581C4#1798911411yescharacteristic
582C4#1798918211yescharacteristic
583C4#1798928411yescharacteristic
584C4#1798930611yescharacteristic
585C2—81no—
586C2#2261396011yescharacteristic
587C2#2262004111yescharacteristic
588C2#2262015311yescharacteristic
589C2#2262191311yescharacteristic
590C2#2262219411yescharacteristic
591C2#2262245911yescharacteristic
592C2#2262247911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2035C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2035D73
hypocenterC73#27061
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. #22611352C2semidirect
4. #22611308C2semidirect
6. #22610700C2semidirect
7. #22610668C2semidirect
9. #22610526C2semidirect
10. #22610518C2semidirect
11. #22610183C2semidirect
14. #22609763C2semidirect
312. C73#27061semidirect

Maximal subgroups

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

subgroupindex
#226143332
#226129622
#226113522
#226113082
#226111312
#226107002
#226106682
#226105412
#226105262
#226105182
#226101832
#226101282
#226100272
#226097632
C73×#20352
#2706173